Lecture Notes in Control and Information Sciences 389 Editors: M. Thoma, F. Allgöwer, M. Morari
Rafael Bru and Sergio Romero-Vivó (Eds.)
Positive Systems Proceedings of the third Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2009) Valencia, Spain, September 2–4, 2009
ABC
Series Advisory Board P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis
Editors Rafael Bru Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camí de Vera s/n, 46022 Valencia Spain E-mail:
[email protected] Sergio Romero-Vivó Instituto Universitario de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camí de Vera s/n, 46022 Valencia Spain E-mail:
[email protected] ISBN 978-3-642-02893-9
e-ISBN 978-3-642-02894-6
DOI 10.1007/978-3-642-02894-6 Lecture Notes in Control and Information Sciences
ISSN 0170-8643
Library of Congress Control Number: Applied for c 2009
Springer-Verlag Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 543210 springer.com
Preface
Nowadays, the researchers into Control Theory and its Applications as well as Matrix Analysis are well aware and recognize the importance of Positive Systems. This volume contains the proceedings of the “Third Multidisciplinary Symposium on Positive Systems: Theory and Applications (POSTA09)” held in Valencia, Spain, September 2–4, 2009. At present, this is the only world congress whose main topic is focused on this field. After this third event, we think that we have established the basis of a regular triennial event supported in the task of the previous organizing committees and hope to have met the requirements of their expectations. This POSTA09 meeting has been organized by members of the “Universitat Polit`ecnica de Val`encia”, who have taken their doctor’s degree in the subjects of the conference. We are grateful to all of them for their enthusiasm, effort and especially their collaboration, without the help of which the outcome would have never been the same. Also, we highly appreciate Christian Commault for having his experience at our disposal. Besides that, we wish to thank the International Programme Committee components and additional referees for their outstanding work in the review process of the contributions. We are happy to say that their constructive suggestions and positive comments have improved the quality of the presentations. We are very much obliged to the following organizations for having financially backed this congress: “Ministerio de Educaci´ on y Ciencia”, “Conselleria d’Educaci´ o de la Generalitat Valenciana”, “Universitat Polit`ecnica de Val`encia (UPV)” and “Sociedad Espa˜ nola de Matem´ atica Aplicada (SEMA)”. We extend our gratitude to the “International Linear Algebra Society (ILAS)”, “Institut Universitari de Matem` atica Multidisciplin` aria (IMM)” and the ´ “Algebra Lineal, An´ alisis Matricial y Aplicaciones (ALAMA)” network for having endorsed this meeting. Finally, we would like to thank all of the participants for their attendance in spite of these hard times. Their presence here reasserts the success of this conference and we hope to see them successively in the future. Valencia, September 2009
Rafael Bru Sergio Romero-Viv´o
Organization
Program Committee Chairman Rafael Bru
Universitat Polit`ecnica de Val`encia, Spain
Members Georges Bastin Luca Benvenuti Vincent Blondel Rafael Cant´ o Carmen Coll Bart De Moor Alberto De Santis Elena De Santis Lorenzo Farina Stephane Gaubert Alessandro Giua Jean-Luc Gouz´e Diederich Hinrichsen Tadeusz Kaczorek Ulrich Krause Volker Mehrmann Ventsi Rumchev Maria Pia Saccomani Elena S´ anchez Jan H. van Schuppen Robert Shorten Anton A. Stoorvogel
Universit´e Catholique de Louvain, Belgium Universit` a di Roma “La Sapienza”, Italy Universit´e Catholique de Louvain, Belgium Universitat Polit`ecnica de Val`encia, Spain Universitat Polit`ecnica de Val`encia, Spain Katholieke Universiteit Leuven, Belgium Universit` a di Roma “La Sapienza”, Italy Universit` a dell’Aquila, Italy Universit` a di Roma “La Sapienza”, Italy INRIA, Ecole Polytechnique, France Universit` a di Cagliari, Italy INRIA Sophia Antipolis, France Universit¨ at Bremen, Germany Warsaw University of Technology, Poland Universit¨ at Bremen, Germany Technische Universit¨ at Berlin, Germany Curtin University of Technology, Australia Universit` a di Padova, Italy Universitat Polit`ecnica de Val`encia, Spain CWI, Amsterdam, The Netherlands The Hamilton Institute, Co. Kildare, Ireland University of Twente, The Netherlands
VIII
Ana M. Urbano Maria Elena Valcher Paul Van Dooren Joseph Winkin
Organization
Universitat Polit`ecnica de Val`encia, Spain Universit` a di Padova, Italy Universit´e Catholique de Louvain, Belgium Universit´e Notre-Dame de la Paix, Belgium
Organizing Committee Bego˜ na Cant´ o Rafael Cant´ o Beatriz Ricarte Sergio Romero-Viv´o
Universitat Universitat Universitat Universitat
Polit`ecnica Polit`ecnica Polit`ecnica Polit`ecnica
de de de de
Val`encia, Val`encia, Val`encia, Val`encia,
Spain Spain Spain Spain
Additional Referees Esteban Bailo Gregory Batt Bego˜ na Cant´ o Bart De Schutter Zong Woo “Victor” Geem Josep Gelonch Bernd Heidergott Julien Hendrickx On´esimo Hern´andez-Lerma Alexandros Karatzoglou Jerzy Klamka Thomas G. Kurtz Hongwei Lin James H. Liu
Maya Mincheva Francisco Pedroche Juan Manuel Pe˜ na Harish Pillai Beatriz Ricarte Sergio Romero-Viv´o Bartek Roszak Boris Shapiro Hal L. Smith Juan Ramon Torregrosa Elena Virnik Yimin Wei Eva Zerz
Contents
Plenary Talks Reputation Systems and Nonnegativity . . . . . . . . . . . . . . . . . . . . . . Cristobald de Kerchove, Paul Van Dooren
3
Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paul Leonard Salceanu, Hal L. Smith
17
Reachability Analysis for Different Classes of Positive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maria Elena Valcher
29
Invited Sessions On the Positive LQ-Problem for Linear Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charlotte Beauthier, Joseph J. Winkin
45
The Importance of Being Positive: Admissible Dynamics for Positive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luca Benvenuti, Lorenzo Farina
55
Detectability, Observability, and Asymptotic Reconstructability of Positive Systems . . . . . . . . . . . . . . . . . . . . . . . Tobias Damm, Cristina Ethington
63
Stability Radii of Interconnected Positive Systems with Uncertain Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diederich Hinrichsen
71
X
Contents
Linear Operators Preserving the Set of Positive (Nonnegative) Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Olga M. Katkova, Anna M. Vishnyakova
83
Convergence to Consensus by General Averaging . . . . . . . . . . . . Dirk A. Lorenz, Jan Lorenz
91
Stability and D-stability for Switched Positive Systems . . . . . . 101 Oliver Mason, Vahid S. Bokharaie, Robert Shorten On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Pham Huu Anh Ngoc Eigenvalue Localization for Totally Positive Matrices . . . . . . . . 123 Juan Manuel Pe˜ na Positivity Preserving Model Reduction . . . . . . . . . . . . . . . . . . . . . . 131 Timo Reis, Elena Virnik The Minimum Energy Problem for Positive Discrete-Time Linear Systems with Fixed Final State . . . . . . . . . . . . . . . . . . . . . . . 141 Ventsi Rumchev, Siti Chotijah A Rollout Algorithm for Multichain Markov Decision Processes with Average Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Tao Sun, Qianchuan Zhao, Peter B. Luh Analysis of Degenerate Chemical Reaction Networks . . . . . . . . 163 Markus Uhr, Hans-Michael Kaltenbach, Carsten Conradi, J¨ org Stelling k-Switching Reachability Sets of Continuous-Time Positive Switched Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Maria Elena Valcher
Contributed Papers Inverse-Positive Matrices with Checkerboard Pattern . . . . . . . . 185 Manuel F. Abad, Mar´ıa T. Gass´ o, Juan R. Torregrosa Some Remarks on Links between Positive Invariance, Monotonicity, Strong Lumpability and Coherency in Max-Plus Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Mourad Ahmane, Laurent Truffet
Contents
XI
Stability Analysis and Synthesis for Linear Positive Systems with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Mustapha Ait Rami Linear Programming Approach for 2-D Stabilization and Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Mohammed Alfidi, Abdelaziz Hmamed, Fernando Tadeo An Algorithmic Approach to Orders of Magnitude in a Biochemical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Eric Benoˆıt, Jean-Luc Gouz´e Structural Identifiability of Linear Singular Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Bego˜ na Cant´ o, Carmen Coll, Elena S´ anchez On Positivity of Discrete-Time Singular Systems and the Realization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Rafael Cant´ o, Beatriz Ricarte, Ana M. Urbano Multi-Point Iterative Methods for Systems of Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Alicia Cordero, Jos´e L. Hueso, Eulalia Mart´ınez, Juan R. Torregrosa Identifiability of Nonaccessible Nonlinear Systems . . . . . . . . . . . 269 Leontina D’Angi , Maria Pia Saccomani, Stefania Audoly, Giuseppina Bellu Trajectory Tracking Control of a Timed Event Graph with Specifications Defined by a P-time Event Graph . . . . . . . . . . . . . 279 Philippe Declerck, Abdelhak Guezzi Tropical Scaling of Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . 291 St´ephane Gaubert, Meisam Sharify Scrutinizing Changes in the Water Demand Behavior . . . . . . . . 305 Manuel Herrera, Rafael P´erez-Garc´ıa, Joaqu´ın Izquierdo, Idel Montalvo Characterization of Matrices with Nonnegative Group-Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Alicia Herrero, Francisco J. Ram´ırez, N´estor Thome Robust Design of Water Supply Systems through Evolutionary Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Joaqu´ın Izquierdo, Idel Montalvo, Rafael P´erez-Garc´ıa, Manuel Herrera
XII
Contents
Applications of Linear Co-positive Lyapunov Functions for Switched Linear Positive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Florian Knorn, Oliver Mason, Robert Shorten A Problem in Positive Systems Stability Arising in Topology Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Florian Knorn, Rade Stanojevic, Martin Corless, Robert Shorten Control of Uncertain (min,+)-Linear Systems . . . . . . . . . . . . . . . . 349 Euriell Le Corronc, Bertrand Cottenceau, Laurent Hardouin On a Class of Stochastic Models of Cell Biology: Periodicity and Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Ivo Marek Implementation of 2D Strongly Autonomous Behaviors by Full and Partial Interconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Diego Napp Avelli, Paula Rocha Ordering of Matrices for Iterative Aggregation - Disaggregation Methods . . . . . . . . . . . . . . . . . . . . . . 379 Ivana Pultarov´ a The Positive Servomechanism Problem under LQcR Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Bartek Roszak, Edward J. Davison Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Reputation Systems and Nonnegativity Cristobald de Kerchove and Paul Van Dooren
Abstract. We present a voting system that is based on an iterative method that assigns a reputation to n + m items, n objects and m raters, applying some filter to the votes. Each rater evaluates a subset of objects leading to an n × m rating matrix with a given sparsity pattern. From this rating matrix a formula is defined for the reputation of raters and objects. We propose a natural and intuitive nonlinear formula and also provide an iterative algorithm that linearly converges to the unique vector of reputations and this for any rating matrix. In contrast to classical outliers detection, no evaluation is discarded in this method but each one is taken into account with different weights for the reputations of the objects. The complexity of one iteration step is linear in the number of evaluations, making our algorithm efficient for large data set.
1 Introduction Many measures of reputation have been proposed under the names of reputation, voting, ranking or trust systems and they deal with various contexts ranging from the classification of football teams to the reliability of each individual in peer to peer systems. Surprisingly enough, the most used method for reputation on the Web amounts simply to average the votes. In that case, the reputation is, for instance, the average of scores represented by 5 stars in YouTube, or the percentage of positive transactions in eBay. Therefore such a method trusts evenly each rater of the system. Besides this method, many other algorithms exploit the structure of networks generated by the votes: raters and evaluated items are nodes connected by votes. A great part of these methods use efficient eigenvector based techniques or trust Cristobald de Kerchove and Paul Van Dooren Universit´e catholique de Louvain (UCL), Department of Applied Mathematics, Avenue Georges Lemaˆıtre, 4 B-1348 Louvain-la-Neuve Belgium, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 3–16. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
4
C. de Kerchove and P. Van Dooren
propagation over the network to obtain the reputation of every node [7, 9, 13–17]. They can be interpreted as a distribution of some reputation flow over the network where reputations satisfy some transitivity: you have a high reputation if you have several incoming links coming from nodes with a high reputation. The average method, the eigenvector based techniques and trust propagation may suffer from noise in the data and bias from dishonest raters. For this reason, they are sometimes accompanied by statistical methods for spam detection [10, 19], like in the context of web pages trying to boost their PageRank scores by adding artificial incoming links [2, 8]. Detected spam is then simply removed from the data. This describes the three main strategies for voting systems: simple methods averaging votes where raters are evenly trusted, eigenvector based techniques and trust propagation where reputations directly depend on reputations of the neighbours, and finally statistical measures to classify and possibly remove some of the items. Concerning the Iterative Filtering (IF) systems which we introduce here, we will make the following assumption: Raters diverging often from other raters’ opinion are less taken into account. We label this the IF-property and will formally define it later on. This property is at the heart of the filtering process and implies that all votes are taken into account, but with a continuous validation scale, in contrast with the direct deletion of outliers. Moreover, the weight of each rater depends on the distance between his votes and the reputation of the objects he evaluates: typically weights of random raters and outliers decrease during the iterative filtering. The main criticism one can have about the IF-property is that it discriminates “marginal” evaluators, i.e., raters who vote differently from the average opinion for many objects. However, IF systems may have different basin of attraction, each corresponding to a group of people with a coherent opinion. Votes, raters and objects can appear, disappear or change making the system dynamical. This is for example the case when we consider a stream of news like in [5]: news sources and articles are ranked according to their publications over time. Nowadays, most sites driven by raters involve dynamical opinions. For instance, the blogs, the site Digg and the site Flickr are good places to exchange and discuss ideas, remarks and votes about various topics ranging from political election to photos and videos. We will see that IF systems allow to consider evolving voting matrices and then provide time varying reputations.
2 Iterative Filtering Systems We first consider the case where the votes are fixed, i.e., the voting matrix does not change over time, and all objects are evaluated by all raters, i.e., the voting matrix is full. With these assumptions, we present the main properties of IF systems and then we restrict ourselves to the natural case of quadratic IF systems where the reputations are given by a linear combination of the votes and the weights of the raters are based on the Euclidean distance between the reputations and the votes.
Reputation Systems and Nonnegativity
5
Let X ∈ n×m be the voting matrix, r ∈ n be the reputation vector of the objects and w ∈ m be the weight vector of the raters. The entry Xi j is the vote to object i given by rater j and the vector x j , the jth column of X, represents the votes of rater j: X = [x1 . . . xm ] . The bipartite graph formed by the objects, the raters and their votes is represented by the n × m adjacency matrix A, i.e., Ai j = 1 if object i is evaluated by rater j, and 0 otherwise. For the sake of simplicity, we assume in this section that every object has been evaluated by all raters Ai j = 1 for all i, j.
(1)
The general case, where the bipartite graph is not complete, will be handled later. The belief divergence d j of rater j is the normalized distance between his votes and the reputation vector r (for a particular choice of norm) 1 d j = x j − r2 . n
(2)
Let us already remark that when the bipartite graph is not complete, i.e., Eq. (1) is not satisfied, then the number of votes varies from one rater to another. Therefore the normalization of the belief divergence d j in Eq. (2) will change depending on this number. Before introducing IF systems, we define the two basic functions of these systems: (1) the reputation function
F:
n
→
m
: F(w) = r,
that gives the reputation vector depending on the weights of the raters and implicitly on the voting matrix X; (2) The filtering function
G:
m
→
n
: G(d) = w,
that gives the weight vector for the raters depending on the belief divergence d of each rater defined in Eq. (2). We formalize the so-called IF-property described in the introduction that claims that raters diverging often from the opinion of other raters are less taken into account. We will make the reasonable assumption that raters with identical belief divergence receive equal weights. Hence, we can write ⎤ ⎡ g(d1 ) ⎥ ⎢ (3) G(d) = ⎣ ... ⎦ . g(dm ) We call the scalar function g the discriminant function associated with G.
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C. de Kerchove and P. Van Dooren
A filtering function G satisfies the IF-property if its associated discriminant function g is positive and decreases with d. Therefore, the IF-property merely implies that a decrease in belief divergence d j for any rater j corresponds to a larger weight w j . Eq. (3) indicates that every rater has the same discriminant function g, but we could also consider personalized functions g j penalizing differently the raters. In [4] three choices of function g are shown to have interesting properties g(d) = d −k , g(d) = e−k d ,
(4) (5)
g(d) = 1 − k d.
(6)
All discriminant function g are positive and decrease with d for positive k and therefore satisfy the IF-property. However k must be small enough to keep g positive in Eq. (6) and hence to avoid negative weights. Definition 1. IF systems are systems of equations in the reputations rt of the objects and the weights wt of the raters that evolve over discrete time t according to the voting matrix X rt+1 = F(wt ), wt+1 = G(dt+1 )
(7) 1 with dt+1 = x j − rt+1 2 j n
(8)
for j = 1, . . . , m and some initial vector of weights w0 . Definition 1 does not imply any convergence properties, nor robustness to initial conditions. The system (7-8) can have several converging solutions and it allows the existence of cycles in the iterative processes. The fixed points of (7-8) satisfy r∗ = F(w∗ ), w∗ = G(d∗ )
(9) 1 with d ∗j = x j − r∗ 2 n
(10)
for j = 1, . . . , m. Let us remark that IF systems can be interpreted as a particular iterative search method to find the stable fixed points of Eq. (9-10). IF systems are a simple iterative scheme for this system with the advantage to be easily extended to take into account dynamical voting matrices Xt with t ≥ 0. In this paper, we focus on IF systems where we fix the reputation function F appearing in Eq. (7,9) and the norm . given in the definition of the belief divergence in Eq. (2). Definition 2. Quadratic IF systems are IF systems where the reputation function F and the belief divergence are respectively given by
Reputation Systems and Nonnegativity
7
w , 1T w
(11)
1 d = (XT − 1rT )◦2 1, n
(12)
F(w) = X
where 1 is the vector of all 1’s and (XT − 1rT )◦2 is the componentwise product (XT − 1rT ) ◦ (XT − 1rT ). In that definition, the reputation function F(w) is naturally given by taking the weighted average of the votes and the belief divergence d (given in the matrix form) is defined using the Euclidian norm. Therefore Eq. (12) are quadratic equations in r and amount to consider an estimate of the variances of the votes for every rater according to a given reputation vector r. For any positive vector w, the reputation vector r then belongs to the polytope P = {r ∈
n
|r =
m
m
j=1
j=1
∑ λ j x j with ∑ λ j = 1 and λ j ≥ 0}.
(13)
From Eq. (11), the iterations and the fixed point in Eq. (7,9) are given by quadratic equations in r and w rt+1 (1T wt ) = Xwt , ∗
∗
∗
r (1 w ) = Xw . T
(14) (15)
The next theorem establishes the correspondence between the iterations of quadratic IF systems and some steepest descent methods minimizing some energy function. The fixed points in Eq. (14,15) are then the stationary points of that energy function. Theorem 1. (see [4]) The fixed points of quadratic IF systems with integrable discriminant function g, are the stationary points of the energy function E(r) =
m
∑
d j (r)
g(u) du,
(16)
j=1 0
where d j is the belief divergence of rater j that depends on r. Moreover one iteration step in quadratic IF systems corresponds to a steepest descent direction with a particular step size (17) rt+1 = rt − α t ∇r E(rt ), with α t =
n . 2(1T wt )
3 Iterative Filtering with Affine Discriminant Function We look at the quadratic IF system with the discriminant function g defined in Eq. (6) where the iterations are given by
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C. de Kerchove and P. Van Dooren
wt , 1T wt wt+1 = G(dt+1 ) = 1 − k dt+1, rt+1 = F(wt ) = X
(18) (19)
starting with equal weights w0 = 1. By substituting w, the fixed point of the system is given by a system of cubic equations in r∗ k (X − r∗ 1T )(1 − (XT − 1(r∗ )T )◦2 1) = 0, n
(20)
with r∗ in the polytope P defined in Eq. (13). Theorem 2 claims that r∗ is unique in P if k is such that the weights are strictly positive for all vectors of reputations r ∈ P. This result uses the associated energy function that we define for affine IF systems.
3.1 The Energy Function The energy function in Eq. (16) associated with system (18,19) is given by E(r) = −
1 T w w + constant, 2k
(21)
where w depends on r according the function G(r). We will see later that this energy function decreases with the iterations, i.e., (E(rt ))t≥0 decreases, and under some assumption on k, it converges to the unique minimum. The iterations in system (18,19) can be written as a particular minimization step on the function E,
1 rt+1 = argmin − G(r)T G(rt ) . r 2k Therefore, we have for all t that (wt+1 )T (wt ) ≥ (wt )T (wt ).
3.2 Uniqueness The following theorem proves that the stable point of quadratic IF systems with g defined in Eq. (6) is unique, under some condition on parameter k. This result follows directly from the energy function E that is a fourth-order polynomial equation. Theorem 2. (see [4]) The system (18,19) has a unique fixed point r∗ in P if k < min d−1 ∞ . r∈P
Reputation Systems and Nonnegativity
9
The Energy function
The Energy function
−4.5
−5.5
−7 0
−.5
−1 0 0.5 1
1
0.8
0.6
0.4
0.2
0.5
0
1
1
(a)
0.8
0.6
0.4
0.2
0
(b)
The Energy function
The Energy function
0
−.4
−.8 0
−0.5
−1 0 0.5 1
1
0.8
0.6
0.4
0.2
0
0.5 1
(c)
1
0.8
0.6
0.4
0.2
0
(d)
Fig. 1 Four energy functions with two objects and increasing values of k. We have in the unit square: (a) a unique minimum; (b) a unique minimum but other stationary points are close to the boundary; (c) a unique minimum and other stationary points; (d) a unique maximum.
3.3 Convergence of the Method We analyze the convergence of system (18,19) that reaches the minimum of the energy function E in P. Let rt and rt+1 be two subsequent points of the iterations given by some search method. Then the next point rt+1 is obtained by choosing a vector v and a scalar γ such that rt+1 = rt + γ v.
(22)
This corresponds to some line search on the scalar energy function e(y) = E(rt + y v)
(23)
that is a polynomial of degree 4. We have that e(0) is the energy at rt and e(γ ) is the energy at rt+1 . Finally it is useful for the sequel to define the scalar that minimizes e given by β = arg min E(rt + y v). (24) t y with r +y v∈P
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C. de Kerchove and P. Van Dooren
System (18,19) provides a steepest descent method with a particular step size. The direction v and the scalar γ in Eq. (22) are v = −∇r E(rt )
and
γ = αt =
n , 2 (1T wt )
so that we recover Eq. (17). This particular step size α t can be compared to the step size β that minimizes E in the same direction given in Eq. (24). We have that the particular step size α t is generally smaller than β in numeric simulations, meaning that the step stops before reaching the minimum of the energy function E in the direction v. The sequence (E(rt ))t≥0 can be shown to decrease so that we have the following convergence result. Theorem 3. (see [4]) The steepest descent method given by system (18,19) converges to the unique fixed point in P if k < min d−1 ∞ . r∈P
There exist greater values of k such that the minimum of E remains unique and the previous methods converge to this minimum. By increasing k, we allow the maxima of E to appear in the polytope P, see Fig. 1(c). Then, we need to verify during the iterations if (rt ) remains in the basin of attraction of E. Theorem 4. (see [4]) If the energy function E in Eq. (21) has a minimum, then the system (18,19) is locally convergent and its asymptotic rate of convergence is linear. Let us remark that for a singular matrix X, the rate of convergence will be faster. In particular, when X is a rank 1 matrix, we have X = r∗ 1T (every object receives m identical votes from the raters) and the method converges in one step. When we take greater values of k maxima of the function E may appear in P. However if the sequence (1T wt ) remains positive, the sequence (E(rt )) remains decreasing and converges to a stationary point of E. In order to avoid saddle points and maxima, we need to avoid to reach the minimum. The idea of increasing k is to make the discriminant function g more penalizing and therefore to have a better separation between honest and dishonest raters. We refer to [4] for more details on this.
4 Sparsity Pattern and Dynamical Votes This section extends some previous results to the case where the voting matrix has some sparsity pattern, that is when an object is not evaluated by all raters. Moreover we analyze dynamical voting matrices representing votes that evolve over time.
Reputation Systems and Nonnegativity
11
4.1 Sparsity Pattern In general, the structure of real data is sparse. We hardly find a set of raters and objects with a vote for all possible pairs. An absence of vote for object i from rater j will imply that the entry (i, j) of the matrix X is equal to zero, that is, by using the adjacency matrix A, if Ai j = 0, then Xi j = 0. These entries must not be considered as votes but instead as missing values. Therefore the previous equations presented in matrix form require some modifications that will include the adjacency matrix A. We write the new equations and their implications using the order of the previous section. Let us already mention that some theorems will be simply stated without proof. Whenever their extensions with an adjacency matrix A are straightforward. The belief divergence for IF systems in Eq. (2) becomes dj =
1 x j − a j ◦ r, nj
where a j is the jth column of the adjacency matrix A and n j is the jth entry of the vector n containing the numbers of votes given to each item, i.e., n = AT 1. On the other hand, the scalar n remains the total number of objects, i.e., the number of rows in A. Therefore, when A is full, then n = n1. Eq. (11-12) for quadratic IF systems can be replaced by the following ones: the reputation function, that remains the weighted average of the votes, is given in matrix form by [Xw] , F(w) = [Aw] [·] is the componentwise division. Let us remark that every entry of Aw must [·] be strictly positive. This means that every object is evaluated by at least one rater with nonzero weight. Then all possible vectors of reputations r are include in the polytope where
P¯ = {r ∈
n
| ri =
m
m
j=1
j=1
∑ λ j x j with ∑ λ j ai j = 1 and λ j ≥ 0}.
The third equation (12) for the belief divergence with the Euclidian norm is changed into T (X − AT ◦ 1rT )◦2 1 d= . (25) [AT 1]
12
C. de Kerchove and P. Van Dooren
With these modifications, the iterations and the fixed point in Eq. (7,9) are given by quadratic equations in r and w (A ◦ rt+1 1T ) wt = Xwt ∗ T
∗
∗
(A ◦ r 1 ) w = Xw .
(26) (27)
Hence we expect an energy function to exist and Theorem 1 is generalized by the following theorem. Theorem 5. (see [4]) The fixed points of quadratic IF systems with integrable discriminant function g, are the singular points of the energy function E(r) =
1 n
m
∑ nj
d j (r)
j=1
g(u) du,
(28)
0
where d j is the belief divergence of rater j that depends on r. Moreover one iteration step in quadratic IF systems corresponds to a dilated steepest descent direction with a particular step size (29) rt+1 = rt − α t ◦ ∇r E(rt ) with α t =
n [1] 2 [Awt ] .
The number of votes n j gives somehow a weight of importance for the mini d
mization of the surface 0 j g(u) du. Therefore a rater with more votes receives more attention in the minimization process.
4.2 Affine Quadratic IF Systems The system for the discriminant function g(d) = 1 − k d is given by rt+1 = F(wt ) =
[Xw] , [Aw]
wt+1 = G(dt+1 ) = 1 − k dt+1,
(30) (31)
with the belief divergence defined in Eq. (25). The energy function is given by E(r) = −
1 T w [w ◦ n] + constant, 2 kn
(32)
where w depends on r according to the function G(r). Theorem 2 remains valid for the system (30-31) and the arguments are similar. The steepest descent method adapted to the system (30-31) converges with the property that the sequence (E(rt )) decreases. The proofs are closely related to the ones presented in Theorems 3.
Reputation Systems and Nonnegativity
13
Fig. 2 Trajectory of reputations (circles) for a 5-periodic voting matrix
Theorem 6. (see [4]) The steepest descent method given by system (30,31) converges to the unique fixed point in P¯ if k < min d−1 ∞ . r∈P¯
The choice of k can be made larger to better separate honest from dishonest raters. Theorem 4 remains valid with a few modifications in its proof to take into account the adjacency matrix A. Theorem 7. (see [4]) If the energy function E in Eq. (32) has a minimum, then (30,31) is locally convergent and its asymptotic rate of convergence is linear. This section shows that most of the earlier analysis can still be applied when we introduce a sparsity pattern in the voting matrix.
4.3 Dynamical Votes We consider in this section the case of time-varying votes. Formally, we have discrete sequences (Xt )t≥0 , (At )t≥0
14
C. de Kerchove and P. Van Dooren
of voting matrices and adjacency matrices evolving over time t. Hence the IF system (7,8) takes into account the new voting matrix Xt+1 in the functions Ft+1 and Gt+1 that become time-dependent: rt+1 = Ft+1 (wt ), wt+1 = Gt+1 (dt+1 ). The system (30,31) for dynamical voting matrices is then given by (30,31) t+1 t X w t+1 t r = Ft+1 (w ) = t+1 t , [A w ] wt+1 = Gt+1 (dt+1 ) = 1 − k dt+1,
(33) (34)
(35) (36)
with the belief divergence dt+1 defined as in Eq. (25) after replacing X and r by Xt+1 and rt+1 . We already now that for subsequent constant matrices Xt with T1 ≤ t ≤ T2 , the iterations on rt and wt of system (35,36) tend to fixed vectors r∗ and w∗ provided that k is not too large. In [4] we give stronger results for the case of 2-periodic voting sequences.
5 Concluding Remarks The general definition of Iterative Filtering systems provides a new framework to analyze and evaluate voting systems. We emphasized the need for a differentiation of trusts between the raters unlike what is usually done on the Web. The originality of the approach lies in the continuous validation scale for the votes. Next, we assumed that the set of raters is characterized by various possible behaviors including raters who are clumsy or partly dishonest. However, the outliers being in obvious disagreement with the other votes remain detectable by the system as shown in the simulations in the cases of alliances, random votes and spammers. Our paper focuses on the subclass of quadratic IF systems and we show the existence of an energy function that allows us to link a steepest descent to each step of the iteration. It then follows that the system minimizes the belief divergence according to some norm defined from the choice of the discriminant function. This method was illustrated in [4] using two data sets: (i) the votes of 43 countries during the final of the EuroVision 2008 and (ii) the votes of 943 movie lovers in the website of MovieLens. It was shown that the IF method penalizes certain types of votes. In the first set of data, this yielded a difference in the ranking used by Eurovision and the ranking obtained by our method, in the sense that countries trading votes with e.g. neighboring countries, would get a smaller weight. The second set of data was used to verify the desired property mentioned in the introduction: raters diverging often from other raters’ opinion are less taken into account. We see two application areas of voting systems: first, the general definition of IF systems offers the possibility to analyze various systems depending on the context
Reputation Systems and Nonnegativity
15
and the objectives we aim for; second, the experimental tests and the comparisons are crucial to validate the desired properties (including dynamical properties) and to discuss the choice of the IF systems. Acknowledgements. This paper presents research results of the Belgian Programme on Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office, and a grant Action de Recherche Concert´ee (ARC) of the Communaut´e Franc¸aise de Belgique. The scientific responsibility rests with its authors.
References 1. Akerloff, G.: The Market for Lemons: Quality Uncertainty and the Market Mechanism. Quaterly Journal of Economics 84, 488–500 (1970) 2. Baeza-Yates, R., Castillo, C., L´opez, V.: PageRank Increase under Different Collusion Topologies. In: First International Workshop on Adversarial Information Retrieval on the Web (2005), http://airweb.cse.lehigh.edu/2005/baeza-yates.pdf 3. de Kerchove, C., Van Dooren, P.: Reputation Systems and Optimization. Siam News (March 14, 2008) 4. de Kerchove, C., Van Dooren, P.: Iterative Filtering in Reputation Systems (submitted, 2009) 5. Del Corso, G.M., Gull´ı, A., Romani, F.: Ranking a stream of news. In: Proceedings of the 14th international conference on World Wide Web (2005) 6. Ginsburgh, V., Noury, A.: Cultural Voting. The Eurovision Song Contest. Mimeo (2004) 7. Guha, R., Kumar, R., Raghavan, P., Tomkins, A.: Propagation of Trust and Distrust. In: Proceedings of the 13th International Conference on World Wide Web, pp. 403–412 (2004) 8. Gy¨ongyi, Z., Garcia-Molina, H.: Link spam alliances. In: VLDB 2005: Proceedings of the 31st international conference on Very large data bases, pp. 517–528 (2005) 9. Kamvar, S., Schlosser, M., Garcia-molina, H.: The Eigentrust Algorithm for Reputation Management in P2P Networks. In: Proceedings of the 12th International Conference on World Wide Web, pp. 640–651 (2003) 10. Kotsovinos, E., Zerfos, P., Piratla, N.M., Cameron, N., Agarwal, S.: Jiminy: A Scalable Incentive-Based Architecture for Improving Rating Quality. In: Stølen, K., Winsborough, W.H., Martinelli, F., Massacci, F. (eds.) iTrust 2006. LNCS, vol. 3986, pp. 221–235. Springer, Heidelberg (2006) 11. Laureti, P., Moret, L., Zhang, Y.-C., Yu, Y.-K.: Information Filtering via Iterative Refinement. EuroPhysic Letter 75, 1006–1012 (2006) 12. McLachlan, G., Krishnan, T.: The EM algorithm and extensions. John Wiley & Sons, New York (1996) 13. Mui, L., Mohtashemi, M., Halberstadt, A.: A Computational Model of Trust and Reputation. In: Proceedings of the 35th Annual Hawaii International Conference, pp. 2431– 2439 (2002) 14. O’Donovan, J., Smyth, B.: Trust in recommender systems. In: Proceedings of the 10th International Conference on Intelligent User Interfaces, pp. 167–174 (2005) 15. Page, L., Brin, S., Motwani, R., Winograd, T.: The PageRank Citation Ranking: Bringing Order to the Web. Stanford Digital Library Technologies Project (1998)
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16. Richardson, M., Agrawal, R., Domingos, P.: Trust Management for the Semantic Web. In: Fensel, D., Sycara, K., Mylopoulos, J. (eds.) ISWC 2003. LNCS, vol. 2870, pp. 351– 368. Springer, Heidelberg (2003) 17. Theodorakopoulos, G., Baras, J.: On Trust Models and Trust Evaluation Metrics for Ad Hoc Neworks. IEEE Journal on Selected Areas in Communications 24(2), 318–328 (2006) 18. Yu, Y.-K., Zhang, Y.-C., Laureti, P., Moret, L.: Decoding information from noisy, redundant, and intentionally distorted sources. Physica A 371(2), 732–744 (2006) 19. Zhang, S., Ouyang, Y., Ford, J., Make, F.: Analysis of a Lowdimensional Linear Model under Recommendation Attacks. In: Proceedings of the 29th annual International ACM SIGIR conference on Research and development in information retrieval, pp. 517–524 (2006)
Lyapunov Exponents and Uniform Weak Normally Repelling Invariant Sets Paul Leonard Salceanu and Hal L. Smith
Abstract. Let M be a compact invariant set contained in a boundary hyperplane of the positive orthant of n for a discrete or continuous time dynamical system defined on the positive orthant. Using elementary arguments, we show that M is uniformly weakly repelling in directions normal to the boundary in which M resides provided all normal Lyapunov exponents are positive. This result is useful in establishing uniform persistence of the dynamics.
1 Introduction Dynamical systems models in population biology are typically defined on the nonnegative cone in Euclidean space. In order to establish persistence of some or all components (species) in the model, it is often necessary to show that a compact invariant set on the boundary of the cone is an isolated invariant set and that it is repelling, at least in some directions normal to M. See [4, 5, 7, 15, 19, 20] for recent work in the theory of persistence, sometimes called permanence. In this paper, building on the work of [4, 15] and [11, 14], we show that Lyapunov exponents can be used to establish the requisite repelling properties for both discrete and continuous time systems. This is well known when M is a fixed point or periodic orbit but not so when the dynamics on M is more complicated. We use only elementary arguments rather than appealing to the multiplicative ergodic theorem [1, 2, 4, 15]. This extends our earlier work in [12–14] which covered only the discrete case. The use of Lyapunov exponents in the study of biological models was pioneered by Metz [8], Metz et. al. [9], who proposed that the dominant Lyapunov exponent gives the best measure of invasion fitness, and by Rand et. al. [10] who used it to characterize the invasion “speed” of a rare species. See also the more recent review Paul Leonard Salceanu and Hal L. Smith School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA, e-mail:
[email protected], e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 17–27. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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P.L. Salceanu and H.L. Smith
by Ferriere and Gatto [3] which deals with computational aspects. Roughly, a positive dominant Lyapunov exponent corresponding to a potential invading species in the environment set by a resident species attractor implies that the invader can successfully invade. Our results will give a mathematically rigorous interpretation of this for the nonlinear dynamics. Ashwin et al. [2] use “normal” Lyapunov exponents and invariant measures to answer the following question: if f : M → M is a smooth map on a smooth finite dimensional manifold, N is a lower dimensional submanifold for which f (N) ⊆ N, and A ⊆ N is an attractor for f|N , is A an attractor for f , or it is an unstable saddle?
2 Main Results Due to our need to use both subscripts and occasionally superscripts for sequences, we adopt the notation x = (x(1) , ..., x(m) )T ∈ m . Let |x| = ∑ |x(i) | denote the norm m: on m and d(z, M) for the distance of z ∈ m to M ⊂ m . Denote by m + = {x ∈ x(i) ≥ 0, ∀i} the nonnegative cone in m . We write x ≥ 0 when x ∈ m + ; if A = (ai j ) is an n × n matrix, then A ≥ 0 if ai j ≥ 0 for all i, j. Observe that |x + y| = |x| + |y| m for vectors x, y ∈ m + . We let Z+ = + . We consider the discrete dynamical system zn+1 = F(zn ), z0 ∈ Z+
(1)
and the continuous dynamical system z (t) = F(z(t)), z0 := z(0) ∈ Z+
(2)
on the nonnegative cone Z+ . It is assumed that (1) and (2) generate a semi-dynamical system on Z+ . In case of (1), F : Z+ → Z+ is continuous; in case (2), F : Z+ → m satisfies z ∈ Z+ , z(i) = 0 ⇒ F (i) (z) ≥ 0 and sufficient regularity properties such that solutions of (2) exist and are unique. We assume that m = p + q, p, q ≥ 1 and that Z+ is decomposed as follows z = (x, y) ∈ Z+ ≡
m +
=
p +×
q +
Compatible with this decomposition, assume that F(z) = ( f (z), g(z)). Define X = {z = (x, y) ∈ Z+ : y = 0}. We assume: X and Z+ \ X are positively invariant sets.
(3)
Positive invariance of X for both (1) and (2) means that g(x, 0) = 0, (x, 0) ∈ X. If F satisfies additional smoothness hypotheses, then it would follow from positive invariance of X that (1) and (2) can be expressed as
Lyapunov Exponents and Normal Repellers
and, respectively as
19
xn+1 = f (zn ) yn+1 = A(zn )yn
x = f (z) y = A(z)y
(4)
(5)
where the matrix function z → A(z) is continuous and satisfies: A(z) ≥ 0,
z∈X
(6)
in case of (4), and Ai j (z) ≥ 0, i = j,
z∈X
(7)
in case of (5). Rather than assume the required smoothness of F, we simply assume hereafter that (6) holds for (4) and that (7) holds for (5). Let + denote either + or + . When we write t ∈ + , that means we consider both discrete and continuous cases. To make the notation more general, let also zt , yt etc. denote z(t), y(t) etc., when t ∈ + . Let (φ (t))t∈ + be the dynamical system generated by (4) (for t ∈ + ) or by (5) (for t ∈ + ), respectively. Let O + (z) := {φ (t, z) : t ∈ + }, which we will refer to as the positive orbit through z. Let I denote the identity matrix. Let P(n, z) and P(t, z) denote the fundamental matrix solutions for un+1 = A(φ (n, z))un
(8)
v (t) = A(φ (t, z))v(t).
(9)
P(n + 1, z) = A(φ (n, z))P(n, z), P(0, z) = I
(10)
and for They satisfy:
for discrete time and d P(t, z) = A(φ (t, z))P(t, z), P(0, z) = I dt for continuous time. In either case, it follows from (6), (7) that P(t, z) ≥ 0, ∀ z ∈ X, ∀ t ∈ + .
(11)
(12)
Let M ⊆ X be compact and positively invariant set. We envision that in typical applications, M will be an invariant set in the interior of the face X of the cone Z+ . In this paper, we will focus on the behavior of solutions near M in Z+ \ X. Following Arnold [1], P(n, z) (or P(t, z)) is a matrix co-cycle generated by (8) (or by (9)). It is trivial to check that P has the following (co-cycle) property: P(t2 , φ (t1 , z))P(t1 , z) = P(t1 + t2 , z), ∀ z ∈ + , ∀ t1 ,t2 ∈ + .
(13)
20
P.L. Salceanu and H.L. Smith
Hereafter, when we take t ∈ + , we refer to (4), and when we take t ∈ + , we refer to (5). Following [1, 2, 6], for any z ∈ M and η ∈ q we define the normal Lyapunov exponent λ (z, η ) as 1 λ (z, η ) = lim sup ln |P(t, z)η |, t ∈ + . t→∞ t
(14)
As noted in [1], λ (z, η ) ∈ {−∞} ∪ and λ (z, η ) = λ (z, aη ), ∀ a ∈ \ {0}. We only consider the case that η ∈ q+ because in that case z + (0, η ) ∈ Z+ and (0, η ) represents a normal vector to M at z. The co-cycle property (13) can be used to show that λ (z, η ) = λ (φ (s, z), P(s, z)η ), s ≥ 0. (15) Definition 1. We call the compact positively invariant set M a uniformly weak normally repelling set if there exists ε > 0 such that lim sup d(φ (t, z), M) > ε , ∀z ∈ Z+ \ X. t→∞
Equivalently, in view of (3), there exists a neighborhood V of M in Z+ such that / V. ∀z ∈ V \ X, ∃ t = t(z) > 0, φ (t, z) ∈ We stress that M may be an attractor relative to the dynamics restricted to the positively invariant set X but we are concerned with the behavior of solutions near M in the positively invariant set Z+ \ X. In [14], we used the terminology “M a uniformly weak repeller” for the definition above; we believe the current terminology gives a more accurate description. The adjective “uniform” reflects that ε is independent of z; “weak” reflects that limit superior, rather than limit inferior, appears in the definition; “normal” indicates that we are only interested in the behavior of solutions in Z+ \ X. First, we give a lemma adapted from [11, 14] that gives an alternative formulation for the “positivity” of Lyapunov exponents. Let U = {η ∈
q +
: |η | = 1}.
Lemma 1. Let K ⊂ X be compact. Assume that ∀ (z, η ) ∈ K × U, ∃ + \ {0} τ = τ (z, η ) such that |P(τ , z)η | > 1.
(16)
Then ∃ c > 1, ∃ V a bounded neighborhood of K in Z+ , such that if L ⊆ V is a positively invariant set, then L ⊂ X and ∀ (z, η ) ∈ L × U, ∃ (ν p ) p ⊆ + , ν p → ∞, |P(ν p , z)η | > c p , ∀ p ≥ 1.
(17)
If, in addition, K is positively invariant, then (16) is equivalent to
λ (z, η ) > 0, ∀ (z, η ) ∈ K × U.
(18)
Lyapunov Exponents and Normal Repellers
21
Proof. Let W = K ×U (so W is compact) and wˆ = (ˆz, ηˆ ) ∈ K ×U. From (16) we have that there exists τˆ = τˆ (ˆz, ηˆ ) ∈ + \ {0} such that |P(τˆ , zˆ)ηˆ | > 1. The function (z, η ) → |P(τˆ , z)η | being continuous, there exist δwˆ > 0, cwˆ > 1 such that |P(τˆ , z)η | > cwˆ , ∀ w = (z, η ) ∈ Bδwˆ (w) ˆ := {w˜ ∈ Z+ × U| |w˜ − w| ˆ < δwˆ }.
(19)
Since W is compact, there exists a finite set {w1 , ..., wk } ⊆ W such that W ⊂ C := ∪ki=1 Bδ i (wi ), where for every i = 1, ..., k, δwi is the quantity corresponding to wi , w coming from (19) (i.e., for every i = 1, ..., k, (19) is satisfied with wˆ replaced by wi ). To simplify notation, let τi := τ (wi ), δi := δwi , i = 1, ..., k. Also, let c := min cwi (hence c > 1) and τ = max τi . Thus, from (19) we have that
i
i
|P(τi , z)η | > c, ∀ w = (z, η ) ∈ Bδi (wi ), ∀ i = 1, ..., k.
(20)
Now let V ⊂ Z+ be a bounded neighborhood of K such that V × U ⊆ C and let L ⊆ V be positively invariant. We prove that L ⊂ X arguing by contradiction: suppose L \ X = 0. / Let a = (ax , ay ) ∈ L \ X. Since |ay | > 0, we can define α := ay /|ay |. Note that α ∈ U. We will show that ∃ (ν p ) p ⊂ + , ν p → ∞, such that |P(ν p , a)α | > c p , ∀ p ≥ 1.
(21)
by inductively constructing the sequence (ν p ) p . Thus, there exists i ∈ {1, ..., k} such that (a, α ) ∈ Bδi (wi ). Then, from (20) we have |P(ν1 , a)α | > c, where ν1 = τi . Now suppose |P(ν p , a)α | > c p for some p ≥ 1. Let α˜ = P(ν p , a)α /|P(ν p , a)α |. Since L \ X is positively invariant, φ (ν p , a) ∈ L \ X, hence φ (2) (ν p , a) > 0, where φ (2) (t, z) denotes the vector formed with the last q components of φ (t, z). So P(ν p , a)α =
1 1 (2) P(ν p , a)ay = φ (ν p , a) > 0. |ay | |ay |
Thus, α˜ ∈ U. There exists j ∈ {1, ..., k} such that (φ (ν p , a), α˜ ) ∈ Bδ j (w j ). Then again, from (20) we have |P(τ j , φ (ν p , a))α˜ | > c, which implies |P(τ j , φ (ν p , a))P(ν p , a)α | > c p+1 . This means, using (13), that |P(ν p+1 , a)α | > c p+1 , where we define ν p+1 = ν p + τ j . Note that, by construction, ν p → ∞ as p → ∞. Hence (21) holds. Then, we have that |φ (2) (ν p , a)| = |P(ν p , a)ay | > c p |ay |, ∀ p ≥ 1, which implies that |φ (2) (ν p , a)| → ∞ as p → ∞. But this is a contradiction to L being bounded. Hence, L ⊂ X. Now (17) can be proved identically as for (21), using that L ⊂ X is positively invariant and that P(t, z) ≥ 0, ∀ z ∈ X, ∀ t ∈ + .
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P.L. Salceanu and H.L. Smith
Now assume that K is also positively invariant. The implication (18) ⇒ (16) is trivial. For the converse, using (17) and the fact that ν p ≤ pτ , ∀ p ≥ 1, we have, for all (z, η ) ∈ K × U, that |P(ν p , z)η |1/ν p > c p/ν p ≥ c1/τ ⇒ Hence
1 1 ln |P(ν p , z)η | > ln c, ∀ p ≥ 1. νp τ
1 1 λ (z, η ) = lim sup ln(|P(t, z)η |) ≥ ln c > 0. τ t→∞ t
This completes our proof.
In the next result we establish sufficient conditions for M to be a uniformly weak normally repelling set. Let
Ω (M) = ∪z∈M ω (z),
(22)
where ω (z) represents the omega limit set of z. Theorem 1. Let M ⊂ X be a nonempty compact and positively invariant. M is a uniformly weak normally repelling set if
If and
λ (z, η ) > 0, ∀ (z, η ) ∈ M × U.
(23)
∀ (z, η ) ∈ M × U, ∀ t ∈ + , P(t, z)η = 0
(24)
λ (z, η ) > 0, ∀(z, η ) ∈ Ω (M) × U
(25)
then (23) holds. Proof. First we show that (23) implies that M is a uniformly weak normally repelling set. For this, we argue by contradiction: suppose M is not a uniformly weak normally repelling set. So, there exists a sequence (˜zm )m ⊆ Z+ \ X such that lim sup d(φ (t, z˜m ), M) < 1/m, ∀ m ≥ 1. t→∞
Hence there exists a sequence (τm )m ⊂ + such that, for each m ≥ 1, we have d(φ (t, z˜m ), M) < 1/m, ∀ t ≥ τm .
(26)
Let zm = (xm , ym ) = φ (τm , z˜m ). Using the positive invariance of Z+ \ X, we have that ym > 0, ∀ m ≥ 1.
(27)
From the semiflow property of φ and from (26) we get d(φ (t, zm ), M) < 1/m, ∀ t ∈ + , m ≥ 1.
(28)
Lyapunov Exponents and Normal Repellers
23
Using (23), we obtain from Lemma 1 (applied with K = M) that there exists V a bounded neighborhood of M in Z+ , having the property that any positively invariant set contained in V is a subset of X. Then there exists m ∈ such that Bm := {z ∈ Z+ |d(z, M) ≤ 1/m} is contained in V . The set L = {φ (n, zm )|n ≥ 0} is positively / But invariant and, according to (28), it is contained in Bm . Also (see (27)) L \ X = 0. this is a contradiction, according to Lemma 1. Hence, M is a uniformly weak repeller. Now we prove the final assertion. Let (a, α ) ∈ M × U. Using 2) and the fact that ω (a) ⊂ X is compact and invariant, we can again apply Lemma 1, now with K = ω (a). So let Va be a neighborhood of ω (a) and c > 1 as in the above mentioned lemma. Since φ (t, a) → ω (a) as t → ∞, there exists τa ∈ + such that φ (t, a) ∈ Va , ∀t ≥ τa . Let L = {φ (t, a)|t ≥ τa }. Then L is a positively invariant set contained in Va . Let α˜ = P(Na , a)α /|P(Na , a)α |. Note that α˜ is well defined, due to (24), and that α˜ ∈ U. So, from (17), there exists a sequence ν p → ∞ such that |P(ν p , φ (τa , a))α˜ | > c p , ∀ p ≥ 1. Thus, using (13) we get |P(ν p + τa , a)α | > c p |P(τa , a)α |, ∀ p ≥ 1. We can find a p large enough such that to have c p |P(τa , a)α | > 1. So, we proved that ∀ (z, η ) ∈ M × U, ∃ τ ∈ + \ {0} such that |P(τ , z)η | > 1, which is equivalent to (23), by Lemma 1. This completes our proof.
Note that (24) is automatically satisfied in the continuous case. In the discrete case, it is equivalent to A(z)η = 0, ∀ z ∈ M, ∀ η ∈ U. As it will be seen below, when the matrix A(z) satisfies stronger positivity conditions, then the Lyapunov exponents are independent of the unit vector η . Let ||A|| = sup{|Aξ | : |ξ | = 1} denote the norm of an n × n matrix. For matrices A, B, we write A ≤ B if ai j ≤ bi j , ∀i, j; inequality A 0 means all entries of A are positive. Proposition 1. Let z ∈ X have compact orbit closure O + (z). In the discrete case, assume that ∃N, P(n, z0 ) 0, n ≥ N, z0 ∈ O + (z). (29) In the continuous case, assume that
Then
A(z0 ) is irreducible, z0 ∈ O + (z).
(30)
1 λ (z, η ) = lim sup ln ||P(t, z)||, ∀ η ∈ U. t→∞ t
(31)
In particular, if (29), respectively (30), holds for each z ∈ M, then (31) holds for every z ∈ M. Proof. First we give the proof for the discrete case (t ∈ ). Let P(n) := P(n, z). We have that ∀ n ∈ , ∃ kn , pn ∈ , with 0 ≤ pn ≤ N − 1, such that n = kn N + pn . Let ˜ Bs := P(N, zsN ), ∀ s ≥ 0. Here, z0 = z. Hence Bs 0, ∀ s ≥ 0. Let P(n) = Bkn · · · B0 . ˜ So P(n) = P(pn , zkn N )P(n). First, we want to apply Theorem 3.4. in [16] for the
24
P.L. Salceanu and H.L. Smith
sequence of matrices B0 , ..., Bs , .... Since O + (z) is compact, it follows that there exist constant matrices C, D 0 such that D ≥ Bs ≥ C, ∀ s ≥ 0. Let δ = min(Ci j ) and γ = max(Di j ). So, the following hold:
i, j
i, j
a) min(Bs )i j ≥ δ > 0, ∀ s ≥ 0; i, j
b) max(Bs )i j ≤ γ < ∞. i, j
Thus, hypotheses of [16, Theorem 3.4] hold and (see exercise 3.6 in [16]) we have that ˜ li P(n) → ci j > 0 as n → ∞, (32) ˜ lj P(n) ˜ i the ith column of P(n). ˜ Then (32) for some ci j independent of l. Denote by P(n) implies that ˜ i| |P(n) lim (33) = ci j . ˜ j| n→∞ |P(n) Let ei ∈ U be the unit vector whose ith component equals one, and the other components are zero. Then, using (33), we get, for any i ∈ {1, ..., q}, that 1 1 ˜ λ (z, ei ) = lim sup ln |P(pn , zkn N )P(n)e i | ≤ lim sup (ln ||P(pn , zkn N )||+ n→∞ n n→∞ n ˜ i |. ˜ i |) = lim sup 1 ln |P(n) + ln |P(n) n→∞ n
(34)
On the other hand, 1 1 ˜ i| ln |P(kn N, z)ei | = lim sup ln |P(n) kn N n→∞ kn N n 1 ˜ i | = lim sup 1 ln |P(n) ˜ i |. ln |P(n) = lim sup n→∞ kn N n n→∞ n
λ (z, ei ) ≥ lim sup n→∞
(35)
Thus, from (34) and (35) we have that 1 ˜ i |. λ (z, ei ) = lim sup ln |P(n) n→∞ n But, for any i, j ∈ {1, ..., q} we have that ˜ i 1 ˜ i | = lim sup 1 ln |P(n) | |P(n) ˜ j| lim sup ln |P(n) ˜ j| n |P(n) n→∞ n n→∞ ˜ i| 1 1 |P(n) ˜ j| ln ln | P(n) = lim sup + ˜ j| n n |P(n) n→∞ 1 ˜ j |. = lim sup ln |P(n) n→∞ n Thus, let
(36)
Lyapunov Exponents and Normal Repellers
25
1 ˜ i |, ∀ i = 1, ..., q. c = λ (z, ei ) = lim sup ln |P(n) n→∞ n
(37)
q
Let η ∈ U. There exist p1 , ..., pq ∈ [0, 1] such that η = ∑ pi ei . Then, using (37), we i=1
obtain q 1 1 λ (z, η ) = lim sup ln |P(n)η | = lim sup ln | ∑ pi P(n)ei | n→∞ n n→∞ n i=1
≥ lim sup n→∞
1 q 1 pi ln |P(n)ei | = lim sup ln |P(n)ei | = c. ∑ n i=1 n→∞ n
(38)
On the other hand, we have q
λ (z, η ) = λ (z, ∑ pi ei ) ≤ max λ (z, ei ) = c, i=1
i=1,..,q
(39)
where we used the following two properties of Lyapunov exponents (see [1] page 114): 1) λ (z, η1 + η2 ) ≤ max{λ (z, η1 ), λ (z, η2 )}, and 2) λ (z, aη ) = λ (z, η ), ∀ a ∈ \ {0}. From (38) and (39) we obtain λ (z, η ) = c. It is clear that 1 c = λ (z, η ) ≤ lim sup ln ||P(n)||. n→∞ n
(40)
Now, we want to show the opposite inequality. Because all norms on a finite dimensional normed linear space are equivalent, there exist constants a, b > 0 such that a||B||1 ≤ ||B|| ≤ b||B||1 for all matrices B, where ||B||1 = max |Bi |. Then i
1 1 ˜ lim sup ln ||P(n)|| ≤ lim sup ln ||P(pn , zkn N )|| · ||P(n)|| n n n→∞ n→∞ ˜ = lim supn→∞ n1 ln ||P(n)|| 1 ˜ ≤ lim sup ln(b||P(n)|| 1) n→∞ n 1 1 ˜ ˜ k )||1 , ln ||P(n = lim sup ln ||P(n)|| 1 = lim k→∞ nk n→∞ n
(41)
for some sequence (nk )k ⊆ , nk → ∞ as k → ∞. There exists j ∈ {1, ..., q} such ˜ k )||1 = |P(n ˜ k ) j | for infinitely many k s. Hence, there exists a subsequence that ||P(n ˜ n˜ k )||1 = |P( ˜ n˜ k ) j |, ∀ k. Then, from (41) we have that (n˜ k )k of (nk )k such that ||P( 1 1 ˜ j | = c. ˜ n˜ k ) j | ≤ lim sup 1 ln |P(n) lim sup ln ||P(n)|| ≤ lim ln |P( n k→∞ n ˜ n→∞ n→∞ n k
(42)
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P.L. Salceanu and H.L. Smith
1 Now, (40) and (42) imply λ (z, η ) = lim sup ln ||P(n)||. Since η ∈ U was arbitrarily n→∞ n chosen, the proof for the discrete case is complete. Now let us consider the continuous case (t ∈ + ). Again, considering z ∈ X fixed, we can denote P(t, z), in short, by P(t). Denote by [·] : → the greatest integer function. The same argument used to prove (36) leads to
λ (z, η ) = lim sup t→∞
1 ln |P([t])η |. [t]
(43)
1 Let n := [t]. Then λ (z, η ) = lim sup ln |P(n)η |. Let Bn := P(1, zn ), ∀ n ∈ . Then n→∞ n P(n) = Bn−1 · · · B0 . Our hypotheses on matrix A(z) guarantee that Bn 0, ∀ n ≥ 0 (see [17, Theorem 1.1]). Now the same proof as for the discrete case, applied with ˜ = P(n)), carries over and leads to N = 1 (hence kn = n, pn = 0 and P(n) 1 1 lim sup ln |P(n)η | = lim sup ln ||P(n)||. n→∞ n n→∞ n But
1 1 1 lim sup ln ||P(n)|| = lim sup ln ||P([t])|| = lim sup ln ||P(t)|| n [t] n→∞ t→∞ t→∞ t
Indeed, the left side is clearly less than or equal to the right hand side and the opposite inequality is obtained as in (34). This completes our proof. Theorem 1 shows that M is a uniformly weak normally repelling set provided λ (z, η ) > 0 for all z ∈ M and all η ∈ U. Furthermore, under a mild hypothesis, it suffices to show λ (z, η ) > 0 for z ∈ Ω (M) = ∪z∈M ω (z) and η ∈ U. Proposition 1 gives conditions for λ (z, η ) to be independent of η ∈ U for all z ∈ M. In this case, using (15), we see that λ (z) = λ (φ (s, z)), s ≥ 0 is constant on forward orbits. We assume hereafter that λ (z) = λ (z, η ) depends only on z ∈ M. As a consequence, if the hypotheses of Theorem 1 hold, and if Ω (M) consists of a finite number of periodic orbits Oi , i = 1, 2, . . . , p, then it suffices to show that λ (zi ) > 0 for some choice zi ∈ Oi , i = 1, 2, . . . , p. In this case, only finitely many exponents must be computed. See [3, 11, 14] where λ (zi ) is related to the spectral radii of a certain Floquet matrix. According to the multiplicative ergodic theorem [1, 6], if M is invariant and there exists an ergodic F-invariant Borel probability measure μ on M, then λ (z) is a constant on M, almost surely. Unfortunately, for our results almost sure positivity of the Lyapunov exponent does not suffice.
References 1. Arnold, L.: Random Dynamical Systems. Springer, Heidelberg (1998) 2. Ashwin, P., Buescu, J., Stewart, I.: From attractor to chaotic saddle: a tale of transverse instability. Nonlinearity 9, 703–737 (1996)
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3. Ferriere, R., Gatto, M.: Lyapunov Exponents and the Mathematics of Invasion in Oscillatory or Chaotic Populations. Theor. Population Biol. 48, 126–171 (1995) 4. Garay, B.M., Hofbauer, J.: Robust Permanence for Ecological Differential Equations, Minimax, and Discretizations. SIAM J. Math. Anal. 34, 1007–1039 (2003) 5. Hirsch, M.W., Smith, H.L., Zhao, X.-Q.: Chain transitivity, attractivity and strong repellors for semidynamical systems. J. Dynamics and Diff. Eqns. 13, 107–131 (2001) 6. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, New York (1995) 7. Magal, P., Zhao, X.-Q.: Global Attractors and Steady States for Uniformly Persistent Dynamical Systems. SIAM J. Math. Anal. 37, 251–275 (2005) 8. Metz, J.A.J.: Fitness. Evol. Ecol. 2, 1599–1612 (2008) 9. Metz, J.A.J., Nisbet, R.M., Geritz, S.A.H.: How Should We Define “Fitness” for General Ecological Scenarios? Tree 7, 198–202 (1992) 10. Rand, D.A., Wilson, H.B., McGlade, J.M.: Dynamics and Evolution: Evolutionarily Stable Attractors, Invasion Exponents and Phenotype Dynamics. Philosophical Transactions: Biological Sciences 343, 261–283 (1994) 11. Salceanu, P.L.: Lyapunov exponents and persistence in dynamical systems, with applications to some discrete-time models. Phd. Thesis, Arizona State University (2009) 12. Salceanu, P.L., Smith, H.L.: Persistence in a Discrete-time, Stage-structured Epidemic Model. J. Difference Equ. Appl. (to appear, 2009) 13. Salceanu, P.L., Smith, H.L.: Persistence in a Discrete-time Stage-structured Fungal Disease Model. J. Biol. Dynamics 3, 271–285 (2009) 14. Salceanu, P.L., Smith, H.L.: Lyapunov Exponents and Persistence in Discrete Dynamical Systems. Discrete and Continuous Dynamical Systems-B (to appear, 2009) 15. Schreiber, S.J.: Criteria for Cr Robust Permanence. J. Differ. Equations 162, 400–426 (2000) 16. Seneta, E.: Non-negative Matrices, an Introduction to Theory and Applications. Halsted Press, New York (1973) 17. Smith, H.L.: Monotone Dynamical Systems: an introduction to the theory of competitive and cooperative systems. Amer. Math. Soc. Surveys and Monograghs 41 (1995) 18. Smith, H.L., Zhao, X.-Q.: Robust Persistence for Semidynamical Systems. Nonlinear Anal. 47, 6169–6179 (2001) 19. Thieme, H.R.: Mathematics in Population Biology. Princeton University Press, New Jersey (2003) 20. Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
Reachability Analysis for Different Classes of Positive Systems Maria Elena Valcher
Abstract. In this survey paper, reachability properties for discrete-time positive systems, two-dimensional discrete state-space models and discrete-time positive switched systems are introduced and characterized. Comparisons among the results obtained in these three settings are presented, thus enlightening which results can be easily extended and what aspects, at present time, are still challenging open problems.
1 Introduction Since the early seventies, positive systems have been the object of a noteworthy interest in the literature. Positive linear systems [14] naturally arise in various fields, such as bioengineering (compartmental models), economic modeling, behavioral science, and stochastic processes (Markov chains or hidden Markov models). Generally speaking, these systems provide the natural framework for modeling physical systems whose describing variables necessarily take nonnegative values. It is clear, however, that apart from the nonnegativity constraint, various additional features may be relevant when capturing the system dynamics. These instances led to the introduction of different classes of positive systems, in particular, (one-dimensional) positive systems, two-dimensional (2D) positive systems and switched positive systems. Even though for each of this class of systems several theoretical problems have been thoroughly investigated, a common research topic for all these classes of systems has been the analysis of structural properties and, in particular, of reachability. The aim of this paper is that of providing a brief survey on the reachability characterizations obtained within these three settings, with special attention to the discrete time cases (even though similar analyses have been performed in the continuoustime cases). Specifically, reachability of discrete-time positive systems will be the Maria Elena Valcher Dip. Ingegneria dell’Informazione, Universit`a di Padova, Italy, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 29–41. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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object of Sect. 2, and it has been investigated, just to quote some of the available references, in [5–7, 10–13, 24]; reachability of discrete two-dimensional positive systems will be discussed in Sect. 3, and it has been the object of [1, 2, 17, 18, 21, 22]; finally, reachability of discrete-time positive switched systems will be addressed in Sect. 4 (see [9, 20, 26–28]). Notation. The (i, j)th entry of a matrix A is denoted by [A]i, j . In the special case of a vector v, its ith entry is [v]i . The symbol + denotes the semiring of nonnegative real numbers. A matrix A (in particular, a vector) with entries in + is said to be nonnegative (A ≥ 0). If A ≥ 0, but at least one entry is positive, A is said to be positive (A > 0). We let ei denote the ith vector of the canonical basis in n (where n is always clear from the context). Given a vector v, the nonzero pattern of v is the set of indices corresponding to its nonzero entries, namely ZP(v) := {i : [v]i = 0}. A vector v ∈ n+ is an ith monomial vector if ZP(v) = ZP(ei ) = {i}. A monomial matrix is a nonsingular square positive matrix whose columns are (distinct) monomial vectors. The Hurwitz products of two n × n matrices A1 and A2 are inductively defined [15] as A1 i
j
A2 = 0,
A1
i
0
A2 = Ai1 ,
A1
i
j
A2 = A1 (A1
when either i or j is negative, if i ≥ 0, A1 0 i−1
j
j
A2 ) + A2 (A1
j
A2 = A2 ,
i
j−1
A2 ),
if j ≥ 0, if i, j > 0.
Notice that ∑i+ j= A1 i j A2 = (A1 + A2) . Basic definitions and results about cones may be found, for instance, in [3, 4]. We recall here only those facts that will be used within this paper. A set K ⊂ n is said to be a cone if α K ⊂ K for all α ≥ 0; a cone is convex if it contains, with any two points, the line segment between them. A cone K is said to be polyhedral if it can be expressed as the set of nonnegative linear combinations of a finite set of generating vectors. This amounts to saying that a positive integer k and an n × k matrix C can be found, such that (s.t.) K coincides with the set of nonnegative combinations of the columns of C. In this case, we adopt the notation K := Cone(C). To efficiently introduce our results, we also need some definitions borrowed from the algebra of non-commutative polynomials [25]. Given the alphabet Ξ = {ξ1 , ξ2 , . . . , ξ p }, the free monoid Ξ ∗ with base Ξ is the set of all words w = ξi1 ξi2 · · · ξik , k ∈ , ξih ∈ Ξ . The integer k is called the length of w and is denoted by |w|, while |w|i represents the number of occurrences of ξi in w. If w˜ = ξ j1 ξ j2 · · · ξ j p is another element of Ξ ∗ , the product is defined by concatena/ the empty tion ww˜ = ξi1 ξi2 · · · ξim ξ j1 ξ j2 · · · ξ j p . This produces a monoid with ε = 0, word, as unit element. Clearly, |ww| ˜ = |w| + |w| ˜ and |ε | = 0. ξ1 , ξ2 , . . . , ξ p is the algebra of polynomials in the noncommuting indeterminates ξ1 , ξ2 , . . . , ξ p . For every family of p matrices in n×n , A := {A1 , A2 , . . . , A p }, the map ψ defined on {ε , ξ1 , ξ2 , . . . , ξ p } by the assignments ψ (ε ) = In and ψ (ξi ) = Ai , i ∈ p, uniquely extends to an algebra morphism of ξ1 , ξ2 , . . . , ξ p into n×n
Reachability Analysis for Different Classes of Positive Systems
31
(as an example, ψ (ξ1 ξ2 ) = A1 A2 ∈ n×n ). If w is a word in Ξ ∗ (i.e. a monic monomial in ξ1 , ξ2 , . . . , ξ p ), the ψ -image of w is denoted by w(A1 , A2 , . . . , A p ).
2 Reachability of Discrete-Time Positive Systems A (discrete-time) positive system is a state-space model x(k + 1) = Ax(k) + Bu(k),
k = 0, 1, 2, . . . ,
(1)
where x(k) and u(k) denote the n-dimensional state variable and the m-dimensional n×m input variable, respectively, at the time instant k, while A ∈ n×n + and B ∈ + . Under the nonnegativity constraint on the system matrices, the state trajectories of the system are constrained within the positive orthant, provided that the initial condition x(0) and the input sequence u(k), k ∈ + , are nonnegative. Reachability property for this type of systems focuses only on nonnegative states, reached by means of nonnegative inputs, and hence it is defined as follows. Definition 1. Given the positive system (1), a state x f ∈ n+ is said to be reachable if there exist k f ∈ and a nonnegative input sequence u(k), k = 0, 1, . . . , k f − 1, that transfers the state of the system from the origin at k = 0 to x f at time k = k f . The positive system (1) is monomially reachable if every monomial vector (equivalently, every canonical vector ei , i ∈ n) is reachable, and reachable if every state x f ∈ n+ is reachable. It is easily seen that monomial reachability is a necessary and sufficient condition for reachability. Necessity is obvious. On the other hand, if each canonical vector ei , i ∈ n, is reachable at some time ki by means a nonnegative input sequence ui (k), then each of them is reachable at k f := maxi ki by means of a suitably right-shifted version, say u¯ i (k), of the sequence ui (k). Consequently, every positive vector x f can be reached at time k f by means of the nonnegative input sequence ∑ni=1 u¯ i (k)[x f ]i . This simple remark allows to convert the reachability problem in the easier monomial reachability problem. An algebraic characterization of monomial reachability, and hence of reachability, can be easily obtained by resorting to the reachability matrix of the system. Definition 2. The reachability matrix at time k of system (1) is Rk (A, B) := [B | AB | . . . | Ak−1 B]. As the expression of the state at time k, starting from the zero initial condition x(0) and under the (nonnegative) soliciting input u(·), is given by
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M.E. Valcher
⎡
u(0) u(1) .. .
⎤
⎢ ⎥ ⎢ ⎥ x(k) = [B | AB | . . . | Ak−1 B] ⎢ ⎥, ⎣ ⎦ u(k − 1) it is clear that the monomial vector ei is reachable at time k if and only if the reachability matrix Rk (A, B) includes an ith monomial column. Therefore Proposition 1. For the n-dimensional positive system (1), the following facts are equivalent ones: • the system is reachable; • the system is monomially reachable; • there exists k f ∈ such that the reachability matrix Rk f (A, B) includes an n × n monomial matrix. All the results reported up to now are quite straightforward. A quite nontrivial step, instead, was taken by Coxson, Larson and Schneider [10] in proving that if a positive system (1) is reachable then the index k f , in the third item of the previous proposition, can always be chosen equal to the system dimension n. Proposition 2. The n-dimensional positive system (1) is reachable if and only if the reachability matrix Rn (A, B) includes an n × n monomial matrix. If we define the reachability index of a reachable positive system (1) as the smallest k f ∈ such that Rk f (A, B) = n+ , the previous proposition tells us that the reachability index cannot exceed the system dimension. The proof of this result is rather involved and it resorts to the precious graph-theoretic approach to the study of the structural properties of positive systems. Indeed, to every n-dimensional system with m inputs (1) we may associate [7, 8, 29] a digraph (directed graph) D(A, B), with n vertices, indexed by 1, 2, . . . , n, and m sources s1 , s2 , . . . , sm . There is an arc ( j, i) from j to i if and only if [A]i j > 0, and there is an arc (s j , i) from the source s j to vertex i if and only if [B]i j > 0. A sequence s j → i0 → i1 → · · · → ik−1 , starting from the source s j , and passing through the vertices i0 , . . . , ik−1 , is an s-path from s j to ik−1 (of length k) provided (s j , i0 ), (i0 , i1 ), . . . , (ik−2 , ik−1 ) are all arcs of D(A, B). It is easily seen that there is a path of length k from s j to some vertex i if and only if the (i, j)th entry of Ak−1 B is positive. Clearly, leaving from some source s j , after k steps one can reach several distinct vertices. This corresponds to saying that the jth column of Ak−1 B has, in general, more than one nonzero entry. We say that an s-path of length k from s j deterministically reaches some vertex i, if no other vertex of the digraph can be reached in k steps starting from s j . If so, we refer to such an s-path as to a deterministic path (of length k) to i. Again, it is obvious that a vertex i can be deterministically reached from some source s j by means of a path of length k if and only if the jth column of Ak−1 B is a ith monomial vector. So, we have realized that monomial reachability (and hence reachability) of a positive system (1) can be easily tested by simply verifying that for each vertex
Reachability Analysis for Different Classes of Positive Systems
33
i ∈ n there is a source and a deterministic path from that source reaching the vertex i. Example 1. Consider the positive system (1) with ⎡ ⎤ ⎡ 01000 0 ⎢1 0 0 0 0⎥ ⎢1 ⎢ ⎥ ⎢ ⎥ A=⎢ B=⎢ ⎢0 0 0 1 0⎥ ⎢1 ⎣0 0 0 0 1⎦ ⎣0 00000 0
⎤ 0 0⎥ ⎥ 0⎥ ⎥. 0⎦ 1
By inspecting the associated digraph D(A, B) one easily sees that, starting from s2 , vertex 5 can be reached deterministically in one step, 4 in two steps and 3 in three steps. On the other hand, starting from s1 , vertex 1 can be reached deterministically in two steps, while 2 in three steps. So, the system is reachable. - 4 - 3 5 D(A, B) 6 s1 @ ? R @ s2 2 1 6 Fig. 1 Graph description of the system of Example 1
Coxson and Larson proved [10] (by using a slightly different terminology, though) that if there exists a deterministic path from some source s j to some vertex i, then there exists a deterministic path from s j to i of length not greater than n, the number of vertices. This led to Proposition 2. This graph-theoretic interpretation turned out to be very profitable, as it allowed to derive canonical forms for reachable positive systems. A first result about canonical forms was derived in [29]. More refined results were later obtained by Bru and co-workers in [5, 7] (see, also, [6], where the concept of reachability index of a positive system (1) was generalized and characterized in graph-theoretic terms).
3 Reachability of Discrete 2D Positive Systems A (discrete) two-dimensional (2D) positive system is a 2D state-space model described by the following first order state-updating equation [15]: x(h + 1, k + 1) = A1 x(h, k + 1) + A2x(h + 1, k) + B1 u(h, k + 1) + B2u(h + 1, k),
(2)
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M.E. Valcher
where the n-dimensional local states x(·, ·) and the m-dimensional inputs u(·, ·) take nonnegative values, A1 and A2 are nonnegative n × n matrices, B1 and B2 are nonnegative n × m matrices, and the initial conditions are assigned by specifying the (nonnegative) values of the state vectors on the separation set C0 := {(h, k) : h, k ∈ , h + k = 0}, namely by assigning all local states of the initial global state X0 := {x(h, k) : (h, k) ∈ C0 }. All input sequences involved have supports included in the half-plane {(h, k) ∈ × : h + k ≥ 0}. For this class of systems (even when no positivity constraint is introduced) reachability represents a rather articulate concept [15, 16]. This is an immediate consequence of the fact that, when defining this concept, we may either refer to the local states or to the global states Xt := {x(h, k) : (h, k) ∈ Ct }, which collect all local states lying on the separation set Ct := {(h, k) : h, k ∈ , h + k = t}. Definition 3. A 2D positive system (2) is said to be • locally reachable if, upon assuming X0 = 0, for every x∗ ∈ n+ there exist (h, k) ∈ × , h + k > 0, and a nonnegative input sequence u(·, ·) such that x(h, k) = x∗ . When so, we will say that x∗ is reachable in h + k steps; • globally reachable if, upon assuming X0 = 0, for every global state X ∗ with entries in n+ , there exist N ∈ + and a nonnegative input sequence u(·, ·) such that the global state XN coincides with X ∗ . When so, we will say that X ∗ is reachable in N steps. If all local (global) states are reachable, system (2) is locally (globally) reachable, and the smallest number of steps which allows to reach every nonnegative local (global) state represents its local (global) reachability index ILR (IGR ). Clearly, as in the standard (nonpositive) case, global reachability ensures local reachability, while the converse is not true.
3.1 Local Reachability of 2D Positive Systems In order to characterize local reachability, we first introduce the reachability matrix in k steps [15] of the 2D positive system (2), i.e. Rk (A1 , A2 , B1 , B2 ) = [B1 B2 A1 B1 A1 B2 + A2 B1 A2 B2 A21 B1 (A1 1 1 A2 )B1 + A21B2 . . . Ak−1 2 B2 ] i−1 j i j−1 = (A1 A2 )B1 + (A1 A2 )B2 i, j≥0, 0 0, can be reached in a single step, by setting σ1 (0) = 1 (and u(0) = x1 ). On the other hand, for T every x f = x1 x2 ≥ 0, with x2 > 0, there exists a sufficiently large k ∈ + , k ≥ 2, such that
0 1 2 ... k − 1 x f ∈ Cone = Cone(Rk (σ2 )), 1 1 1 ... 1 where σ2 (i) = 2, for every i ∈ [0, k − 1]. In particular, from Eq. (7) together with that x1 (k) ≤ (k − 1)x2 (k). Thus, x f can the expression of Rk (σ2 ) we may" deduce #
be reached in a minimum of k = xx12 + 1 steps. As a particular case, when x1 = 0 (hence k = 1) it is sufficient to set u(0) = x2 ; if k > 1, then x f can be reached by x1 setting u(0) = k−1 and u(k − 1) = x2 − u(0), where the nonnegativity of u(k − 1) is ensured by the definition of k. So, x f can be reached in k steps. This ensures that 2 = Cone(R (σ )) ∪ ∪+∞ Cone(R (σ )) , and hence the system is reachable. 1 1 2 k + k=0 However, since every nonnegative vector in 2+ which is not a 1st monomial vector can only be reached by steadily setting the switching sequence to the value 2, we may deduce that: 1) for every finite k ∈ + , Cone(R1 (σ1 ))∪Cone(Rk (σ2 )) = 2+ , thus proving that Proposition 5 gives only a sufficient condition; 2) there is no upper bound on supx∈2 min{k : ∃ σ with |σ | = k such that x ∈ + Cone(Rk (σ ))} = IR . Thus the system is reachable, but IR is not finite. Luckily, monomial reachability is easily captured. Proposition 6. [27] The switched positive system (6) is monomially reachable if and only if ∃ N ∈ such that the reachability matrix in N steps RN = [w(A1 , A2 , . . . , A p )B1w(A1 , A2 , . . . , A p )B2. . .w(A1 , A2 , . . . , A p )B p ] includes an n × n monomial submatrix.
w∈Ξ ∗ 0≤|w|≤N−1
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At this point a natural question arises: if the system is monomially reachable and we let N denote the minimum positive integer such that RN includes an n × n monomial matrix, what is the maximum value that N may reach? This amounts to defining a “monomial reachability index” and to searching for an upper bound on it. It turns out [27] that the upper bound is 2n − 1 and it is strict, meaning that examples can be given of (single-input) positive switched systems whose monomial reachability index has just that value.
References 1. Bailo, E., Bru, R., Gelonch, J., Romero, S.: On the reachability index of positive 2-D systems. IEEE Transactions on Circuits and Systems II 5(10), 1–7 (2006) 2. Bailo, E., Gelonch, J., Romero, S.: An upper bound on the reachability index for a special class of positive 2-D systems. Electronic Journal of Linear Algebra 18, 1–12 (2009) 3. Barker, G.P.: Theory of cones. Lin. Alg. Appl. 39, 263–291 (1981) 4. Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. Academic Press, New York (1979) 5. Bru, R., Caccetta, L., Rumchev, V.G.: Monomial subgraphs of reachable and controllable positive discrete–systems. International Journal of Applied Mathematics and Computer Science 15, 159–166 (2005) 6. Bru, R., Coll, C., Romero, S., S´anchez, E.: Reachability indices of positive linear systems. Electronic Linear Algebra 11, 88–102 (2004) 7. Bru, R., Romero, S., S´anchez, E.: Canonical forms for positive discrete-time linear control systems. Linear Algebra & its Appl. 310, 49–71 (2000) 8. Brualdi, R.A., Ryser, H.J.: Combinatorial matrix theory. Cambridge Univ. Press, Cambridge (1991) 9. Conner, L.T., Stanford, D.P.: The structure of the controllable set for multimodal systems. Linear Algebra & its Appl. 95, 171–180 (1987) 10. Coxson, P.G., Larson, L.C., Schneider, H.: Monomial patterns in the sequence Ak b. Lin. Alg. Appl. 94, 89–101 (1987) 11. Coxson, P.G., Shapiro, H.: Positive reachability and controllability of positive systems. Lin. Alg. Appl. 94, 35–53 (1987) 12. Fanti, M.P., Maione, B., Turchiano, B.: Controllability of linear single-input positive discrete time systems. Int. J. of Control 50, 2523–2542 (1989) 13. Fanti, M.P., Maione, B., Turchiano, B.: Controllability of multi-input positive discrete time systems. Int. J. of Control 51, 1295–1308 (1990) 14. Farina, L., Rinaldi, S.: Positive linear systems: theory and applications. Series on Pure and Applied Mathematics. Wiley-Interscience, New York (2000) 15. Fornasini, E., Marchesini, G.: Doubly indexed dynamical systems. Math. Sys. Theory 12, 59–72 (1978) 16. Fornasini, E., Marchesini, G.: Global properties and duality in 2-D systems. Systems & Control Letters 2(1), 30–38 (1982) 17. Fornasini, E., Valcher, M.E.: On the positive reachability of 2D positive systems. In: Farina, L., Benvenuti, L., De Santis, A. (eds.) Positive Systems. LNCIS, pp. 297–304 (2003)
Reachability Analysis for Different Classes of Positive Systems
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18. Fornasini, E., Valcher, M.E.: Controllability and reachability of 2D positive systems: a graph theoretic approach. IEEE Trans. Circuits and Systems, Part I: Regular Papers 52(3), 576–585 (2005) 19. Ge, S.S., Sun, Z., Lee, T.H.: Reachability and controllability of switched linear discretetime systems. IEEE Trans. Aut. Contr. 46(9), 1437–1441 (2001) 20. Conner Jr., L.T., Stanford, D.P.: The structure of the controllable set for multi-modal systems. Linear Algebra & its Appl. 95, 171–180 (1987) 21. Kaczorek, T.: Reachability and controllability of 2D positive linear systems with state feedback. Control and Cybernetics 29(1), 141–151 (2000) 22. Kaczorek, T.: Positive 1D and 2D systems. Springer, London (2002) 23. Maeda, H., Kodama, S.: Positive realization of difference equations. IEEE Trans. Circ. Sys. CAS-28, 39–47 (1981) 24. Rumchev, V.G., James, D.J.G.: Controllability of positive linear discrete time systems. Int. J. of Control 50, 845–857 (1989) 25. Salomaa, A., Soittola, M.: Automata theoretic aspects of formal power series. Springer, Heidelberg (1978) 26. Santesso, P., Valcher, M.E.: Reachability properties of discrete-time positive switched systems. In: Proceedings of the 45th Conference on Decision and Control (CDC 2006), San Diego (CA), pp. 4087–4092 (2006) 27. Santesso, P., Valcher, M.E.: Monomial reachability and zero-controllability of discretetime positive switched systems. Systems and Control Letters 57, 340–347 (2008) 28. Stanford, D.P., Conner Jr., L.T.: Controllability and stabilizability in multi-pair systems. SIAM J. Contr. Optim. 18(5), 488–497 (1980) 29. Valcher, M.E.: Controllability and reachability criteria for discrete time positive systems. Int. J. of Control 65, 511–536 (1996) 30. Xie, G., Wang, L.: Reachability realization and stabilizability of switched linear discretetime systems. J. Math. Anal. Appl. 280, 209–220 (2003)
On the Positive LQ-Problem for Linear Discrete Time Systems Charlotte Beauthier and Joseph J. Winkin
Abstract. The finite horizon Linear-Quadratic (LQ) optimal control problem with nonnegative state constraints (denoted by LQ+ ) is studied for positive linear systems in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, necessary and sufficient conditions are reported for the LQ+ -optimal control to be given by the standard LQ-optimal state feedback law. Sufficient conditions are also reported for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover a positivity criterion for the LQ-optimal closed-loop system is derived for positive systems with a positively invertible (dynamics) generator.
1 Introduction An important question in system and control theory is the invariance of the nonnegative orthant of the state space for linear systems. When they satisfy that property, such systems are called positive linear systems. They encompasse controlled dynamical models where all the variables, i.e. the state and output variables, should remain nonnegative for any nonnegative initial conditions and input functions. An overview of the state of the art in positive systems theory is given e.g. in [9], [16] and [19]. Typical examples of positive systems are economics models, chemical processes or age-structured populations (see e.g. [9, 10]). Charlotte Beauthier and Joseph J. Winkin University of Namur (FUNDP), Department of mathematics, Rempart de la Vierge, 8, 5000 Namur, Belgium, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 45–53. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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Several system theoretic problems have already been investigated for positive systems. In particular the LQ problem with nonnegative control constraints has been studied for (general) linear systems : see e.g. [13] and references therein for the LQ problem with positive controls and [16] for the minimal energy positive control problem for reachable positive systems. When synthesizing a feedback control law for a positive system, it is meaningful, from the modeling point of view, to aim at keeping the positivity property of the open-loop system for the designed closed-loop system. In [8], using a controllable block companion transformation, sufficient conditions on the weighting matrices of a quadratic cost criterion are established to ensure that the closed-loop system is positive. This idea was generalized in [14] in order to remove the restrictive positivity assumption that was required on such transformation. Here we report some results concerning the finite-horizon positive linear quadratic (LQ+ ) problem for positive linear time systems, [4]. The infinite horizon LQ+ problem in continuous time is studied in [17] by means of a Newton-type iterative scheme which is inspired by the one developed in [11], and the LQ+ problem for positive linear continuous time systems is studied in [3]. Due to the lack of space, only hints or short proofs are given for the main results and numerical examples are omitted. More details are available in [4].
2 Problem Statement Let X and Y be matrices in IR p×q . The property that, for all i = 1, . . . , p and for all j = 1, . . . , q, xi j ≥ yi j , (xi j > yi j , respectively), is denoted by X ≥ Y , ( X Y , respectively). A matrix M ∈ IR p×q is said to be nonnegative, (strictly positive, respectively) if M ≥ 0, (M 0, respectively). In particular, these notations and definitions obviously apply to the case q = 1, i.e. to vectors x ∈ IR p . Consider the following linear time-invariant system description in discrete time, denoted by [A, B] : xi+1 = A xi + B ui, i = 0, . . . , N − 1, x0 = xˆ0 ≥ 0,
(1)
where the state xi and the control ui are in IRn and IRm respectively, A and B are real matrices and xˆ0 ∈ IRn denotes any fixed initial state. Definition 1. The system [A, B] given by (1) is said to be positive if, for all initial conditions xˆ0 ≥ 0 and for all controls (ui )N−1 i=0 ≥ 0, the state trajectories are nonnegative, i.e. for all i = 0, . . . , N, xi ≥ 0. The following characterizations are well-known (see e.g. [9], [16]). Proposition 1. The system [A, B] is positive if and only if A and B are nonnegative matrices. The finite horizon positive LQ problem in discrete time, which will be denoted by LQN+ , consists of minimizing the quadratic functional : $ % 1 N−1 N−1 1/2 2 2 T J(xˆ0 , (ui )i=1 ) := (2) ∑ (R ui + C xi ) + xN S xN 2 i=0
On the Positive LQ-Problem for Linear Discrete Time Systems
47
for a given positive linear system described by (1), where the initial state xˆ0 ≥ 0 is fixed, under the constraints ∀ i ∈ {0, . . . , N}, xi ≥ 0,
(3)
where N is a fixed final time, R ∈ IRm×m is a symmetric positive definite matrix, C belongs to IR p×n and S ∈ IRn×n is a symmetric positive semidefinite matrix. In other words, the LQN+ problem consists of minimizing a quadratic functional for a given positive system while requiring that the state trajectories be nonnegative for any fixed nonnegative initial state, whence the positivity property should be kept for the optimal state trajectories. In this framework, it is not required that the input function (ui )N−1 i=0 be nonnegative.
3 Optimality Conditions Applying the discrete time maximum principle with state constraints (see e.g. [12]), i.e. the Karush-Kuhn-Tucker optimality conditions, yields a characterization, together with a computational procedure, for an LQN+ optimal control. Theorem 1 (Optimality conditions based on the Maximum Principle). a) The LQN+ problem has a solution (ui )N−1 i=0 if and only if there exist multipliers λi such that ui = −R−1 BT pi , i = 0, . . . , N − 1, where [xTi pTi ]T ∈ IR2n is the solution of the recurrent Hamiltonian equation
x 0 xi = H i+1 − , i = N − 1, . . ., 0 pi pi+1 λi
A−1 A−1 B R−1 BT with x0 = xˆ0 , pN = S xN − λN , where H = is the CT C A−1 AT + CT CA−1 B R−1 BT Hamiltonian matrix, and for all i = 0, . . . , N, xi ≥ 0, λi ≥ 0 and λiT xi = 0 (complementarity condition). b) By using the matrix form of the recurrent Hamiltonian equation, (ui )N−1 i=0 is solution of the LQN+ problem if and only if there exist multiplier matrices Λi such that ui = Ki (xˆ0 ) xi := −R−1BT Yi Xi−1 xi , i = 0, . . . , N − 1, where
Xi X 0 = H i+1 − , i = 0, . . . , N − 1 Yi Yi+1 Λi with the final condition XN = I and YN = S − ΛN , and for all i = 0, . . . , N
Λi X0−1 xˆ0 ≥ 0, xˆT0 X0−T ΛiT Xi X0−1 xˆ0 = 0 and
(complementarity condition)
Xi X0−1 xˆ0 ≥ 0.
(4) (5) (6)
Hints a) This result follows directly from the Karush-Kuhn-Tucker optimality conditions with state constraints (by using the discrete-time analogue of e.g. [12, Theorem 4.1]), for necessity, and from the fact that the functional (2) is convex and the
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dynamics and inequality constraints (1) and (3) are defined by linear functions, for sufficiency. b) This proof is a straightforward extension of the one of [7, Theorem 167, pp. 6366]. The main fact is the invertibility of the matrices Xi , which can be proved by using an evaluation lemma, as in [7, Corollary 134, p. 61] : see [2]. 2 Remark 1. a) The terminology used here is borrowed from [7]. The optimality conditions in Theorem 1 (a) are also called Euler-Lagrange equations in the classical optimization litterature. Observe also that H is a symplectic matrix. b) A priori, in view of conditions (4)-(6), the function Ki (xˆ0 ) in Theorem 1 (b) clearly depends upon the choice of the initial state xˆ0 . Stronger conditions are needed in order to make it independent of the initial state, i.e. such that the optimal control law be of the state feedback type ui = Ki xi . Such conditions are reported next. Proposition 2. The conditions (4)-(6) are satisfied for all initial states xˆ0 ≥ 0 if and only if the following conditions hold for all i = 0, . . . , N :
and
Λi X0−1 ≥ 0,
(7)
ΛiT Xi + XiT Λi = 0
(8)
Xi X0−1 ≥ 0.
(9)
The proof of this result is based on the following lemma. Lemma 1. A matrix M ∈ IRn×n is a skew-symmetric matrix, i.e. M = −M T , if and only if for all x ≥ 0, xT M x = 0. (10) Proof. Proof of Proposition 2: The fact that conditions (4) and (6) hold for all xˆ0 ≥ 0 is obviously equivalent to conditions (7) and (9). By Lemma 1, condition (5) holds for all xˆ0 ≥ 0 if and only if the matrix X0−T ΛiT Xi X0−1 is skew-symmetric, or equivalently ΛiT Xi is a skew-symmetric matrix, i.e. (8) holds. Remark 2. a) Conditions (7)-(9) can be hard to check in general. However they obviously hold with Λi = 0 in an important particular case. See Corollary 1 below. b) The optimality conditions in Theorem 1 and Proposition 2 also hold for linear systems (1) that are not positive. However the positivity assumption plays a crucial role for obtaining the criteria reported in Sect. 4. In view of the analysis above, it is easy to get conditions such that the LQN+ problem has a solution. These conditions are based on the standard problem which will be denoted by LQN and which consists of minimizing the quadratic functional (2) for a given positive linear system described by (1) (without any nonnegativity constraint on the state trajectory). It is well-known that its solution is given by ui = Ki xi := −R−1 BT Yi Xi−1 xi , i = 0, . . . , N − 1 where [XiT YiT ]T ∈ IR2n×n is the solution of the matrix recurrent Hamiltonian equation,
X XN I Xi = H i+1 , = . (11) Yi Yi+1 YN S
On the Positive LQ-Problem for Linear Discrete Time Systems
49
Equivalently the solution of the LQN problem is given, for all i = 0, . . . , N − 1, by ui = −R−1 BT Pi+1 [I + B R−1 BT Pi+1 ]−1 xi , where Pi is the solution of the recurrent Riccati equation (RRE), i = N, . . . , 1, (see e.g. [7]) : −Pi−1 = CT C + AT Pi A − AT Pi B(I + R−1BT Pi B)−1 R−1 BT Pi A, PN = S.
(12)
Corollary 1 (Optimality conditions based on admissibility). The solution of the (standard) LQN problem is solution of the LQN+ problem for all xˆ0 ≥ 0 if and only if the LQN optimal state trajectories are admissible, i.e. nonnegative for all xˆ0 ≥ 0, or equivalently, one of the following equivalent conditions holds : a) The standard closed-loop matrix A + B Ki is a nonnegative matrix for all i = 0, . . . , N − 1, i.e. ∀ k, l, ∀ i = 0, . . . , N − 1, (B R−1 BT Pi )kl ≤ akl .
(13)
b) The matrix solution of the matrix recurrent Riccati equation (12) is such that for all i = 0, . . . , N, Xi X0−1 ≥ 0. Hints Corollary 1 follows from Theorem 1 and Proposition 2 by applying the discrete time version of a known characterization of the positivity of homogeneous linear time-varying systems in continuous time, see e.g. [1] and [15]. In addition, the solution of the LQN problem is given as in Theorem 1 where the multiplier ma2 trices Λi are identically equal to zero.
4 Positivity Criteria In this section, the LQN problem is studied with the aim of finding conditions on the problem data such that the standard closed-loop system is positive, i.e. such that the conditions of Corollary 1 hold. This can be interpreted as solving an inverse LQN+ problem.
4.1 Minimal Energy Control In the sequel, σ (A) and ρ (A) denote the spectrum and the spectral radius of a matrix A, respectively. The matrix norm that is used here is the one induced by the euclidian vector norm. Consider the particular problem of minimal energy control with penalization of the final state, i.e. the LQN problem (1)-(2) where C is equal to zero. By computing the expression of Pi in terms of the matrix solution of the recurrent Hamiltonian equation, we obtain the following result : Theorem 2. If A 0 and if the spectral radius ρ (S) of the final state penalty matrix is sufficiently small such that
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⎧ λmin (R) (1 − σ ) ⎪ , if σ < 1 ⎪ ⎪ ⎪ B2 ⎪ ⎨ λ (R) (σ − 1) min , if σ > 1 ρ (S) = max μi < γ := ⎪ B2 σ N μi ∈σ (S) ⎪ ⎪ ⎪ λmin (R) ⎪ ⎩ , if σ = 1 B2 N
(14)
where σ := σmin (A) σmax (A), with σmin (A) (σmax (A), respectively ) denoting the smallest (the largest, respectively) singular value of A and λmin (R) := min{λ : λ ∈ σ (R)}, then the LQN closed-loop system is positive and therefore the solution of the LQN problem is solution of the LQN+ problem. Hints The positivity constraint on the closed-loop matrix can be written in terms of the solution Pi of the RRE (see condition (13)), where B ≥ 0. In addition, Pi = Yi Xi−1 = (AT )N−i S [I + G(N, i) S]−1 AN−i , where N−i−1
G(N, i) :=
∑
(A−1 )i−N+k+1 B R−1 BT (AT )N−i−k−1 ,
k=0
B2 1 − σ N−i ρ (S) ρ (S) ≤ . Thus, if (14) holds, λmin (R) 1 − σ γ ρ (S) , whence, by choosing ρ (S) sufficiently small, then S [I + G(N, i) S]−1 ≤ 1 − ρ (S) γ condition (13) will hold, since ∀ k, l, akl > 0 and the sequences ((AT )N−i )Ni=0 and (AN−i )Ni=0 are bounded. 2 and, for σ = 1, G(N, i) S ≤
Remark 3. a) If σ ≥ 1 and if the time horizon N is increased, ρ (S) has to be decreased accordingly for condition (14) to be satisfied with a fixed matrix R. This reveals a tradeoff between positivity and stability of the closed-loop system in a receding horizon approach. b) The minimal energy control problem with nonnegative controls and with a final state equality constraint is solved in [16, Subsection 3.4.1] for reachable systems. Here we use a penalization term in the cost instead of a final state constraint, it is not assumed that the system is reachable and it is not required that the input function (ui )N−1 i=0 be nonnegative.
4.2 Nonnegative Hamiltonian Matrix Theorem 3. If the Hamiltonian matrix H and the penalty matrix S are nonnegative and if the solution of the matrix recurrent Hamiltonian equation is such that X0−1 ≥ 0, then the LQN closed-loop system is positive and therefore the solution of the LQN problem is solution of the LQN+ problem. Proof. Multiplying the matrix recurrent Hamiltonian equation (11) on the right by X0−1 gives
On the Positive LQ-Problem for Linear Discrete Time Systems
Xi+1 Xi X0−1 = H X0−1 Yi Yi+1
with
51
−1
XN X0−1 X0 = . YN X0−1 S X0−1
It follows by induction that, for all i = 0, . . . , N, Xi X0−1 ≥ 0. Then, by using Corollary 1 (b), one gets the conclusion.
4.3 Monomial Systems A nonnegative matrix M is monomial if M is a diagonal matrix up to a permutation, i.e. M = D P = diag[mi ]ni=1 P, where D is a positive definite diagonal matrix and P is a permutation matrix, or equivalently M −1 ≥ 0, see e.g. [5] and [18]. Definition 2. A positive system [A, B], described*by (1), is said + to be monomial m diag[bi ]i=1 if A is a monomial matrix and B is of the form B = . 0(n−m)×m
Definition 3. Let L and M be monomial matrices. L and M are said to be strucs turaly similar, denoted by L = M, if and only if there exist positive definite diagonal matrices D1 and D2 such that L = D1 M D2 . s
It is easy to check that “=” is an equivalence relation on the set of monomial matrices. The following straightforward result will be needed below : Lemma 2. Let L and M be monomial matrices. Let P be a permutation matrix such that L = D1 P and M = P D2 where D1 and D2 are positive definite diagonal matrices. Then L and M are structuraly similar. Theorem 4. Consider a monomial system described by (1) and the quadratic cost (2) where C, R and S are diagonal matrices. Then the LQN closed-loop system is positive and therefore the solution of the LQN problem is solution of the LQN+ problem. Proof. By using the explicit form of H where A = D P, PT = P−1 and the fact s that CT C P−1 = P−1CT C (see Lemma 2), one gets : −1
P D−1 0 D−1 B R−1 BT H= ¯ −1 D + CT C D¯ D−1 B R−1 BT 0 P−1 CT C DD where D¯ is a positive definite diagonal matrix. Therefore H is nonnegative. Now, by using the matrix recurrent Hamiltonian equation with this expression of H, it can be shown by induction that, for all i = 0, . . . , N − 1, Xi = (P−1 )N−i DX,i and Yi = (P−1 )N−i DY,i , where DX,i and DY,i are positive definite diagonal matrices. Therefore, X0 = (P−1 )N D0 where D0 is a positive definite diagonal matrix. Hence X0 is a monomial matrix and X0−1 ≥ 0. It follows by Theorem 3 that the LQN closedloop system is positive.
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5 Concluding Remarks The results reported in this paper have been obtained for the finite horizon LQ+ problem in discrete time. Their possible extensions to the infinite horizon problem are currently under investigation. The continuous time case was studied in [3]. Another perspective for this work is the case where the positivity of the open-loop system is not required. Indeed, one can observe that the analysis and the results in Sect. 3 and Subsect. 4.1 can be generalized to the case where the system [A, B] is not positive.
Acknowledgements. The authors wish to thank the reviewers of this paper for their helpful suggestions and comments.
References 1. Angeli, D., Sontag, E.D.: Monotone control systems. IEEE Transactions on Automatic Control 48(10), 1684–1698 (2003) 2. Beauthier, C.: Le Probl`eme lin´eaire quadratique positif. M´emoire de DEA (Master Thesis), FUNDP, Namur (2006) 3. Beauthier, C., Winkin, J.J.: Finite horizon LQ-optimal control for continuous time positive systems. In: Proceedings of the Eighteenth International symposium on Mathematical Theory of Networks and Systems (MTNS 2008), Virginia Tech. Blacksburg, Virginia, USA (2008) 4. Beauthier, C., Winkin, J.J.: LQ-optimal control of positive linear systems (submitted 2009) 5. Berman, A., Plemmons, R.J.: Inverses of nonnegative matrices. Linear and Multilinear Algebra 2, 161–172 (1974) 6. Bixby, R.E.: Implementation of the simplex method: the initial basis. ORSA Journal on Computing 4(3) (1992) 7. Callier, F.M., Desoer, C.A.: Linear System Theory. Springer, New York (1991) 8. Castelein, R., Johnson, A.: Constrained optimal control. IEEE Transactions on Automatic Control 34(1), 122–126 (1989) 9. Farina, L., Rinaldi, S.: Positive Linear Systems. John Wiley, New York (2000) 10. Godfrey, K.: Compartmental Models and Their Applications. Academic Press, London (1983) 11. Guo, C.-h., Laub, A.J.: On a Newton-like Method for Solving Algebraic Riccati Equations. SIAM J. Matrix Anal. Appl. 21(2), 694–698 (2000) 12. Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Review 37(2), 181–218 (1995) 13. Heemels, W.P.M.H., Van Eijndhoven, S.J.L., Stoorvogel, A.A.: Linear quadratic regulator problem with positive controls. Int. J. Control 70(4), 551–578 (1998) 14. Johnson, A.: LQ state-constrained control. In: Proceedings of the IEEE/IFAC Joint Symposium on Computer-Aided Control System Design, pp. 423–428 (1994) 15. Kaˇczorek, T.: Externally and internally positive time-varying linear systems. Int. J. Appl. Math. Comput. Sci. 11(4), 957–964 (2001)
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16. Kaˇczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 17. Laabissi, M., Winkin, J., Beauthier, C.: On the positive LQ-problem for linear continoustime systems. In: Proceedings of the 2nd Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2006), Grenoble, France. LNCIS, pp. 295–302. Springer, Heidelberg (2006) 18. Plemmons, R.J., Cline, R.E.: The generalized inverse of a nonnegative matrix. In: Proceedings of the American Mathematical Society, vol. 31(1) (1972) 19. Van Schuppen, J.H.: Control and System Theory of Positive Systems. Lecture Notes (2007)
The Importance of Being Positive: Admissible Dynamics for Positive Systems Luca Benvenuti and Lorenzo Farina
Abstract. Positive linear systems display peculiar dynamics due to the positivity constraints on input, state and output variables. In this paper we review such peculiarities for externally and internally positive linear systems. The properties of externally positive systems are shown in terms of poles and zeros location and input–output response, and those of internally positive systems in terms of eigenvalues location. Open problems are also presented. The presentation style of this paper is very informal, aiming to convey to the reader just a taste of the “importance of being positive”.
1 Introduction Positivity constraints on system’s variables is often found in engineering applications. In fact, this is the case of any variable representing any possible type of resource measured by a quantity such as time [11, 29], money and goods [17, 19], buffer size and queues [27], data packets flowing in a network [23], human, animal and plant populations [20], concentration of any substance [13, 15, 28] including mRNAs, proteins and molecules [8], electric charge [2, 6, 12], and light intensity levels [3, 22]. Moreover, also probabilities are positive quantities, so that also hidden Markov models [1, 26] and phase-type distributions models are subject to positivity constraints [24, 25]. Such constraints impose limitations on the input-output dynamics and also on the parameters describing the system’s behavior, that is on poles, zeros and eigenvalues location. These limitations are quite stringent and often allow to characterize qualitatively the behavior of the system and therefore dramatically simplify its analysis. Luca Benvenuti and Lorenzo Farina Dipartimento di Informatica e Sistemistica “A. Ruberti”, Sapienza Universit`a di Roma, via Ariosto 25, 00185 Roma, Italy, e-mail:
[email protected],
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In this paper we briefly review the most important features of positive SISO linear systems by considering externally and internally positive systems.
2 External and Internal Positivity Given a discrete-time SISO linear system, its “internal” representation is given by x(k + 1) = Ax(k) + bu(k) y(k) = cx(k) and its “external” representation (transfer function) by H(z) = c(zI − A)−1b where H(z) is the Z-transform of the system impulse response h(k). In the case of a continuous-time SISO linear system, the “internal” representation is x(t) ˙ = Ax(t) + bu(t) y(t) = cx(t) and the “external” representation (transfer function) is H(s) = c(sI − A)−1b where H(s) is the Laplace transform of the system impulse response h(t). A system is said to be externally positive if for any nonnegative input and zero initial state, the output is always nonnegative. By contrast, a system is said to be internally positive if for any nonnegative input and nonnegative initial state, the state trajectory and the output are always nonnegative. It is plain that an internally positive system is also externally positive whereas the converse do not always hold. The problem of finding conditions for an externally positive system to be also internally positive is the so called positive realization problem which is reviewed in the tutorial paper [4]. In the next sections we will illustrate the properties of externally positive linear systems in terms of poles and zeros location and input–output response and those of internally positive linear systems in terms of eigenvalues location.
3 Dynamic Properties of Externally Positive Linear Systems We first review two time-domain properties of externally positive linear systems.
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The first property refers to nonnegativity of the impulse response which stems immediately from the definition itself. In fact, the Dirac delta function is a nonnegative input. As shown in [14], this property can be exploited in the design of a feedback control system that maximizes the size of an unknown-but-bounded disturbance without violating any hard constraint. Another interesting property is the monotonicity of the step response since it implies that no undershoot and overshoot are present. This may be a desired property in the design of a control system in several applications such as a manipulator performing pick and place operations close to a wall, filling a tank with a fluid in minimum time without spilling over, or temperature control in a hazardous environment [7, 9]. We consider now two frequency-domain properties of externally positive linear systems, the first related to transfer function poles location, the second to the location of zeros. The first property refers to the location of the dominant poles, where a dominant pole (or eigenvalue) is that of maximum modulus [maximal real part] for discretetime systems [continuous-time systems]. The property simply states that one of the dominant pole is real. Moreover, for discrete-time systems, such real dominant pole is nonnegative. In fact, if only complex dominant poles are present, then the long term behavior of the impulse response would be oscillating thus contradicting the nonnegativity assumption. The same reasoning applies for negative real dominant poles for discrete-time systems [10]. It’s worth noting that complex pairs can be dominant poles as illustrated by the following examples. Consider the discrete-time system with transfer function H(z) =
z − cos ϕ 2 + z − 1 z2 − (2 cos ϕ ) z + 1
(1)
whose poles are 1 and e±iϕ and are all dominant. The transfer function has a pair of dominant complex poles and its impulse response h(k) = 2 + cos[(k − 1) ϕ ] is clearly nonnegative for all k ≥ 1. As an additional example, consider the continuous-time system with transfer function H(s) =
2 1 + 2 s + 1 s + 2s + 2
(2)
whose poles are −1 and −1 ± i and are all dominant. The transfer function has a pair of dominant complex poles and its impulse response h(t) = e−t (2 + sin(t)) is clearly nonnegative for all t ≥ 0. The second property refers to the location of real zeros. More precisely, the real zeros of a continuous-time [discrete-time] externally positive linear system are
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smaller in value [in modulus] than the dominant real pole. In fact, the transfer function of the system is H(s) =
∞
h(t)e−st dt
0
and since h(t) is nonnegative and e−st is nonnegative for any real value of s, then H(s) cannot be zero for any real value of s in the radius of convergence of the Laplace transform, that is greater than the real dominant pole [10]. Finally, we end this section by stating a simple but useful criterion for checking BIBO stability for externally positive systems. In fact, an externally positive continuous-time linear system is BIBO stable if and only if all the coefficients of the transfer function denominator have the same sign. To prove the sufficiency1 of this assertion, consider that if the system is not BIBO stable, then it has a positive dominant real pole thus arriving at a contradiction since a monic polynomial with positive coefficients cannot be zero when evaluated at the positive dominant real pole of the system (Frobenius pole).
4 Dynamic Properties of Internally Positive Linear Systems It is well known that positivity imposes a specific sign pattern on the dynamic system matrix A, in particular, an internally positive discrete-time linear system is fully characterized by having a nonnegative matrix A, while an internally positive continuous–time linear system by having a Metzler matrix A (i.e. having nonnegative off-diagonal entries). Moreover an internally positive linear system is such that vectors b and c are nonnegative. Nonnegativity of the matrix A enforces strong conditions on the admissible location of the eigenvalues of an internally positive discrete-time linear system [5, 21]. We begin with discussing the location of dominant eigenvalues described by the celebrated Perron-Frobenius theorem (see [21]). 2 In fact, the Perron-Frobenius theorem, which holds for irreducible nonnegative matrices, can be used to gain insight also into the case of a generic nonnegative matrix, since this matrix can be reduced, by reordering state variables, to a block triangular matrix with irreducible diagonal blocks. Moreover the spectrum of such a matrix is the union of the spectra of the diagonal blocks, so that the dominant eigenvalues of a nonnegative square matrix A of dimension n are among the roots of
λ k − ρ (A)k = 0 1 2
(3)
Necessity of such condition is known to hold also for non externally positive systems. The case of nonnegative nilpotent matrices is not considered since, in this case, all the eigenvalues are located at the origin of the complex plane.
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for some (possibly more than one) values of k = 1, ..., n. In particular, one of the dominant eigenvalues is positive real 3 and any other dominant eigenvalue has a degree 4 not greater than that of the dominant positive real eigenvalue. From this result it follows that there not exists an internally positive linear system having the externally positive system transfer function (1) when ϕ /π irrational. In fact there is no integer k such that equation (3) holds, that is eikϕ − 1 = 0. The admissible location of the dominant eigenvalues of an internally positive continuous-time linear system can be obtained by noting that for any Metzler matrix A there exists a nonnegative value α such that A + α I is a nonnegative matrix. As a consequence, the spectrum of a Metzler matrix is that of the nonnegative matrix A + α I leftward shifted by α . Therefore, the dominant eigenvalue of an internally positive continuous-time linear system is unique (possibly multiple) and real5 . This results clearly shows that there is no internally positive system having the externally positive system transfer function considered in Example 2. Let us now consider the limitation imposed by internal positivity on the whole spectrum of the system. We first recall a fundamental result on eigenvalue location for nonnegative matrices. This result will be used hereafter to study the spectrum of discrete and continuous-time internally positive systems. ρ Denote by Θn the set of points in the complex plane that are eigenvalues of nonnegative n × n matrices with Frobenius eigenvalue ρ . For example, the region ρ ρ Θ2 consists of points on the segment [−ρ , ρ ] and the region Θ3 consists of points in the interior and on the boundary of the triangle with vertices ρ , ρ e2π i/3, ρ e4π i/3 ρ and on the segment [−ρ , ρ ]. A complete characterization of the regions Θn = ρΘn1 for any n has been given by Karpelevich [16] (see also [18], and Theorem 1.2 at page 168 of reference [21]). The region Θn1 is symmetric with respect to the real axis, is included in the disc |z| ≤ 1, and intersects the circle |z| = 1 at points e2π ia/b , where a and b run over the relatively prime integers satisfying 0 ≤ a ≤ b ≤ n. The boundary of Θn1 consists of these points and of curvilinear arcs connecting them in circular order. The endpoints of an arc are e2π ia1 /b1 and e2π ia2 /b2 (b1 ≤ b2 ) and each arc is given by the following parametric equation:
λ b2 (λ b1 − s)[n/b1 ] = (1 − s)[n/b1] λ b1 [n/b1 ] where the real parameter s runs over the interval 0 ≤ s ≤ 1. Since the dynamic matrix A of an internally positive discrete-time linear system is ρ nonnegative, then its spectrum is contained in the region Θn where n is the internal dimension of the system and ρ its Frobenius eigenvalue. For the sake of illustration, ρ ρ the regions Θ3 and Θ4 are depicted in Figure 1. The spectrum of an internally positive continuous-time linear system can be characterized using the previous result. In fact, since the spectrum of a Metzler matrix 3 4 5
This eigenvalue is called the Frobenius eigenvalue The degree of an eigenvalue is the size of the largest diagonal block containing it in the Jordan canonical form of the matrix. This eigenvalue is called the Frobenius eigenvalue
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L. Benvenuti and L. Farina Im ½(A) e2¼i/3
½(A) e 2¼i/3
i½(A)
Im
Re
Re
{ ½(A)
½(A) e4¼i/3
½(A)
½(A)
{½(A)
½(A) e4¼i/3
-i½(A)
Fig. 1 The regions bounding the spectrum of a generic third-order (left) and fourth order (right) internally positive discrete-time linear system.
A is that of the nonnegative matrix A + α I leftward shifted by α , then the spectrum of a generic Metzler matrix is obtained by considering a value of α arbitrarily large. Consequently, the spectrum of an internally positive continuous-time linear system of dimension n and with Frobenius eigenvalue ρ is contained in the angular region with opening angle 2π π− n symmetric with respect to the real axis with vertex located at ρ . For the sake of illustration, the region for n = 3 is depicted in Figure 2. This result shows again that the externally positive system with transfer function (2) cannot have a finite dimension internally positive representation.
Lp
Uh (D)
Fig. 2 The region bounding the spectrum of a generic third-order internally positive continuous-time positive system.
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5 Open Problems There is a considerable number of open issues related to positive linear systems. Examples of useful results that seem to be very hard to obtain are: 1. determine the zeros location for externally positive linear systems; 2. infer nonnegativity of the whole impulse response directly from the systems parameters, or at least 3. determine directly from the systems parameters the minimum number of samples of the impulse response to be checked in order to infer nonnegativity of the whole impulse response; 4. fully characterize the frequency response of externally positive linear systems.
References 1. Anderson, B.D.O.: From Wiener to hidden Markov models. IEEE Control Systems, 41– 51 (June 1999) 2. Benvenuti, L., Farina, L.: Discrete–time filtering via charge routing networks. Signal Processing 49(3), 207–215 (1996) 3. Benvenuti, L., Farina, L.: The design of fiber–optic filters. IEEE/OSA Journal of Lightwave Technology 19(9), 1366–1375 (2001) 4. Benvenuti, L., Farina, L.: A tutorial on the positive realization problem. IEEE Transactions on Automatic Control 49(5), 651–664 (2004) 5. Benvenuti, L., Farina, L.: Eigenvalue regions for positive systems. Systems & Control Letters 51, 325–330 (2004) 6. Benvenuti, L., Farina, L., Anderson, B.D.O.: Filtering through combination of positive filters. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 46(12), 1431–1440 (1999) 7. Darbha, S., Bhattacharyya, S.P.: On the synthesis of controllers for a non-overshooting step response. IEEE Transactions on Automatic Control 48(5), 797–800 (2003) 8. De Jong, H.: Modeling and simulation of genetic regulatory systems: a literature review. Journal of Computational Biology 9(1), 67–103 (2002) 9. Deodhare, G., Vidyasagar, M.: Design of non-overshooting feedback control systems. In: Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, USA (1990) 10. Farina, L., Rinaldi, S.: Positive linear systems: theory and applications. In: Pure and Applied Mathematics. Wiley–Interscience, New York (2000) 11. Gaubert, S., Butkovic, P., Cuninghame-Green, R.: Minimal (max,+) realization of convex sequences. SIAM Journal on Control and Optimization 36, 137–147 (1998) 12. Gersho, A., Gopinath, B.: Charge-routing networks. IEEE Transactions on Circuits and Systems 26(1), 81–92 (1979) 13. Jacquez, J.A.: Compartmental analysis in biology and medicine, 2nd edn. University of Michigan Press, Ann Arbor (1985) 14. Jayasuriya, S.: On the determination of the worst allowable persistent bounded disturbance for a system with constraints. Journal of Dynamic Systems, Measurement, and Control 117, 126–133 (1995)
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15. Kajiya, F., Kodama, S., Abe, H.: Compartmental Analysis - Medical Applications and Theoretical Background, Karger, Basel (1984) 16. Karpelevich, F.I.: On the characteristic roots of matrices with nonnegative elements. Izv. Akad. Nauk SSSR Ser. Mat. 15, 361–383 (1951) (in Russian); In Eleven Papers Translated from Russian, American Mathematical Society 140 (1988) 17. Krause, U.: Positive nonlinear systems in economics. In: Maruyama, T., Takahashi, W. (eds.) Nonlinear and Convex Analysis in Economic Theory. Lecture Notes in Economics and Mathematical Systems, vol. 419, pp. 181–195. Springer, Heidelberg (1995) 18. Ito, H.: A new statement about the theorem determining the region of eigenvalues of stochastic matrices. Linear Algebra and its Applications 246, 241–246 (1997) 19. Leontieff, W.W.: The Structure of the American Economy 1919–1939. Oxford University Press, New York (1951) 20. Leslie, P.H.: On the use of matrices in certain population mathematics. Biometrika 35, 183–212 (1945) 21. Minc, H.: Nonnegative Matrices. John Wiley & Sons, New York (1987) 22. Moslehi, B., Goodman, J.W., Tur, M., Shaw, H.J.: Fiber–optic lattice signal processing. In: Proceedings of the IEEE, vol. 72, pp. 909–930 (1984) 23. Mounier, H., Bastin, G.: Compartmental modelling for traffic control in communication networks. IEEE Transactions on Automatic Control (2002) (submitted) 24. O’Cinneide, C.A.: Characterization of phase–type distributions. Stochastic Models 6, 1–57 (1990) 25. O’Cinneide, C.A.: Phase–type distributions: open problems and a few properties. Stochastic Models 15, 731–757 (1999) 26. Rabiner, L.R.: A tutorial on hidden Markov models and selected applications in speech recognition. In: Proceedings of the IEEE, vol. 77, pp. 257–286 (1989) 27. Saaty, T.L.: Elements of Queueing Theory. McGraw-Hill, New York (1961) 28. Segel, I.H.: Enzyme Kinetics. Whiley, New York (1975) 29. Xi, N., Tarn, T.J., Bejczy, A.K.: Intelligent planning and control for multi robot coordination: an event based approach. IEEE Transactions on Robotics and Automation 12, 439–452 (1996)
Detectability, Observability, and Asymptotic Reconstructability of Positive Systems Tobias Damm and Cristina Ethington
Abstract. We give a survey of detectability, observability and reconstructability concepts for positive systems and sketch some applications to the analysis of stochastic equations.
1 Introduction There have been a number of contributions to the definition of observability and detectability for different classes of linear control systems recently, see [3, 4, 8, 10, 14, 21, 24]. A common feature of these papers is the distinction between properties which are equivalent for deterministic linear time-invariant systems (compare also [20, 23]). In particular, the following properties are of interest. Usually, one defines a system to be detectable (resp. observable), if all its unstable (resp. nontrivial) modes produce a non-zero output, i.e. if vanishing of the output y(t) = 0 for all t implies that the state x(t) converges to zero (resp. is equal to zero). In the deterministic case, it follows that a system is detectable if and only if the dual system is stabilizable, which again is equivalent to the existence of an asymptotically stable linear dynamic state observer. Here this property will be called asymptotic reconstructability. Moreover, there are equivalent algebraic criteria, the so called Hautus-test [15] (or Popov-Belevich-Hautus test), which play an important rˆole in the discussion of algebraic Lyapunov and Riccati equations. For many classes of stochastic systems, however, these properties fall apart and their usefulness differs. Therefore different concepts have been developed. Several authors (e.g. [5, 9, 12, 13, 21]) have chosen mean-square stabilizability of the dual Tobias Damm and Cristina Ethington Department of Mathematics, TU Kaiserslautern. D-67663 Kaiserslautern, Germany, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 63–70. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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system as a defining property for detectability. Some authors (e.g. [3]), call this property MS-detectability. However, as discussed in [8], this choice is not wellmotivated. First no clear interpretation with respect to dynamical properties of the underlying stochastic control system has been given (as mentioned e.g. in [21]). Second, there is no simple equivalent algebraic criterion like the Hautus-test for deterministic systems. Third, in applications to generalized algebraic Lyapunov and Riccati equations only the generalized Hautus-test is used, which is weaker than stabilizability of the dual system (e.g. [13]). In [8] a generalized version of the Hautus test was given and shown (for stochastic differential equations) to be equivalent to the system being detectable (in the sense that all unstable modes produce non-zero output). This property is called e.g. Wdetectability in [4] or β -detectability in [7]. In the present note, we suggest abstract definitions and characterizations of detectability, observability and asymptotic reconstructability in terms of positive operators and clarify some relations between these notions. We then show how these definitions are related to the dynamical properties of different classes of systems.
2 Resolvent Positive Operators We summarize some results on resolvent positive operators which have been collected e.g. in [1] or [6, 7]. Let H be some finite-dimensional real vector-space, ordered by a closed, solid, pointed convex cone H+ . On H we consider a scalar product ·, ·. By H+∗ we denote the dual cone, and for a linear operator T : H → H we denote the adjoint operator by T ∗ . Definition 1. A linear operator T : H → H is called positive, T ≥ 0, if T (H+ ) ⊂ H+ . It is called resolvent positive if there exists an α0 ∈ such that for all α ≥ α0 the resolvent (α I − T )−1 is positive, i.e. (α I − T )−1 (H+ ) ⊂ H+ . There are many other equivalent characterizations of resolvent positivity (e.g. [6, 11]). Here, the following will be relevant. Proposition 1. A linear operator T : H → H is resolvent positive, if and only if it is exponentially positive, which means that eT s ≥ 0 for all s ≥ 0. Example 1. (i) Let H = n be ordered by the cone H+ = n+ . A matrix A ∈ n×n , regarded as a mapping A : H → H , is positive, if and only if all its entries are nonnegative. It is resolvent positive, if and only if all off-diagonal entries of A are nonnegative, i.e. A is a Metzler matrix. This is, by definition, equivalent to saying that −A is a Z-matrix. We call A stable, if σ (A) ⊂ − . Hence, again by definition, A is resolvent positive and stable, if and only if −A is an M-matrix M-matrix (see [2, 16, 18]).
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(ii)
65
Let H = H n denote the space of n × n symmetric matrices ordered by the cone H+ = H+n of nonnegative definite matrices. Then for any A ∈ n×n , the operator ΠA : H n → H n , ΠA (X) = A∗ XA is positive, whereas both the continuous-time Lyapunov operator LA : H n → H n , LA (X) = AT X + XA, and the discrete-time Lyapunov operator (also Stein operator) SA : H n → H n , SA (X) = A∗ XA − X, are resolvent positive but, in general, not positive.
Note that in both examples the cone H+ is self-dual, i.e. H+ = H+∗ . The following result goes back to [19]. Proposition 2. Let T : H → H be resolvent positive and set β = max Re σ (T ). Then there exists an eigenvector V ∈ H+ , V = 0, such that TV = β V . Moreover, the following are equivalent (where X < 0 means X ∈ int H+ ): (a) (b) (c)
β (T ) < 0, ∃X < 0 : T (X) < 0, ∀Y < 0 : ∃X < 0 : T (X) = Y .
3 Detectability and Observability Definition 2. Let T : H → H be resolvent positive and Y ∈ H+∗ . Consider the positive linear system X˙ = T (X), y(t) = X(t),Y . The solution of the initial value problem with X(0) = X0 ∈ H+ will be denoted by X(t, X0 ). We call the pair (T,Y ) (i) detectable, if y(t, X0 ) = 0 for all t ≥ 0 implies X(t, X0 ) → 0 for t → ∞. (ii) observable, if y(t, X0 ) = 0 for all t ≥ 0 implies X0 = 0. The following result is a positive analogue of the Hautus-criterion. Proposition 3. The pair (T,Y ) as in the previous definition is detectable, if and only if V,Y > 0 for any eigenvector V ∈ H+ of T corresponding to an eigenvalue λ ≥ 0, (b) observable, if and only if V,Y > 0 for any eigenvector V ∈ H+ of T corresponding to an arbitrary eigenvalue.
(a)
Proof. (a) Assume that the criterion in (a) does not hold, i.e. there exist λ ≥ 0, X0 ∈ H+ \ {0}, so that T (X0 ) = λ X0 and X0 ,Y = 0. Since X(t, X0 ) = eλ t X0 , we have y(t, X0 ) = 0 for all t, but X(t, X0 ) → 0 as t → ∞. Hence the system is not detectable. Vice versa, assume that the criterion holds and for some X0 ∈ H+ \{0} and all t→∞ t ≥ 0 we have y(t, X0 ) = 0. Then we have to show that X(t, X0 ) −→ 0. Let X+ = cl conv{X(t, X0) t ≥ 0} denote the closed convex hull of the positive orbit of X(t, X0 ). Let further X = X+ − X+ be the minimal subspace of H containing X+ . Then X+ is a closed solid pointed convex cone in X . By construction, both
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X+ and X are invariant with respect to X˙ = T (X). That means T (X ) ⊂ X , and the restriction T X is resolvent positive with respect to X+ . Let βX be the spectral bound of T X . By Proposition 2 there exists an eigenvector VX ∈ X+ ⊂ H+ , such that T (VX ) = βX VX . Since X(t, X0 ),Y = 0 for all t ≥ 0 we conclude V,Y = 0 for all V ∈ X . In particular VX ,Y = 0. It follows now from the detectability criterion that βX < 0, which implies asymptotic stability of X(t, X0 ) for all X0 ∈ X . (b) If the criterion does not hold, then – as in (a) – we have a nontrivial solution X(t, X0 ) = eλ t ∈ H+ with y(t, X0 ) = 0. Conversely, if y(t, X0 ) = 0 for all t ≥ 0, then – as in (a) – we have the T -invariant subspace X with X ,Y = {0}. If X0 = 0 then X contains an eigenvector, i.e. the criterion is violated.
4 Asymptotic Reconstructability Note that detectability and observability in the sense of Definition 2 do not imply any means to reconstruct the state of the system. In fact, it is obvious that the measurement y = X,Y in general will not be sufficient to distinguish two different solutions of the system. Example 2. Consider H = 2 ordered by 2+ . For T (X) = X, the differential equation X˙ = T (X) defines a positive system, and for any Y > 0 the pair (T,Y ) is observable. But y(t, X0 ) = et X0 ,Y just depends on X0 ,Y . If for instance Y = [1, 1]T , then y(t, [0, 1]T ) = y(t, [1, 0]T ) for all t. Note that Y > 0 always implies In this case Y ⊥ ∩ H+ = {0}. The observability. ⊥ smaller the dimension of span Y ∩ H+ the smaller also the number of nonobservable modes is likely to be. When using positive systems e.g. on H n to analyze stochastic systems on n , this is a natural requirement. We therefore aim at a concept of asymptotic reconstructability which also has this property. To this end we consider a different condition on the pair (T,Y ), which can easily be formulated for arbitrary ordered vector spaces. For the important special cones introduced in Example 1, we show how it leads to positive asymptotic observer equations. Definition 3. Let T : H → H be resolvent positive and Y ∈ H+∗ . We call the pair (T,Y ) asymptotically reconstructable if T ∗ (Z) − Y < 0 for some Z > 0. Let us note a simple general implication. Lemma 1. If (T,Y ) is asymptotically reconstructable then it is detectable. Proof. If T ∗ (Z) − Y < 0 for some Z ∈ int H+∗ . If TV = λ V for some eigenpair (λ ,V ) ∈ × H+ \ {0} satisfying V,Y = 0 then 0 > V, T ∗ (Z) − Y = T (V ), Z = λ V, Z . Since V, Z ≥ 0, we have λ < 0.
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The converse implication does not hold in general as was demonstrated in Example 2. Now let us consider two special cases.
4.1 Asymptotic Reconstructability on Ên This case may look a bit artificial, but it illustrates the concept. Let H = n and H+ = n+ . Then T is a Metzler-matrix and Y ∈ n+ . Let diag(Y ) denote the diagonal matrix whose diagonal contains the entries of Y and assume that y = diag(Y )X. Consider the extended system X˙ = T X , y = diag(Y )X X˙ˆ = T Xˆ + K diag(Y )Xˆ − y , where Xˆ is the state of the observer parametrized by the diagonal matrix K. For the error E = Xˆ − X we have E˙ = (T + K diag(Y ))E =: TK E .
(1)
Lemma 2. There exists a diagonal matrix K, so that σ (TK ) ⊂ − , if and only if (T,Y ) is asymptotically reconstructable. Proof. Note that TK is Metzler and β (TK ) = β (TK∗ ) If T ∗ Z − Y < 0 for some Z ∈ int H+∗ , then we set K = − diag(Z)−1 , so that TK∗ Z = T ∗ Z − Y < 0, i.e. β (TK ) < 0 by Proposition 2. Conversely, if β (TK ) < 0 TK∗ Z˜ − Y = T ∗ Z˜ + diag(Y )K Z˜ < 0 for some Z˜ ∈ int H+∗ . It is easy to see that diag(Y )K Z˜ > −α Y for some α > 0. Hence T ∗ Z − Y < 0 for ˜ α. Z = Z/ Remark 1. For general systems with A ∈ n×n , C ∈ n×n the existence of a matrix K so that σ (A + KC) ⊂ − is equivalent to detectability in the usual sense, i.e. rank(sI − AT ,CT ) = n for all s ∈ + . But the characterization in Lemma 2 requires the assumptions that A is Metzler and that C = diag(Y
) is diagonal 1 −2 00 and nonnegative. For instance, the pair (A,C) = , , with Y = 0 0 −1 00 T is not detectable, but AZ − Y <
0 for
Z = [1, 1] . Similarly, the pair (A,C) = 0 0 −1 0 −1 , with Y = is detectable, but AZ −Y < 0 for any Z > 0. 0 −1 00 0 Definition 3 thus is specific for positive systems.
4.2 Asymptotic Reconstructability on H n Let H = H n and H+ = H+n . Let y = Y X and consider the extended system
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X˙ = T (X) , y = Y X ˆ + KY Xˆ + XY ˆ K T − y − yT , X˙ˆ = T (X) where K ∈
n×n .
(2)
For the error E = Xˆ − X we have E˙ = T (E) + KY E + EY K T =: TK (E) .
Lemma 3. There exists a matrix K ∈ (T,Y ) is asymptotically reconstructable.
n×n ,
(3)
so that σ (TK ) ⊂ − , if and only if
Proof. Note that TK is resolvent positive for all K and thus σ (TK ) ⊂ − if and only if TK∗ (Z) = T ∗ (Z) + ZKY + Y K T Z < 0 for some Z > 0. If T ∗ (Z) − Y < 0 then we may just set K = − 12 Z −1 to get TK∗ (Z) < 0. Vice versa, assume that TK∗ (Z) < 0 with Z > 0. Let U = [U1 ,U2 ] ∈ n×n be orthogonal with the columns of U1 spanning KerY . Then U1T YU1 = 0 together with U1T TK∗ (Z)U1 > 0 implies U1T T ∗ (Z)U1 < 0. For α > 0 we have
α U1T T (Z)U1 α U1T T ∗ (Z)U2 U T (T ∗ (α Z) − Y )U = , α U2T T ∗ (Z)U1 α U2T T ∗ (Z)U2 − U2T YU2 where U2T YU2 > 0. If α is sufficiently small the Schur-complement −U2T YU2 + α U2T T ∗ (Z)U2 − U2T T ∗ (Z)U1 (U1T T ∗ (Z)U1 )−1U1T T ∗ (Z)U2 , is negative, proving T (α Z) −Y < 0. Since α Z > 0, this proves that (T,Y ) is asymptotically reconstructable. As a consequence we obtain a simple characterization of detectability. Corollary 1. Let A ∈ n×n , C ∈ p×n . The pair (A,C) is detectable in the usual sense, i.e. rank(sI − AT ,CT ) = n, if and only if there exists a positive definite matrix Z, so that AZ + ZAT − CT C < 0.
5 Some Applications We sketch two set-ups where our concepts can be applied directly. An in-depth analysis of these examples is beyond the scope of this contribution.
5.1 Stochastic Differential Equations Consider the Itˆo-type stochastic differential equation d ξ = Aξ dt + A0ξ dw ,
η = Cξ .
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Then X = E(ξ ξ T ) satisfies X˙ = AX + XAT + A0 XAT0 = T (X), which is a positive system on H n . If from the measurements η it is possible to estimate E(ηη T ) = CXCT , then also the output y = trace XCT C = X,Y is available. Then the system is detectable, if and only if the pair (T,Y ) with Y = CT C is detectable. If further we assume that even the output Y X = CT CX is available, then we can set up the observer Eq. (2) for the second moments X. Note, however, that this requires different measurements than just η = Cξ and may be unrealistic, but it is exactly the underlying assumption in the concept of MS-detectability (compare [8]).
5.2 Markov Jump Linear Systems Here we consider n-dimensional systems of the form (e.g. [22])
ξ˙ (t) = A(θ (t))ξ (t) ,
η (t) = C(θ (t))ξ (t) ,
(4)
where θ (t) is a Markov process in continuous time on a finite sample space S = {1, 2, . . . , N}. The distributions p ∈ [0, 1]N with p j = P(θ = j) of the process are subject to the transition equation p(t) ˙ = Λ p(t) where eΛ is a stochastic matrix. For T X j = E(x(t)x(t) δθ (t), j ), j = 1, . . . , N, where δi, j is Kronecker’s delta, we have the coupled set of Lyapunov equations (cf. [17]) N
X˙ j = ATi Xi + Xi Ai + ∑ λi j X j = T j (X) ,
j = 1, . . . , N , (λi j ) = Λ .
(5)
j=1
Here we may consider the space H = (H n )N ordered by H+ = (H+n )N . Then T = (T j , . . . , TN ) is resolvent positive, and (C1T C1 , . . . ,CNT CN ) = Y ∈ H+∗ . Detectability, observability and reconstructability properties can now immediately be transferred to the system (4) like in the previous subsection.
6 Summary and Outlook Positive linear systems on matrix algebras are used as auxiliary systems for various linear control problems. The notions of detectability, observability and asymptotic reconstructability can be formulated conveniently in terms of resolvent positive operators. While the interpretation of the first two is clear for arbitrary spaces H , we have clarified the meaning of asymptotic reconstructability only for (H , H+ ) = ( n , n+ ) and for (H , H+ ) = (H n , H+n ). A general analysis will be an issue for further research. Similarly, a unified treatment of different classes of stochastic systems (e.g. from networked control) shall be carried out in more detail.
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References 1. Berman, A., Neumann, M., Stern, R.: Nonnegative Matrices in Dynamic Systems. John Wiley and Sons Inc., New York (1989) 2. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. In: Classics in Applied Mathematics, vol. 9. SIAM Publications, Philadelphia (1994) 3. Costa, E., do Val, J.: On the detectability and observability of discrete-time Markov jump linear systems. Syst. Control Lett. 44(2), 135–145 (2001) 4. Costa, E., do Val, J., Fragoso, M.: A new approach to detectability of discrete-time infinite Markov jump linear systems. SIAM J. Control Optim. 43(6), 2132–2156 (2005) 5. Da Prato, G., Ichikawa, A.: Stability and quadratic control for linear stochastic equations with unbounded coefficients. Boll. Unione Mat. Ital., VI. Ser., B 6, 987–1001 (1985) 6. Damm, T.: Stability of linear systems and positive semigroups of symmetric matrices. In: Benvenuti, L., Santis, A.D., Farina, L. (eds.) Positive Systems. LNCIS, vol. 294, pp. 207–214. Springer, Heidelberg (2003) 7. Damm, T.: Rational Matrix Equations in Stochastic Control. LNCIS, vol. 297. Springer, Heidelberg (2004) 8. Damm, T.: On detectability of stochastic systems. Automatica 43(5), 928–933 (2007) 9. Drˇagan, V., Halanay, A., Stoica, A.: A small gain theorem for linear stochastic systems. Syst. Control Lett. 30, 243–251 (1997) 10. Drˇagan, V., Morozan, T.: Stochastic observability and applications. IMA J. Math. Control & Information 21(3), 323–344 (2004) 11. Elsner, L.: Quasimonotonie und Ungleichungen in halbgeordneten R¨aumen. Linear Algebra Appl. 8, 249–261 (1974) 12. Fragoso, M.D., Costa, O.L.V., de Souza, C.E.: A new approach to linearly perturbed Riccati equations in stochastic control. Appl. Math. Optim. 37, 99–126 (1998) 13. Freiling, G., Hochhaus, A.: On a class of rational matrix differential equations arising in stochastic control. Linear Algebra Appl. 379, 43–68 (2004) 14. H¨ardin, H.M., Van Schuppen, J.H.: Observers for linear positive systems. Linear Algebra Appl. 425(2-3), 571–607 (2007) 15. Hautus, M.L.J.: Controllability and observability conditions of linear autonomous systems. Indag. Math. 31, 443–448 (1969) 16. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) 17. Morozan, T.: Parametrized Riccati equations for controlled linear differential systems with jump Markov perturbations. Stochastic Anal. Appl. 16(4), 661–682 (1998) ¨ 18. Ostrowski, A.M.: Uber die Determinanten mit u¨ berwiegender Hauptdiagonale. Comm. Math. Helv. 10, 69–96 (1937) 19. Schneider, H.: Positive operators and an inertia theorem. Numer. Math. 7, 11–17 (1965) 20. Sontag, E.D.: Mathematical Control Theory, Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York (1998) 21. Tessitore, G.: Some remarks on the detectability condition for stochastic systems. In: Da Prato, G. (ed.) Partial differential equation methods in control and shape analysis. Lect. Notes Pure Appl. Math., vol. 188, pp. 309–319. Marcel Dekker, New York (1997) 22. Ugrinovskii, V.: Randomized algorithms for robust stability and guaranteed cost control of stochastic jump parameter systems with uncertain switching policies. J. Optim. Th. & Appl. 124(1), 227–245 (2005) 23. Van Willigenburg, L., De Koning, W.: Linear systems theory revisited. Automatica 44, 1686–1696 (2008) 24. Zhang, W., Chen, B.-S.: On stabilizability and exact observability of stochastic systems with their applications. Automatica 40, 87–94 (2004)
Stability Radii of Interconnected Positive Systems with Uncertain Couplings Diederich Hinrichsen
Abstract. We analyze interconnections of finitely many (exponentially) stable positive systems which interact via uncertain couplings of arbitrarily prescribed structure. We view the couplings as structured perturbations of the direct sum of the subsystems and derive computable formulas for the corresponding stability radius with respect to some costumary perturbation norms.
1 Introduction A basic problem in the stability analysis of large-scale systems is to determine under which conditions on the magnitude of the couplings the stability of the subsystems implies the stability of the overall system. In the literature one can find many sufficient criteria which guarantee the stability of the interconnected system given the stability of its subsystems, see [15] and the references therein. The purpose of this paper is to derive results which are, in a certain sense, necessary and sufficient. We restrict ourselves to time-invariant linear couplings. We consider composite systems Σcomp with a prescribed interconnection structure. If Σcomp consists of N subsystems Σi , the interconnection structure is given by specifying that the output of Σ j is coupled a nonnegative matrix E = (ei j ) ∈ N×N + to the input of Σi if and only if ei j > 0. Gershgorin [6] had the idea of viewing any square matrix as a perturbation of the corresponding diagonal matrix. Analogously we regard the coupled system Σcomp as a perturbation of the (“nominal”) blockdiagonal system Σ = ⊕Σi which describes the collection of decoupled subsystems. This view of the couplings as perturbations is particularly appropriate in our context, since the interconnections between the subsystems are considered as uncertain. However, we deviate from Gershgorin’s Diederich Hinrichsen Zentrum f¨ur Technomathematik, Universit¨at Bremen, Postfach 330 440, 28334 Bremen, Germany, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 71–81. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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approach in two respects: First, we do not suppose that we know exactly the “diagonal elements” (the decoupled subsystems) but allow for some parametric uncertainty of these systems; secondly we do not allow arbitrary off-diagonal perturbations (couplings between any two subsystems) but only those which preserve the given coupling structure. In this latter respect our approach ressembles that of Brualdi [2] who used the zero pattern of a given matrix in order to sharpen Gershgorin’s and Brauer’s inclusion theorems, see [13]. Our aim is to determine the stability radius of the system Σ with respect to the given perturbation/coupling structure. A standard tool for the robustness analysis of uncertain systems with structured perturbations is μ -analysis [3]. However, μ -analysis provides, in general, only estimates for stability radii, see [7], [8]. In contrast, our objective is to obtain exact formulas. Moreover, a reformulation of our problem in terms of blockdiagonal perturbations, as usual in μ -analysis, would blow up the dimension of the matrices to multiples of the overall system dimension. For large scale systems this may cause severe computational difficulties. Therefore we aim at reduced order formulas for the stability radii which involve only matrices of order N (the number of interconnected subsystems). In most cases N will be significantly smaller than the dimension of the overall system. To obtain precise results, a specific investigation of the special class of uncertain systems, the particular perturbation structure and perturbation norm is necessary. In this paper we deal especially with large scale systems composed of positive subsystems. For an analysis of spectral value sets and stability radii of composite systems consisting of general time-invariant linear systems, see [13]. It is well known that stability radii probems are greatly simplified under positivity assumptions. For positive systems with full block uncertainty the real and the complex stability radii coincide and can be computed by a simple formula [9], [17], whereas for general systems the two radii are different and the computation of the real stability radius is an intricate problem, see [7, §5.3]. Meanwhile many papers have appeared which extend the early results of [9], [17] to other classes of positive systems and different types of uncertainty. For instance, block-diagonal and positive-affine perturbations have been considered in [8], infinite dimensional positive systems in [4], [5], positive delay systems in [16], [12] and positive Volterra systems in [14]. However, up till now, there are no papers on the stability radius problem for interconnected positive systems with uncertain couplings of prescribed structure. The paper is organized as follows. After the preliminaries in the next section we present a detailed description of the interconnected system in Section 3. All the new results of this paper are contained in Section 4. We end the paper with a conjecture the proof or disproof of which remains an open question.
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2 Preliminaries In this section we introduce some basic concepts and fix the notation. The symbols , , + , denote the sets of positive integers, real numbers, non-negative real numbers and complex numbers, respectively. For any N ∈ we set N := {1, 2, . . . , N}. By n×m we denote the set of n by m matrices with entries in where = or . If A = (ai j ) ∈ n×m we define |A| := (|ai j |) and for real matrices A = (ai j ), B = (bi j ) ∈ n×m we write A ≤ B if a ≤ b for all i ∈ n, j ∈ m. If A is square then σ (A) and ij ij ρ (A) = max{|λ |; λ ∈ σ (A)} denote its spectrum and spectral radius, respectively. The spectral radius has the following monotonicity property, see [10, §8.1]: ∀A ∈ n×n , B ∈
n×n +
:
|A| ≤ B ⇒ ρ (A) ≤ ρ (|A|) ≤ ρ (B).
(1)
) we denote the set of triples of matrices (A, B,C) with A ∈ n×n , B ∈ q×n , n, l, q ∈ . The open left half-plane is denoted by − = {s ∈ ; Re s < 0} and A ∈ n×n is called stable if σ (A) ⊂ − . We use the conventions n,l,q ( n×l ,C ∈
By
0−1 = ∞,
∞−1 = 0,
inf 0/ = ∞.
(2)
In this section we suppose that (A, B,C) ∈ n,l,q () and Δ ⊂ l×q is a -linear subspace provided with a norm ·Δ . We consider perturbations of the form A ; A(Δ ) = A + BΔ C,
Δ ∈ Δ.
(3)
Definition 1. Given a system (A, B,C) ∈ n,l,q () and given a perturbation space (Δ , ·Δ ), the stability radius of A with respect to perturbations of the form (3) is defined by (4) rΔ (A, B,C) = inf{Δ Δ ; Δ ∈ Δ , σ (A(Δ )) ⊂ − }. It is easily seen that the infimum in (4) is in fact a minimum if rΔ (A, B,C) is finite. In this case the stability radius is the norm of a smallest perturbation in Δ which destabilizes A. rΔ (A, B,C) = 0 if and only if σ (A) ⊂ − . rΔ (A, B,C) = ∞ if and only if σ (A + BΔ C) ⊂ − for all Δ ∈ Δ . For arbitrary perturbation spaces (Δ , ·Δ ) the determination of rΔ (A, B,C) is a difficult problem and, in general, only estimates are available. In the full-block case, however, where Δ = l×q and the stability radius rΔ (A, B,C) is denoted by r (A, B,C), computable formulas are available for both the complex stability radius r (A, B,C) and the real stability radius r (A, B,C) (if (A, B,C) ∈ n,l,q ( ) and · Δ is the spectral norm), see [7, §5.3]. A linear system x(t) ˙ = Ax(t), A ∈ n×n is said to be positive if the positive orthant n is invariant for the corresponding flow, i.e., eAt ≥ 0 for all t ≥ 0. In this case the + matrix A is called a Metzler matrix. A ∈ n×n is a Metzler matrix if and only if all the off-diagonal entries of A are nonnegative, i.e., tIn + A ≥ 0 for some t ≥ 0. A norm · on n is said to be monotone if it satisfies
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|x| ≤ |y| ⇒ x ≤ y,
x, y ∈ n .
Every p-norm · p on n , 1 ≤ p ≤ ∞ is monotone. The operator norm · associated with a pair of monotone vector norms needs not be monotone. However, we have the following properties, see [17]. Lemma 1. Suppose that m , n are provided with monotone norms ·m , ·n , and · denotes the corresponding operator norm on m×n . Then there exists u ∈ n+ , un = 1 such that Pum = P. 1. For every P ∈ m×n + 2. If P ∈ m×n , Q ∈ m×n and |P| ≤ Q, then P ≤ |P| ≤ Q. + 3. If P ∈ m×n is of rank one, then P = |P| . The following proposition illustrates that the stability radius problem is substantially simplified if the system (A, B,C) ∈ n,l,q ( ) is positive, i.e. A is a Metzler matrix and B,C ≥ 0. Proposition 1. [17] Suppose A ∈ n×n is a stable Metzler matrix, (B,C) ∈ n×l + × and l , q are provided with monotonic norms. Then, with respect to the induced operator norm · L (q ,l ) on l×q , = , , q×n + ,
r (A, B,C) = r (A, B,C) = r+ (A, B,C) = CA−1 B−1 . L (l ,q ) where r+ (A, B,C) = inf{Δ L (q ,l ) ; Δ ∈
l×q + ,
(5)
σ (A(Δ )) ⊂ − }.
3 Interconnected Systems To develop a framework for the analysis of interconnected positive systems with uncertain couplings we need some additional notation, see [13]. In the following, l, n, q, are finite N-tuples l = (l1 , . . . , lN ), n = (n1 , . . . , nN ), q = (q1 , . . . , qN ), l j , n j , q j , N ∈ . By l×q we denote the set of N × N block matrices ⎤ ⎡ Δ11 · · · Δ1N ⎢ .. ⎥ , Δ ∈ li ×q j , (i, j) ∈ N × N. [Δi j ] = [Δi j ]i, j∈N = ⎣ ... (6) ij . ⎦
ΔN1 · · · ΔNN
Suppose that E ∈ N×N is a given nonnegative matrix with entries ei j ≥ 0. A block matrix Δ = (Δi j ) ∈ Δ := l×q is said to be of structure E if ei j = 0 implies Δi j = 0. Let Δ E ⊂ Δ be the vector space of all the block matrices Δ ∈ l×q of structure E. Given X = (xi j ) ∈ h×k ,Y = (yi j ) ∈ h×k where h, k are positive integers, the Hadamard product of X and Y , denoted by X ◦ Y , is defined by X ◦ Y = (xi j yi j ) ∈ h×k . For k ∈ the kth Hadamard power of X is defined by X ◦k := (x◦k i j ) where k if x = 0 and x◦k := 0 if x = 0. Given a matrix X = (x ) ∈ N×N and := x x◦k ij ij ij ij ij ij a block matrix Y = [Y jk ] j,k∈N ∈ l×q , the Hadamard block product of X and Y is,
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by definition, the block matrix X ◦ Y := [xi jYi j ]i, j∈N ∈ l×q . Note that for every Δ = (Δi j ) ∈ l×q we have E ◦ Δ = [ei j Δi j ]i, j∈N ∈ Δ E . Given N subsystems (Ai , Bi ,Ci ) ∈ ni ,li ,qi (), i ∈ N, consider the system
Σ:
x(t) ˙ = Ax(t) + Bu(t),
y(t) = Cx(t)
(7)
where A, B, C are the block-diagonal matrices A = ⊕Ni=1 Ai ∈ n×n , B = ⊕Ni=1 Bi ∈ n×l , C = ⊕Ni=1Ci ∈ q×n . Σ is the direct sum of the N subsystems
Σi :
x˙i (t) = Ai xi (t) + Biui (t),
yi (t) = Ci xi (t),
i ∈ N.
(8)
The transfer matrix of Σ is the direct sum of the transfer matrices of the subsystems: G(s) = C(sI− A)−1B = ⊕Ni=1 Gi (s),
Gi (s) = Ci (sIni − Ai )−1 Bi , i ∈ N.
(9)
Introducing, for a given Δ = (Δi j ) ∈ Δ := l×q , the couplings ui (t) =
∑ ei j Δi j y j (t),
i∈N
(10)
j∈N
one obtains the coupled system equations x˙i (t) = Ai xi (t) + Bi
∑ ei j Δi jC j x j (t),
i∈N
(11)
j∈N
which altogether describe the interconnected system ⎡ ⎤ ⎡ ⎤ x˙1 x1 ⎢ .. ⎥ ⎢ .. ⎥ ΣΔ : ⎣ . ⎦ = (A + B(E ◦ Δ )C) ⎣ . ⎦ . x˙N
(12)
xN
In this description the scaling matrix E has a double role. On the one hand it defines the structure of the admissible couplings: The subsystem Σ j is coupled to the subsystem Σi (by the uncertain coupling ei j Δi j ) if and only if ei j > 0. On the other hand the positive entries ei j of E provide weights for the blocks Δi j . Since these weights cannot, in general, be absorbed by the matrices Bi and/or C j , they provide an additional scaling flexibility.
4 Stability Radii In this section we derive computable formulas for the stability radii of the blockdiagonal system x˙ = Ax with respect to structured perturbations of the form A ; A(Δ ) := A + B(E ◦ Δ )C,
Δ ∈ Δ,
(13)
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see (12). The perturbed matrix A(Δ ) defined in (13) is the system matrix of the coupled system ΣΔ . We continue to use the set-up of the previous section. Let ·Y j be a norm on Y j := q j , ·YDj its dual norm, see [10], and ·Ui a norm on Ui := li . The corresponding operator norms on li ×q j (resp. qi ×l j ) are denoted by ·L (Y j ,Ui ) (resp. ·L (U j ,Yi ) ). For any Δ = (Δi j ) ∈ l×q and M = (Mi j ) ∈ q×l we define Δ˜ := Δi j L (Y j ,Ui ) ∈
N×N , +
M˜ := Mi j L (U j ,Yi ) ∈
Lemma 2. If M = (Mi j ) ∈ q×l and E ∈
N×N +
˜ ρ ((E ◦ Δ )M) ≤ ρ ((E ◦ Δ˜ )M),
N×N . +
(14)
then
Δ = (Δi j ) ∈ l×q .
(15)
Proof. Let Y =(Yi j ), Yi j =((E ◦ Δ )M)i j = ∑Nk=1 eik Δik Mk j then by [13, Lemma 4.1]
ρ (Y ) ≤ ρ (Y˜ ) where Y˜ = (Yi j )L (U j ,Ui ) ). ˜ i j for i, j ∈ Since Yi j L (U j ,Ui ) ≤ ∑Nk=1 eik Δik L (Yk ,Ui ) Mk j L (U j ,Yk ) = ((E ◦ Δ˜ )M) N, we have 0 ≤ Y˜ ≤ (E ◦ Δ˜ )M˜ and hence (15) follows from (1). Let N (·) be any vector norm on N×N and define the perturbation norm on Δ = l×q by Δ Δ = N (Δ˜ ), Δ ∈ l×q . (16) Lemma 3. If M = ⊕Nj=1 M j ∈ q×l is block-diagonal and E ∈ every matrix δ = (δi j ) ∈ N×N , there exists Δ ∈ l×q such that +
N×N , +
˜ ρ ((E ◦ Δ )M) = ρ ((E ◦ Δ˜ )M) and Δ˜ = (δi j ).
then, for
(17)
If M j ≥ 0 for all j ∈ N, then Δ can be chosen to be nonnegative. Proof. For j ∈ N let u j ∈ l j and y j ∈ q j be such that u j U j = y∗j YDj = 1 and y∗j M j u j = M j L (U j ,Y j ) . Define Du := diag(u1 , . . . , uN ) ∈ l×N ,
Dy∗ := diag(y∗1 , . . . , y∗N )N×q .
Then M˜ = diag(y∗1 M1 u1 , . . . , y∗N MN uN ) = Dy∗ MDu and so (E ◦ δ ) M˜ = (E ◦ δ )Dy∗MDu , ρ (E ◦ δ )M˜ = ρ (Du (E ◦ δ )Dy∗ M) .
(18)
Setting Δi j := δi j ui y∗j ∈ li ×q j we obtain a block matrix Δ := (Δi j ) ∈ l×q satisfying Δi j L (Y j ,Ui ) = δi j for all i, j ∈ N, i.e. Δ˜ = δ , and E ◦ Δ = (ei j Δi j ) = ei j δi j ui y∗j = Du (E ◦ δ )Dy∗ , hence ρ ((E ◦ Δ )M) = ρ (Du (E ◦ δ )Dy∗ M) = ρ (E ◦ δ )M˜ = ρ (E ◦ Δ˜ )M˜ by (18). This proves the first part of the lemma. If, additionally, M j ≥ 0 for all j ∈ N then the
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above vectors u j , y j can be chosen nonnegative by Lemma 1. But then Δ = (Δi j ) = (δi j ui y∗j ) ≥ 0. This proves the second part of the lemma. Now suppose we are given the following data: (Ai , Bi ,Ci ) ∈ ni ,li ,qi (), Gi (s) = Ci (sI − Ai )−1 Bi , i ∈ N, E ∈
N×N + ,
A = ⊕Ni=1 Ai , σ (A) ⊂ − , B = ⊕Ni=1 Bi , C = ⊕Ni=1Ci , G(s) = ⊕Ni=1 Gi (s).
(19)
For any G ∈ N×N we define
μ N (G; E) = inf{N (δ ); δ ∈
N×N , +
ρ ((E ◦ δ )G) ≥ 1}.
(20)
The following theorem reduces the stability radius problem for the complex |n|dimensional interconnected system ΣΔ of dimension |n| = ∑m i=1 ni to an N-dimensional problem with non-negative data. Theorem 1. Suppose (19) and let Δ = l×q be provided with the norm (16), then the stability radius of A with respect to perturbations of the form (13) is given by r (A, B,C; E) := inf{Δ Δ ; Δ ∈ l×q , σ (A(Δ )) ⊂ − } = min μ N (D(ıω ); E). ω ∈
(21) where D(s) = diag(G1 (s)L (U1 ,Y1 ) , . . . , GN (s)L (UN ,YN ) ). If all the subsystems (Ai , Bi ,Ci ), i ∈ N are positive and the spaces Ui ,Yi are provided with monotone norms then r (A, B,C; E) = r (A, B,C; E) = r+ (A, B,C; E) = μ N (D(0); E).
(22)
where r (A, B,C; E), r+ (A, B,C; E) are defined analogously to r (A, B,C; E). Proof. By [7, Lemma 5.2.7] the following equivalence holds for all s ∈ ρ (A) and all Δ ∈ l×q : s ∈ σ (A + B(E ◦ Δ )C) ⇔ 1 ∈ σ ((E ◦ Δ )G(s)).
(23)
, = D(s), it follows from Lemma 2 that 1 ∈ σ ((E ◦ Δ )G(s)) implies Since G(s) ρ ((E ◦ δ )D(s))} ≥ 1 for δ := Δ˜ . Conversely, if ρ := ρ ((E ◦ δ )D(s))} ≥ 1 for some δ ∈ N×N then there exists by Lemma 3 Δ ∈ l×q such that ρ ((E ◦ Δ )G(s))} = ρ + ˜ and Δ = δ , whence Δ Δ = N (δ ). Let λ ∈ σ ((E ◦ Δ )G(s)) satisfy |λ | = ρ ≥ 1, then 1 ∈ σ ((E ◦ (λ −1 Δ ))G(s)) and λ −1 Δ ∈ l×q is of norm λ −1 Δ Δ ≤ N (δ ). Therefore it follows from the continuity of the spectrum that r (A, B,C; E) = min inf{Δ Δ ; Δ ∈ Δ , 1 ∈ σ ((E ◦ Δ )G(ıω ))} ω ∈
= min inf{N (δ ); δ ∈ ω ∈
= min μ N (D(ıω ); E). ω ∈
N×N , +
ρ ((E ◦ δ )D(ıω ))} ≥ 1}
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This proves (21). Now assume that (A, B,C) is a positive system and the spaces Ui ,Yi are provided with monotone norms. Then Gi (ıω )L (Ui ,Yi ) ≤ Gi (0)L (Ui ,Yi ) for all i ∈ N, ω ∈ (see [17, §4]) and hence ρ ((E ◦ δ )D(ıω )) ≤ ρ ((E ◦ δ )D(0)) for all δ ∈ N×N by (1). It follows that minω ∈ μ N (D(ıω ); E) = μ N (D(0); E) + whence r (A, B,C; E) = μ N (D(0); E) by (21). Trivially we have r ≤ r ≤ r+ . Hence it suffices by (20) and (23) to prove that for every δ ∈ N×N such that ρ ((E ◦ + δ )D(0))} ≥ 1 there exists Δ ∈ l×q satisfying (17). But this follows by application + of Lemma 3 to the nonnegative block-diagonal matrix M =: ⊕Ni=1 Gi (0) ≥ 0. Remark 1. Theorem 1 generalizes known results for single systems (N = 1) to interconnected systems with an arbitrarily prescribed coupling structure. In fact, let 1×1 = . Then (A, B,C) = N = 1 and choose E = (1) ∈ 1×1 + , N (δ ) = |δ | for δ ∈ (A1 , B1 ,C1 ) and, omitting the subindices, we have D(s) = G(s)L (l ,q ) = C(sI − A)−1 BL (l ,q ) , D(0) = CA−1 BL (l ,q ) , r (A, B,C, E) = r (A, B,C), and
μ N (D(s); E) = {|δ |; δ ∈
+,
ρ (δ D(s))} ≥ 1} = 1/G(s)L (l ,q ) , s ∈ \ σ (A).
Hence (21) and (22) specialize to r (A, B,C) = minω ∈ 1/G(ıω )L (l ,q ) and r (A, B,C) = r (A, B,C) = r+ (A, B,C) = 1/CA−1BL (l ,q ) , respectively. The latter formula coincides with (5) and the first one coincides with formula (22) in [7, § 5.3.2]. We will now apply the previous theorem to specific norms N on N×N in order to derive explicit and computable formulas for the stability radius r (A, B,C; E) of interconnected positive systems. For this we need to determine μ N (D(0); E). Since D(0) = diag(d1 , . . . , dN ), di = Gi (0)L (Ui ,Yi ) is diagonal, we have (E ◦ δ )D(0) = δ ◦ (ED(0)) for δ = (δi j ) ∈ N×N and hence by (20) +
μ N (D(0), E) = inf{N (δ ); δ ∈ We first consider the maximum norm on
N×N , +
(24)
N×N :
N (δ ) = δ max := max |δi j |, i, j∈N
ρ (δ ◦ (ED(0)))} ≥ 1}.
δ = (δi j ) ∈
N×N
.
(25)
Corollary 1. Suppose (19) and let Δ = l×q be provided with the norm (16) where N is the maximum norm (25). If all the subsystems (Ai , Bi ,Ci ), i ∈ N are positive and the spaces Ui ,Yi are provided with monotone norms then r (A, B,C; E) = r (A, B,C; E) = r+ (A, B,C; E) = 1/ρ (ED(0)).
(26)
Proof. If δ ∈ N×N satisfies ρ (δ ◦ (ED(0)))} ≥ 1 then δ max ρ (ED(0)) ≥ 1 + by (1) and hence μ N (D(0), E) ≥ 1/ρ (ED(0)) by (20). On the other hand, if 1 denotes the N × N-matrix with all entries equal to one, then δ := (1/ρ (ED(0)))1 satisfies δ max = 1/ρ (ED(0)) and ρ (δ ◦ (ED(0)))} = 1 whence μ N (D(0), E) ≤
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1/ρ (ED(0)) by (24). We conclude that μ N (D(0), E) = 1/ρ (ED(0)) and so (26) follows from (22). We now consider operator norms on N×N which are induced by weakly monotone and permutation invariant norms on N (e.g. p-norms, 1 ≤ p ≤ ∞). A norm ·N on N is called permutation invariant if (xi )N = (xπ (i) )N for all (xi ) ∈ N and all permutations π of N. It is called weakly monotone if x N ≥ yN for all x, y ∈ N such that, for all i ∈ N, either yi = xi or yi = 0. Definition 2. [11, Def.5.7.20] The maximum cycle geometric mean of a nonnegis ative matrix G = (gi j ) ∈ N×N + c(G) := max
k
∏ gi j i j+1
1/k
(27)
j=1
in which k + 1 is identified with 1 and the maximum is taken over all sequences of distinct indices i1 , . . . , ik ≤ N and over all k ∈ N. Given two nonnegative matrices G, H ∈ see [11, Theorem 5.7.21].
N×N +
the following inequalities hold,
c(G) ≤ ρ (G) ≤ Nc(G) and ρ (G ◦ H) ≤ c(G)ρ (H).
(28)
Corollary 2. Suppose (19) and let Δ = l×q be provided with the norm (16) where N is any operator norm induced by a permutation invariant and weakly monotone norm on N . If all the subsystems (Ai , Bi ,Ci ), i ∈ N are positive and the spaces Ui ,Yi are provided with monotone norms then r (A, B,C; E) = r (A, B,C; E) = r+ (A, B,C; E) = 1/c(ED(0)).
(29)
Proof. If δ ∈ N×N satisfies ρ (δ ◦ (ED(0)))} ≥ 1 then N (δ ) ≥ ρ (δ ) ≥ + 1/c(ED(0)) by (28), hence μ N (D(0), E) ≥ 1/c(ED(0)) by (24) and equality holds if c(ED(0)) = 0. To prove the converse inequality, suppose c(ED(0)) > 0 and let i1 , . . . , ik ≤ N be distinct indices such that ∏kj=1 gi j i j+1 = c(G)k for G := ED(0). Deby δi j i j+1 = 1/c(G), j ∈ k and δi j = 0 for (i, j) ∈ N 2 \ {(i j , i j+1 ); j ∈ fine δ ∈ N×N + k}. Then Δ has at most one non-zero entry (= 1/c(G)) in each row so that δ xN ≤ (1/c(G))xN by weak monotonicity and permutation invariance. Moreover H := δ ◦ (ED(0)) satisfies c(H)k ≥ ∏kj=1 hi j i j+1 = ∏kj=1 δi j i j+1 gi j i j+1 = c(G)−k c(G)k = 1. Therefore ρ (δ ◦ (ED(0))) ≥ c(H) ≥ 1 by (28) and N (δ ) ≤ (1/c(G)), whence μ N (D(0), E) ≤ 1/c(ED(0)). This shows μ N (D(0), E) = 1/c(ED(0)) and concludes the proof. Finally, we consider the case where N is a mixed norm of the form N (δ ) := max i∈N
where 1 < p < ∞ is given.
∑ |δi j | p
j∈N
1/p
,
δ∈
N×N
(30)
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Corollary 3. Suppose (19), 1 < p, p∗ < ∞ and 1/p + 1/p∗ = 1. Let Δ = l×q be provided with the norm (16) where N is given by (30). If all the subsystems (Ai , Bi ,Ci ), i ∈ N are positive and the spaces Ui ,Yi are provided with monotone norms then −1/p∗ ∗ ∗ . (31) r (A, B,C; E) = r (A, B,C; E) = r+ (A, B,C; E) = ρ (E ◦p D(0) p ) Proof. Suppose that δ ∈ N×N satisfies ρ (δ ◦(ED(0)))}=1. By Perron-Frobenius + theory there exists y ∈ N+ , y = 0 such that δ ◦ (ED(0))y = y, i.e. ∑ j∈N δi j d j ei j y j = yi for i ∈ N. If 1 < p < ∞ then by H¨older’s inequality yi ≤
∑ |δi j | p
j∈N
1/p
∑ |d j ei j y j | p
∗
1/p∗
.
j∈N
∗ ∗ ∗ ∗ Setting zi = yip we obtain from (30) that zi ≤ N (δ ) p ∑ j∈N d pj eipj z j for i ∈ N. By ∗ ∗ ∗ [2, Thm. 2.1.11] this implies ρ (E ◦p D(0) p ) ≥ 1/N (δ ) p , hence μ N (D(0), E) ≥ ∗ ∗ ∗ [ρ (E ◦p D(0) p )]−1/p by (24). ∗ ∗ ∗ To prove the converse inequality, let ρ := [ρ (E ◦p D(0) p )]1/p and z ∈ N+ , z = 0 ∗ ∗ ∗ ∗ ∗ ∗ such that E ◦p D(0) p z = ρ p z, i.e. ∑ j∈N d pj eipj z j = ρ p zi for i ∈ N. Let yi :=
1/p∗
zi , i ∈ N. Since · p is the dual norm of · p∗ there exists, for every i ∈ N, a vector δ (i) = (δi j ) j∈N of p-norm δ (i) p = 1/ρ such that for i ∈ N 1
1
∗
∗
∑ δi j d j ei j y j = ρ (d j ei j y j ) j∈N p∗ = ρ ∑ d pj eipj y pj
j∈N
j∈N
∗
1/p∗
=
1 p∗ p∗ 1/p∗ ρ yi . ρ
Hence δ := (δi j ) ∈ N×N is of norm N (δ ) = maxi∈N δ (i) p = 1/ρ and satisfies + δ ◦ (ED(0))y = y, whence ρ (δ ◦ (ED(0))) ≥ 1. This proves μ N (D(0), E) = 1/ρ = ∗ ∗ ∗ [ρ (E ◦p D(0) p )]−1/p and so the corollary follows from Theorem 1. Concluding Remark In the robust control of large scale systems it is important to make sure that the closed loop interconnected system remains stable if the links between certain sub˘ systems break down or change with time. Accordingly, Siljak’s concept of connective stability requires robustness against time-varying perturbations/couplings [15]. However, counterparts to the above results for time-varying perturbations (of prescribed structure E) are not yet available. To determine stability radii with respect to time-varying perturbations is known to be a difficult problem [18]. However, under positivity assumptions, the problem may become tractable. I conclude the paper with the following conjecture: Formulas (26), (29), and (31) also hold true if the perturbations are allowed to be time-varying. The proof or disproof of this conjecture is an open problem.
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References 1. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. In: Classics in Applied Mathematics, vol. 9. SIAM Publications, Philadelphia (1994) 2. Brualdi, R.A.: Matrices, eigenvalues, and directed graphs. Linear Multilinear Algebra 11(2), 143–165 (1982) 3. Doyle, J.C.: Analysis of feedback systems with structured uncertainties. IEE Proc. Part D 129, 242–250 (1982) 4. Fischer, A.: Stability radii of infinite-dimensional positive systems. Math. Control Signals Syst. 10, 223–236 (1997) 5. Fischer, A., Hinrichsen, D., Son, N.K.: Stability radii of Metzler operators. Vietnam J. of Mathematics 26, 147–163 (1998) ¨ 6. Gershgorin, S.A.: Uber die Abgrenzung der Eigenwerte einer Matrix. Izvestia Akad. Nauk SSSR, Ser. Fis-Mat. 6, 749–754 (1931) 7. Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Springer, Berlin (2005) 8. Hinrichsen, D., Son, N.K.: μ -analysis and robust stability of positive linear systems. Appl. Math. and Comp. Sci. 8, 253–268 (1998) 9. Hinrichsen, D., Son, N.K.: Stability radii of positive discrete-time systems. In: Proc. 3rd Int. Conf. Approximation and Optimization in the Caribean, Aportaciones Mat., Comunicaciones, Puebla, M´exico, vol. 24, pp. 113–124 (1995). Sociedad Matem´atica Mexicana, M´exico (1998) 10. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990) 11. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991) 12. Hua, G., Davison, E.J.: Real stability radii of linear time-invariant time-delay systems. scl. 50, 209–219 (2003) 13. Karow, M., Hinrichsen, D., Pritchard, A.J.: Interconnected systems with uncertain couplings: explicit formulae for μ -values, spectral value sets and stability radii. SIAM J. Control Optim. 45(3), 856–884 (2006) 14. Ngoc, P.H.A.: Stability radii of positive linear Volterra-Stieltjes equations. Journal of Differential Equations 243, 101–122 (2007) ˇ 15. Siljak, D.D.: Large Scale Dynamic Systems. Stability and Structure. Series in System Science and Engineering. North-Holland, New York (1978) 16. Son, N., Ngoc, P.: Robust stability of positive linear time-delay systems under affine parameter perturbations. Acta Math. Vietnamica 24, 353–372 (1999) 17. Son, N.K., Hinrichsen, D.: Robust stability of positive continuous time systems. Numer. Functional Anal. Optim. 17, 649–659 (1996) 18. Wirth, F.: On the calculation of real time-varying stability radii. Int. J. Robust & Nonlinear Control 8, 1043–1058 (1998)
Linear Operators Preserving the Set of Positive (Nonnegative) Polynomials Olga M. Katkova and Anna M. Vishnyakova
Abstract. This note deals with linear operators preserving the set of positive (nonnegative) polynomials. Numerous works of prominent mathematicians in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description was not formulated explicitly, it is almost lost for possible applications. In the paper we formulate and prove these classical results and give some applications. For example, we prove that there are no linear ordinary differential operators of order m ∈ with polynomial coefficients which map the set of nonnegative (positive) polynomials of degree ≤ ( m2 + 1) into the set of nonnegative polynomials. This result is a generalization of a Theorem by Guterman and Shapiro.
1 Introduction and Statement of Results Real polynomial P is called positive (nonnegative) polynomial if P(x) > 0 (P(x) ≥ 0) for every x ∈ . Positive (nonnegative) polynomials arise in different branches of mathematics, physics, engineering and other sciences (see [3] for a number of open problems connected with the topic, see also a recent paper [2]). Numerous works of prominent mathematicians devoted to linear operators preserving the set of positive (nonnegative) polynomials and connected questions (see, for example, [1], [6]-[12] and the references therein). These classical works in fact contain the exhaustive description of linear operators preserving the set of positive (nonnegative) polynomials. In spite of this, since this description is not formulated explicitly, it is almost lost for possible applications. In this short note we formulate and prove these classical results and give some applications. Olga M. Katkova and Anna M. Vishnyakova Department of Mathematics, Kharkov National University, 4 Svobody sq., 61077 Kharkov, Ukraine, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 83–90. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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We will use two standard notations: [x] will denote the set of real polynomials and m [x] = {Q ∈ [x] : deg Q ≤ m}, m = 0, 1, . . .. The following theorem is a classical result. Unfortunately we can not name its authors, but it is based on the ideas which arose in works by Euler, Gauss and Sylvester. Theorem 1. Let A : [x] → [x] be a linear operator. Let Pk (x) := A(xk ), k = 0, 1, . . .. 1. The operator A preserves the set of nonnegative polynomials if and only if all principal minors of the infinite matrix A (x) := (Pi+ j (x))∞ i, j=0 are nonnegative for every x ∈ . 2. The operator A : 2m [x] → [x] maps the set of nonnegative polynomials of degree ≤ 2m into the set of nonnegative polynomials if and only if all principal minors of the (m + 1) × (m + 1) matrix A (x, m) := (Pi+ j (x))m i, j=0 are nonnegative for every x ∈ . The following theorem is a corollary of Theorem 1 and a remark of Guterman and Shapiro ([4], see also [5]). Theorem 2. Let A : [x] → [x] be a linear operator. Denote by Pk (x) := A(xk ), k = 0, 1, . . .. 1. The operator A preserves the set of positive polynomials if and only if all principal minors of the infinite matrix A (x) := (Pi+ j (x))∞ i, j=0 are nonnegative for every x ∈ and P0 (x) > 0 for every x ∈ . 2. The operator A : 2m [x] → [x] maps the set of positive polynomials of degree ≤ 2m into the set of positive polynomials if and only if all principal minors of the (m + 1) × (m + 1) matrix A (x, m) := (Pi+ j (x))m i, j=0 are nonnegative for every x ∈ and P0 (x) > 0 for every x ∈ . With the help of Theorems 1 and 2 we obtain the following two results. Theorem 3. Given a natural number l and two real sequences {αk }∞ k=0 and k k k+2l {βk }∞ , k= k=0 define a linear operator A : [x] → [x] by A(x ) = αk x + βk x 0, 1, . . .. The operator A preserves the set of nonnegative polynomials if and only if ∞ all principal minors of the infinite matrices A1 := (αi+ j )∞ i, j=0 and A2 := (βi+ j )i, j=0 are nonnegative. Theorem 4. Consider an arbitrary natural number m, a sequence of natural numbers {n(k)}∞ k=0 , and a double sequence of polynomials {Q j (x, k)} j∈,k=0,1,2... . Suppose that Q j (x, k) ≡ 0 for k + j < 0 and Q−m (0, m) = 0. Define a linear operator n(k) A : [x] → [x] by A(xk ) = ∑ j=−m Q j (x, k)xk+ j , k = 0, 1, . . . . Then the operator A does not map the set of nonnegative (positive) polynomials from 2( m +1) [x] into 2 the set of nonnegative polynomials. The last theorem is a generalization of the following surprising result by Guterman and Shapiro.
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Theorem 5. [4, Theorem A] Let A : [x] → [x] be a linear ordinary differential operator of order m ≥ 1 with polynomial coefficients Q = (q0 (x), q1 (x), . . . qm (x)), qm (x) ≡ 0:
A( f ) = q0 (x) f (x) + q1 (x) f (x) + . . . + qm (x) f (m) (x), f ∈ [x]. Then for any coefficient sequence Q the operator A does not map the set of nonnegative (positive) polynomials from 2m [x] into the set of nonnegative polynomials. Moreover, the following more precise theorem is a direct corollary of Theorem 4. Theorem 6. Let A : [x] → [x] be a linear ordinary differential operator of order m ≥ 1 with polynomial coefficients Q = (q0 (x), q1 (x), . . . qm (x)), qm (x) ≡ 0:
A( f ) = q0 (x) f (x) + q1 (x) f (x) + . . . + qm (x) f (m) (x), f ∈ [x]. Then for any coefficient sequence Q the operator A does not map the set of nonnegative (positive) polynomials from 2( m +1) [x] into the set of nonnegative polyno2 mials. Note that the degree 2 m2 + 1 in the previous Theorem could not be decreased: for every m ∈ there exist linear ordinary differential operators A : [x] → [x] of order m which map the set of nonnegative polynomials from 2 m2 [x] into the set of nonnegative polynomials. As a simple example of such operator one could take A( f ) = f (m) (or A( f ) = f + f (m) ). Some important examples of such differential operators were described by R.Remak. Theorem 7. ([11], see also [10, VII, Problem 38]). Let A : [x] → [x] be a linear ordinary differential operator of order 2m, m ∈ , with constant coefficients a0 , a1 , . . . a2m : A( f ) =
a1 a2m (2m) a0 f (x) + f (x) + . . . + f (x), f ∈ [x]. 0! 1! (2m)!
Then the operator A maps the set of nonnegative polynomials from 2m [x] into the m set of nonnegative polynomials if and only if the quadratic form ∑m i=0 ∑ j=0 ai+ j ξi ξ j is nonnegative definite. Using the method which was applied to prove Theorems 1, 6 and 7 we obtain the following result. Theorem 8. Let A : [x] → [x] be a linear ordinary differential operator of order m ≥ 1 with polynomial coefficients Q = (q0 (x), q1 (x), . . . qm (x)): A( f ) =
q0 (x) q1 (x) qm (x) (m) f (x) + f (x) + . . . + f (x), f ∈ [x]. 0! 1! m!
Then the operator A maps the set of nonnegative polynomials from 2 m2 [x] into the set of nonnegative polynomials if and only if all principal minors of the following
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O.M. Katkova and A.M. Vishnyakova m
2 ( m2 + 1) × ( m2 + 1) matrix Aq (x) := (qi+ j (x))i, j=0 are nonnegative for every x∈ .
2 Proof of Theorems 1 and 2 Proof of Theorem 1. 1. It is well known that a real polynomial is nonnegative if and only if it can be represented as a sum of squares of two real polynomials (see, for example, [10, VI, Problem 44] or [1, Chapter 1, §1]). Whence a linear operator A preserves the set of nonnegative polynomials if and only if A(Q2 ) is a nonnegative polynomial for every Q ∈ [x], that is for every n = 0, 1, 2 . . . and for every set of real numbers ξ0 , ξ1 , . . . , ξn a polynomial A (ξ0 + ξ1 x + . . . + ξnxn )2 is nonnegative. We have A (ξ0 + ξ1 x + . . . + ξn xn )2 =
n
∑
ξi ξ j A(xi+ j ) =
i, j=0
n
∑
Pi+ j (x)ξi ξ j .
i, j=0
Hence A preserves the set of nonnegative polynomials if and only if for every n = 0, 1, 2 . . . and for every x ∈ the quadratic form ∑ni, j=0 Pi+ j (x)ξi ξ j is nonnegative and the statement of the theorem follows from the well-known criterion of nonnegative definiteness. 2. To prove this statement it is sufficient to note that a real polynomial from 2m [x] is nonnegative if and only if it could be represented as a sum of squares of two real polynomials from m [x] and to apply the reasonings mentioned above. Proof of Theorem 2. Using continuity arguments we obtain that if A preserves the set of positive polynomials then all principal minors of the infinite matrix A (x) are nonnegative for every real x, moreover A(x0 ) = P0 (x) is a positive polynomial. Suppose now that all principal minors of A (x) are nonnegative for every x ∈ and P0 (x) > 0 for every x ∈ . By theorem 1 we obtain that A preserves the set of nonnegative polynomials. Let P be a positive polynomial. Denote by μ := minx∈ P(x) (obviously μ > 0). Since Q(x) := P(x) − μ2 is a positive polynomial, then A(Q) = A(P) − μ2 P0 is a nonnegative polynomial, thus A(P) is a positive polynomial.
3 Proof of Theorems 3, 4, 6 and 8 Proof of Theorem 3. Suppose that the operator A preserves the set of nonnegative polynomials. Then by Theorem 1 all principal minors of the infinite matrix A (x) := ∞ αi+ j xi+ j + βi+ j xi+ j+2l i, j=0 are nonnegative for every x ∈ . Let us choose an arbitrary natural number n and an arbitrary set of integers 0 ≤ s1 < s2 < . . . < sn and consider the n × n principal minor of the matrix A (x) formed by rows and columns with numbers s1 , s2 , . . . , sn . We have
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n det αsi +s j xsi +s j + βsi +s j xsi +s j +2l i, j=1 = n x2(s1 +s2 +...+sn ) det αsi +s j + βsi+s j x2l i, j=1 ≥ 0 for every x ∈ . Hence for every x ∈ \ {0} n det αsi +s j + βsi +s j x2l
i, j=1
It follows that
(1)
≥ 0.
n det αsi +s j i, j=1 ≥ 0.
So we obtain that all principal minors of the matrix A1 := (αi+ j )∞ i, j=0 are nonnegative. Analogously we have n det αsi +s j xsi +s j + βsi +s j xsi +s j +2l i, j=1 = (2) n x2(s1 +s2 +...+sn +nl) det αsi +s j x−2l + βsi+s j i, j=1 ≥ 0 for every x ∈ . Hence for every x ∈ \ {0} n det αsi +s j x−2l + βsi +s j
i, j=1
≥ 0.
Tending x → +∞ we conclude that n det βsi +s j i, j=1 ≥ 0. So we obtain that all principal minors of the matrix A2 := (βi+ j )∞ i, j=0 are nonnegative. Suppose now that all principal minors of two infinite matrices A1 := (αi+ j )∞ i, j=0 and A2 := (βi+ j )∞ are nonnegative. Let us consider two linear operators A1 , i, j=0 k k k k+2l , k= A2 : [x] → [x] defined by A1 (x ) = αk x , k = 0, 1, . . . and A2 (x ) = βk x 0, 1, . . .. Using Theorem 1 it is easy to check that both linear operators preserve the set of nonnegative polynomials. Hence the given operator A = A1 + A2 also preserves the set of nonnegative polynomials. Proof of Theorem 4. Suppose that the operator A maps the set of nonnegative (positive) polynomials from 2( m +1) [x] into the set of nonnegative polynomials . 2
Denote by s := m2 + 1. By Theorem 1 all principal minors of the matrix A (x, s) := s A(xi+ j ) i, j=0 are nonnegative for every x ∈ . Suppose m = 2t, t ∈ . Then s = t + 1. The following 2 × 2 principal minor of A (x,t + 1) formed by rows and columns with numbers t − 1 and t + 1 is nonnegative:
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n(2t) - n(2t−2) Q (x, 2t − 2)x2t−2+ j ∑ j=−2t Q j (x, 2t)x2t+ j - ∑ j=−2t j det -≥0 n(2t) n(2t+2) ∑ j=−2t Q j (x, 2t)x2t+ j ∑ j=−2t Q j (x, 2t + 2)x2t+2+ j for every x ∈ such a way
. Since Q j (x, k) ≡ 0 for k + j < 0 we rewrite the last inequality in
n(2t) - n(2t−2) Q (x, 2t − 2)x2t−2+ j ∑ j=−2t Q j (x, 2t)x2t+ j - ∑ j=−2t+2 j det -≥0 n(2t) n(2t+2) ∑ j=−2t Q j (x, 2t)x2t+ j ∑ j=−2t Q j (x, 2t + 2)x2t+2+ j Let us set x = 0 in this inequality. We have - Q−2t+2 (0, 2t − 2) Q−2t (0, 2t) - ≥ 0, det 0 Q−2t (0, 2t) that is equivalent to (Q−2t (0, 2t))2 ≤ 0. Since m = 2t the last inequality contradicts the assumption Q−m (0, m) = 0. Suppose m = 2t − 1, t ∈ . Then s = t. The following 2 × 2 principal minor of A (x,t) formed by rows and columns with numbers t − 1 and t is nonnegative: - n(2t−2) Q (x, 2t − 2)x2t−2+ j n(2t−1) Q (x, 2t − 1)x2t−1+ j ∑ j=−2t+1 j - ∑ j=−2t+1 j det - n(2t−1) -≥0 n(2t) - ∑ j=−2t+1 Q j (x, 2t − 1)x2t−1+ j ∑ j=−2t+1 Q j (x, 2t)x2t+ j for every x ∈ such a way
. Since Q j (x, k) ≡ 0 for k + j < 0 we rewrite the last inequality in
- n(2t−2) Q (x, 2t − 2)x2t−2+ j n(2t−1) Q (x, 2t − 1)x2t−1+ j ∑ j=−2t+1 j - ∑ j=−2t+2 j det - n(2t−1) -≥0 n(2t) 2t+ j - ∑ j=−2t+1 Q j (x, 2t − 1)x2t−1+ j Q (x, 2t)x ∑ j=−2t+1 j Let us set x = 0 in this inequality. We have - Q−2t+2 (0, 2t − 2) Q−2t+1 (0, 2t − 1) - ≥ 0, det 0 Q−2t+1 (0, 2t − 1) that is equivalent to (Q−2t+1 (0, 2t − 1))2 ≤ 0. Since m = 2t − 1 the last inequality contradicts the assumption Q−m (0, m) = 0. Proof of Theorem 6. Suppose that there exists a coefficient sequence Q such that the operator A( f ) = q0 (x) f (x) + q1 (x) f (x) + . . . + qm (x) f (m) (x), qm (x) ≡ 0, maps the set of nonnegative (positive) polynomials from 2( m +1) [x] into the set 2
of nonnegative polynomials. Let us choose α ∈ such that qm (α ) = 0 and consider a linear operator B : [x] → [x] of the form B( f ) := q0 (x + α ) f (x) + q1 (x + α ) f (x) + . . . + qm (x + α ) f (m) (x). Obviously the operator B also maps the set of nonnegative (positive) polynomials from 2( m +1) [x] into the set of nonnegative 2
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polynomials. Namely, let f (x) be an arbitrary nonnegative (positive) polynomial. Then g(x) = f (x − α ) is also a nonnegative (positive) polynomial, so A(g)(y) is nonnegative for every y ∈ , and thus A(g)(x + α ) = B( f )(x) is a nonnegative polynomial. The operator B satisfies the conditions of Theorem 4 with n(k) ≡ 0, thus it does not map the set of nonnegative (positive) polynomials from 2( m +1) [x] into 2 the set of nonnegative polynomials. Proof of Theorem 8. Obviously a polynomial f (x) is nonnegative if and only if a polynomial h(x) = f (x + x0 ) is nonnegative (where x0 is an arbitrary real number). So, as we noted in the proof of Theorem 6, the operator A maps the set of nonnegative polynomials from 2 m2 [x] into the set of nonnegative polynomials if and only if for every nonnegative polynomial f ∈ 2 m2 [x] and for every x ∈ A( f )(0) =
q1 (x) q0 (x) qm (x) (m) f (0) + f (0) + . . . + f (0) ≥ 0. 0! 1! m!
A real polynomial from 2 m2 [x] is nonnegative if and only if it could be represented as a sum of squares of two real polynomials from m2 [x]. Whence A( f )(0) ≥ 0 for every nonnegative polynomial f ∈ 2 m2 [x] if and only if A(g2 )(0) ≥ 0 for every g ∈ m2 [x], that is for every set of real numbers ξ0 , ξ1 , . . . , ξ m2 and for every m x ∈ a polynomial A (ξ0 + ξ1 x + . . . + ξ m2 x 2 )2 (0) is nonnegative. We have m A (ξ0 + ξ1 x + . . . + ξ m2 x 2 )2 (0) =
m2
∑
ξi ξ j qi+ j (x) ≥ 0.
i, j=0
Hence A maps the set of nonnegative polynomials from 2 m2 [x] into the set of nonnegative polynomials if and only if for every x ∈ the quadratic form m2 qi+ j (x)ξi ξ j is nonnegative. The statement of theorem follows from the well∑i, j=0 known criterion of nonnegative definiteness. Acknowledgements. The authors express their deep gratitude to the organizers and participants of the Workshop ”P´olya-Schur-Lax problems: hyperbolicity and stability preservers” (2007, Palo-Alto, USA) for the fruitful discussions and interesting new problems. The authors are very grateful to referees for valuable comments and suggestions.
References 1. Akhiezer, N.I.: The classical moment problem and some related questions in analysis. In: Kemmer, N. (ed.) Translated from the Russian, 253 pages. Hafner Publishing Co., New York (1965) 2. Borcea, J.: Classifications of linear operators preserving elliptic, positive and nonnegative polynomials. arXiv:0811.4374
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3. Borcea, J., Br¨and´en, P., Csordas, G., Vinnikov, V.: P´olya-Schur-Lax problems: hyperbolicity and stability preservers, http://www.aimath.org/pastworkshops/polyaschurlax.html 4. Guterman, A., Shapiro, B.: On linear operators preserving the set of positive polynomials. JFPTA 3(2), 411–429 (2008) 5. Guterman, A., Shapiro, B.: A note on positivity preserveres. Math. Res. Lett. 15 (2008) ¨ 6. Hamburger, H.: Uber eine Erweiterung des Stieltjesschen Momentenproblems. Parts I, II, III. Math. Ann. 81, 235–319; ibid. 82 (1921), 20–164, 168–187 (1920) ¨ 7. Hurwitz, A.: Uber definite Polynome. Math. Ann. 73, 173–176 (1913) 8. Krein, M.G., Nudelman, A.A.: The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development. Translated from the Russian by D. Louvish. In: Translations of Mathematical Monographs, Vol. 50, 417 pages. American Mathematical Society, Providence (1977) ¨ 9. P´olya, G., Schur, I.: Uber zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math. 144, 89–113 (1914) 10. P´olya, G., Szeg¨o, G.: Problems and Theorems in Analysis. Reprint of the 1976 English translation. Classics in Mathematics, vol. II. Springer, Berlin (1998) 11. Remak, R.: Bemerkung zu Herrn Stridsbergs Beweis des Waringschen Theorems. Math. Ann. 72, 153–156 (1912) 12. Schur, I.: Bemerkungen zur Theorie der beschr¨ankten Bilinearformen mit unendlichvielen Ver¨aderlichen. J. f¨ur Math. 140 (1911)
Convergence to Consensus by General Averaging Dirk A. Lorenz and Jan Lorenz
Abstract. We investigate sufficient conditions for a discrete nonlinear non-homogeneous dynamical system to converge to consensus. We formulate a theorem which is based on the notion of averaging maps. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.
1 Introduction We consider the problem of consensus formation under the action of general nonlinear averaging maps (or general means). We consider a set of agents n = {1, . . . , n} where each of them has coordinates in a d-dimensional opinion space S ⊂ d . The individual coordinates of agent i at time t ∈ are labeled xi (t) ∈ S, and x(t) ∈ Sn ⊂ ( d )n is called the profile at time t ∈ . Hence, we study discrete dynamical systems in ( d )n of the following form x(t + 1) = ft (x(t))
(1)
where ft : Sn → Sn . We denote the component functions by fti . Accordingly we use upper indices for the number of the agents and lower indices for the dimension of the opinion space, e.g. xik denotes the k-th component of the opinion of the i-th agent. We assume that the maps ft are averaging maps (see below), and we are interested in conditions that ensure, that the solution converges to a consensus, i.e. there is γ such that xi (t) → γ for every i. With matrices At ∈ n×n we get a linear example Dirk A. Loreny Institute for Analysis and Algebra, TU Braunschweig, 38092 Braunschweig, Germany, e-mail:
[email protected] Jan Lorenz Chair of Systems Design, Department of Management, Technology, and Economics, ETH Z¨urich, Z¨urich, Switzerland, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 91–99. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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by ft (x) = At x. To recover averaging maps (as defined below) At is row-stochastic for all t ∈ . This problem is solved in special cases, e.g. when A is independent of t [9], in the case of finitely many A [10] or when the At ’s have positive diagonals, are type-symmetric and have positive minima uniformly bounded from below for all t [3, 6]. Here we treat the non-linear case similar to [5, 7, 8].
2 General Averaging Maps In this section we define averaging maps (or general means, respectively). First we consider d = 1 and hence, an appropriate opinion space S is an interval. A general mean is a function g : Sn → S such that the sandwich inequality mini∈n xi ≤ g(x) ≤ maxi∈n xi holds. An example is the power mean for real p = 0 and positive x: 1 1 Pp (x) = ( ((x1 ) p + · · · + (xn ) p )) p n
The power mean includes the arithmetic (p = 1) and harmonic (p = −1) means. √ n The geometric mean x1 . . . xn is approached for p → 0, and max{x1 , . . . , xn } and min{x1 , . . . , xn } for p → ∞ respectively p → −∞. Further on, there are weighted means: for nonnegative numbers α1 , . . . , αn which sum up to one there is the weighted arithmetic mean ∑ni=1 αi xi and the weighted geometric mean ∏ni=1 xαi i . Another generalization is the f -mean. For a continuous and injective function f : S → (which is thus invertible on its range) the f -mean of x1 , . . . , xn is Pf (x) = f (−1) (
1 n ∑ f (xi )). n i=1
The power mean is represented here as f (x) = x p , the geometric mean as f (x) = log(x). Of course, more means can be defined by means of means. Now we extend the definition of a general mean to d-dimensional opinion vectors for d ≥ 2. So let S ⊂ d be an appropriate opinion space. All means mentioned for the case d = 1 can be generalized to higher dimensions by taking them componentwise. Further on, one may define different one-dimensional means in each component. What is a proper generalization of the one-dimensional sandwich inequality? We are going to answer this question with the help of generalized barycentric coordinate maps, but first we discuss two straightforward generalizations. First, a function g : Sn → S is called a convex-hull mean if it fulfills the convex-hull sandwich inclusion: g(x) ∈ convi∈n xi . Obviously, the componentwise weighted arithmetic mean fulfills this property. But, e.g., the componentwise geometric mean does not (see Figure 1). Another possibility is, to use cubei∈n xi := [mini∈n xi , maxi∈n xi ] instead of the convex hull. Notice that max and min are componentwise and that the interval
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is multidimensional. So cube represents the smallest closed hypercube in d which covers all vectors x1 , . . . , xn . Figure 1 shows an example of the convex hull and the cube of a set of points in 2 . A function g : Sn → S is called a cube mean if it fulfills the cube sandwich inclusion: g(x) ∈ cubei∈n xi .
4
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Fig. 1 A set of six points in 2≥0 . The gray area is their convex hull (left), all possible weighted geometric means (center) and their cube (right).
The convex-hull means and the cube means can be generalized further with the help of a generalized barycentric coordinate map. To motivate the construction note that the cube is indeed the convex hull of the 2d points which lie at its vertices. Hence, we could write cubei∈n xi = convi∈2d yi (x) with an appropriate map y : n →
which takes n points in and maps them to the 2d possible combinations of taking componentwise max and min. To formalize this further we call y : Sn → Sm a generalized barycentric coordinate map if for every k ∈ n it holds that xk ∈ convi∈m yi (x). It is now natural to call convi∈m yi (x) the y-convex hull of x. So, a y-convex hull is a set-valued function from Sn to the compact and convex subsets of S. We call a set y-convex, if it is the union of the y-convex hulls of all n of its points. A function g is a y-convex hull mean if g(x) ∈ convi∈m yi (x). Note that the convex hull is obtained from the y-convex hull with m = n and y the identity. The cube is obtained with m = 2d and an appropriate mapping y. Many other examples fit into this setting: the smallest interval for any basis of d [1, Example 2], or the smallest polytope with faces parallel to a set of k ≥ d + 1 hyperplanes [1, Example 3] containing x1 , . . . , xn (the generalized barycentric coordinates are then the extreme points of the polytope, perhaps with multiplicities e.g. if the number of extreme points is smaller than n). Now, we define the central notion of this paper. 2d
Definition 1. Let S ⊂ d , y : Sn → Sm be a generalized barycentric coordinate map such that S is y-convex. A mapping f : Sn → Sn is called a y-averaging map, if for every x ∈ Sn it holds convi∈m yi ( f (x)) ⊂ convi∈m yi (x).
(2)
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Furthermore, a proper y-averaging map is a y-averaging map, such that for every x ∈ Sn which is not a consensus, the above inclusion is strict.
3 Convergence to Consensus To state a sufficient condition on the maps ft which ensure that the solution of (1) converge to consensus we need a little more notation. Let d(x,C) denote the distance of a point x and a compact set C. The Hausdorff metric for compact non-empty sets is defined as dH (B,C) := max{sup d(b,C), sup d(c, B)}. b∈B
c∈C
If B ⊂ C it holds dH (B,C) := supb∈B d(b,C). The next notion we need, is ‘equiproper averaging map’: Definition 2. Let y be a generalized barycentric coordinate map and let F be a family of proper y-averaging maps. F is called equiproper, if for every x ∈ Sn which is not a consensus, there is δ (x) > 0 such that for all f ∈ F dH convi∈m yi ( f (x)), convi∈m yi (x) > δ (x). (3) We can state the following lemma: Lemma 1. Let ft be a sequence of averaging maps forming an equiproper family of averaging maps such that ft → g pointwise. Then g is a proper averaging map. Proof. First we show that g is an averaging map. Take x ∈ Sn and let ε > 0, due to convergence of ( ft )i to gi and continuity of y there is t0 such that for all t > t0 it holds yi ( ft (x)) − yi (g(x)) < ε . Due to yi ( ft (x)) ∈ convi∈m yi (x) it follows that the maximal distance of yi (g(x)) to convi∈m yi (x) is less than ε , and thus yi (g(x)) ∈ convi∈m yi (x) because convi∈m yi (x) is closed. Now we show that g is proper. To this end, let x ∈ Sn be not a consensus. / convi∈m yi (g(x)). (Note We have to show that there is z∗ ∈ convi∈m yi (x) but z∗ ∈ ∗ n m that z ∈ S, while x ∈ S and y(x) ∈ S .) We know that there is for each t ∈ an z(t) ∈ convi∈m yi (x) with z(t) ∈ / convi∈m yi ( ft (x)). According to the equiproper property it can be chosen such that the distance of z(t) to convi∈m yi ( ft (x)) is bigger than δ (x)/2 > 0 for all t ∈ . Further on, we know that the set difference convi∈m yi ( ft (x))\convi∈m yi (x) is non empty and bounded, thus there is a subsequence ts such that z(ts ) converges to an z∗ ∈ convi∈m yi (x). Because of the construction it also holds z∗ ∈ / convi∈m yi (g(x)). Lemma 2. Let x(t) be a solution of (1) and let C(t) = convi∈m yi (x(t)). There exists a subsequence ts such that x(ts ) → c for s → ∞ and it holds ci ∈ C = ∩t C(t) for every i. Proof. Due to compactness of C(0)n there exists a convergent subsequence of x(t) and we call its limit c. It remains to show that ci ∈ C. Let T ∈ and ε > 0.
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Take s large enough, to have x(ts ) − c ≤ ε . If we take s even larger, we have xi (ts ) ∈ C(T ) which shows ci ∈ C(T ) since C(T ) is closed. Hence, ci ∈ C(T ) for every T . Before we state our main theorem we need one more notion: We call a family F of continuous functions equicontinuous if for every ε > 0 there exists δ > 0 such that for every f ∈ F it holds that x − x ≤ δ implies f (x) − f (x ) ≤ ε . Note that δ is chosen independently of f . Now, the main theorem is as follows: Theorem 1. Let S ⊂ d , y be a generalized barycentric coordinate map such that S is y-convex, and F be an equicontinuous family of equiproper y-averaging maps on Sn . Then it holds for any sequence ( ft )t∈ with ft ∈ F and any x(0) ∈ Sn that the solution of (1) converges to a consensus. Proof. In the first step one shows that C from Lemma 2 fulfills C = convi∈m yi (c). This step uses that, due to the Theorem of Arzel`a-Ascoli and Lemma 1, there is a uniformly convergent subsequence of ft with a proper y-averaging map g as a limit. Now one shows that convi∈m yi (g(c)) = convi∈m yi (c) which implies that c is a consensus since g is proper. For details we refer to [5]. It remains to show that the whole sequence x(t) converges to c = (γ , . . . , γ ). This can be seen as follows: For ε > 0 there exists s0 such that for s > s0 we have yi (x(ts )) − γ ≤ ε . Moreover, for t > ts0 we have x(t) ∈ C(ts0 ) and hence xi (t) = ∑ j a j y j (x(ts0 )) is convex combination. We conclude xi (t) − γ = ∑ a j (y j (x(ts0 )) − γ ) ≤ ∑ y j (x(ts0 )) − γ ≤ mε j
(4)
j
and hence, x(t) → c = (γ , . . . , γ ).
The proof follows the lines of the main theorem in [4] which now appears as a corollary since there the system x(t + 1) = f (x(t)) with just one proper averaging map f is considered. Some more corollaries can be deduced. Corollary 1. Let F = { f1 , . . . , fm } be a finite family of proper y-averaging maps on Sn ⊂ ( d )n , with S an appropriate opinion space. Let F be uniformly continuous. Then it holds for a sequence ( ft )t∈ with ft ∈ F and x(0) ∈ Sn that the solution of (1) converges to consensus. Proof. Since the family is finite, it is uniformly continuous and equiproper.
Corollary 2. Let F be a family of averaging maps on Sn ⊂ ( d )n , with S an appropriate opinion space. Let F be uniformly equicontinuous and at least one element in F is proper and all proper elements of F are equiproper. Let ( ft )t∈ be a sequence with ft ∈ F and ts be a subsequence such that fts is proper. Then, for x(0) ∈ Sn , the solution of (1) converges to consensus. Proof. Theorem 1 for the sequence fts gives subsequential convergence to consensus. An estimate similar to (4) gives convergence of the whole sequence.
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The proof of the next corollary uses similar techniques. Corollary 3. Let F be a family of averaging maps on Sn ⊂ ( d )n , with S an appropriate opinion space. Let F be uniformly equicontinuous. Let ( ft )t∈ be a sequence with ft ∈ F and (ts )s∈ be a subsequence (with t0 = 0) such that the fs := fts+1 −1 ◦ fts+1 −2 ◦ · · · ◦ fts +1 ◦ fts are equiproper. Then, for x(0) ∈ Sn , the solution of (1) converges to consensus. In the spirit of [1] we state another generalization of Theorem 1. The generalization deals with deformations of the hull. To this end, let S, T ⊂ d be compact and φ : T → S be a homeomorphism. For a generalized barycentric coordinate map y : Sn → Sm we define the y, φ -hull as φ −1 (convi∈m yi (φ (x))). Now, a y, φ -averaging map g is defined analogous to Definition 1:
φ −1 (convi∈m yi (φ (g(x)))) ⊂ φ −1 (convi∈m yi (φ (x))). Note, that the y, φ -hull is not necessarily convex, see [1, Example 6]. The extension of the notions ‘proper’ and ‘equiproper’ is straightforward. For the proof of the following theorem we refer to [5]. Theorem 2. Let φ : T → S be continuous with Lipschitz continuous inverse and let y be a generalized barycentric coordinate map such that S is y-convex. Let G be a family of equicontinuous, equiproper y, φ -averaging maps on T n . Then it holds for any sequence (gt )t∈ with gt ∈ G and any x(0) ∈ T n that the solution of x(t + 1) = gt (x(t)) converges to a consensus.
4 Examples and Counterexamples We give some examples that illustrate the role of the different assumptions in Theorem 1. → ( ≥0 )2 with 3 1 1 2 3 1 1 2 if x1 + x2 > 10, 4x + 4x , 4x + 4x 1 2 f (x , x ) := 1 3 2 1 1 1 2 3 (x ) 4 (x ) 4 , (x ) 4 (x ) 4 otherwise.
Example 1. Let f : (
2 ≥0 )
This averaging map converges to consensus but is not continuous. For example for x(0) = (1, 9) it converges to (3, 3) but for x(0) = (1 + ε , 9) it converges to (5 + ε /2, 5 + ε /2). So continuity is not necessary for convergence. Example 2. Let f : ( )3 → ( )3 with 1 2 1 3 1 (x , x , 2 x + 2 min{x1 , x2 }) if x3 < min{x1 , x2 }, f (x1 , x2 , x3 ) := (x1 , x1 , x1 ) otherwise.
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Starting with x(0) = (2, 3, 1) the discrete dynamical system x(t + 1) = f (x(t)) will converge to (2, 3, 2), although it is a proper averaging map. But it is not continuous at all points where x3 = min{x1 , x2 } and x1 = x2 . Example 3. Let ft : ( )2 → ( )2 with ft (x1 , x2 ) := (1 − 4−t )x1 + 4−t x2 , 4−t x1 + (1 − 4−t )x2 . It is easy to see that theses ft ’s are proper and that for t ≥ 1 and x(1) = (0, 1) it holds that x1 (t) < 13 and x2 (t) > 23 . Obviously, { ft |t ∈ } is not equiproper because ft converges to the identity as t → ∞. Example 4. Let ft : ( )2 → ( )2 with ft (x1 , x2 ) := (1 − 1t )x1 + 1t x2 , x2 . This example is not equiproper, because ft converges to the identity for t → ∞. But for t ≥ 2 and any initial values (x1 (2), x2 (2)) ∈ ( )2 the system x(t + 1) = 1 1 2 2 ft (x(t)) has the solution x(t) = ( t−1 x (2) + t−2 t−1 x (2) , x (2)) and thus converges to consensus at x2 (2) for all initial values. Of course, equicontinuity is not necessary. Example 1 gives a one-element family of not equicontinuous averaging maps which converge. Now, we show that a family of uniformly continuous proper averaging maps is not enough to ensure convergence. The example is inspired by bounded confidence. Example 5 (Vanishing confidence). Let ft : ( ft )i (x) :=
n
→
n
with
∑nj=1 Dt (|xi − x j |)x j ∑nj=1 Dt (|xi − x j |)
and Dt : ≥0 → ≥0 . Now, ft is an averaging map for any choice of Dt . Further on, ft is continuous if Dt is, and ft is proper if Dt is strictly positive. The HegselmannKrause model [2, 7] with homogeneous bound of confidence ε > 0 comes out for Dt being a non-continuous cutoff function 1 if y ≤ ε Dt (y) = 0 otherwise. y t
We chose Dt (y) := e−( ε ) as a sequence of functions which has the cutoff function as a limit function. So, Dt is continuous but {Dt |t ∈ } is not equicontinuous. Rough estimates show that with x(0) = (0, 8), ε = 1 the process x(t) = ft (x(t)) does not converge to consensus although only proper averaging maps are involved.
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5 Comparison with a Theorem of Moreau Theorem 1 is similar to a theorem of Moreau [8, Theorem 2]. We cite it here to discuss similarities and differences. It incorporates changing communication networks into self-maps with averaging properties. Theorem 3 (Moreau). Let ( ft )t∈ be a sequence of self maps on Sn ⊂ ( d )n , with S convex and closed. Let (N(t))t∈ be a communication regime where all networks have positive diagonals: Moreover, assume thatthere is T ∈ such that for t0 +T
all t0 ∈ the network inc( ∑ N(t)) has only one essential class1 . t=t0
Further on, it should exist for each network with positive diagonal N, each x ∈ Sn and each agent k ∈ n a compact set ek (x, N) such that 1. For all t ∈ it holds ( ft )k (x) ∈ ek (x, N), 2. ek (x, N) ⊂ ri convi∈nb(k,N) {xi }, 3. ek (x, N) depends continuously on x. Then it holds for x(0) ∈ Sn that the solution of (1) converges to a consensus. The theorem has been significantly reformulated in comparison with the original to fit it in our vocabulary. Especially the original theorem is about “uniform global attractivity of the system with respect to the set of equilibrium solutions x1 = · · · = xn = constant” which is equivalent to convergence to consensus for every x(0) ∈ Sn . Items 1 and 2 in the assumptions of Theorem 3 are similar to a ‘proper convex hull averaging map with respect to the current network’. It is averaging due to conv, and proper due to ri (actually ri is a stronger assumption than proper). The continuity assumption in item 3 shows similarity to the assumption of equicontinuity in Theorem 1. Equiproper from Theorem 1 finds its analog in Theorem 3 in the fact that in item 1 it holds ( ft )k (x) ∈ ek (x, N) and ek is independent of t. Especially, the assumption that the ek ’s are in the relative interior of convex hulls is more strict than the assumption of properness. But, the theorems can not be compared directly. Moreau’s Theorem allows changing communication topologies and poses assumptions on this. Our theorem does not deal with communication networks, but with arbitray switching update maps from an equiproper set. Example 6. Let g1 , g2 , g3 , √ g4 : ( d )3 → d with g1 (x) := max{x1 , x2 , x3 }, g2 (x) := 3 1 2 3 1 1 2 3 1 2 3 3 (x + x + x ), g3 (x) := x x x and g4 (x) := min{x , x , x } be general multidimensional means (all computations componentwise) and f σ1 σ2 σ3 : ( d )3 → ( d )3 with f σ1 σ2 σ3 := (gσ1 , gσ2 , gσ3 ) be averaging maps. Now it is easy to verify, that F := { f σ1 σ2 σ3 | (σ1 , σ2 , σ3 ) ∈ {1, 2, 3, 4}3 but 1 and 4 not both in (σ1 , σ2 , σ3 )} is an equicontinuous and equiproper set of averaging maps w.r.t cube. Thus, for any sequence ft with elements from F and x(0) ∈ ( d )3 it holds that x(t + 1) = ft (x(t)) 1
With inc we denote the incidence matrix, i.e. inc(A)i, j = 1 if Ai, j = 0 and 0 otherwise.
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converges to consensus due to Theorem 1. Theorem 3 is not applicable because item 2 does not hold for all elements of F.
References 1. Angeli, D., Bliman, P.A.: Stability of leaderless multi-agent systems. Extension of a result by Moreau. Mathematics of Control, Signals & Systems 18(4), 293–322 (2006) 2. Hegselmann, R., Ulrich Krause, U.: Opinion Dynamics Driven by Various Ways of Averaging. Computational Economics 25(4), 381–405 (2004) 3. Hendrickx, J.M., Blondel, V.D.: Convergence of Different Linear and Non-Linear Vicsek Models. CESAME research report 2005.57 (2005) 4. Krause, U.: Compromise, consensus, and the iteration of means. Elemente der Mathematik 63, 1–8 (2008) 5. Lorenz, D.A., Lorenz, J.: On conditions for convergence to consensus. arXiv.org/abs/0803.2211 (March 2008) 6. Lorenz, J.: A Stabilization Theorem for Dynamics of Continuous Opinions. Physica A 355(1), 217–223 (2005) 7. Lorenz, J.: Repeated Averaging and Bounded Confidence – Modeling, Analysis and Simulation of Continuous Opinion Dynamics. PhD thesis, Universit¨at Bremen (March 2007) 8. Moreau, L.: Stability of Multiagent Systems with Time-Dependent Communication Links. IEEE Transactions on Automatic Control 50(2) (2005) 9. Seneta, E.: Non-Negative Matrices and Markov Chains, 2nd edn. Springer, Heidelberg (1981) 10. Wolfowitz, J.: Products of Indecomposable, Aperiodic, Stochastic Matrices. In: Proceedings of the American Mathematical Society Eugene, vol. 15, pp. 733–737 (1963)
Stability and D-stability for Switched Positive Systems Oliver Mason, Vahid S. Bokharaie and Robert Shorten
Abstract. We consider a number of questions pertaining to the stability of positive switched linear systems. Recent results on common quadratic, diagonal, and copositive Lyapunov function existence are reviewed and their connection to the stability properties of switched positive linear systems is highlighted. We also generalise the concept of D-stability to positive switched linear systems and present some preliminary results on this topic.
1 Introduction While the stability properties of positive linear time-invariant (LTI) systems have been thoroughly investigated and are now completely understood, the theory for nonlinear, uncertain and time-varying positive systems is considerably less welldeveloped. In fact, many natural and fundamental questions on the stability of such systems remain unanswered. It is clear that for many practical applications there is a need to extend the theory for positive LTI systems to broader and more realistic system classes incorporating nonlinearities and time-varying parameters. Another separate and interesting line of recent research has focussed on extending the stability properties of positive LTI systems to positive descriptor systems [11]. Our principal focus in the present paper is on extending the stability theory of positive LTI systems to switched positive linear systems [9]. We review recent work on the stability of these systems, highlighting the connection between various notions of stability and the existence of corresponding types of common Lyapunov function. We also consider an extension of the concept of D-stability to positive Oliver Mason, Vahid S. Bokharaie and Robert Shorten Hamilton Institute, National University of Ireland Maynooth, Co. Kildare, Ireland, e-mail:
[email protected],
[email protected],
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switched linear systems, present some preliminary results for this question and highlight some directions for future research.
2 Notation and Background Throughout, denotes the field of real numbers, + denotes the set of non-negative real numbers, n stands for the vector space of all n-tuples of real numbers and m×n is the space of m × n matrices with real entries. For x in n , x denotes the i th i component of x, and the notation x 0 (x ! 0) means that xi > 0 (xi ≥ 0) for 1 ≤ i ≤ n. The notations x ≺ 0 and x # 0 are defined in the obvious manner. We write AT for the transpose of A ∈ n×n and for a symmetric P in n×n the notation P > 0 means that the matrix P is positive definite. Throughout the paper, in an abuse of notation, for LTI systems we shall use the term stability to denote asymptotic stability. Also, when referring to switched linear systems, stability shall be used to denote asymptotic stability under arbitrary switching [9]. For a positive LTI system x(t) ˙ = Ax(t)
(1)
where A ∈ n×n is a Metzler matrix (meaning that the off-diagonal entries of A are non-negative), the equivalences we collect in the following result are well known. Proposition 1. [4] Let A ∈ are equivalent:
n×n
be a Metzler matrix. The following statements
(a)The LTI system (1) is stable; (b)A is Hurwitz, meaning that its eigenvalues lie in the open left half plane; (c)There exists P > 0 such that AT P + PA < 0; (d)There exists a diagonal matrix D > 0 such that AT D + DA < 0; (e)There exists a vector v 0 in n with Av ≺ 0; (f) For any diagonal matrix D > 0, the system x(t) ˙ = DAx(t) is stable. While the equivalence of (a), (b) and (c) in the previous result also holds for any LTI system, properties (d), (e) and (f) are specific to positive LTI systems. With regard to point (e), as A is Hurwitz and Metzler if and only if AT is Hurwitz, an equivalent condition for stability for positive LTI systems is the existence of v 0 satisfying AT v ≺ 0. Such a v can be used to define a copositive linear Lyapunov function V (x) = vT x for the system (1). The property described in (f) is known as D-stability and establishes that stability of positive LTI systems is robust with respect to parametric uncertainties given by diagonal scaling. Later in the paper, we shall be concerned with investigating the connection between concepts similar to those in (e) and (f) for switched positive linear systems. Before this, in the following section, we shall review some recent work on the stability of switched positive linear systems.
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3 Lyapunov Functions and Stability for Switched Positive Linear Systems It is well known that a switched positive linear system of the form x(t) ˙ = A(t)x(t)
A(t) ∈ {A1 , A2 }
(2)
can be unstable for certain choices of switching sequence even when the individual system matrices A1 , A2 are asymptotically stable [9]. This observation has led to great interest in the stability of such systems under arbitrary switching regimes. A key result in this connection is that stability of (2) is equivalent to the existence of a common Lyapunov function for the individual component LTI systems [9]. In the light of Proposition 1, three classes of Lyapunov function naturally suggest themselves for positive switched linear systems: • Common Quadratic Lyapunov Functions (CQLFs): V (x) = xT Px where P = PT > 0 and ATi P + PAi < 0 for i = 1, 2; • Common Diagonal Lyapunov Functions (CDLFs): V (x) = xT Dx where D = diag(d1 , . . . , dn ), D > 0 and ATi D + DAi < 0 for i = 1, 2; • Common Linear Copositive Lyapunov Functions (CLLFs): V (x) = vT x where v 0 and ATi v ≺ 0 for i = 1, 2. In the interests of brevity, we shall abuse notation slightly and say that the matrices A1 , A2 have a CQLF, CDLF or CLLF rather than always referring to the associated LTI systems. Recall the following well-known necessary condition for the stability of positive switched linear systems (in fact this is a necessary condition for stability for general switched linear systems)[9]. Lemma 1. Let A1 , A2 ∈ n×n be Metzler and Hurwitz. Suppose that the associated switched positive linear system (2) is stable. Then for any real γ ≥ 0, A1 + γ A2 is Hurwitz. Common Quadratic Lyapunov Functions (CQLFs) In [5], the relationship between the existence of CQLFs, the stability of all matrices of the form A1 + γ A2 with γ ≥ 0, and the stability of the system (2) was considered. For 2-dimensional systems, the following result was established. Theorem 1. Let A1 , A2 ∈ ments are equivalent:
2×2
be Hurwitz and Metzler. Then the following state-
(a)A1 , A2 have a CQLF; (b)The switched system (2) is stable; (c)A1 + γ A2 is Hurwitz for all real γ ≥ 0. Further, the equivalence of (b) and (c) can be extended to the case of an arbitrary finite number of positive LTI systems. Formally, it was shown in [5] that given ˙ = A(t)x(t), Metzler, Hurwitz matrices A1 , . . . , Ak in 2×2 , the switched system x(t)
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A(t) ∈ {A1 , . . . , Ak } is stable if and only if A1 + γ2 A2 + · · · + γk Ak is Hurwitz for all real γ2 ≥ 0, . . . γk ≥ 0. The equivalence of (a), (b) and (c) fails immediately for 3-dimensional systems. Moreover, the equivalence of (b) and (c) is not true for arbitrary dimensions [5]. In fact, in a very recent paper [3], a 3-dimensional example of an unstable switched system for which A1 + γ A2 was Hurwitz for all γ ≥ 0 was explicitly described. In connection with CQLF existence and the stability of positive switched linear systems, it has been shown in [7] for 2 and 3 dimensional systems that if rank(A2 − A1 ) = 1, and A2 , A1 are both Hurwitz, then the associated LTI systems always possess a CQLF and the switched linear system (2) is stable. Common Diagonal Lyapunov Functions (CDLFs) As stable positive LTI systems have diagonal Lyapunov functions, it is natural to ask under what conditions families of such systems will possess a common diagonal Lyapunov function. In the paper [6], the following result was derived for systems with irreducible system matrices (for the definition of irreducible matrices, see [1]). Theorem 2. Let A1 , A2 ∈ n×n be irreducible, Metzler and Hurwitz. A1 , A2 have a CDLF if and only if A1 + DA2 D is Hurwitz for all diagonal matrices D > 0. The above result allows us to establish a connection between the existence of a CDLF and a form of robust stability for switched positive linear systems. First of all, note that for A1 , A2 irreducible, Metzler and Hurwitz, Theorem 2 shows that if A1 , A2 have a CDLF, then so do D1 A1 D1 , D2 A2 D2 for any choice of diagonal matrices D1 > 0, D2 > 0. Hence the existence of a CDLF guarantees the stability of the positive switched linear system x(t) ˙ = A(t)x(t) A(t) ∈ {D1 A1 D1 , D2 A2 D2 }
(3)
for any diagonal matrices D1 > 0, D2 > 0. Conversely, if A1 , A2 do not have a CDLF, then it follows from Theorem 2 that there is some diagonal matrix D > 0 such that A1 + DA2 D is not Hurwitz. This then immediately implies from Lemma 1 that the switched system (3) is not stable with D1 = I, and D2 = D. This discussion establishes the following result. Proposition 2. Let A1 , A2 ∈ n×n be irreducible, Metzler and Hurwitz. The switched system (3) is stable for any diagonal matrices D1 > 0, D2 > 0 if and only if A1 , A2 have a CDLF. Common Linear Copositive Lyapunov Functions (CLLFs) It is also possible to establish the stability of positive switched linear systems using copositive linear Lyapunov functions. As noted in [2], traditional Lyapunov functions may give conservative stability conditions for positive switched systems as they fail to take into account that trajectories are naturally constrained to the positive orthant. The existence of a CLLF for a pair of Metzler, Hurwitz matrices A1 , A2 is equivalent to the feasibility of the linear inequalities v 0, AT1 v ≺ 0, AT2 v ≺ 0. For the most part, we shall be concerned with the feasibility of the related system of inequalities v 0, A1 v ≺ 0, A2 v ≺ 0 as this is more relevant to the extension
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of the concept of D-stability for switched positive linear systems that interests us. Conditions for the feasibility of this system of inequalities (for compact sets of matrices) have been given in terms of P-matrix sets in the paper [10]. It is important for what follows to make clear the distinction between the existence of a common v 0 satisfying ATi v ≺ 0 for i = 1, 2 (CLLF existence), and the existence of a common v 0 such that Ai v ≺ 0 for i = 1, 2. For switched systems (in contrast with the LTI case), these two conditions are not equivalent. This can be seen from the following simple 2 × 2 example. Example 1. A1 =
−1 2 1 −3
, A2 =
−6 6 2 −6
It can be verified algebraically that Ai v ≺ 0 for i = 1, 2 where v = (5 2)T . However, it is easy to show that there can be no v 0 satisfying ATi v ≺ 0 for i = 1, 2. An algebraic condition for CLLF existence was derived in [8]. In the interests of brevity, we shall not explicitly state this result here but rather state the following technical result which follows from Theorem 3.1 in that paper. This fact shall prove useful in our later discussion. Lemma 2. Let A1 , A2 ∈ n×n be Metzler and Hurwitz. Suppose that there is no non-zero v ! 0 in n with A1 v # 0, A2 # 0. Then there is some diagonal D > 0 such that A1 + DA2 is singular.
4 Switched Positive Linear Systems and D-Stability: The 2-d Case In this and the following section, we shall investigate the following generalisation of the notion of D-stability to positive switched linear systems. Definition 1. Let A1 , A2 ∈ n×n be Metzler and Hurwitz. The associated switched positive linear system (2) is said to be D-stable if for any diagonal matrices D1 , D2 ∈ n×n with D > 0, D > 0, the system 1 2 x(t) ˙ = A(t)x(t)
A(t) ∈ {D1 A1 , D2 A2 }
(4)
is stable. For positive LTI systems, Proposition 1 shows that stability and D-stability are equivalent. Our first observation, in Example 2, is to note that this equivalence is not true in the switched case. First of all, we note the following simple necessary condition for D-stability, which follows immediately from Lemma 1. Lemma 3. Let A1 , A2 ∈ n×n be Metzler and Hurwitz. Suppose that the associated switched positive linear system (2) is D-stable. Then for any diagonal matrix D > 0, A1 + DA2 is Hurwitz.
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Example 2. Consider the Metzler, Hurwitz matrices in 2×2 −2 0 −1 5 A1 = , A2 = 1 −4 0 −1 It is straightforward to verify that A1 + γ A2 is Hurwitz for all γ ≥ 0. Hence by Theorem 1, the associated switched system is stable. On the other hand, choosing 20 0 D= 0 0.5 it is easily verified that A1 + DA2 is not Hurwitz. Hence by Lemma 3 the associated switched system is not D-stable. The above example illustrates that for switched positive linear systems, the concepts of stability and D-stability are not equivalent, in contrast to the LTI system case. In the following result, we show that the necessary condition given in Lemma 3 is also sufficient for D-stability for 2-dimensional systems. Theorem 3. Let A1 , A2 ∈ 2×2 be Metzler and Hurwitz. The positive switched linear system (2) is D-stable if and only if A1 + DA2 is Hurwitz for all diagonal matrices D > 0. Proof. Lemma 3 has already established the necessity of this condition. For sufficiency let D1 > 0, D2 > 0 be diagonal matrices and let γ ≥ 0 be any non-negative real number. By hypothesis, A1 + γ D−1 1 D2 A2 is Hurwitz for γ > 0 and it is trivially true for γ = 0. However, this matrix is also Metzler and hence by point (f) of Proposition 1, D1 A1 + γ D2 A2 = D1 (A1 + γ D−1 1 D2 A2 ) is also Hurwitz. It now follows immediately from Theorem 1 that the switched system (4) associated with D1 A1 , D2 A2 is stable. As this is true for any diagonal D1 > 0, D2 > 0, the system (2) is D-stable. The next result establishes a connection between the existence of a common solution to the inequalities v 0, Ai v ≺ 0 for i = 1, 2 and D-stability for (2). Corollary 1. Let A1 , A2 ∈
2×2
be Metzler and Hurwitz. Then:
(i) If there is some v 0 with A1 v ≺ 0, A2 v ≺ 0 then the system (2) is D-stable; (ii)If (2) is D-stable then there exists some non-zero v ! 0 with A1 v # 0, A2 v # 0. Proof. (i) Suppose there is some v 0 with Ai v ≺ 0 for i = 1, 2. Then for any diagonal D > 0, DA2 v ≺ 0 and (A1 + DA2 )v ≺ 0. Moreover, A1 + DA2 is Metzler. Hence, from point (e) of Proposition 1, it follows that A1 + DA2 is Hurwitz. Theorem 3 now implies that the switched system (2) is D-stable. (ii) If (2) is D-stable, then Theorem 3 implies that A1 + DA2 is Hurwitz for all diagonal D > 0. It now follows from Lemma 2 that there must exist some non-zero v ! 0 with A1 v # 0, A2 v # 0. Note that the sufficient condition for D-stability presented in point (i) of Corollary 1 is not necessary as demonstrated by the following example.
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Example 3. Consider the Metzler, Hurwitz matrices A1 , A2 given by: −2 1 −3 1 A1 = , A2 = 2 −2 2 −1 Using Theorem 4.1 of [8] it is straightforward to show that there is no vector v 0 with A1 v ≺ 0, A2 v ≺ 0. On the other hand, it can be verified algebraically that for any diagonal D > 0, A1 + DA2 is Hurwitz and hence the switched system (2) is D-stable by Theorem 3.
5 D-Stability in Higher Dimensions In this section, we present a result extending Corollary 1 to higher dimensional positive switched linear systems. While the following result is stated for switched systems with two constituent systems, the argument can easily be amended to derive a corresponding result for an arbitrary number of constituent systems. Theorem 4. Let A1 , A2 ∈
n×n
be Metzler and Hurwitz. Then:
(i) If there is some v 0 with A1 v ≺ 0, A2 v ≺ 0 then the system (2) is D-stable; (ii)If (2) is D-stable then there exists some non-zero v ! 0 with A1 v # 0, A2 v # 0. Proof. (i) The first step in proving (i) is to show that the existence of such a v is sufficient for the stability of the switched system (2). With this in mind, suppose that there exists some v 0 satisfying A1 v ≺ 0, A2 v ≺ 0, and let a (piecewise-constant) switching signal σ : + → {1, 2} be given such that A(t) = Aσ (t) for all t ≥ 0. Furthermore, let 0 = t0 ,t1 ,t2 , . . . ,tk , . . . , be the switching times or points of discontinuity of σ . As is standard for switching systems [9], we assume that there is some nonvanishing dwell-time τ > 0 such that tk+1 − tk ≥ τ for all k ≥ 0. Let xσ (., x0 ) denote the unique, piecewise C1 solution of (2) corresponding to the initial condition x0 and the switching signal σ . Also, for i = 1, 2, let x(i) (., x0 ) denote the unique solution of the stable positive LTI system x˙ = Ai x corresponding to the initial state x0 . Note the following readily verifiable facts concerning the solutions of the positive LTI systems with system matrices A1 , A2 . (a)For i = 1, 2, if x0 ! 0, x1 ! 0 satisfy x0 # x1 , then x(i) (t, x0 ) # x(i) (t, x1 ) for all t ≥ 0. This simply records the well-known fact that positive LTI systems are monotone; (b)For i = 1, 2, as dtd x(i) (0, v) = Ai v ≺ 0, it follows that there is some δ > 0 such that x(i) (t, v) ≺ v for 0 ≤ t ≤ δ . Combining (a) and (b) we see immediately that for 0 ≤ t ≤ δ , and i = 1, 2, x(i) (t + δ , v) = x(i) (t, x(i) (δ , v)) # x(i) (t, v) # v. Simply iterating this process, it is easy to see that for i = 1, 2, x(i) (t, v) # v for all t ≥ 0.
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Now consider the solution xσ (., v) of (2) corresponding to the initial condition v and the switching signal σ . The argument in the previous paragraph guarantees that for 0 ≤ t ≤ t1 , xσ (t, v) # v (as the dynamics in this interval are given by one of the constituent positive LTI systems). But in the second interval [t1 ,t2 ), the system dynamics are again given by a positive LTI system with xσ (t1 , v) # v as initial condition. Hence from the previous argument combined with point (a) above, we can conclude that for t1 ≤ t ≤ t2 , xσ (t, v) # v. Continuing in this way, we can easily see that for all t ≥ 0, we have xσ (t, v) # v. As the switching signal σ was arbitrary, we can conclude that xσ (t, v) # v holds for all switching signals. It is now straightforward to show that the solutions of (2) are uniformly bounded. In fact, for any x0 ! 0 such that x0 ∞ ≤ K1 , if vmin = min{v1 , . . . , vn }, then x0 # (K1 /vmin )v. It now follows that for all t ≥ 0, xσ (t, x0 ) # (K1 /vmin )v and hence that xσ (t, x0 )∞ ≤ K1 (vmax /vmin ) for all t ≥ 0 where vmax = max{v1 , . . . , vn }. Now if there is some v 0 with Ai v ≺ 0 for i = 1, 2, then (Ai + ε I)v ≺ 0 for sufficiently small positive ε > 0. Therefore, the trajectories of the switched system corresponding to A1 + ε I, A2 + ε I are also uniformly bounded for small enough positive ε > 0. This immediately implies that the original system (2) is globally asymptotically stable. To complete the proof of (i), note that for any positive definite diagonal matrices D1 , D2 , the matrices D1 A1 and D2 A2 are Metzler and Hurwitz. Moreover, if Ai v ≺ 0 for i = 1, 2, then Di Ai v ≺ 0 for i = 1, 2. The above argument now immediately implies that the system (4) is stable and hence the original system (2) is D-stable as claimed. The result given by (ii) follows immediately from Lemma 3 and Lemma 2. Note that the result given by (i) provides a condition for stability of (2) that is distinct although related to the condition given by CLLF existence.
6 Concluding remarks In this paper, we have discussed a number of problems relating to the stability properties of switched positive linear systems. In particular, we have reviewed recent work on common quadratic, copositive and diagonal Lyapunov functions for these systems and on the relationship between the existence of such functions and various notions of stability for switched positive systems. We have also discussed the notion of D-stability for positive switched systems and presented separate necessary and sufficient conditions for D-stability for n-dimensional systems. More detailed and complete results have also been given for 2-dimensional systems. A number of interesting directions for future research emerge from the work described here. For instance, it would be interesting to investigate the possibility of Theorem 3 extending to dimensions higher than 2, even for some restricted system class. Also, the question of whether stability and D-stability are equivalent for any subclass of positive switched linear systems arises naturally. It is straightforward to
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show that this is true for upper (or lower) triangular positive systems, for example, but are there any more interesting such classes? Acknowledgements. The authors are very grateful to the organisers of the session on Stability and Control of Positive Systems for their kind invitation to contribute to the session. This work has been supported by the Irish Higher Education Authority (HEA) PRTLI Network Mathematics grant, by Science Foundation Ireland (SFI) grant 08/RFP/ENE1417 and by SFI PI Award 07/IN.1/1901.
References 1. Berman, A., Plemmons, R.J.: Non-negative Matrices in the Mathematical Sciences. SIAM Classics in Applied Mathematics (1994) 2. Camlibel, M.K., Schumacher, J.M.: Copositive Lyapunov Functions. In: Unsolved Problems in Mathematical Systems and Control Theory. Princeton University Press, Princeton (2004), http://press.princeton.edu/math/blondel 3. Fainshil, L., Margaliot, M., Chigansky, P.: Positive Switched Linear Systems are not Uniformly Stable, even for n = 3. IEEE Transactions on Automatic Control (to appear 2009) 4. Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Pure and Applied mathematics. John Wiley & Sons, Inc., New York (2000) 5. Gurvits, L., Shorten, R., Mason, O.: On the Stability of Switched Positive Linear Systems. IEEE Transactions on Automatic Control 52(6), 1099–1103 (2007) 6. Mason, O., Shorten, R.: On the Simultaneous Diagonal Stability of a Pair of Positive Linear Systems. Linear Algebra and its Applications 413, 13–23 (2006) 7. Mason, O., Shorten, R.: Quadratic and Copositive Lyapunov Functions and the Stability of Positive Switched Linear Systems. In: Proceedings of American Control Conference, New York (2007) 8. Mason, O., Shorten, R.: On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems. IEEE Transactions on Automatic Control 52(7), 1346–1349 (2007) 9. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability Theory for Switched and Hybrid Systems. SIAM Review 49(4), 545–592 (2007) 10. Song, Y., Seetharama-Gowda, M., Ravindran, G.: On Some Properties of P-matrix Sets. Linear Algebra and its Applications 290, 237–246 (1999) 11. Virnik, E.: Stability Analysis of Positive Descriptor Systems. Linear Algebra and its Applications 429, 2640–2659 (2008)
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems Pham Huu Anh Ngoc
Abstract. An explicit criterion for positive linear Volterra-Stieltjes differential systems is given. Then new explicit criteria for uniform asymptotic stability and exponential asymptotic stability of positive linear Volterra-Stieltjes differential systems are presented. Finally, a crucial difference between the uniform asymptotic stability and the exponential asymptotic stability of linear Volterra-Stieltjes differential systems is shown.
1 Preliminaries Let := or . For an integer l, q ≥ 1, l denotes the l−dimensional vector space over K. Inequalities between real matrices or vectors will be understood componentwise, i.e. for two real matrices A = (ai j ) and B = (bi j ) in l×q , we write A ≥ B iff ai j ≥ bi j for i = 1, · · · , l, j = 1, · · · , q. We denote by l×q + the set of all nonnegative matrices A ≥ 0. Similar notations are adopted for vectors. For x ∈ n and P ∈ l×q we define |x| = (|xi |) and |P| = (|pi j |). A norm · on n is said to be monotonic if x ≤ y whenever x, y ∈ n , |x| ≤ |y|. Every p-norm on n , 1 ≤ p ≤ ∞, is monotonic. Throughout the paper, if not stated otherwise, the norm of a matrix P ∈ l×q is understood as its operator norm associated with a given pair of monotonic vector norms on l and q , that is P = max{Py; y = 1}. Then one has P ∈ l×q , Q ∈
l×q + , |P|
≤Q ⇒
P ≤ |P| ≤ Q,
(1)
see, e.g. [7]. In what follows, we denote γ := {z ∈ : Re z ≥ γ } with given γ ∈ . For any matrix A ∈ n×n the spectral abscissa of A is denoted by μ (A) = max{Re λ : λ ∈ σ (A)}, where σ (A) := {s ∈ ; det(sIn − A) = 0} is the spectrum of A. A matrix Pham Huu Anh Ngoc Institute of Mathematics, Ilmenau Technical University, Weimarer Straße 25, 98693 Ilmenau, DE, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 111–121. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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A ∈ n×n is called a Metzler matrix if all off-diagonal elements of A are nonnegative. Let J be an interval of . A matrix function η (·) : J → l×q is called increasing on J if η (θ2 ) ≥ η (θ1 ) for θ1 , θ2 ∈ J, θ1 < θ2 .
2 Explicit Criterion for Positive Linear Volterra-Stieltjes Differential Systems Consider a linear Volterra-Stieltjes differential system of the form x(t) ˙ = Ax(t) +
t 0
d[B(s)]x(t − s),
for a.a. t ∈
+,
(2)
where A ∈ n×n is a given matrix and B(·) : + → n×n is a given matrix function of locally bounded variation on + . Furthermore, we always assume that B(·) is normalized to be right-continuous on + and vanishes at 0. From the theory of integro-differential systems (see e.g. [1]), it is well-known that there exists a unique locally absolutely continuous matrix function R(·) : + → n×n such that ˙ = AR(t) + R(t)
t 0
d[B(s)]R(t − s),
a.a. t ∈
+,
R(0) = In .
Then R(·) is called the resolvent of (2). Moreover, for given f ∈ L1loc ( following nonhomogeneous system x(t) ˙ = Ax(t) +
t 0
d[B(s)]x(t − s) + f (t),
a.a. t ∈
t 0
R(t − s) f (s)ds,
t∈
+,
+,
+,
has a unique solution x(·) satisfying the initial condition x(0) = x0 ∈ represented by the variation of constants formula x(t) = R(t)x0 +
(3) n ),
the
(4) n
and it is
(5)
see e.g. [1, p. 81]. Definition 1. Let σ ∈ + and ϕ ∈ C([0, σ ], n ). A vector function x(·) : + → is called a solution of (2) through (σ , ϕ ) if x(·) is absolutely continuous on any compact subinterval of [σ , +∞) and satisfies (2) for almost all t ∈ [σ , +∞) and x(t) = ϕ (t), ∀t ∈ [0, σ ]. We denote it by x(· ; σ , ϕ ). n
Remark 1. By the fact mentioned above on solution of the nonhomogeneous system (4), it is easy to check that for fixed σ ∈ + and given ϕ ∈ C([0, σ ], n ), there exists a unique solution of (2) through (σ , ϕ ) and it is given by
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems
x(t + σ ; σ , ϕ ) = R(t)ϕ (σ ) +
t 0
R(t − u)
In the above, it is understood that
u+σ u
u+σ u
113
. d[B(s)]ϕ (u + σ − s) du, t ∈
d[B(s)]ϕ (u + σ − s) = 0 when σ = 0.
+.
(6)
Definition 2. We say that (2) is positive, if for any σ ≥ 0 and any ϕ ∈ C([0, σ ], n ), ϕ ≥ 0, the corresponding solution x(· ; σ , ϕ ) is also nonnegative, that is x(t; σ , ϕ ) ≥ 0, ∀t ≥ σ . We are now in the position to prove the first main result of this paper. Theorem 1. The system (2) is positive if and only if A ∈ and B(·) is an increasing matrix function on + .
n×n
is a Metzler matrix
Proof. (The ”if” part) Let σ ≥ 0 and ϕ ∈ C([0, σ ], n ), ϕ ≥ 0. Then x(·; σ , ϕ ) is given by (6). Since B(·) is increasing on + and ϕ ≥ 0, it follows that u+σ u
d[B(s)]ϕ (u + σ − s) ≥ 0, ∀u ≥ 0.
Thus x(·; σ , ϕ ) ≥ 0,t ∈ + , provided R(t) ≥ 0,t ∈ + . It remains to show that x(t; 0, x0 ) = R(t)x0 ≥ 0, t ∈ + for any x0 ≥ 0. Note that x(t) := x(t; 0, x0 ), t ∈ + satisfies x(t) = e x0 + At
t
e
A(t−u)
0
u 0
d[B(s)]x(u − s) du, t ∈
+.
Fix a > 0. We consider the operator T : C([0, a],
ϕ → T ϕ (t) := eAt x0 +
t 0
n
) −→ C([0, a],
eA(t−u)
u 0
n
)
d[B(s)]ϕ (u − s) du, t ∈ [0, a].
By induction, it is easy to show that for ϕ1 , ϕ2 ∈ C([0, a], n ) and k ∈ , we have k k T k ϕ2 (t) − T k ϕ1 (t) ≤ Mk!t ϕ2 − ϕ1 , ∀t ∈ [0, a], where M := M1 M2 and M1 := maxs∈[0,a] eAs , M2 := Var(B; 0, a). This implies that T k is a contraction for k ∈ sufficiently large. Fix k0 ∈ sufficiently large, by the contraction mapping principal, there exists a unique solution of the equation x = T x in C([0, a], n ). Moreover, the sequence (T mk0 ϕ0 )m∈ , with an arbitrary ϕ0 ∈ C([0, a], n ) converges to this solution in the space C([0, a], n ). Choose ϕ0 ∈ C([0, a], n ), ϕ0 ≥ 0. Since A ∈ n×n is a Metzler matrix, B(·) is increasing on + , it follows that T mk0 ϕ0 ≥ 0, ∀m ∈ . Thus x(t; 0, x0 ) ≥ 0, ∀t ∈ [0, a]. Since a > 0 is arbitrary, x(t; 0, x0 ) ≥ 0, ∀t ∈ + . (The ”only if” part) We first show that A is a Metzler matrix. Let {e1 , e2 , ..., en } be the standard basis of n . Fix j ∈ {1, 2, ..., n}. Let x(t) := x(t; 0, e j ),t ∈ + . Then x(·) ≥ 0, satisfies (2) a.a. t ∈ + . Fix i ∈ {1, 2, ..., n} with i =
j and k ∈ . Since x(·) is absolutely continuous on [0, 1/k], we have x(t) = e j + 0t x(s)ds,t ˙ ∈ [0, 1/k]. It follows that there exists tk ∈ [0, 1/k] such that
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P.H.A. Ngoc
x(t ˙ k ) = Ax(tk ) +
tk 0
d[B(s)]x(tk − s)ds and eTi x(t ˙ k ) ≥ 0.
˙ k ) = eTi Ae j ≥ 0 for i = j. Thus A is a Metzler matrix. This gives limk→+∞ eTi x(t Let B(·) = (bi j (·)). We now prove that bi j (·) is increasing on + for every i, j ∈ {1, 2, ..., n}. To do so, let ψ ∈ C([0, σ ], ), ψ ≥ 0, ψ (σ ) = 0 with given σ > 0. For fixed i0 ∈ {1, 2, ..., n} and let us define ϕ := (ϕ1 , ..., ϕn )T ∈ C([0, σ ], n ), where ϕi = ψ if i = i0 otherwise ϕi = 0. Set x(t) := x(t; σ , ϕ ) = (x1 (t), x2 (t), ..., xn (t))T ≥ 0, ∀t ≥ σ . Note that x(·) satisfies (2) a.a. t ∈ [σ , σ + 1/k] for given k ∈ . Since x1 (·) is absolutely continuous on [σ , σ + 1/k], x1 (·) ≥ 0, x1 (σ ) = 0, there exists tk ∈ [σ , σ + such that x(·) satisfies (2) at tk and x˙1 (tk ) ≥ 0. This yields limk→+∞ x˙1 (tk ) =
1/k] σ 0 ψ (σ − s)d[b1i
0 (s)] ≥ 0. Thus, the linear functional defined by L : C([0, σ ], ) → , ψ → Lψ := 0σ ψ (σ − s)d[b1i0 (s)], is positive. Taking into account [6, Lemma 3.4], we conclude that b1i0 (·) is increasing on [0, σ ]. Since σ > 0 is arbitrary, b1i0 (·) is increasing on + . By a similar way, we can show that bi j (·) is increasing on + for any i, j ∈ {1, 2, ..., n}. This completes the proof. The following is immediate from Theorem 1. Corollary 1. Let A ∈ n×n be a given matrix and let B(·) : + → n×n be a given continuous matrix function. Then, a linear Volterra integro-differential system of the convolution type x(t) ˙ = Ax(t) +
t 0
B(t − s)x(s)ds,
t∈
is positive if and only if A is a Metzler matrix and B(t) ∈
+,
n×n +
(7)
for all t ≥ 0.
3 Stability of Positive Linear Volterra-Stieltjes Systems In this section, we offer new and novel criteria for uniform asymptotic stability and exponential asymptotic stability of positive linear Volterra-Stieltjes differential systems of the form (2). Throughout this section, we assume that +∞ 0
|dB(t)| < +∞.
˜ := Then the Laplace-Stieltjes transform of B(·) : B(z) for z ∈ 0 .
(8)
+∞ −zs dB(s), is well-defined 0 e
3.1 Explicit Criterion for uniform asymptotic stability Definition 3. (i) The zero solution of (2) is said to be uniformly stable (US) if for each ε > 0, there exists δ > 0 such that
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems
ϕ ∈ C([0, σ ],
n
115
), ϕ < δ ⇒ x(t; σ , ϕ ) < ε , ∀t ≥ σ .
(ii) The zero solution of (2) is said to be uniformly asymptotically stable (UAS) if it is US and if there exists δ0 > 0 such that ∀ε > 0, ∃ T (ε ) > 0 :
ϕ ∈ C([0, σ ],
n
), ϕ < δ0 ⇒ x(t; σ , ϕ ) < ε , ∀t ≥ T (ε ) + σ .
If the zero solution of (2) is US (UAS) then we also say that (2) is US (UAS), respectively. Denote
Δ (z) := zIn − A −
+∞
e−zt dB(t),
0
z ∈ 0 ,
(9)
the characteristic matrix of (2). Theorem 2. [7, Th. 5.3] Let (8) hold. Then the following statements are equivalent (i) det Δ (z) = 0, ∀z ∈ 0 ; (ii) the resolvent R(·) of (2) belongs to L1 ( + , n×n ); (iii) the system (2) is UAS. Theorem 3. Suppose that (8) holds and (2) is positive. Then, (2) is UAS if and only if
μ (A +
+∞
0
dB(t)) < 0.
(10)
Proof. Assume that μ (A + 0+∞ dB(t)) < 0. Since (2) is positive, A ∈ n×n is a Metzler matrix and B(·) is an increasing matrix function on + , by Theorem 1.
For an arbitrary z ∈ 0 , we have 0+∞ e−zt dB(t) ≤ 0+∞ e− Rezt dB(t) ≤ 0+∞ dB(t). a standard property of Metzler matrices (see e.g. [6, Th. 2.1 (iv)]), μ (A +
By +∞ −zt dB(t)) ≤ μ (A + 0+∞ dB(t)) < 0. It follows that z ∈ σ (A + 0+∞ e−zt dB(t)). 0 e That is, det Δ (z) = 0. By Theorem 2, (2) is UAS. Conversely, suppose (2) is UAS. Then, det Δ (z) = 0 ∀z ∈ 0 , by Theorem 2. Consider the real function defined by f (θ ) = θ − μ (A + 0+∞ e−θ t dB(t)), θ ∈ [0, +∞). Clearly, f is continuous and limθ →+∞ f (θ ) = +∞. We show that f (0) > 0. Seeking a contradiction, assume that f (0) ≤ 0. Then there
is λ1 ≥ 0 such that f (λ1 ) = 0. That is, λ1 = μ (A + 0+∞ e−λ1t dB(t)). Since A + 0+∞ e−λ1t dB(t) is a Metzler matrix,
λ1 ∈ σ (A + 0+∞ e−λ1t dB(t)), by the Perron-Frobenius theorem, see e.g. [5, Th. 2.1].
Thus, det Δ (λ1 ) = 0. This is a contradiction. Hence, μ (A+ 0+∞ dB(t)) = − f (0) < 0, as required. This completes the proof. The following is immediate from Theorems 1, 3. Corollary 2. Let A ∈ n×n be a Metzler matrix and let B(·) : given nonnegative continuous matrix function. If +∞ 0
B(s)ds < +∞,
then the system (7) is UAS if and only if μ (A +
+∞ 0
B(t)dt) < 0.
+
→
n×n
be a
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P.H.A. Ngoc
Remark 2. Necessity of positivity The assumption of positivity of (2) in Theorem 3 cannot, in general, be omitted. To see this, we consider the scalar equation x(t) ˙ = Ax(t) + where A = 0, B(·) : s
B(s) := 0
+
→
t 0
d[B(s)]x(t − s)
for a.a. t ≥ 0,
(11)
is defined for s ≥ 0, by
b(τ )d τ ;
b:
+
⎧ 2 3e 2τ ⎪ ⎨ e2 −1 e , τ ∈ [0, 1) b(τ ) = e2−4 e2τ , τ ∈ [1, 2) ⎪ ⎩ −1 τ ∈ [2, ∞). 0,
→ ,
An easy computation yields +∞ 0
|dB(s)| < +∞,
A+
+∞ 0
dB(s) = −
e2 0, g(2) = 2 − 2
2 0
e−2s b(s)ds = −1 +
1 e2 − 1
< 0,
and hence there exists t ∗ ∈ (0, 2) such that g(t ∗ ) = 0. Now Theorem 2 implies that (11) is not uniformly asymptotically stable. Note that since B(·) is not increasing on + , (11) is not positive, by Theorem 1.
3.2 Explicit Criterion for Exponential Asymptotic Stability Definition 4. The zero solution of (2) is said to be exponentially asymptotically stable (EAS) if there exist M ≥ 1, α > 0 such that ∀σ ≥ 0, ∀ϕ ∈ C([0, σ ],
n
), ∀t ≥ σ :
x(t; σ , ϕ ) ≤ Me−α (t−σ ) ϕ .
If the zero solution of (2) is EAS then we also say that (2) is EAS.
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems
117
Theorem 4. Suppose that (8) holds and (2) is positive. Then the following statements are equivalent (i) (2) is EAS; (ii) (2) is UAS and - +∞ αs (12) e dB(s)∃α >0: - < +∞. 0
(iii) (12) holds and μ (A +
+∞ 0
dB(t)) < 0.
Proof. Note that (ii) ⇔ (iii) is immediate from Theorem 3. It remains to show that (i) ⇔ (ii). (ii) ⇒ (i) S TEP 1: We show that ∃ K, ε > 0 ∀t ∈
: R(t) ≤ Ke−ε t .
+
Since B(·) is increasing on + , we have | 0+∞ e−zs dB(s)| ≤ 0+∞ eα s dB(s), for any z ∈ −α . Thus, if det Δ (z) = 0 for some z ∈ −α then |z| ≤ A+ 0+∞ e−zs dB(s) ≤ A + 0+∞ eα s dB(s) := T0 , by (1). Hence det Δ (z) = 0 for z ∈ with − α ≤ ˚ Re z ≤ 0 and | Im z| ≥ T0 + 1. Since / det Δ (·) is analytic on −α , it has at most 0 a finite number of zeros in D := z ∈ − α /2 ≤ Re z ≤ 0, | Im z| ≤ T0 + 1 . In addition, since (2) is UAS, det Δ (z) = 0 ∀z ∈ 0 , by Theorem 2. It follows that c0 := sup {Re z : z ∈ , det Δ (z) = 0} < 0. Choose ε ∈ (0, min{−c0 , α }). Then, it is easy to check that R1 (·) := eε · R(·) and Δ1 (·) := Δ (· − ε ) are, respectively, the resolvent and the characteristic matrix of the equation y(t) ˙ = (A + ε In)y(t) + where F(·) is defined by F(s) :=
t 0
d[F(s)]y(t − s),
t∈
+,
(13)
s ετ 0 e d[B(τ )], s ≥ 0. By (12), we have
+∞ 0
|dF(s)| < +∞.
Since det Δ (z) = 0 for all z ∈ −ε , it follows that det Δ1 (z) = 0 for all z ∈ 0 . Applying Theorem 3 to (13), we get R1 (·) ∈ L1 ( + ; n×n ). This im plies that 0t d[F(s)]R1 (t − s) ∈ L1 ( + , n×n ), by a standard property of convosee e.g. [1, page 96]. From the resolvent equation, R˙ 1 (t) = AR1 (t) +
lutions, t ˙ 1 (·) ∈ L1 ( + , n×n ). Since d[F(s)]R + , it follows that R 1 (t − s), a.e. on 0 1 n×n R˙ 1 (·), R1 (·) ∈ L ( + , ), we have R1 (t) → 0 as t → +∞. In particular, R1 (·) ∈ L∞ ( + , n×n ) which yields R(t) ≤ Ke−ε t ∀t ∈ + , for some K > 0. S TEP 2: We show that ∀σ ≥ 0, ∀ϕ ∈ C([0, σ ],
n
), ∀t ≥ σ :
x(t; σ , ϕ ) ≤ Me−ε (t−σ ) ϕ ,
for some M > 0. By Step 1, R(t) ≤ Ke−ε t ∀t ∈ mula (6), it remains to show that
+.
Taking into account the for-
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P.H.A. Ngoc
∃ K1 > 0, ∀(σ , ϕ ) ∈ + × C([0, σ ]; n ) ∀t ≥ σ : - t u+σ . −ε (t−σ ) - R(t − u) d[B(s)] ϕ (u + σ − s) duϕ . - 0 - ≤ K1 e u Let VB (s) := Var(B; 0, s), s ∈ + . Since B(·) is increasing on + , (12) is equivalent to 0+∞ eα s d[VB (s)] < +∞. Then we have for t ∈ + , - t u+σ . - t - R(t − u) d[B(s)]ϕ (u + σ − s) duKe−ε (t−u) -≤ u
0
- -
u+σ u
t d[B(s)]ϕ (u + σ − s)-du ≤ Kϕ e−ε (t−u)
≤ Ke−ε t ϕ +∞ 0
0
αs
t
e(ε −α )u
0
u+σ
e d[VB (s)] ≤
u
u+σ
u
0
eα s d[VB (s)]du ≤ Ke−ε t ϕ
K α −ε
+∞ 0
d[VB (s)]du
+∞
e(ε −α )u du
0
e d[VB (s)] e−ε (t−σ ) ϕ . αs
(i) ⇒ (ii) S TEP 1: We show, by induction, that - +∞ m - < +∞. ∀m∈ : t dB(t) - 0 -
(14)
In what follows, we may consider without restriction of generality, the norm U := ∑nij=1 |ui j | for U := (ui j ) ∈ n×n . Since (2) is EAS, we have R(t) ≤ M1 e−β t ,
t∈
+,
(15)
ˆ be the Laplace transform of for some β , M1 > 0. Choose α ∈ (0, β ). Let R(·) ˆ R(·). Then, (15) implies that R(·) is analytic on −α . Taking Laplace transforms ˆ = In , z ∈ 0 . Thus, to two sides of the differential equation in (3), we get Δ (z)R(z) ˆ ˆ det R(0) = 0. Since the function z → det R(z) is continuous at z = 0, there exists ˆ = 0 for all z ∈ Bα0 (0). Thus R(·) ˆ −1 exists on Bα0 (0). α0 ∈ (0, α ) such that det R(z) ˆ −1 . ˆ Since the entries of R(·) are analytic on Bα0 (0), so must be the entries of R(·) Therefore ˆ −1 , V (z) := zIn − A − R(z) ˜ for z ∈ 0 . By a standard property of is analytic on Bα0 (0). Note that V (z) = B(z) the Stieltjes-Laplace transform, we have ∀m ∈
∀s ∈ Bα0 (0) ∩ ˚ 0 :
V (m) (s) = (−1)m
+∞ m −st dB(t). 0 t e
(16)
Set M := V (0). For m = 1, seeking a contradiction, we suppose that ∃T >1:
T 0
(t − 1)dB(t)dt > M.
(17)
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems
119
Choose δ0 > 0 sufficiently small such that ∀h ∈ (0, δ0 ) ∀t ∈ [0, T ] :
1 − e−ht ≥ t − 1. h
(18)
Since B(·) is increasing on + , we have for h > 0 sufficiently small, n +∞ n T - V (h) − V (0) 1 − e−ht - = ∑ dB (t) ≥ (t − 1)dB pq(t) pq ∑ h h p,q=1 0 p,q=1 0 - T -. (t − 1)dB(t) =-
(19)
0
By invoking (17), (18), (19) and continuity of the norm, we arrive at the contradiction - - V (h) − V (0) - - T -≥- > M. (t − 1)dB(t) M = V (0) = lim h h→0+ 0 Therefore, (14) holds for m = 1. If (14) holds for m, then it can be shown analogously as in the previous paragraph for m = 1 that (14) holds for m + 1 by replacing V (·) by V (m) (·). This proves Step 1. S TEP 2: We show that (12) holds. Note that by (14), (16), we have for every m ∈ , V (m) (0) = lim (−1)m s→0+
+∞ 0
t m e−st dB(t) = (−1)m
+∞
t m dB(t).
0 (k)
V (0) k Since V (·) is analytic on Bα0 (0), the Maclaurin’s series ∑∞ k=0 k! s , is for some α1 > 0, absolutely convergent in Bα1 (0). Therefore, by (14)-(16), ∞
αk ∑ k!1 k=0
+∞ 0
t k dB pq (t) =
+∞
(k)
|V pq (0)| ∑ k! α1 k < +∞, k=0
for p, q ∈ {1, 2, ..., n} and so, in view of increasing property of B(·), we have - +∞ n +∞ α1 t -= ∑ e dB(t) eα1t dB pq (t) = 0
n
∑
p,q=1
$ 0
p,q=1 0
% % $ n +∞ k +∞ α1 (α1 t)k k ∑ k! dB pq(t) = ∑ ∑ k! 0 t dB pq(t) < +∞. p,q=1 k=0 k=0
+∞ +∞
This completes the proof.
Remark 3. Uniform asymptotic stability versus exponential asymptotic stability
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By definition, it is easy to see that the exponential asymptotic stability of (2) (not necessarily positive) implies its uniform asymptotic stability. However, the converse of this statement does not hold even for positive systems. To see this, we consider the scalar Volterra-Stieltjes equation x(t) ˙ = −x(t) +
t 0
d[B(s)]x(t − s)
(20)
1 where B : + → , s → B(s) := 0s 2(τ +1) 2 d τ . Note that (20) is positive, by Theorem 1. First, we show that (20) is uniformly asymptotically stable. Since (20) is positive and +∞ +∞ 1 1 −1 + dB(s) = −1 + ds = − , 2 2(s + 1) 2 0 0
(20) is uniformly asymptotically stable, by Theorem 3. Secondly, since ∀ γ > 0 ∃ T0 = T0 (γ ) ≥ 0 ∀t ≥ T0 : eγ t
1 > 1, 2(t + 1)2
we conclude ∀γ > 0 :
+∞ 0
eγ s dB(s) =
+∞ 0
eγ s
1 ds = +∞. 2(s + 1)2
Therefore, (20) is not exponentially asymptotically stable, by Theorem 4. Finally, if B(·) in (20) is defined by B(t) :=
1 − e−t , 2
t ≥ 0,
then it is easy to see that (20) is exponentially asymptotically stable, by Theorem 4. Acknowledgements. Dedicated to Professor Nguyen Khoa Son on the occasion of his 60th birthday. The author is supported by the Alexander von Humboldt Foundation.
References 1. Gripenberg, G., Londen, S.O., Staffans, O.: Volterra Integral and Functional Equations. Cambridge Univeristy Press, New York (1990) 2. Murakami, S.: Exponential asymptotic stability for scalar linear Volterra equations. Differential Integral Equations 4, 519–525 (1991) 3. Naito, T., Shin, J.S., Murakami, S., Ngoc, P.H.A.: Characterizations of positive linear Volterra differential equations. Integral Equations and Operator Theory 58, 255–272 (2007)
On Positivity and Stability of Linear Volterra-Stieltjes Differential Systems
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4. Ngoc, P.H.A.: On exponential asymptotic stability of linear Volterra-Stieltjes differential systems (in preparation, 2009) 5. Ngoc, P.H.A., Naito, T., Shin, J.S.: Characterizations of postive linear functional differential equations. Funkcialaj Ekvacioj 50, 1–17 (2007) 6. Ngoc, P.H.A., Naito, T., Shin, J.S., Murakami, S.: On stability and robust stability of positive linear Volterra equations. SIAM Journal on Control and Optimization 47, 975– 996 (2008) 7. Ngoc, P.H.A., Murakami, S., Naito, T., Shin, J.S., Nagabuchi, Y.: On positive linear Volterra-Stieltjes equations. Integral Equations and Operator Theory (in press) (2009)
Eigenvalue Localization for Totally Positive Matrices Juan Manuel Pe˜na
Abstract. We survey eigenvalue localization results for totally positive matrices and we show its potential application in several problems. We first recall some localization results for the real eigenvalues of the real matrices and which can be considered an alternative to Gerschgorin disks, because they provide a sharper information in cases where the intervals corresponding to Gerschgorin disks are not sharp. An exclusion interval for the real eigenvalues of the real matrices is also recalled. The applications of these results are illustrated. On the other hand, a new lower bound for the minimal eigenvalue of a totally positive matrix is announced, and it improves the information derived from the Gerschgorin disks. We also survey some applications of stochastic totally positive matrices to birth and death processes and to Computer Aided Geometric Design. Birth and death processes occur in many chemical phenomena. In some recent problems in Computer Aided Geometric Design it is very important to know a good lower bound for the minimal eigenvalue of stochastic totally positive matrices. We apply to these problems the localization results mentioned above. Finally, we present a class of totally positive matrices to which we can apply a direct method for the eigenvalue computation recently obtained.
1 Introduction and Basic Results A matrix such that all its minors are nonnegative is called totally positive (TP). If all minors are positive, then we say that the matrix is strictly totally positive (STP). TP matrices present many applications (see [1, 9, 18]) in several fields such us Statistics, Approximation Theory, Computer Aided Geometric Design (C.A.G.D.) or Economy. Juan Manuel Pe˜na Departamento de Matem´atica Aplicada/IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 123–130. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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In this paper we revisit some results on eigenvalue localization and relate them with some problems on TP matrices. Let us start by recalling the following basic facts which can be found, for instance, in the following results of [1]: Corollary 6.6, Theorem 6.2 and Theorem 3.3. Theorem 1. Let A be an n × n TP matrix. Then: (i) All the eigenvalues of A are nonnegative. If A is STP, then they are also distinct (and positive). (ii)Given the n × n diagonal matrix J := diag(1, −1, 1, . . ., (−1)n−1 ), if A is in addition nonsingular, then the matrix JA−1J is TP. Theorem 1 (i) assures the positivity of the eigenvalues of a nonsingular TP matrix and, taking into account that J −1 = J, Theorem 1 (ii) shows that the inverse of a nonsingular TP matrix has the spectrum of another TP matrix. Therefore the results on eigenvalue localization of TP matrices presented in this paper can be also applied to their inverses. As for the eigenvectors of an STP matrix, it is also well-known (cf. Theorem 6.3 of [1]) that, if its eigenvalues are λ1 > λ2 > · · · > λn and they are associated to eigenvectors x1 , . . . , xn , then xk has exactly k − 1 variations of signs. In Section 2 we first recall some results on real eigenvalue localization. These results are an alternative to the classical Gerschgorin disks and we also illustrate their use. We also announce a result for bounding the minimal eigenvalue of a nonsingular TP matrix which improves the information provided by the Gerschgorin disks. Results of Section 2 will have application to the problems commented in Section 3, which deals with TP stochastic matrices. Finally, Section 4 presents a class of TP matrices to which we can apply a direct method for the eigenvalue computation recently obtained.
2 Eigenvalue Localization Results As recalled in Sect. 1, the eigenvalues of a TP matrix are nonnegative and so real. Let us start this section by presenting some results which are considered alternative to the classical Gerschgorin disks (see [14, 19–22] for more related results). Let A = (aik )1≤i,k≤n be a real matrix. We shall use the following notations: for each i = 1, . . . , n ri+ := max{0, ai j | j = i},
ri− := min{0, ai j | j = i},
(1)
The following result providing inclusion intervals for the real eigenvalues of a real matrix was proved in Theorem 3.5 of [19]. Theorem 2. Let A = (aik )1≤i,k≤n be a real matrix; let ri+ , ri− be as in (1); and let λ be a real eigenvalue of A. Then
λ ∈ S :=
n 1 i=1
[aii − ri+ − ∑ |ri+ − aik | , aii − ri− + ∑ |ri− − aik |]. k=i
k=i
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Now we introduce the following notations: for each i = 1, . . . , n s+ i := max{0, min{ai j | j = i}},
s− i := min{0, max{ai j | j = i}}.
(2)
The following result providing an exclusion interval for the real eigenvalues of a real matrix was proved in Proposition 2.9 of [19]. − Theorem 3. Let A = (aik )1≤i,k≤n be a real matrix, let s+ i , si be as in (2) and let λ be a real eigenvalue of A. Then $ %
λ∈ / E :=
n
n
− max { ∑ ai j − ns+ i }, min { ∑ ai j − nsi } . i=1,...,n
i=1,...,n j=1
j=1
Let us illustrate the use of the previous results with the matrix ⎛ ⎞ k ··· k ⎜ .. ⎟ , k > 0, A = ⎝ ... .⎠ k ··· k which has eigenvalues 0 (with multiplicity n − 1) and nk (with multiplicity 1). The intervals provided by the Gerschgorin discs are [(−n + 2)k, nk]. Since ri+ = k and ri− = 0, the intervals provided by Theorem 2 are [0, nk]. Besides, since s+ i = k and = 0, the exclusion interval provided by Theorem 3 is (0, nk). In conclusion, the s− i combination of both theorems provides exactly the eigenvalues 0 and nk. We finish this section announcing a result (Theorem 4.4 of [23]) for bounding the minimal eigenvalue of a nonsingular TP matrix which improves the information provided by the Gerschgorin disks. It uses the following notation for index subsets: given i ∈ {1, . . . , n} let Ji := { j | | j − i| is odd}. Theorem 4. Let A be a nonsingular TP matrix, and let λmin (> 0) be its minimal eigenvalue. For each i ∈ {1, . . . , n}, let Ji be the index subset defined above. Then:
λmin ≥ min{aii − i
∑ ai j }.
j∈Ji
A problem where this bound can be applied will be presented in the next section.
3 Eigenvalue Localization for Stochastic TP Matrices and Applications Let us recall that a nonnegative matrix is called row stochastic (or simply stochastic) if all its row sums are 1. The following result was proved in Proposition 3.2 of [21] and provides an upper bound of the real eigenvalues different from 1 of a stochastic matrix in terms of the least off-diagonal element.
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Theorem 5. Let A = (aik )1≤i,k≤n be a stochastic matrix and let s+ and w be the least off-diagonal and diagonal entries of A, respectively. If λ is a real eigenvalue of A, then either λ = 1 (with algebraic multiplicity 1 if s+ > 0) or 2w − 1 ≤ λ ≤ 1 − ns+. Let us now present two different applications of stochastic TP matrices related to birth and death processes and to Computer Aided Geometric Design (C.A.G.D.), respectively. Birth and death processes occur in many chemical phenomena, such as in aerosol chemistry (see [16]), in applications of industrial chemistry (see [24]), or in stochastic models of chemical reactions (see [17]). A birth and death process is a stationary Markov process whose state space is the nonnegative integers and whose transition probability matrix (Pi j (t))i, j=0,1,2,··· , t ≥ 0, with Pi j (t) = Pr{x(t) = j | x(0) = i} satisfies the conditions (as t → 0) Pi,i+1 (t) = λit + o(t), Pi,i−1 (t) = μit + o(t) and Pi,i (t) = 1 − (λi + μi )t + o(t), where λi > 0 for all i ≥ 0, μi > 0 for all i ≥ 1 and μ0 ≥ 0. By the results of [11] (see also [12]), the matrices associated to these processes are TP. Moreover they are STP for all t > 0. In fact, in [12] it was proved that, for all i1 < i2 < · · · < in and j1 < j2 < · · · < jn , the determinants ⎛ ⎞ Pi1 , j1 (t) · · · Pi1 , jn (t) ⎜ .. ⎟ det ⎝ ... (3) . ⎠ Pin , j1 (t) · · · Pin , jn (t) have the following interpretation. Suppose that n labelled particles start out in states i1 , . . . , in and execute the process simultaneously and independently. Then the determinant (3) is equal to the probability that at time t the particles will be found in states j1 , . . . , jn respectively without any two of them ever having been coincident (in the same state) in the intervening time. See also the related papers [10–13]. Let us now present the second application, where we see that localization results for the minimal eigenvalue of a nonsingular TP matrix have an application in C.A.G.D. We start with some basic definitions of this field. Let us recall that the collocation matrix of a system ofunivariate real functions (u0 (t), . . . , um (t)) at the u , . . . , u j≤m m 0 points (ti )ri=0 in R is given by M := (u j (ti ))0≤ 0≤i≤r . In C.A.G.D. it is t0 , . . . ,tr convenient that the system of functions is blending, that is , ui (t) ≥ 0 for all t and i and ∑m i=1 ui (t) = 1 for all t. Obviously, (u0 , . . . , um ) is blending if and only if all its collocation matrices are stochastic. A basis (u0 , . . . , um ) is totally positive (TP) when all its collocation matrices are TP. A blending basis that is TP is said to be a normalized totally positive (NTP) basis. A basis is NTP if and only if all its collocation matrices are TP and stochastic. It is well-known (cf. [18]) that the bases providing shape preserving representations are precisely the NTP bases. By Theorem 4.2 of [3], a space with a normalized totally positive basis always has a unique normalized B-basis, which is the basis with optimal shape preserving
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properties. For instance the Bernstein basis is the normalized B-basis in the case of the space of polynomials of degree at most n on a compact interval [2] and the B-splines form the normalized B-basis of their corresponding space [3]. All normalized totally positive bases satisfy the property known as the progressive iterative approximation property, which assure convergence of an interpolation process (see [5, 15]). In [5] it was proved that the normalized B-basis is the NTP basis with the fastest convergence rates for interpolating a curve and that the tensor product of normalized B-bases also present the fastest convergence rates for interpolating a surface. The proofs also show that the smallest eigenvalue of a nonsingular collocation matrix of the normalized B-basis is always greater than or equal to the smallest eigenvalue of the corresponding collocation matrix of another NTP basis. In fact, the convergence rate for the progressive iterative approximation property of the NTP basis depends on the minimal eigenvalue of the corresponding collocation matrix. This matrix is TP and stochastic and this explains why bounding this minimal eigenvalue for TP matrices is so important in C.A.G.D. Let us mention some key facts which allowed us to prove the mentioned result about the fastest convergence rates of the normalized B-basis. The matrix of change of basis K between an NTP basis (u0 , . . . , um ) and the normalized B-basis (b0 , . . . , bm ) (i.e., K such that (u0 , . . . , um ) = (b0 , . . . , bm )K) is TP and stochastic, and then it provides (if we factorize this matrix as a product of nonnegative bidiagonal and stochastic matrices) a corner cutting algorithm (see p. 240 of [7] and [8]) from the control polygon of the given NTP basis to the control polygon of the NTP B-basis. Since K is TP and stochastic, by Theorem 2.6 of [18] we can write K = Fn−1 Fn−2 · · · F1 G1 · · · Gn−2 Gn−1 , with
1 ⎜0 1 ⎜ ⎜ .. .. ⎜ . . ⎜ ⎜ 0 1 Fi = ⎜ ⎜ α 1 − αi+1,1 i+1,1 ⎜ ⎜ .. ⎝ . and
⎞
⎛
..
. αn,n−i 1 − αn,n−i
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⎛ ⎞ 1 0 ⎜ .. .. ⎟ ⎜ ⎟ . . ⎜ ⎟ ⎜ ⎟ 1 0 ⎜ ⎟ ⎜ ⎟, 1 − α1,i+1 α1,i+1 Gi = ⎜ ⎟ ⎜ ⎟ . . .. .. ⎜ ⎟ ⎜ ⎟ ⎝ 1 − αn−i,n αn−i,n ⎠ 1
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where, ∀ (i, j), 0 ≤ αi, j < 1. The previous factorization has been a key property used in the mentioned proof.
4 A Class of TP Matrices with a Direct Method for Eigenvalue Computation In [6] the class of r-convexity preserving matrices for all r ≤ k, important in many applications (see [4]), is considered and a direct method to compute their k largest eigenvalues is presented. The computational cost of the corresponding method is O(kn2 ) elementary operations for computing the k largest eigenvalues of an n × n matrix. The good stability properties of the method are also commented in that paper. In the particular case of an r-convexity preserving matrix for all r ≤ n − 1 we provide a direct method of O(n3 ) elementary operations to compute all its eigenvalues. Here we show that a class of TP matrices satisfies the previous hypotheses. Let us start by recalling the basic notations. Let k be a nonnegative integer. A vector v = (v1 , v2 , . . . , vn )T ∈ Rn is said to be k-convex if Δ k vi ≥ 0 for all i ∈ {1, . . . , n − k}, where
Δ k vi :=
k ∑ j (−1)k− j vi+ j . j=0 k
A vector v ∈ Rn is said to be k-concave if the vector −v is k-convex. Observe that a vector is 0-convex if and only if it is nonnegative and a vector is 1-convex if and only if it is monotonically increasing. A matrix A is said to be k-convexity preserving if for any k-convex vector v, the vector Av is also k-convex. Let us observe that A is 0-convexity preserving if and only if it transforms nonnegative vectors into nonnegative vectors, which is equivalent to A ≥ 0. A matrix A is 1-convexity preserving if and only if it is monotonicity preserving. Pk−1 will denote the set of k-convex and k-concave vectors. We shall denote by E the lower triangular matrix ⎞ ⎛ 1 0 ... ... 0 ⎛ ⎞ .. ⎟ ⎜ 1 0 ... 0 ⎜−1 1 . . . .⎟ ⎟ ⎜ ⎜ .. . . . . .. ⎟ ⎟ ⎜ ⎜. . . .⎟ (4) E := ⎜ E −1 = ⎜ 0 −1 1 . . . ... ⎟ . ⎟, ⎟ ⎜ ⎝1 . . . 1 0⎠ ⎟ ⎜ . . . . ⎝ .. . . . . . . 0⎠ 1 ... 1 1 0 . . . 0 −1 1 For each j ∈ {1, . . . , n}, let E j be the following n × n matrix: E1 := E and for j ≥ 2, ⎞ ⎞ ⎛ ⎛ I j−1 | 0 I j−1 | 0 ⎠, ⎠. ⎝ E −1 E j := ⎝ j = 0 | E 0 | E −1
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where I j−1 is the ( j − 1) × ( j − 1)− identity matrix and E is the (n − j + 1) × (n − j + 1)-matrix given by (4). Corollary 4.5 of [4], which is recalled below, provides a class of n × n TP matrices which are r-convexity preserving matrix for all r ≤ n − 1. So, we can apply to them the direct method of [6] for computing their eigenvalues. Theorem 6. Let A be a nonsingular totally positive matrix such that APr ⊆ Pr , r = 0, . . . , n − 1. Then A is a r-convexity preserving matrix for all r and A is similar to an upper triangular matrix of the form: ⎛ ⎞ λ1 . . . . . . . . . ⎜ . ⎟ ⎜ 0 λ2 . . . .. ⎟ −1 ⎜ ⎟ (E1 · · · En−1 ) A(E1 · · · En−1 ) = ⎜ . . ⎟ ⎝ .. . . . . . . .. ⎠ 0 . . . 0 λn with λ1 ≥ λ2 ≥ · · · ≥ λn > 0. In [4] we also have a source of examples of TP matrices satisfying the hypothesis of the previous result. Acknowledgements. Supported by the Spanish Research Grant MTM2006-03388 and by Gobierno de Arag´on and Fondo Social Europeo.
References 1. Ando, T.: Totally Positive Matrices. Linear Algebra Appl. 90, 165–219 (1987) 2. Carnicer, J.M., Pe˜na, J.M.: Shape preserving representations and optimality of the Bernstein basis. Advances in Computational Mathematics 1, 173–196 (1993) 3. Carnicer, J.M., Pe˜na, J.M.: Totally positive bases for shape preserving curve design and optimality of B-splines. Computer Aided Geometric Design 11, 633–654 (1994) 4. Carnicer, J.M., Pe˜na, J.M.: Generalized convexity preserving transformations. Computer Aided Geometric Design 13, 179–197 (1996) 5. Delgado, J., Pe˜na, J.M.: Progressive iterative approximation and bases with the fastest convergence rates. Computer Aided Geometric Design 24, 10–18 (2007) 6. Delgado, J., Pe˜na, J.M.: Computation of the eigenvalues of convexity preserving matrices. Applied Mathematics Letters (to appear, 2009) 7. Goodman, T.N.T., Micchelli, C.A.: Corner cutting algorithms for the B´ezier representation of free form curves. Linear Algebra Appl. 99, 225–252 (1988) 8. Goodman, T.N., Said, H.B.: Shape preserving properties of the generalized Ball basis. Computer Aided Geometric Design 8, 115–121 (1991) 9. Karlin, S.: Total Positivity. Stanford University Press, Stanford (1968) 10. Karlin, S., McGregor, J.: A characterization of birth and death processes. Proc. Nat. Acad. Sci. 45, 375–379 (1959) 11. Karlin, S., McGregor, J.: Coincidence properties of birth and death processes. Pacific. J. Math. 9, 1109–1140 (1959)
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12. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific. J. Math. 9, 1141–1164 (1959) 13. Karlin, S., McGregor, J.: Classical diffusion processes and total positivity. J. Math. Anal. Appl. 1, 163–183 (1960) 14. Li, H.-B., Huang, T.-Z., Li, H.: On some subclasses of P-matrices. Numer. Linear Algebra Appl. 14, 391–405 (2007) 15. Lin, H., Bao, H., Wang, G.: Totally positive bases and progressive iteration approximation. Computer & Mathematics with Applications 50, 575–586 (2005) 16. Losert-Valiente Kroon, C.M., Ford, I.J.: Stochastic Birth and Death Equations to Treat Chemistry and Nucleation in Small Systems. In: 17th International Conference: Nucleation and Atmospheric Aerosols, pp. 332–336. Springer, Galway (2007) 17. Mitrophanov, A.Y.: Note on Zeifman’s bounds on the rate of convergence for birth-death processes. J. Appl. Probab. 41, 593–596 (2004) 18. Pe˜na, J.M.: Shape preserving representations in Computer Aided-Geometric Design. Nova Science Publishers, Newark (1999) 19. Pe˜na, J.M.: A class of P-matrices with applications to the localization of the eigenvalues of a real matrix. SIAM J. Matrix Anal. Appl. 22, 1027–1037 (2001) 20. Pe˜na, J.M.: On an alternative to Gerschgorin circles and ovals of Cassini. Numer. Math. 95, 337–345 (2003) 21. Pe˜na, J.M.: Exclusion and inclusion intervals for the real eigenvalues of positive matrices. SIAM J. Matrix Anal. Appl. 26, 908–917 (2005) 22. Pe˜na, J.M.: Refining Gerschgorin disks through new criteria for nonsingularity. Numerical Linear Algebra with Applications 14, 665–671 (2007) 23. Pe˜na, J.M.: Eigenvalue bounds for some classes of P-matrices. Preprint 24. Ross, J.V., Pollett, P.K.: Extinction times for a birth-death process with two phases. Math. Biosci. 202, 310–322 (2006)
Positivity Preserving Model Reduction Timo Reis and Elena Virnik
Abstract. We propose a model reduction method for positive systems that ensures the positivity of the reduced model. Our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Stability is preserved and an error bound in the H∞ -norm is provided.
1 Introduction We consider linear time-invariant systems in continuous-time x(t) ˙ = Ax(t) + Bu(t), x(0) = x0 , y(t) = Cx(t) + Du(t),
(1)
where A ∈ n×n , B ∈ n×p , C ∈ n×n , D ∈ p×p are real constant coefficient matrices. The state x, input u and output y are real-valued vector functions. We focus on (internally) positive systems. These are systems whose state and output variables take only nonnegative values at all times t for any nonnegative initial state and any nonnegative input, see, e.g., [5, 7, 9]. We consider the problem of model reduction which preserves the positivity of a system. We generalize the model reduction methods of standard balanced truncation and singular perturbation balanced truncation such that positivity is preserved. Our technique uses a linear matrix inequality (LMI) approach and we show that, stability is preserved and an error bound in the H∞ -norm is provided. We present the results for continuous-time systems, however, we indicate that all results can be extended to the discrete-time case, [12]. Timo Reis and Elena Virnik Institut f¨ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 131–139. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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This paper is organized as follows. In Section 2, we introduce the notation, define the positivity concept and give a well-known characterization. We review the basics of balanced truncation and introduce positivity-preserving generalizations in Section 3. The applicability of the proposed methods is demonstrated by means of an example in Section 4.
2 Preliminaries Let the matrix quadruple [ A , B , C , D] denote the system (1). The function G(s) = C(sI − A)−1 B + D is called transfer function of the system and s is called frequency variable of the transfer function. Conversely, [ A , B , C , D] is called realization of G. Let H∞ be the space of all transfer functions that are analytic and bounded in the open right half-plane + . The continuous-time H∞ -norm is defined by G∞ = sup G(s)2 , s∈+
(2)
respectively, where · 2 denotes the spectral matrix norm. A matrix A ∈ n×n is called c-stable if all its eigenvalues are located in the open left complex half-plane. We call a realization of a continuous-time system [ A , B , C , D] stable, if A is c-stable. A matrix A ∈ m×n is called nonnegative (positive) and we write A ≥ 0 (A > 0) if all entries are nonnegative (positive). A matrix A ∈ n×n is called Z-matrix if its off-diagonal entries are non-positive. A matrix for which −A is a Z-matrix we call −Z-matrix. For a matrix A we have that eAt ≥ 0 for all t ≥ 0 if and only if A is a −Z-matrix, see, e.g., [11]. Let B ∈ n×n B ≥ 0 with spectral radius ρ (B). A matrix A of the form A = α I − B, with α > 0, and α ≥ ρ (B) is called M-matrix. If α > ρ (B) then A is a nonsingular M-matrix, if α = ρ (B) then A is a singular M-matrix. The class of M-matrices is a subclass of the Z-matrices. Accordingly, a matrix A ∈ n,n for which −A is an M-matrix is called a −M-matrix. Note that for a nonsingular M-matrix A, we have A−1 ≥ 0 [11]. A symmetric matrix A is called positive (semi)definite and we write (A ! 0) A 0 if for all x = 0 we have (xT Ax ≥ 0) xT Ax > 0. If this holds for −A then A is called negative (semi)definite and we write (A # 0) A ≺ 0. For matrices A, B we write (A # B) A ≺ B if (B − A ! 0) B − A 0. Next we define the class of positive systems and state a well-known characterization, see, e.g., [5, 7]. Definition 1. System (1) is called positive if for all t ∈ + the state x(t) and the output y(t) are nonnegative for any nonnegative initial state x0 and any measurable input function u : → p with u(τ ) ≥ 0 for all τ ∈ [0,t]. Theorem 1. System (1) is positive if and only if A is a −Z-matrix and B,C, D ≥ 0.
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3 Balanced Truncation for Positive Systems Consider the system [ A , B , C , D ] in continuous-time and assume that A is c-stable. The method of balanced truncation is based on Lyapunov equations. It generates c-stable reduced order systems. However, positivity is not preserved in general. Instead, we consider model reduction based on Lyapunov inequalities AP + PAT + BBT # 0,
AT Q + QA + CT C # 0,
(3)
with diagonal matrices P, Q ! 0. In the following we show that for positive systems the equations (3) are solvable. Moreover, we show the existence of a positive diagonal transformation T ∈ n×n such that for the transformed system [ Ab , Bb , Cb , Db ] given by Ab = T −1 AT,
Bb = T −1 B,
Cb = CT,
and Db = D,
(4)
ATb Qb + Qb Ab + CbT Cb # 0,
(5)
Qb = diag(Σ , 0no , Σo , 0nco )
(6)
the corresponding Lyapunov inequalities Ab Pb + Pb ATb + BbBTb # 0, are fulfilled for Pb = diag(Σ , Σc , 0no , 0nco ), with 0 ≺ Σc ∈
nc ×nc ,
0 ≺ Σo ∈
no ×no
and
Σ = diag(σ1 , σ2 , . . . , σk ) for some σ1 ≥ σ2 ≥ . . . ≥ σk > 0.
(7)
We will call [ Ab , Bb , Cb , Db ] a positive balanced realization. Note that the indices no and nc may be zero. Theorem 2. Consider the c-stable continuous-time positive standard system (1). Then, there exists a diagonal matrix T 0 such that the positive system given by (4) [ Ab , Bb , Cb , Db ] is positive balanced, i.e. there exist diagonal matrices Pb ! 0, Qb ! 0 as in (6), such that the Lyapunov inequalities in (5) hold. Proof. A −M-matrix is diagonally stable, i.e., there exist diagonal positive definite matrices X,Y such that AX + XAT ≺ 0 and AT Y + YA ≺ 0, see, e.g. [1, Theorem 1]. In particular, there exist diagonal matrices P ! 0, Q ! 0 such that (3) holds. We define a permutation matrix Π such that
Π T P Π = diag(X11 , X22 , 0no , 0nco ),
Π T Q Π = diag(Y11 , 0nc ,Y33 , 0nco )
with the additional property that X11 = diag(x1 , . . . , xk ),
Y11 = diag(y1 , . . . , yk )
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satisfy x1 y1 ≥ x2 y2 ≥ . . . ≥ xk yk > 0. Setting 1
−1 4 ) , Inc , Ino , Inco ), T¯ = diag((X11Y11
T = Π T¯ ,
we have that Pb = T −1 PT −T , Qb = T T QT have the desired form. The transformed system is given by [ Ab , Bb , Cb , Db ] as defined in (4). Since Ab is a −Zmatrix and Bb ,Cb , Db ≥ 0, the transformed system is again positive by Theorem 1. The numbers σ1 , . . . , σk play the role of Hankel singular values for conventional balanced truncation [6]. Consider a partition
A11 A12 B Ab = , Bb = 1 , Cb = C1 C2 , (8) A21 A22 B2 where A11 ∈ × and either = k or < k such that σ+1 < σ . The matrices B and C are partitioned accordingly. By means of balanced realizations, reduced-order models ˙ (t) = A2 2x(t) + Bu(t), 2 x2 (9) 2x(t) + Du(t) 2 y2(t) = C2 2 B, 2 D 2 C, 2 are defined by can now be constructed, where A, 2 = A11 , B2 = B1 , C2 = C1 , D 2 = D. A
(10)
An alternative method for the construction of reduced-order models is 2 = A11 − A12 A−1 A21 , B2 = B1 − A12A−1 B2 , A 22 22 −1 −1 2 2 C = C1 − C2 A22 A21 , D = D − C2A22 B2 .
(11)
For the reduced-order models, we have the following result. Theorem 3. Let [ A , B , C , D ] be a realization of G(s) that is c-stable. Moreover, let Ab , Bb , Cb , Db be constructed as in (4), such that (5) holds for Pb ! 0, Qb ! 0 2 , B2 , C2 , D 2 ] be the realization that is either constructed via (10) or as in (6). Let [ A 2 , B2 , C2 , D 2 2 ] is positive and the transfer function G(s) (11). Then, the system [ A = −1 2 2 2 2 C(sI − A) B + D satisfies 2 ∞≤2 G − G
k
∑
σi .
(12)
i=+1
Proof. It suffices to show the positivity of the reduced-order systems. For a proof of the error bound in (12), we refer to [8]. 2≥0 The reduced-order system defined in (10) is again positive, since B2 ≥ 0, C2 ≥ 0, D 2 and A is a −M-matrix as a submatrix of a −M-matrix. The positivity of the reduced-order system defined in (11) can be seen as follows: 2 is preserved, since it is a Schur complement of A The −M-matrix property of A
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[11]. Furthermore, since A22 is also a −M-matrix, we have A−1 22 ≤ 0, and hence, 2D 2 C, 2 ≥ 0. B, The main difference between the reduced-order models (10) and (11) is that the 2 model (10) is exact for s = ∞ meaning that G(∞) = G(∞), whereas (11) is exact at s = 0. In balanced truncation based on Lyapunov equations, the first method is called standard balanced truncation, whereas the second method is called singular perturbation balanced truncation. Note that for the computation of positive reduced-order models, there is no need to compute a balanced realization explicitely. Instead, for solutions P = diag(p1 , . . . , pn ) ! 0
and
Q = diag(q1 , . . . , qn ) ! 0
(13)
of (3), indices i1 , . . . , in have to be found such that we have pi1 qi1 ≥ . . . ≥ pi qi > pi+1 qi+1 ≥ . . . ≥ pin qin .
(14)
Then, a reduced-order model (9) can be obtained in the following way: Let I1 = [i1 , . . . , i ], I2 = [i+1 , . . . , in ] and A¯ 12 = A(I1 , I2 ), B¯ 1 = B(I1 , [1, . . . , m]), A¯ 11 = A(I1 , I1 ), A¯ 22 = A(I2 , I2 ), B¯ 2 = B(I2 , [1, . . . , m]), A¯ 21 = A(I2 , I1 ), C¯1 = C([1, . . . , p], I1 ), C¯2 = C([1, . . . , p], I2 ).
(15)
and either define the reduced-order model by 2 , B2 , C2 , D 2 ] = [ A¯ 11 , B¯ 1 , C¯1 , D ] [A
(16)
or by 2 , B2 , C2 , D 2] [A =[ A¯ 11 −A¯ 12 A¯ −1 A¯ 21 , B¯ 1 −A¯ 12 A¯ −1 B¯ 2 , C¯1 −C¯2 A¯ −1 A¯ 21 , D−C¯2 A¯ −1 B¯ 2 ]. 22
22
22
(17)
22
These systems are linked to the reduced-order models (10) and (11), respectively, via a positive diagonal state-space transformation. Therefore, the positivity as well as the error bound (12) are still valid. Summarizing, we present the reduction procedure in algorithmic form. Algorithm 4 Positivity-preserving model reduction by a) standard balanced truncation b) singular perturbation balanced truncation. Given a positive system G = [ A, B, C, D ], compute a positive reduced-order model 2 = [ A, 2 B, 2 D 2 C, 2 ]. G 1. Solve the Lyapunov inequalities (5) for diagonal P and Q as in (13). 2. Form distinct indices i1 , . . . , in such that pik qik ≥ pil qil for k > l. 3. Choose ∈ {1, . . . , n} such that pi qi > pi+1 qi+1 and define the vectors of indices I1 = [i1 , . . . , i ], I2 = [i+1 , . . . , in ]. 4. Build the reduced-order a) by (16) or b) by (17).
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u1
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f 13
h2
f o,2
f o,1
Rn−1
... ...
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f 23
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f n−1,n
R3
h3 ...
hn
f o,n
...
f o,3
Rn
Fig. 1 System of n water reservoirs
One can see that the error of the reduced-order model can be estimated by 2 ∞≤2 G − G
n
∑
√
pik qik .
(18)
k=+1
Let us finally give a remark on the Lyapunov inequalities (3). It is clear that their solutions are not unique and one should√ look for solutions P = diag(p1 , . . . , pn ), Q = diag(q1 , . . . , qn ) such that PQ has a large number of small diagonal elements. This yields components of the state which are candidates to truncate. A good heuristic for this is the minimization of the trace of P and Q [4]. For getting even sharper bounds, the Lyapunov inequalities can be solved once more while now minimizing the sum of those diagonal elements of P and Q corresponding to the candidates for truncation.
4 Example In this section we present a numerical example to demonstrate the properties of the discussed model reduction approach for positive systems. The numerical tests R Version 7.4.0 on a PC with an Intel(R) Pentium(R) 4 CPU were run in MATLAB 3.20GHz processor. Example 1. Consider a system of n water reservoirs such as schematically shown in Fig. 1. All reservoirs R1 , . . . , Rn are assumed to be located on the same level. The base area of Ri and its fill level are denoted by ai and hi , respectively. The first reservoir R1 has an inflow u which is the input of the system, and for each i ∈ {1, . . . , n}, Ri has an outflow fo,i through a pipe with diameter do,i . The output of the system is assumed to be the sum of all outflows. Furthermore, each Ri and R j are connected by a pipe with diameter di j = d ji ≥ 0. The direct flow from Ri to R j is denoted by fi j . We assume that the flow depends linearly on the difference between the pressures on both ends. This leads to the equations
Positivity Preserving Model reduction
fi j (t) = di2j · c · (hi(t) − h j (t)),
137 2 fa,i (t) = do,i · c · (hi (t) − h j (t)),
where c is a constant that depends on the viscosity and density of the medium and gravity. The fill level of Ri thus satisfies the following differential equation % $ n 1 2 2 ˙hi = c −do,i hi (t) + ∑ di j (h j (t) − hi (t)) + δ1i u(t), ai ai j=1 where δ1i denotes the Kronecker symbol, that is δ1i = 1 if i = 1 and zero otherwise. Then, we obtain system (1) with D = 0 and matrices A = [ai j ]i, j=1,...,n , 2 and B = [bi1 ]i=1,...,n , C = [c1 j ] j=1,...,n with bi1 = δa1i1 , ci1 = c · do,i c −d 2 − ∑nk=1 dik2 ai j = · 2 o,i ai di j
i = j, i = j,
where we define dii = 0. For our illustrative computation, we have constructed the presented compartment model with ten states. We assume that we have two well connected substructures each consisting of five reservoirs, where each reservoir is connected with every other reservoir by a pipe of diameter 1. The substructures are connected with each other by a pipe of diameter 0.2 between reservoirs one and ten. For simplicity reasons, we also set all base areas of the reservoirs to 1 and also c = 1. Solving the Lyapunov inequalities (3), we obtain the values σ1 = σ2 = σ3 = σ4 = 23.3581, σ5 = 14.9864, σ6 = 0.0097, σ7 = σ8 = σ9 = σ10 = 0.0055. We show only the results obtained with singular perturbation balanced truncation. However, qualitatively similar results were obtained via standard balanced truncation. The reduced model with five states is again positive with ⎡ ⎤ −5.0000 1.0000 1.0000 1.0000 0.8022 ⎢ 1.0000 −5.0000 1.0000 1.0000 0.8022 ⎥ ⎢ ⎥ ⎥ Ar = ⎢ ⎢ 1.0000 1.0000 −5.0000 1.0000 0.8022 ⎥ , ⎣ 1.0000 1.0000 1.0000 −5.0000 0.8022 ⎦ 1.2466 1.2466 1.2466 1.2466 −5.0400 ⎡ ⎤ ⎡ ⎤T 0 8.4087 ⎢ 0 ⎥ ⎢8.4087⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ Br = ⎢ 0 ⎥ , Cr = ⎢ ⎢8.4087⎥ , Dr = 0. ⎣ 0 ⎦ ⎣8.4087⎦ 0.1483 7.0113 The frequency responses, i.e. the transfer function at values s = jω , for ω ∈ [0, 10], of the original and of the reduced order model are depicted in the upper diagram of Fig. 2. The lower diagram shows the frequency response of the error system along with the mutual error bound 0.0636.
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original model reduced-order model
0.8
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0
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Frequency plots of error system and error bound 0.07 0.06
error plot error bound
0.05 0.04 0.03 0.02 0.01 0
0
1
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3
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7
Fig. 2 Frequency plot showing original and reduced order model
5 Conclusion In this paper, we have presented a model reduction technique that preserves the positivity of a system in the continuous-time. The presented method can be extended for the discrete-time case. The proposed method is based on the existence of a diagonal solution of the corresponding Lyapunov inequalities, which may be obtained via LMI solution methods. The reduction then may be performed by standard balanced truncation or singular perturbation balanced truncation methods. It is shown that both methods preserve positivity. Furthermore, a numerical example in the continuous-time case is provided and illustrates the functionality of the proposed algorithm. Acknowledgements. Supported by the DFG Research Center M ATHEON in Berlin.
References 1. Araki, M.: Application of M-matrices to the stability problems of composite dynamical systems. J. Math. Anal. Appl. 52, 309–321 (1975) 2. Beck, C., Doyle, J., Glover, K.: Model reduction of multidimensional and uncertain systems. IEEE Trans. Automat. Control 41, 1466–1477 (1996) 3. Berman, A., Plemmons, R.J.: Nonnegative Matrices. In: The Mathematical Sciences, Classics in Applied Mathematics, 2nd edn., vol. 9. Society for Industrial and Applied
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Mathematics (SIAM), Philadelphia (1994) 4. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities. In: System and Control Theory, SIAM Studies in Applied Mathematics, vol. 15. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994) 5. Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and applications. John Wiley and Sons Inc., New York (2000) 6. Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds. Internat. J. Control 39, 1115–1193 (1984) 7. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 8. Liu, Y., Anderson, B.D.O.: Singular perturbation approximation of balanced systems. Internat. J. Control 50, 1379–1405 (1989) 9. Luenberger, D.G.: Introduction to Dynamic Systems. John Wiley and Sons Inc., New York (1979) 10. Reis, T., Virnik, E.: Positivity preserving balanced truncation for descriptor systems. SIAM J. Control Optim. (to appear, 2009) 11. Varga, R.S.: Matrix Iterative Analysis. In: Springer Series in Computational Mathematics, vol. 27. Springer, Berlin (2000) 12. Virnik, E.: Analysis of positive descriptor systems, PhD thesis, TU Berlin (2008)
The Minimum Energy Problem for Positive Discrete-Time Linear Systems with Fixed Final State Ventsi Rumchev and Siti Chotijah
Abstract. The non-negativity of controls in positive linear discrete-time systems usually gives rise to complementarity conditions in the first-order Karush-KuhnTucker optimality conditions - this complicates the analytic solution and usually leads to numerical solutions. The intrinsic relationship between reachable sets and the minimum-energy problem is exploited in this paper to obtain an analytic solution of the minimum-energy control problem for positive linear discrete-time systems with any pair of fixed terminal (initial and final) states.
1 Introduction The minimum-energy problem for time invariant linear systems is a classical problem in control theory. It has nice analytic solutions if no restrictions are imposed on the state and control variables [1, 8]. Positive discrete-time linear systems (PDLS) are defined on cones and not on linear spaces since the control and trajectory are to be non-negative. The non-negativity of control in such systems gives rise to complementarity conditions in the first-order Karush-Kuhn-Tucker optimality conditions [3], which complicates the analytic solution and usually leads to numerical solutions. At the same time the appeal and the advantages of analytic solutions are well appreciated. To the best of our knowledge the only analytical solution to the minimum-energy problem for PDLS with fixed final state is obtained by Kaczorek [6]. The result is developed under some assumptions, among which the assumption of zero initial state seems to be quite restrictive. The relation of the problem with the reachable sets, i.e. the geometry of the problem, is not studied in [6] either. Ventsi Rumchev Department of Mathematics and Statistics, Curtin University of Technology. GPO Box U 1987 Perth, WA 6845, Perth, Australia e-mail:
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Related work for continuous-time systems with non-negative controls is published in [4, 5, 7, 9] but the positivity of the system is not exploited in these papers, except in [7], where conditions that guarantee the positivity of the closed–loop linear quadratic optimal system with free final state are developed. Positivity is an intrinsic property of positive systems and in many cases it helps to simplify the analysis and the results. This paper is concerned with the minimum-energy problem for PDLS with any pair of fixed terminal (initial and final) states. We focus on the scalar case as a more simple case in order to expose the relationship between the reachable sets and the minimum-energy problem. Using the dynamic programming approach [1] we obtain a nice analytic solution to the problem.
2 Problem Formulation and some Preliminaries The minimum-energy problem for scalar positive discrete-time linear systems (PDLS) with fixed final state is formulated as follows [6]. Minimize 1 T −1 (1) J = ∑ u2 (t) 2 t=0 subject to x(t + 1) = ax(t) + bu(t), a, b ≥ 0,
t = 0, . . . , T − 1 u(t) ∈ R+
(2) (3)
where x(t) is the state at time t = 0, . . . , T , u(t) ∈ R+ is the control sequence, the symbol R+ denotes the set of all non-negative real numbers, T is a finite-time horizon, and the initial and final state are given by x(0) = x0 ≥ 0
and
x(T ) = xT ≥ 0.
(4)
The state variables x(t), t = 0, . . . , T , are, clearly, non-negative for any nonnegative initial state x0 ≥ 0, and any (non-negative) control sequence u(t),t=0,. . . ,T-1. Definition 1. The set of all states Rt (0) of PDLS (2)-(3) reachable from the origin in t-steps by admissible (i.e., non-negative) control sequences {u(0), u (1) , . . . , u (t − 1)} is called a t-steps reachable set. It is defined as t−1 Rt (0) = x x = ∑ at−k−1 bu(k); a, b, x0 ≥ 0 and k=0
. u(k) ≥ 0 for k = 0, 1, . . . ,t − 1. t = 0, ..., T − 1,
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with R0 (0) ≡ 0. Any non-negative state can, clearly, be reached by non-negative controls in (at least) one step if b > 0. If b = 0, the control does not affect the system state and no positive state can be reached from the origin. Consequently, we have R+ , b > 0 Rt (0) = {0}, b = 0
Definition 2. The set of all states Rt (x0 ) of PDLS (2)-(3) reachable from the initial state x0 = x(0) in t-steps by non-negative control sequences {u(0), u (1) , . . . , u (t − 1)} is defined as t−1 Rt (x0 ) = x x = at x0 + ∑ at−k−1 bu(k) a, b, x0 ≥ 0 and k=0
. u(k) ≥ 0 for k = 0, 1, . . . ,t − 1 . = at x0 + Rt (0)
(5)
t = 1, 2, . . . , T with R0 (x0 ) ≡ x0 . It is not difficult to see that Rt (x0 ) =
at x0 + R+ = [at x0 , ∞), b > 0 {at x0 }, b=0
So, the reachable set Rt (0) is a particular case of Rt (x0 ) for x0 = 0. The PDLS (2)-(3) is asymptotically stable if and only if 0 ≤ a < 1 [2]. It is stable, if 0 ≤ a ≤ 1. Let Rts (x0 ) denote the reachable sets of a stable PLDS. For b > 0 the reachable sets of a stable system possess the nested property s Rt−1 (x0 ) ⊆ Rts (x0 ),t = 1, 2, ..., T,
(6)
where the inclusion is strict if the system is asymptotically stable, i.e. 0 ≤ a < 1. The inclusion property (6) is in the opposite direction us Rtus (x0 ) ⊂ Rt−1 (x0 ),t = 1, 2, ..., T,
if the PLDS is unstable (a > 1) and b > 0, where Rtus (x0 ) denotes the t-steps reachable set of an unstable PDLS (2)-(3).
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The PDLS (2)-(3) is (a) reachable if and only if b > 0 and then any non-negative state x can be reached from the origin by an admissible (that is a non-negative) control in one step; (b)null-controllable (in finite-time) or controllable-to-the-origin, if and only if a = 0 and then the origin can be reached from any non-negative state by zero control in one step; (c)controllable if and only if it is reachable (b > 0) and null-controllable (a = 0) and then the system can be driven by a (non-negative) control from any non-negative initial state x0 ≥ 0 into any terminal state x ≥ 0 in one step (see [2, 10]).
3 Main Results Theorem 1. Let xT ∈ RT (x0 ). Then, the optimal control sequence that minimizes the cost function (1) in the minimum-energy problem (1)-(4) with fixed final state is given by ⎧ T −(t+1) a xT − aT−t x∗ (t) ⎪ ⎪ , b>0 ⎨ T −1 2(T −(i+1)) , t = 0, 1, 2..., T − 1, (7) u∗ (t) = b a ∑ ⎪ ⎪ i=t ⎩ 0, b=0 where x∗ (t) is the corresponding optimal trajectory, and the optimal value of the cost function (1) is ⎧ 2 ⎪ 1 xT − a T x0 ⎪ ⎪ ⎨ , b>0 T −1 J0∗ = 2 b2 ∑ a2(T −i−1) . (8) ⎪ ⎪ i=0 ⎪ ⎩ 0, b=0 Proof. The hypothesis xT ∈ RT (x0 ) implies that there exists a solution to the twopoint boundary-value problem (2) - (4). In other words, there exists an admissible (non-negative) control sequence {u (t) ≥ 0,t = 0, 1, . . . , T − 1} such that the corresponding trajectory {x0 , x(1), ..., x(T − 1), xT } is feasible (that is non-negative). When b = 0 the PDLS (2)-(3) is not reachable and the reachable set RT (x0 ) consists of the point aT x0 only. Then, the only solution to the minimum-energy problem (1)-(4) is the trivial one: {u∗ (t) = 0, t = 0, 1, . . . , T − 1}, {x0 , ax0 , ..., aT −1 x0 , xT } and J0∗ = 0. Let b > 0. Since by hypothesis xT ∈ RT (x0 ), the two-point boundary-value problem (2)-(4) is consistent and there exists at least one solution. To find the solution that minimises the cost function (1) we apply the dynamic programming procedure [1]. The Bellman equation for the minimum-energy problem (1)-(4) can be written as
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Jt (x) = min u≥0
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. 1 2 u + Jt+1 (x) , u = u (t) , x = x (t) , t = 0, . . ., T − 1 , 2
with JT (x) = 0. Moving backwards we try for t = T − 1, t = T − 2 and formulate the induction hypothesis: 1 Jt (x) = 2
2 xT − aT−t x
(9)
T −1
b2 ∑ a2(T −i−1) i=t
and u (t) =
aT −(t+1) b xT − aT−t x T −1
≥ 0.
(10)
b2 ∑ a2(T −(i+1)) i=t
Let expressions (9) and (10) be true for t = k + 1, that is 2 T −(k+1) x 1 xT − a , Jk+1 (x) = 2 2 T −1 2(T −(i+1)) b ∑ a
(11)
i=k+1
and, respectively, u (k + 1) =
aT −(k+2) b xT − aT−(k+1) x b2
T −1
∑
≥ 0.
(12)
a2(T −(i+1))
i=k+1
We prove that (9) and (10) are true for t = k, that is 2 1 xT − aT−k x Jk (x) = 2 2 T −1 2(T −i−1) b ∑ a i=k
and u (k) =
aT −(k+1) b xT − aT −(k+1) x T −1
≥ 0.
b2 ∑ a2(T −(i+1)) i=k
For t = k the Bellman equation is specified as . 1 2 Jk (x) = min u + Jk+1 (x) . u≥0 2
(13)
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A substitution of state equation (2) into (11) yields ⎧ ⎫ ⎪ ⎪ ⎪ ⎨1 ⎬ T−k−1 (ax + bu) 2 ⎪ − a x 1 T , u2 + Jk (x) = min T −1 ⎪ u≥0 ⎪ 2 2 ⎪ ⎪ ⎩ ⎭ b2 ∑ a2(T −i−1)
(14)
i=k+1
where u = u (k) and x = x(t) is to be specified by the initial condition x(0) = x0 . 2 1 2 1 xT − aT −k−1 (ax + bu) The differentiation of the expression u + with reT −1 2 2 b2 ∑ a2(T −i−1) i=k+1
spect to u results in u (k) =
aT −(k+1) b xT − aT −k x T −1
≥ 0.
(15)
b2 ∑ a2(T −(i+1)) i=k
The substitution of (15) in (14) leads to 2 1 xT − aT −k x Jk (x) = . 2 2 T −1 2(T −i−1) b ∑ a i=k
Thus, the assumptions (9) and (10) are true for t = k, and, therefore, they are true by induction for any t. For t = 0 we have aT −1 b xT − aT x0 u (0) = ≥ 0. T −1 b2 ∑ a2(T −(i+1)) i=0
This concludes the proof of the theorem.
Under the hypotheses of the theorem, the optimal control sequence is given by (7), the corresponding optimal trajectory by t−1
x∗ (t) = at x0 + b ∑ at−1− j u∗ ( j) ,
t = 1, 2, ..., T − 1,
j=0
and the optimal cost function is represented by (8). Remark 1. 1. If xT ∈ / RT (x0 ) then the two-point boundary-value problem (2)(4) is inconsistent and, therefore, the minimum-energy problem with fixed final state (1)-(4) has no solution.
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2. If b > 0 and xT belongs to the boundary of RT (x0 ), that is xT = aT x0 , the optimal control sequence (7) is a zero sequence and the corresponding optimal trajectory becomes x0 , ax0 , . . . , aT −1 x0 , xT . 3. Let the system (2)-(3) be controllable that is let a = 0 and b > 0. Then, it is not difficult to see using limits that expression (7) is reduced to u∗ (t) = 0 for t = 0, 1, 2, . . . , T − 2 and u∗ (T − 1) =
xT , b
and, consequently, the minimal value of the cost function becomes J0∗ (x) =
1 x2T . 2 b2
4. For b > 0 and t = 0 expression (7) becomes aT −1 b xT − aT x0 ∗ u (0) = . T −1 b2 ∑ a2(T −(i+1)) i=0
The above expression for u∗ (0) for x0 = 0 agrees with the result in [6]. The expression [7], however, is obtained for any non-negative pair {x0 , xT } such that xT ∈ RT (x0 ) and, therefore, is more general than that in [6], where (among the other assumptions) the reachability of PDLS and a zero initial state are required. The expression b2
T −1
∑ a2(T −(i+1)) = (b, ab, a2b, ..., aT −1 b) · (b, ab, a2b, ..., aT −1b)T ≥ 0
i=0
is the gramian of PDLS (2) - (3). 5. The optimal control law (7) for b > 0 can be treated as a feedback control since it depends on the current state. As a matter of fact, it can also be represented as an open-loop control that depends on the initial and final states as the corollary below shows. Corollary 1. Under the assumptions of Theorem 1, the optimal control can be represented as an open-loop control namely ⎧ T −(t+1) xT − a T x0 1 − a 2 ⎨a , b>0 u∗ (t) = , t = 0, 1, 2, . . . , T − 1. (16) b 1 − a2T ⎩ 0, b=0 The optimal trajectory corresponding to (16) , then, becomes ⎧ aT −t xT − aT x0 1 − a2t ⎨ t a x0 + , b>0 x∗ (t) = ,t = 0, 1, 2, . . . , T, (1 − a2T ) ⎩ t b=0 a x0 ,
(17)
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and the cost function (1) is given by ⎧ 2 ⎨ 1 xT − a T x0 1 − a 2 , b>0 . J0∗ (x) = 2 b2 (1 − a2T ) ⎩ 0, b=0
(18)
Remark 2. 1. The expressions (16) and (17) clearly tell us that the optimal control sequence and the optimal trajectory are non-negative. 2. It is easy to see from (17) that the optimal trajectory ends at the desired final state that is x∗ (T ) = xT . 3. The optimal control sequence (17) is easy to calculate since u∗ (t + 1) = u∗ (t)/a. 4. If the initial state x0 is zero, expressions (16), (17) and (18) become even simpler u∗ (t) =
aT −(t+1) (1 − a2)xT , b(1 − a2T )
aT −t 1 − a2t xT x (t) = , (1 − a2T ) ∗
t = 0, 1, 2, . . . , T − 1,
t = 0, 1, 2, . . . , T,
and J0∗ (x) =
1 1 − a2 x2T . 2 b2 (1 − a2T )
The meaning of the above expressions is transparent. 5. It is worth noting that the expression (16) for the optimal control sequence and the expression (18) for the minimal value of the cost function are the same as those when no restrictions are imposed on controls. This is because the minimum of J0 as a function of u(0), u(1), . . . , u(T − 1) is achieved at an interior point u∗ (k) > 0, k = 0, 1, 2, . . . , T − 1, for b > 0 and xT ∈ RT (x0 ).
4 Concluding Remarks Using the dynamic programming approach an analytic solution of the minimum energy problem for positive discrete-time linear systems with any pair of fixed terminal states and scalar controls is obtained and analysed in the paper. The relationship between the problem and the geometric properties of the system is revealed and well exploited. The optimal control sequence is represented in two different (equivalent) forms – a feedback form and an open-loop form. The minimum energy problem has a trivial (zero) solution if the positive discrete-time linear system does not possess the reachability property. It does not have a solution if the final state does not belong to the T − steps reachable set RT (x0 ). The optimal solution becomes quite simple if the system is controllable or the initial state is zero. We discuss the minimum-energy
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problem for positive discrete-time linear systems with vector controls and any fixed terminal pair of states in another paper.
References 1. Bertsekas, D.P.: Dynamic Programming and Optimal Control, 3rd edn. Athena Scientific, Belmont (2005) 2. Caccetta, L., Rumchev, V.: A survey of reachability and controllability for positive linear systems. Annals of Operations Research 98, 101–122 (2000) 3. Facchini, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1. Springer, New York (2003) 4. Heemels, W., Van Eijndhoven, S., Stoorvogel, A.: A linear quadratic regulator problem with positive contols. International Journal of Control 70(4), 551–578 (1998) 5. Hu, Y., Zhou, X.Y.: Constrained stochastic LQ-control with random coefficients and application to portfolio selection. SIAM Journal on Control and Optimization 44(2), 441– 466 (2005) 6. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 7. Laabissi, M., Winkin, J., Beauthier, C.: On the positive LQ-problem for linear continuous-time systems. In: Commault, C., Marchand, N. (eds.) Positive Systems. LNCIS, vol. 341, pp. 295–302. Springer, Heidelberg (2006) 8. Lewis, F., Syrmos, V.: Optimal Control, 2nd edn. John Wiley & Sons, New York (1995) 9. Pachter, M.: The linear-quadratic optimal control problem with positive controllers. International Journal of Control 32(4), 589–608 (1980) 10. Rumchev, V., James, G.: Controllability of positive linear discrete-time systems. International Journal of Control 50(3), 845–857 (1989)
A Rollout Algorithm for Multichain Markov Decision Processes with Average Cost Tao Sun, Qianchuan Zhao and Peter B. Luh
Abstract. Many of simulation based learning algorithms have been developed to obtain near optimal policies for Markov decision processes (MDPs) with large state space. However, most of them are for unichain problems. In view that some applications involve multichain processes and it is NP-hard to determine whether a MDP is unichain or not, it is desirable to obtain an algorithm that is applicable to multichain problems as well. This paper presents a rollout algorithm for multichain MDPs with average cost. Preliminary analysis of the estimation error and parameter settings are provided based on the problem structures, i.e., mixing time of transition matrix. Ordinal optimization and Optimal Computing Budget Allocation are also suggested to improve the efficiency of the algorithm.
1 Introduction This paper deals with Markov decision processes (MDPs) with average cost criterion. The entire MDP theory is based on solving the so called optimality equation which describes the optimality principle [15]. There are two classical methods for Tao Sun Center for Intelligent and Networked Systems (CFINS), Department of Automation and TNLIST Lab, Tsinghua University, Beijing 100084, China and China Mobile Research Institute, Beijing, 100053, China, e-mail:
[email protected] Qianchuan Zhao (Corresponding author. Tel.+8610-62783612, Fax.+8610-62796115) Center for Intelligent and Networked Systems (CFINS), Department of Automation and TNLIST Lab, Tsinghua University, Beijing 100084, China, e-mail:
[email protected] Peter B. Luh Center for Intelligent and Networked Systems (CFINS), Department of Automation and TNLIST Lab, Tsinghua University, Beijing 100084, China and Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 06269-2157 USA, e-mail:
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solving the optimality equation of an MDP namely policy iteration and value iteration ([3] and [15]). They are both iterative methods based on solving a series of linear equations. When the state space of the problem is large, it turns out that neither policy iteration nor value iteration can be used to obtain exact solutions because of two reasons: first, the linear equations involved is too large to be solved; second, even the solutions (either a policy or a value function) could not be stored. In practice, instead, approximations are often used and simulation is one way to estimate the policy performance. In this paper, we will focus on how to do policy iteration approximately by simulation. The policy iteration method has two steps: policy evaluation and policy improvement. When simulation is used at the policy evaluation step, the approximate methods are known as simulation based policy iteration or on-line policy iteration (see, e.g., [11]). When storing a policy is impossible, one can only perform policy evaluation and policy improvement for the states of a sample path. Such methods are known as rollout methods ([4] and [9]). The rollout algorithm was developed first for finite horizon total cost MDPs in [4]. The simulation runs to the end of the time horizon to evaluate the goodness of an action. For average cost MDPs, the time horizon is infinite which makes simulation impossible to look ahead to the end of the time horizon as total cost MDPs. Moreover, the optimization for an average cost MDP should consider the underlying structure of the transition probability matrix, i.e., the number of recurrent classes in the state space. A MDP is said to be unichain if there is only one recurrent class in the state space under any policy. If there is some policy that leads to more than one recurrent class, the problem is called a multichain MDP [15]. The multichain MDPs appear in inventory control problems [15] and maintenance problems [16]. Compared with unichain MDPs, the optimization for multichain MDPs should select recurrent classes whose average costs are small. Moreover, the problem to determine whether a MDP is unichain or not is NP-hard [19]. Therefore, it is important to obtain an algorithm applicable for multichain MDPs. However, most of the existing learning or simulation based online optimization algorithms (see, e.g., [7], [8], [11] and [17]) are developed for unichain models. There are only a few algorithms that consider multichain cases in recent years (see, e.g., [1], [14]). In this paper a rollout algorithm which aims to address large multichain MDPs is developed. In Section 2, the formulation of average cost MDP is introduced and the policy iteration for solving multichain MDPs is presented. A multichain rollout algorithm is developed in Section 3 and its properties are presented. In Section 4, the estimation error of the rollout algorithm caused by finite look-ahead horizon of simulation is analyzed through mixing time which reflects the structure of the problem. Some techniques are discussed to improve the efficiency of the algorithm. Section 5 concludes this paper with some remarks.
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2 Formulation of Markov Decision Processes In this section, we will introduce the average cost MDP first. Then, policy iteration method is presented to solve a general multichain MDP. The method is the basis for the multichain rollout algorithm to be developed in Section 3.
2.1 Average Cost Markov Decision Processes We consider a discrete-time Markov chain, X = {Xk , k = 0, 1, ...}, with a finite state space S = {1, ..., |S |}, where | · | denotes the set cardinality. Let A be a finite set of actions and A (i) stand for all feasible actions for the state i. We consider the set of stationary policies denoted by E . A policy μ ∈ E is a time independent mapping μ : S → A . Under policy μ , the action μ (i) ∈ A (i) taken for state i leads to state transition probability from i to j described by p μ (i) (i, j), j = 1, 2, ..., |S |. The Markov chain evolves following the transition probability matrix Pμ , with [Pμ ]i, j = pμ (i) (i, j). We consider the average cost over infinite time horizon. Let ημ be a vector which stands for the performance of the policy μ . The element ημ (i) is the expected average cost starting from the state i: 6 K−1 1 ημ (i) = lim E ∑ f μ (Xk , μ (Xk )) (1) K→∞ K K=0 where X0 = i, i ∈ S . The optimization problem is to find an optimal policy μ that minimizes the average cost for any initial state, i.e., / 0 (2) μ ∗ = arg min ημ . μ∈ E
For a transition matrix P, the limiting matrix is defined as: 1 K−1 k ∑P K→∞ K k=0
P∗ = lim
(3)
By the definition of P∗ in (3), the average cost can be calculated as follows [3]: 1 K−1 k ∑ Pμ f μ = Pμ∗ f μ K→∞ K k=0
ημ = lim
(4)
Moreover, it can be obtained that [3, 15]:
ημ = Pμ ημ
(5)
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2.2 Policy Iteration for Multichain Markov Decision Processes Policy iteration is a classical method to solve an average cost MDP. The method starts from an initial policy and results in an optimal policy by iteratively conducting policy evaluation and policy improvement steps. In the following, we will first present policy iteration for multichain MDPs. For unichain MDPs, the policy iteration can be simplified which is pointed out in the next. Before we proceed, let us define is a function gμ of the states and the policy μ which is known as the “bias” [3, 15]. Let Pμ be a state transition matrix of a policy μ and the cost vector be f μ , then the bias under the policy is defined as: gμ = lim gKμ .
(6)
K→∞
In the above, gKμ ≡
K−1
∑ (Pμk − Pμ∗ ) f μ .
(7)
k=0
The difference g(i) − g( j) between the bias of the two states i, j is the difference between the total costs starting from state i instead of j. Therefore, bias is also called “relative value”[15]. We are ready to introduce the steps of the policy iteration algorithm for multichain cases [3]. Policy Evaluation is to obtain the average cost ημ and the bias gμ of a policy μ by solving a set of linear equations for (η , g, u) as follows:
η = Pμ η , η + g = f μ + Pμ g, g + u = Pμ u
(8) (9) (10)
Policy Improvement updates a policy μ n based on the policy evaluation step. According to the average cost, a new policy μ n+1 will make the states evolve among the recurrent classes with small average costs. Namely, if min{Pμ ημ n } = ημ n , let μ ∈E
μ n+1 = argmin{Pμ ημ n }
(11)
/ 0 μ n+1 = arg min f μ + Pμ gμ n
(12)
μ ∈E
Otherwise, let
μ ∈E¯
In (12), E¯ is the set of policies which satisfy the equation min{Pμ η μn } = η μn . μ ∈E
It can be seen that the policy improvement step for the multichain cases is conducted in an embedded manner. Firstly, select the recurrent class by (11). Then if the policy is not changed, action update will be conducted according to (12). For unichain problems, both the policy evaluation and the policy improvement steps are simplified. This is caused by the fact that the average cost is independent
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of states when the problem is unichain. Namely, for unichain problems, η can be represented by a scaler instead of by a vector as for multichain problems. Therefore, in the policy evaluation step, it does not need to solve (8) and (10) which are used to characterize the vector η . Moreover, in the policy improvement step, the optimization in (11) is not necessary because it is satisfied automatically.
3 A Rollout Algorithm for Multichain Markov Decision Processes In this section, a rollout algorithm for multichain MDP is developed. There are two key points compared with traditional policy iteration. Firstly, the policy evaluation step for a policy is performed by simulation, which avoids the difficulty of solving linear equations. Secondly, the algorithm is performed online and only optimize actions for the states encountered on the sample path. In contrast, traditional policy iteration updates actions for all the states at each iteration. Since the rollout algorithm runs simulations based on an existing policy μ for improvement, such a policy μ is called the “base policy.” In the next, all the state transition matrix P, average cost η , bias g and cost vector f together with the Qfactor defined below are related with policy μ . For symbolic simplifications, μ will be omitted when there are no ambiguities.
3.1 A Rollout Algorithm for Multichain Models The policy improvement equations (11, 12) in the policy iteration process can be conducted state-wisely for multichain problems [15]. Namely, if the action for a state cannot be updated by equation (11), one can turn to (12) for the state to achieve a possible improvement. This procedure is described as below. State-wise Policy Improvement For a state i, select the action which can make the system run under a recurrent class with the lowest average cost. Let a row vector pi,a be the state distribution when taking an action a at a state i. Accordingly, the solution set of the optimization equation (11) for state i is: . ¯ A (i) = a a ∈ arg min {pi,a η } . (13) a∈A (i)
Let aμ (i) be the action taken by the policy μ at the state i. If aμ (i) ∈ / A¯(i), then define an action a(i) ˆ such that a(i) ˆ ∈ A¯(i). Otherwise, perform optimization by equation (12) by choosing an action / 0 a(i) ˆ ∈ arg min f (i, a) + pi,a g (14) a∈A¯(i)
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By the definition of gμ in (6), the Q-factor is defined as: * Q(i, a) = lim
K→∞
f (i, a) + pi,a
K−1
∑ (P
k
+ ∗
−P )f
(15)
k=0
For a state, the equation (14) is to choose a smallest Q-factor. For unichain MDPs, the elements of the average cost vector η are the same. Therefore, A¯(i) = A (i) and the policy improvement step is only performed based on (14). For multichain problems, optimizing actions for states are performed within the set of A¯μ (i) obtained by (13). For every pair of actions a1 , a2 within the set A¯μ (i), we have (pi,a1 − pi,a2 )η = 0. Therefore, Q(i, a1 ) − Q(i, a2) * f (i, a1 ) + pi,a1
= lim
K→∞
*
= lim
K→∞
= lim
K→∞
K−1
k=0
k=0
(16)
+
∑ (Pk − P∗) f − f (i, a2 ) − pi,a2 ∑ (Pk − P∗) f
f (i, a1 ) − f (i, a2 ) + (pi,a1 − pi,a2 )
K−1
∑P
k
f + (pi,a1 − pi,a2 )
k=0
*
= lim
K→∞
K−1
f (i, a1 ) − f (i, a2 ) + (pi,a1 − pi,a2 ) f (i, a1 ) − f (i, a2 ) + (pi,a1 − pi,a2 )
∑P
+ ∗
f
k=0
+
K−1
∑ Pk f + K(pi,a1 − pi,a2 )η
k=0
*
K−1
K−1
+
∑P
k
f
k=0
= QK (i, a1 ) − QK (i, a2 ) + Γ (K, a1 , a2 )
(17)
In (17), QK (i, a) = f (i, a) + pi,a
K−1
∑ Pk f
(18)
k=0
The third term Γ (K, a1 , a2 ) in (17) converges to 0 when K → ∞. This is because,
Γ (K, a1 , a2 ) = (pi,a1 − pi,a2 )
∞
∑
k=K
Pk f = (pi,a1 − pi,a2 )
∞
∑ (Pk − P∗) f
k=K
∞
In the above, ∑ (Pk − P∗ ) f converges to 0 when K → ∞ (p.339 in [15]). k=K
Based on the above analysis, we can get an online rollout algorithm for multichain MDPs. Let μ be the base policy, for a state i encountered on a sample path at any time, the rollout algorithm chooses actions according to the following algorithm: Algorithm 1 The Rollout Algorithm for Multichain MDPs 1) Select recurrent classes with small average costs. Estimate the set A¯(i) defined in (13) by Aˆ(i) which is obtained based on an estimation of pi,a η as:
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. 1 1 ¯ ˆ QK,W (i, a)} A (i) = a a ∈ arg min { a∈A (i) W K where Q¯ K,W (i, a) is computed by simulation: 6 W K w w Q¯ K,W (i, a) = ∑ ∑ f μ (ik (ξ ), μ (ik (ξ ))) i0 = i, a(i0 ) = a w=1
(19)
(20)
k=1
If aμ (i) ∈ / Aˆμ (i), choose an action a(i) ˆ for the state i by: a(i) ˆ ∈ Aˆμ (i). Otherwise, execute step 2). The symbol ξ w stands for a realization of uncertainties on a simulation run and ik (ξ w ) is the state at time k under the realization ξ w . 2) Obtain the estimation Q¯ K (i, a) of QK (i, a) as follows 1 Q¯ K (i, a) ≡ f (i, a) + Q¯ K,W (i, a) W and determine an action a(i) ˆ for the state i such that 0 / a(i) ˆ ∈ arg min Q¯ K (i, a) a∈Aˆ(i)
(21)
In Algorithm 1, W is the number of simulation replications and K is the length of look ahead at each simulation.
3.2 Properties of the Multichain Rollout Algorithm The equation (20) calculates the cost of W simulation runs, the cost of each simulation is the total cost within time horizon K. Each simulation starts from the state i and employs the action a. Moreover, for the states encountered in the subsequent 1 to K time horizon, the action is chosen according to the base policy μ . As a consequence of the law of large numbers, we have the following theorem. Theorem 2. For the simulation based multichain rollout algorithm (Algorithm 1), the following results hold: 1) When W → ∞, Q¯ K (i, a) → QK (i, a). 2) When K → ∞,W → ∞, W1 K1 Q¯ K,W (i, a) → pi,a η . Therefore, Aˆ(s) → A¯(s). 3) For the comparisons of actions a1 , a2 ∈ A¯(s), Q¯ K (i, a1 ) − Q¯ K (i, a2 ) → Q(i, a1 ) − Q(i, a2 ), when K → ∞,W → ∞. It can be seen that the Algorithm 1 is essentially a simulation based one step of policy iteration. Further comments on the multichain rollout algorithm are summarized as below. 1. Compared withe policy iteration, which aims at optimal policies, the algorithm reflects the idea of “Goal Soften”: only evaluate and improve the interested
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states; only conduct a single step of policy iteration. The length of look-ahead K is truncated and is not infinite as the Definition 15. The sample mean can be estimated by a number of replications. To reduce the comparison variances, “common random numbers” (using the same ξ w ) may be applied. 2. The algorithm has the advantage to deal with problems for which the underlining model is not completely known. By Algorithm 1, it can be seen that the algorithm only calculates the cost and does not need to know the explicit information of the problem, i.e., the state transition probabilities. This is the common advantage of learning algorithms. 3. Although Algorithm 1 is developed for multichain problems, the computation effort is similar to unichain cases. This is because the calculation in step 1 can be directly used in step 2 and will not need additional simulation effort. For unichain average cost problems, the average cost is independent of initial states. Therefore, Algorithm 1 will be simplified and only performs step 2.
4 Analysis of Rollout Algorithms Rollout algorithms estimate means by conducting simulation W times and looks ahead K time horizon for each simulation. Therefore, the length of look-ahead and the replication numbers are two aspects of the computational load. They are similar as classical “Exploration and Exploitation”[14] in optimization. In this section, we will analyze these two parameters and investigate how to improve the efficiency of simulations.
4.1 The Effect of Look Ahead The issue of choosing the length of look-ahead exists in many learning algorithms. It is presented that the length of look-ahead should be comparable with the means of the arrival time when compares the difference between two states [7]. Another way is to use the “regenerative point” ([6] and [11]) setting a reference state which can be arrived from any state. The bias is estimated based on the sample paths from the interested state to the reference state. However, the first arrival time between two states may be quite long for large problems. Moreover, there may be no information helping us to find such a reference state. Investigating the look-ahead length is important to identify the estimation error caused by the truncation. In addition, it helps us to know the goodness of the near optimal policies and provides guidance on the computational load allocation. Related results include applying stationary policies to total cost MDPs (see, e.g., [2] and [12]) and applying policies obtained under total cost to average cost criteria ([8] and [12]).
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The step 1 in Algorithm 1 happens only for transient states while the step 2 is executed both for unichain MDPs and multichain MDPs. Therefore, we analyze how the look-ahead length K affects the comparison of Q-factors in step 2. Here, we give some preliminary results obtained under unichain cases. When the Markov chain under a stationary policy is ergodic, there is a single stationary distribution π and lim Pk = P∗ . Starting from state i, after k time units, k→∞
the state distributions are different from the stationary distribution as:
Δi (k) =
1 2
∑ |pki, j − π (i)|,
(22)
j∈S
where pki, j is the probability that state i transits to j after k steps. The “Mixing Time” [5] used to measure the speed of a Markov chain’s convergence and is defined as: Tmix = max min{k : Δi (k) < ε }, i
i∈S
(23)
In the above, ε is a small positive number. For the truncated error of Q-factors, we have the following results related with mixing time. Theorem 3. Let K be the look ahead length, the estimation error of a Q-factor can be quantified as follows: |Q(s, a) − QK (s, a)| ≤ ε g,
if Tmix ≤ K
(24)
Here g is the sup-norm. Proof. By the definition g in (6), the following holds: g − gK =
∞
∞
∑ (Pk − P∗) f = PK ∑ (Pk − P∗) f
k=K
k=0
= P g = (PK − P∗ )g K
In the above, the last equality holds because P∗ g = 0. For the Rollout algorithm: |Q(i, a) − QK (i, a)| = ∑ p( j|i, a) [g( j) − gK ( j)] j∈S 6 K = ∑ p( j|i, a) ∑ p j,i − π (i) g(i) j∈S i∈S ≤ ε g Fast mixing means that the increase of mixing time is polynomial with the logarithm of the number of states [5]. If we know that the state transition matrix of a problem has the property of fast mixing, we may choose a proper K for a small problem. When the problem scale increases, we only need to increase K polynomially.
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The topology of the transitions between states generally has “small world” phenomena which induces fast mixing [18]. It is known that an aperiodic Markov chain converge to stationary distribution exponentially [15] and most of the differences are achieved in the first few transitions. All of these suggests that it may not need to choose a large K.
4.2 Conducting Simulations Efficiently The simulation of stochastic systems has been well investigated, which makes it possible to use existing techniques to improve the efficiency of simulations for rollout algorithms. Here, we present some techniques that have been introduced in [16] to improve the efficiency of simulations for rollout algorithms. The rollout algorithm performs Monte Carlo simulations to estimate Q-factors. The estimation accuracy (confidence interval) improves slowly and no faster than √ O(1/ W ) ([10] and [13]). However, from equation (16) it can be seen that the key of the algorithm is to determine the correct rank of Q-factors and select the best action. From another point of view, rollout algorithms use a crude model to estimate the real ranks of Q-factors. This is because the base policy is obtained by heuristics and is not optimal. However, an “accurate model” employs an optimal policy as the base policy. Therefore, both the rank comparisons and the crude model are consistent with the idea of “ordinal optimization” [13]. Ordinal optimization tells us that to determine “rank” is much easier than to determine the “value”. Compared with the slow convergence of values, ranks converge exponentially [13]. This is intuitively reasonable since it is easier to tell which is larger than to tell how much one is larger than another. Through incorporating ordinal optimization, our idea is that there is no need to estimate Q-factors precisely by too many simulation samples. Besides ordinal optimization, “Optimal Computing Budget Allocation” [10] (OCBA) can be used to improve the efficiency of simulation when the rollout algorithm is implemented. Since each Q-factor is a random variable, the most straightforward way is to obtain a same number of samples to estimate their means. This may not a good choice because some unpromising Q-factors need not to estimate their value as precise as those good ones. By OCBA, a few simulations (samples) for each candidate actions are conducted first to get rough estimations of means and variance. Then, the number of simulations to be conducted in the next will be determined by these estimated quantities and the total number of simulation runs to be allocated. Such procedure will be performed until all the simulation budget is used. In this way, those actions achieve small Q-factors are sampled more than others. The estimation of Q-factors has intrinsic parallelism. Except the action to be evaluated, other factors are the same, e.g., current state, simulation model and the uncertainties generated following common random numbers. Thus, the algorithm can be easily implemented in “Single Program Multiple Data” manner to speed up the optimization.
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5 Conclusions In this paper, a multichain rollout algorithm is developed to address MDPs with large state space. This is an online learning based algorithm for multichain MDPs, for which few of learning algorithm exists in the literature. Through the structural property reflected by the mixing time, preliminary results are obtained under unichain conditions for the estimation errors caused by the truncated look-ahead length. Further research lies on testing the algorithm by numerical examples and applying it to real problems. Acknowledgements. This work was supported by NSFC Grant (60574067, 60736027, 60721003, 60704008) and in part by the Programme of Introducing Talents of Discipline to Universities (National 111 International Collaboration Project) B06002.
References 1. Auer, P., Ortner, R.: Logarithmic online regret bounds for undiscounted reinforcement learning. In: Advances in Neural Information Processing Systems, vol. 19. MIT Press, Cambridge (2007) 2. Bertsekas, D.P.: Dynamic Programming and Optimal Control, 3rd edn., vol. I. Athema Scientific, Belmont (2005) 3. Bertsekas, D.P.: Dynamic Programming and Optimal Control, 3rd edn., vol. II. Athema Scientific, Belmont (2007) 4. Bertsekas, D.P., Casta˜non, D.A.: Rollout algorithms for stochastic scheduling problems. Journal of Heuristics 5, 89–108 (1999) 5. Boyd, S., Diaconis, P., Xiao, L.: Fastest mixing Markov chain on a graph. SIAM Review 46(4), 667–689 (2004) 6. Cao, X.R.: Stochastic learning and optimization– A Sensitivity based approach. Springer, New York (2007) 7. Cao, X.R., Wan, Y.: Algorithms for sensitivity analysis of Markov systems through potentials and perturbation realization. IEEE Transactions on Control Systems Technology 6(4), 482–494 (1998) 8. Chang, H.S., Marcus, S.I.: Approximate receding horizon approach for Markov decision processes: average reward case. Journal of Mathematical Analysis and Applications 286, 636–651 (2003) 9. Chang, H.S., Givan, R., Chong, E.K.P.: Parallel rollout for online solution of partially observable Markov decision processes. Discrete Event Dynamic Systems: Theory and Applications 14, 309–341 (2004) 10. Chen, C.H., Y¨ucesan, J., Lin, E., Chick, S.E.: Simulation budget allocation for further enhancing the efficiency of ordinal optimization. Discrete Event Dynamic Systems: Theory and Applications 10, 251–270 (2000) 11. Fang, H.T., Cao, X.R.: Potential-based on-line policy iteration algorithms for Markov decision processes. IEEE Transactions on Automatic Control 49, 493–505 (2004) 12. Hernandez-Lerma, O., Lasserre, J.B.: Error bounds for rolling horizon policies in discrete-time Markovcontrol processes. IEEE Transactions on Automatic Control 35, 1118–1124 (1990)
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13. Ho, Y.C., Zhao, Q.C., Jia, Q.S.: Ordinal Optimization: Soft Optimization for Hard Problems. Springer, New York (2007) 14. Kearns, M., Singh, S.: Near-optimal reinforcement learning in polynomial time. Machine Learning 49(22), 209–232 (2002) 15. Puterman, M.L.: Markov Decision Process: Discrete Stochastic Dynamic Programming. John Wiley & Sons, New York (1994) 16. Sun, T., Zhao, Q.C., Luh, P.B., Tomastik, R.N.: Optimization of joint replacement policies for multi-part systems by a rollout framework. IEEE Transactions on Automation Science and Engineering 5(4), 609–619 (2008) 17. Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge (1998) 18. Tahbaz-Salehi, A., Jadbabaie, A.: Small World phenomenon, rapidly mixing Markov chains, and average consensus algorithms. In: Proceedings of IEEE Conference on Decision and Control, New Orleans, LA. pp. 276–281 (2007) 19. Tsitsiklis, J.N.: NP-Hardness of checking the unichain condition in average cost MDPs. Operations research letters 35(3), 319–323 (2007)
Analysis of Degenerate Chemical Reaction Networks Markus Uhr, Hans-Michael Kaltenbach, Carsten Conradi and J¨org Stelling
Abstract. Positivity of states and parameters in dynamic models for chemical reaction networks are exploited by Chemical Reaction Network Theory (CRNT) to predict the potential for multistationarity of ‘regular’ networks without knowledge of parameter values. Especially for biochemical systems, however, CRNT’s large application potential cannot be realized because most realistic networks are degenerate in the sense of CRNT. Here, we show how degenerate networks can be regularized such that the theorems and algorithms of CRNT apply. We employ the method in a case study for a bacterial reaction network of moderate size.
1 Introduction Chemical – and biochemical – reaction networks are sets of chemical compounds connected through reactions. The formal analysis of their dynamic properties, for instance, in the area of systems biology remains difficult because often network topologies and kinetic parameters are uncertain or unknown. Hence, there is a general interest in developing formal analysis methods that consider the structure of the induced dynamic system alone [2]. In modeling signal transduction or cell cycle, for example, the connection between the network structure and the existence of multiple positive steady states (multistationarity) is critical. Chemical reaction network dynamics can be represented by a system of ordinary differential equations (ODEs). If, as in this paper, all kinetics are of the mass action form, then the right hand sides of the ODEs are polynomials in the Markus Uhr, Hans-Michael Kaltenbach and J¨org Stelling Dept. Biosystems Science & Engineering, ETH Zurich, 8092 Z¨urich, Switzerland, e-mail:
[email protected],
[email protected] Carsten Conradi Max-Planck-Institute Magdeburg, Magdeburg, Germany, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 163–171. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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concentrations c and rate constants k (bold-face symbols denote vectors and matrices). Importantly, meaningful concentrations and rate constants need to be positive. These positivity constraints and the special structure induced by mass-action kinetics have been exploited to develop the Chemical Reaction Network Theory (CRNT) [3, 4]. It connects network topology and the existence of multiple positive steady states independent of parameter values. CRNT employs a nonnegative integer δ obtained solely from the stoichiometry associated with the network. If, for example, δ = 1 for a network and if that network is ‘regular’ in the sense of CRNT, then the existence of multiple steady state solutions to the polynomial steady state equations requires feasibility of at least one of a (potentially large) number of linear inequality systems. However, realistic networks often have δ > 1, leading to polynomial inequalities. To circumvent this, we previously suggested a decomposition of the overall network into subnetworks with δ = 1. If the resulting subnetwork is regular, then one can use the deficiency one algorithm to establish multistationarity for this subnetwork. To confirm multistationarity for the overall network we present sufficient conditions that allow the extension of multistationarity to the overall network (see [1] and especially [5] for the extension of solutions from the subnetwork to the overall network). Because many realistic networks in biology have only degenerate subnetworks, we here develop approaches for regularization of degenerate reaction networks. Thus the present contribution extends the ideas of [1] and [5] in the following sense: one can now use the deficiency one algorithm to establish multistationarity for the regularized subnetworks, while it is still possible to use the results of [1] and [5] to extend multistationarity from the (regularized) subnetwork to the overall network.
2 Chemical Reaction Network Theory A reaction network in CRNT consists of a set S of chemical species and a set R of reactions, each mapping a multiset of species to another multiset of species. Each such multiset is called a complex, together forming the set C . Denote by m = |S |, n = |C |, and r = |R| the number of species, complexes, and reactions of a network, respectively. Fixing any order of the species, we will further identify a complex by its stoichiometry vector y = [y1 , . . . , ym ]T ∈ m , where yi denotes the number of species i in the complex. C←A→B B + C → 2A
(1)
Consider the example reaction network (1) with species S = {A, B,C}, complexes C = {{A}, {B}, {C}, {B,C}, {A, A}} and reactions R = {A → B, A → C, B + C → 2A}. The complex B + C is represented, e. g., by the vector yB+C = [0, 1, 1]T. The state variables of such a reaction network are the species concentrations ci (t) ∈ ≥0 which give the amount of species i per unit volume at time t. We will write c ≡ c(t) ∈ m ≥0 for the vector of species concentrations. For each reaction j
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with y → y , there is a column n j = (y − y) in the stoichiometric matrix N and a corresponding reaction rate v j . Assuming mass action kinetics, the rate is given by yi v j = k j · cy k j · ∏ m i=1 ci , where the ‘’ sign means ‘equal by definition’. The parameter k j ∈ >0 is the rate constant for that reaction. By collecting all rates v j in a vector v(k, c), we obtain the ODE system describing the concentration dynamics: c˙ = f (k, c) = N · v(k, c).
(2)
Often the matrix N does not have full row rank; let S im(N) and s rank(N) with s < m and W ∈ m×m−s a basis of ker(NT ). Then, one has the conservation relations WT c(t) = const. and thus c(t) ∈ c0 + S for solutions c(t) of (2) with c(0) = c0 , motivating the following original definition of multistationarity: Definition 1 (Multistationarity, cf. [3]). Consider a reaction network with associated ODE system (2). We say that the network admits multistationarity if there exist at least two distinct, positive vectors c∗ and c∗ and a positive vector k with f (k, c∗ ) = 0
(3a)
f (k, c∗ ) = 0
(3b)
WT (c∗ − c∗ ) = 0.
(3c)
An important observation of CRNT is that the dynamics can be written in terms of complexes rather than species. Species concentrations are first mapped onto “complex concentrations” by ψ (c) = ∑y cy · ey with the sum running over all complexes and ey the corresponding canonical basis vector of n . The dynamics is then given by A = Ia IK where IK and Ia can be interpreted as a complex flux and a complex stoichiometry, respectively. Mapping the derivatives of complex concentrations back to species concentration changes by the matrix of stoichiometry vectors Y = [y1 | · · · |yn ] completes the decomposition of f (c) = YIa IK ψ (c) (Fig. 1 and [6]). Importantly, although f is non-linear, the only non-linearity occurring in the decomposition is the map ψ . The matrix IK gives the relation between the reactions and their associated substrate complexes. This can be modeled as a bipartite graph where one set of nodes are the reactions and the other set are the complexes. For every reaction of the network, there is an edge in this graph connecting the reaction with its substrate complex. Every edge is weighted by a rate constant ki . Then, the matrix IK is the adjacency matrix of this weighted bipartite graph. The matrix Ia is the incidence matrix of the network graph defined as follows. Definition 2 (Network Graph). The network graph of a chemical reaction network (S , C , R) is the directed graph G = (C , R) with the complexes as vertices and the reactions as edges. The following three notions are central in the CRNT literature and are based on the network graph [3]. Most importantly, the deficiency of a network captures
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Fig. 1 Commutative diagram of the different decompositions of c˙ = f (c).
n
o aBB BB Ia BB BB
A
n
| O IK || | || | } | r ψ Y aBB | BB v N ||| BB | BB | | }| f m o m
the degrees of freedom in mapping complex concentration changes onto species concentration changes such that the latter are zero. Definition 3 (Linkage Classes). The (strongly) connected components of the network graph are called (strong) linkage classes. A strong linkage class and its complexes are called terminal if there is no reaction mapping one of its complexes out of this class. Definition 4 (Deficiency Space; Deficiency). The deficiency δ of a reaction network is the dimension of its deficiency space:
δ dim [ker(Y) ∩ im(Ia )] . Lemma 1. [7] The deficiency δ satisfies the relation
δ = n − l − s, where n is the number of complexes, l the number of linkage classes and s = rank(N). Note that n − l = rank(Ia ).
3 Degenerate Chemical Reaction Networks with Deficiency One An algorithm for determining whether a network of deficiency δ = 1 admits multistationarity and for computing a pair of steady states is given in [4], the so-called deficiency one algorithm. It is applicable to any regular reaction network; that is, a network satisfying the following conditions (cf. [4] or [1]): (C.1) (C.2) (C.3) (C.4) (C.5)
The deficiency of the network is δ = 1. The deficiency of every linkage class is δ = 0. ker (N) contains a positive vector. Every strong linkage class contains exactly one terminal strong linkage class. Terminal strong linkage classes do not contain any cycles.
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As many reaction networks have δ > 1, an application of the algorithm to the complete network is not possible. Therefore, we suggested an elementary flux mode (EFM) decomposition of a network [1]. Definition 5 (EFM; Stoichiometric Generator ([6] and [1])). The generators of the pointed polyhedral cone ker(YIa ) ∩ r≥0 are called elementary flux modes (EFMs). In particular, an EFM is a vector v ∈ r≥0 that satisfies YIa v = 0 Given two EFMs v, v , we have that / 0 {i | vi = 0} ⊆ i | v i = 0 ⇒ v = 0 or
(4a)
v = α v
(4b)
A stoichiometric generator is an EFM g that additionally satisfies Ia g = 0.
(4c)
Subnetworks defined by stoichiometric generators are guaranteed to satisfy conditions (C.1) – (C.3) and (C.5) but not necessarily condition (C.4) [1]. Biochemical reaction networks describing metabolic reactions, for example, have a prominent source of degeneracy even for a sufficiently detailed degree of modeling: species uptake. To describe open systems, CRNT introduces the zero-complex 0. / For example, 0/ → A + B would be an uptake reaction for the complex A + B with a certain rate. However, if more than one complex is taken up, this method leads to a linkage class with two or more terminal strong linkage classes, thus violating condition (C.4) This motivates our following definition of a degenerate reaction network: Definition 6 (Degenerate; 0-degenerate). / A reaction network is called degenerate iff it violates the condition (C.4). That is, the network has at least one linkage class with more than one terminal strong linkage class. It is called 0-degenerate / if moreover it is generated by a single stoichiometric generator (thus δ = 1) and the only violation of condition (C.4) is due to several uptake reactions from the zero complex, resulting in several terminal strong linkage classes each linked to the zero complex by a single reaction. The zero complex renders the reaction equations for the species uptakes particularly simple because the entries vi of these reactions are independent of the species concentrations. This allows to remove the 0-degeneracy / from the network: individual uptake reactions with rate constants ki are replaced by a single reaction for the uptake of a combined complex as shown in (5). The idea is formalized in Definition 7.
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degenerate
regularized
C
C
k3
k3
k1 q8 A qq / 0/ Mq MMM & k2 2B
/ 0/
k∗
(5)
/ g1 A + g2 B
Definition 7 (Regularized Network). Consider a 0-degenerate / reaction network with n complexes, r reactions, and u > 1 uptake reactions such that the zero complex gives rise to a linkage class with more than one terminal strong linkage class. Let y1 , . . . , yu be the complexes taken up with rate constants k1 , . . . , ku , respectively. The regularized network is then given by deleting all u uptake reactions and introducing a new uptake reaction with rate constant k∗ and uptake complex y¯ g1 y1 + · · · + guyu , where gi are elements of the unique (up to scalar multiplication) generator g of ker(YIa ) ∩ r≥0 . Thus, the regularized network has the same concentration vector c but different numbers of reactions and rate constants ki . To analyze the connection between the dynamical systems derived from the two networks, split k, g, v and N into components belonging to uptake reactions (indicated by ˜ ) and components not belonging to uptake reactions (indicated by ˆ ): regularized network subnetwork ∗ ˜ k k = kˆ kr = kkˆ ˜ g = ggˆ gr = 1gˆ ∗ k˜ vr (kr , c) = vˆ (kk,c) v(k, c) = vˆ (k,c) ˆ ˆ r ˆ ˆ N = y1 . . . yu |N N = g1 y1 + . . . + gu yu |N The following lemma establishes the equivalence of the dynamics of the two systems. Lemma 2 (Regularization Lemma). With f r (kr , c) Nr vr (kr , c) the regularized version of f (k, c) N v(k, c), the following two statements hold: (a) If there exists (k, c) ∈
r×
m
with f (k, c) = 0, then k˜ = α g˜
(6)
(b) For k˜ = α g˜ as above, any c > 0, any α > 0, and any kˆ ∈ r−u we have that (7) f [ α g˜ T , kˆ T ]T , c ≡ f r [ α , kˆ T ]T , c . Proof. (a) Consider (k, c) ∈ r × m with f (k, c) = 0. Then v(k, c) = α g, where g is the unique (up to scalar multiplication, cf. [1]) generator of ker(YIa ) ∩ r≥0 . ˜ = α g˜ . (b) With the decomposition into ˆ T ]T = α [ g˜ T , gˆ T ]T and thus k Then, [ k˜ T , vˆ (k,c) uptake and non-uptake parts,
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ˆ α g˜ = f [ k˜ T , kˆ T ]T , c = y1 . . . yu | N vˆ
ˆ α ≡ f r [ α , kˆ T ]T , c , (8) g1 y1 + . . . + guyu | N vˆ which is identical to f r (kr , c), with k∗ = α .
Corollary 1 (Equivalence of conservation relation). Following the notation in Lemma 2, let c f (t) and c f r (t) be solutions for the systems f and f r , respectively, with the same initial condition c0 . If k˜ = α g˜ , then c f (t) ≡ c f r (t) for all t ≥ 0. Further, with Wr a basis for the left kernel of Nr , (Wr )T c f r (t) ≡ (Wr )T c0 implies (Wr )T c f (t) ≡ (Wr )T c0 . This result motivates the following new definition of multistationarity for 0/ degenerate reaction networks which is a stronger version of Definition 1: Definition 8 (Multistationarity of 0-degenerate / networks). A 0-degenerate / network is called multistationary iff its regularized version admits multistationarity.
4 Application Example We use the upper part of glycolysis (including the pentose phosphate pathway) as an application example for EFM-based analysis of multistationarity. The corresponding network shown in Fig. 2 is an important part in bacterial metabolism. It is the central pathway for growing on monosaccharides (e. g. glucose) and for producing energy for the cell. It is also important for the synthesis of precursors for amino acids and nucleotides. Notably, even for such well-characterized pathways the dynamic features are not fully understood. Our network model consists of 4 uptake reactions, 7 outflows, and 13 enzymecatalyzed reactions. This results in a mass-action model with 53 species, 79 complexes and 78 reactions. The network deficiency is 8, which hampers CRNT-based analysis without prior network decomposition. In analyzing the model, we found 123 EFMs and 88 of these EFMs were stoichiometric generators. The subnetworks that correspond to these generators have between 19 and 34 reactions (see Fig. 2 for an example) and they all are degenerate. However, the only degenerate linkage classes are those containing the uptake reactions and thus the network is 0/ degenerate. We regularized the subnetworks to enable analysis with the deficiency one algorithm. Interestingly, none of the subnetworks corresponding to an EFM can admit multiple steady states. However, as an EFM-based analysis only provides sufficient conditions for multistationarity of the entire network, multistationarity of the overall network is still possible [1]. Moreover, multistationarity in the overall network seems very likely, given its complex structure, which contains several feedback loops (e. g. via the energy carrier ATP).
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NADPH
G6P
PGlac
zwf
pgl pgi
F6P
ATP
PGluc
tktAB
pfkA
fbp
gnd
talB
F16P
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NADPH CO2
Rl5P
fba tktAB
DHAP
tpiA
C2
G3P ATP
rpiA
E4P
R5P
C3 tktAB talB
S7P Fig. 2 Model for the upper part of glycolysis and the pentose phosphate pathway. Black reactions form an EFM subnetwork that can be analyzed with the deficiency one algorithm. Gray reactions are not active in the EFM.
5 Conclusions and Perspectives The application of some of the main methods and theorems of CRNT (in particular the deficiency one algorithm) is currently restricted to regular reaction networks, which limits their applicability to real-world problems. Here, we developed and applied a simple method for model regularization to circumvent the limitations for (at least certain classes of) biochemical reaction networks. Future generalizations of more elements of CRNT, for instance, of the deficiency one theorem are interesting and preliminary results indicate that such extensions of the theory are feasible. Finally, our case-study of a medium-sized network model indicates that this is a promising approach, for instance, to understand the remarkable robustness of biological systems [8]. Acknowledgements. This work was supported in part by the European Union FP6 project ‘BaSysBio’ and the European Union FP7 project ‘UniCellSys’.
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References 1. Conradi, C., Flockerzi, D., Raisch, J., Stelling, J.: Subnetwork analysis reveals dynamic features of complex (bio)chemical networks. Proc. Natl. Acad. Sci. 104(49), 19175–19180 (2007) 2. Doyle, F.J., Stelling, J.: Systems interface biology. J. R. Soc. Interface 10(3), 603–616 (2006) 3. Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors – I. The deficiency zero and deficiency one theorems. Chemical Engineering Science 42(10), 2229–2268 (1987) 4. Feinberg, M.: Chemical reaction network structure and the stability of complex isothermal reactors – II. Multiple steady states for networks of deficiency one. Chemical Engineering Science 43(1), 1–25 (1988) 5. Flockerzi, D., Conradi, C.: Subnetwork analysis for multistationarity in mass action kinetics. Journal of Physics: Conference Series 138, 36 pages (012006) (2008) 6. Gatermann, K., Wolfrum, M.: Bernstein’s second theorem and Viro’s method for sparse polynomial systems in chemistry. Advances in Applied Mathematics 34(2), 252–294 (2005) 7. Gunawardena, J.: Chemical reaction network theory for in-silico biologists. Bauer Center for Genomics Research (2003) 8. Stelling, J., Sauer, U., Szallasi, Z., Doyle, F.J., Doyle, J.: Robustness of cellular functions. Cell 118(6), 675–685 (2004)
k-Switching Reachability Sets of Continuous-Time Positive Switched Systems Maria Elena Valcher
Abstract. In the paper, the k-switching reachability set Rk of a continuous-time positive switched system is introduced, and a necessary and sufficient condition for the chain of this sets {Rk , k ∈ } to stop increasing after some finite index k is given. It is shown that, for special classes of (multiple-input) positive switched systems, reachability always ensures that Rn = n+ , n being the system dimension.
1 Introduction “Switched linear systems” are systems whose describing equations change, according to some switching law, within a (possibly infinite) family of (linear) subsystems. On the other hand, the positivity requirement is often introduced in the system model whenever the physical nature of the describing variables constrains them to take only positive (or at least nonnegative) values. As a result, positive linear systems [1] naturally arise in fields such as bioengineering (compartmental models), economic modeling, behavioral science, and stochastic processes (Markov chains or hidden Markov models). In this perspective, switched positive systems are mathematical models which keep into account two different aspects: the fact that the system dynamics can be suitably described by means of a family of subsystems, each of them formalizing the system laws under specific operating conditions, among which the system commutes, and the nonnegativity constraint the physical variables are subject to. This is the case when trying to describe certain physiological and pharmacokinetic processes, like the insulin-sugar metabolism. Of course, the need for this class of systems in specific research contexts has stimulated an interest in theoretical issues related to them, in particular, reachability/controllability properties [4, 5, 7], and stability issues [2, 3]. Maria Elena Valcher Dip. Ingegneria dell’Informazione, Universit`a di Padova, Italy, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 173–181. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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In this paper, we introduce the concept of k-switching reachability set Rk of a positive switched system. This definition as well as some preliminary results on the chain of sets {Rk , k ∈ } for the class of single-input continuous-time positive switched systems have been derived in [6]. In this paper it is shown that, for special classes of (multiple-input) positive switched systems, reachability always ensures that Rn = n+ , n being the system dimension. Notation. For every k ∈ , we set k := {1, 2, . . . , k}. + is the semiring of nonnegative real numbers, n+ the set of n-dimensional vectors with entries in + , and n×p + the set of n × p matrices with entries in + . A matrix A with entries in + is a nonnegative matrix; if A is nonnegative and A = 0, A is a positive matrix, while if all its entries are positive it is a strictly positive matrix. A Metzler matrix, on the other hand, is a real square matrix, whose offdiagonal entries are nonnegative. The nonzero pattern of a vector v is the set of indices corresponding to its nonzero entries, namely ZP(v) := {i : [v]i = 0}, where [v]i is the ith entry of v. We let ei denote the ith vector of the canonical basis in n (where n is always clear from the context), whose entries are all zero except for the ith which is unitary. We say that a vector v ∈ n+ is an ith monomial vector if ZP(v) = ZP(ei ) = {i}. For any set S ⊆ n, we set eS := ∑i∈S ei .
2 Reachability Property and k-Switching Reachability Sets A (continuous-time) positive switched system is described by the following equation x˙ (t) = Aσ (t) x(t) + Bσ (t)u(t),
t∈
+,
(1)
where x(t) and u(t) denote the n-dimensional state variable and the m-dimensional input, respectively, at the time t, and σ is a switching sequence, taking values in a finite set P = {1, 2, . . . , p} = p. We assume that the switching sequence is piece-wise constant, and hence in every time interval [0,t[ there is a finite number of discontinuities, which corresponds to a finite number (say k, including the initial time) of switching instants 0 = t0 < t1 < · · · < tk−1 < t. For each i ∈ P, the pair (Ai , Bi ) represents a continuous-time positive system, which means that Ai is an n × n Metzler matrix and Bi is an n × m nonnegative matrix. As a first step, we recall the definition of monomial reachability and of reachability for positive switched systems. Definition 1. [5] A state x f ∈ n+ is said to be reachable if there exist some time t > 0, a switching sequence σ : [0,t[→ P and an input u : [0,t[→ m + , that lead the state trajectory from x(0) = 0 to x(t) = x f . A positive switched system is monomially reachable if every monomial vector (equivalently, every canonical vector ei , i ∈ n) is reachable, and reachable if every state x f ∈ n+ is reachable.
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Monomial reachability is a necessary (but, unfortunately, not sufficient) condition for reachability [4] and it admits a rather easy characterization. Proposition 1. [5] Given a positive switched system (1), commuting among p subsystems (Ai , Bi ), i ∈ P, the following conditions are equivalent: i) the system is monomially reachable; ii) ∀ i ∈ n there exists an index j = j(i) ∈ P such that A j ei = αi ei , for some αi ≥ 0, and one column of B j is an ith monomial vector. In the special case of single-input positive switched systems (1), monomial reachability is equivalent to the fact that there exists a relabeling of the p subsystems (Ai , Bi ), i ∈ P, such that the first n subsystems satisfy Ai ei = αi ei
and
B i = β i ei ,
(2)
for suitable αi ≥ 0 and βi > 0. We now introduce the definition of k-switching reachability set. Definition 2. [6] Given a positive switched system (1) and a positive integer k, we define the k-switching reachability set, and denote it by Rk , as the set of states that can be reached in finite time by the system, by making use (of a nonnegative input signal u(·) and) of a switching sequence σ that commutes no more than k − 1 times, meaning that the switching instants of the switching sequence are no more than k (i.e., 0 = t0 < t1 < · · · < t−1 with ≤ k). It is easily seen that Rk is a cone, since if x f belongs to Rk then α · x f surely does, for every α ≥ 0. However, in general, it is neither convex nor polyhedral. Of course, we are interested in investigating how the cone Rk varies, as k varies over the positive integers. To this end, we recall that the state at the time t, starting from the zero initial condition, under the action of the input u(τ ), τ ∈ [0,t[, and of the switching sequence σ : [0,t[→ P, with switching instants 0 = t0 < t1 < · · · < tk−1 < t and switching values i0 , i1 , . . . , ik−1 ∈ P (i.e., i = σ (t) for t ∈ [t ,t+1 [), can be expressed as follows: x(t) = eAik−1 (t−tk−1 ) ...eAi1 (t2 −t1 ) + eAik−1 (t−tk−1 ) ...eAi2 (t3 −t2 ) +
t tk−1
t1 t0 t2 t1
eAi0 (t1 −τ ) Bi0 u(τ )dτ + eAi1 (t2 −τ ) Bi1 u(τ )dτ + . . .
(3)
eAik−1 (t−τ ) Bik−1 u(τ )dτ .
Therefore, a vector x f ∈ n+ belongs to Rk if and only if it can be expressed as in (3), for suitable u(·) ≥ 0,t,t ∈ + , and i ∈ P, ∈ {0, 1, . . ., k − 1}. Even more, it is easily seen that x f ∈ Rk if and only if xf = e
Aik−1 (t−tk−1 )
w+
t tk−1
eAik−1 (t−τ ) Bik−1 u(τ )dτ ,
(4)
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for some 0 < tk−1 < t, some ik−1 ∈ P, a nonnegative signal u(·) and some vector w ∈ Rk−1 . Clearly, Rk ⊆ Rk+1 , and hence R1 ⊆ R2 ⊆ · · · ⊆ Rk ⊆ ..... Moreover, if the above chain of subsets of n+ stops at some stage, namely Rk = Rk+1 for some k ∈ , then it cannot be increased any more [6]. We want to investigate under which conditions an index k ∈ can be found, such that Rk = Rk+1 . To this end, we denote by Rt (Ai , Bi ) the cone of (positive) states which are reachable at time t > 0 (by means of nonnegative inputs) by the single subsystem x˙ (t) = Ai x(t) + Bi u(t). Notice that, differently from what happens to standard linear systems, Rt (Ai , Bi ) typically grows with t [1]. As an immediate consequence of equation (4), we obtain the following identity (5) Rk+1 = ∪i∈P ∪t>0 eAit Rk + Rt (Ai , Bi ) , which leads to the following result. Proposition 2. Given a positive switched system (1), commuting among p singleinput subsystems, the following facts are equivalent: i) there exists k ∈ such that Rk = Rk+1 ; ii) there exists k ∈ such that eAit Rk + Rt (Ai , Bi ) ⊆ Rk , ∀ t > 0, ∀ i ∈ P. Proof. i) ⇒ ii) If Rk = Rk+1 , then the set of states which are reachable (in finite time) coincides with Rk . Clearly, if a state x f > 0 is reachable, then eAit x f + v(i,t) is reachable, too, for every i ∈ P, every t > 0 and every vector v(i,t) ∈ Rt (Ai , Bi ). Indeed, once x f has been reached, it is sufficient to switch to the ith subsystem and apply a suitable input signal for a lapse of time equal to t. This ensures that eAit Rk + Rt (Ai , Bi ) ⊆ Rk , ∀ t > 0, i ∈ P. ii) ⇒ i) If eAit Rk + Rt (Ai , Bi ) ⊆ Rk for every t > 0 and every i ∈ P, then (by (5)), Rk+1 ⊆ ∪i∈P ∪t>0 Rk = Rk , and since the converse inclusion Rk ⊆ Rk+1 is always true, this implies that i) holds. At this stage of our research, it is not clear, yet, whether, for a reachable system (1) an index k can always be found such that Rk = n+ . There are classes of systems, however, for which this is surely true and it turns out that reachability ensures that Rn = n+ . This is the case of single-input positive switched systems of dimension n = 2 or n = 3 [6]. Further classes of systems endowed with these properties will be investigated in the following sections. To conclude the section, we introduce a technical lemma we will use in the following. The lemma makes use of the following notation. Given n ∈ , a set of n × n Metzler matrices {A1 , A2 , . . . , A p } =: {Ai , i ∈ P}, and a subset S ⊆ n, we define the set IS := {i ∈ P : ZP(eAi eS ) = S }. (6) Lemma 1. If the n-dimensional continuous-time positive switched system (1) is reachable, then for every set S ⊆ n the index set IS is non-empty. Proof. Let x f ∈ n+ be a reachable state that can be reached through the switching sequence σ , ordinately taking the values i0 , i1 , . . . , ik−1 ∈ n and set S := ZP(x f ).
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Clearly, x f belongs to Rk for some k ∈ and hence it can be expressed as in (4) for some 0 < tk−1 < t, some ik−1 ∈ P, a nonnegative signal u(·) and some vector w ∈ Rk−1 . If ZP(x f ) = S = ZP(eAik−1 (t−tk−1 ) w), then [5] ik−1 ∈ IS and hence IS = 0. / If ZP(x f ) = S ZP(eAik−1 (t−tk−1 ) w), then it is sufficient to notice that, due to the nonnegativity of all matrix functions and signals involved, t Aik−1 (t−τ ) A e Bik−1 u(τ )dτ = ZP e ik−1 Bik−1 z , 0/ = ZP tk−1
where z is any positive vector satisfying: d ∈ ZP(z) if and only if [u]d (·), the dth entry of the vector function u(·), is positive on a non-zero measure time interval within ]tk−1 ,t[. This ensures that ZP(x f ) = S = ZP(eAik−1 (t−tk−1 ) w) ∪ A A A A ZP(e ik−1 Bik−1 z) = ZP(e ik−1 w) ∪ ZP(e ik−1 Bik−1 z) = ZP(e ik−1 (w + Bik−1 z)), and hence [5] ik−1 ∈ IS .
3 Positive Switched Systems with a Constant State Matrix In this section we consider the special class of positive switched systems which commute among p subsystems having the same system matrix A, but different inputto-state matrices Bi , i ∈ P, and hence described by the differential equation: x(t) ˙ = Ax(t) + Bσ (t)u(t),
t∈
+,
(7)
where A is an n × n Metzler matrix, while the matrices Bi , i ∈ P, are nonnegative. For these systems a strong characterization of reachability can be derived. Proposition 3. Given a continuous-time positive switched system (7), which switches among p subsystems (A, Bi ), i ∈ P, sharing the same system matrix, the following facts are equivalent ones: i) the system is reachable; ii) the system is monomially reachable; iii)A is a nonnegative diagonal matrix, and ∀ i ∈ n there exists an index j = j(i) ∈ P such that one column of B j is an ith monomial vector. If any of the previous equivalent conditions holds, then every positive vector x f belongs to R|ZP(x f )| and hence, in particular, Rn = n+ . Proof. The proof of i) ⇒ ii) is obvious, while ii) ⇒ iii) follows immediately from Proposition 1 in the special case when A j = A for every j ∈ P. So, it remains to be shown that iii) ⇒ i). To this end, assume first that m = 1, namely that we are dealing with singleinput systems. If so, condition iii) can be rephrased as follows: A is a nonnegative
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diagonal matrix (|P| ≥ n), and there exists a relabeling of the subsystems such that the first n vectors Bi are linearly independent monomial vectors. In other words, after a suitable relabeling, we get (for suitable αi ≥ 0 and βi > 0) ⎡ ⎢ ⎢ A=⎢ ⎣
α1
⎡
⎤
⎥ ⎢ ⎢ ⎥ ⎥ , B1 B2 . . . Bn = ⎢ ⎣ ⎦
α2 ..
.
β1
⎤ ⎥ ⎥ ⎥. ⎦
β2 ..
αn
.
(8)
βn
Let, now, x f ∈ n+ be any positive vector. Set ZP(x f ) = {i0 , i1 , . . . , ik−1 }. Notice that all indices i j belong to n. If we assume t = k as final time, t0 = 0 < t1 = 1 < · · · < tk−1 = k − 1 as switching instants, and we assume that the input u(·) takes some suitable positive constant value ui in the time interval [ti ,ti+1 [= [i, i + 1[, then equation (3) becomes x(k) = eA(k−1) +
k k−1
1 0
eA(1−τ ) Bi0 u0 dτ + eA(k−2)
eA(k−τ ) Bik−1 uk−1 dτ = eA(k−1)
+ eA(k−2)
1 0
2 1 1
0
eA(1−τ ) Bi1 dτ · u1 + . . . +
eA(1−τ ) Bi0 dτ · u0
1 0
eA(2−τ ) Bi1 u1 dτ + . . .
eA(1−τ ) Bik−1 dτ · uk−1 .
So, upon noticing that e
A(k−1− j)
1
e 0
A(1−τ )
Bi j d τ = e
A(k−1− j)
1 αi j (1−τ )
e
0
βi j ei j dτ = γi j ei j ,
for some γi j > 0, it follows that x(k) = ∑k−1 j=0 γi j ei j u j . So, it is easily seen that by properly choosing the values u j > 0 we can always ensure that x(k) = x f , and hence i) holds. Notice that the proof also shows that the last part of the proposition statement holds for m = 1. On the other hand, if we are dealing with the multiple-input case and condition iii) holds, we may associate with system (7) a new single-input positive switched system which is the system described as in (7), and that commutes among all subsystems (A, B∗i ), B∗i a column of some B j , j ∈ P. This amounts to considering the dynamic of system (7) under the constraint that at every instant t only one input is active. Clearly, the associated single-input system satisfies condition iii), too, and hence, by the previous part of the proof, it is reachable. But this means that, for system (7), all positive states can be reached by making use of a single input at every instant, and hence (7) is reachable. So, the result is proved.
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4 Positive Switched Systems with a Constant Input-to- State Matrix Let us address, now, the class of positive switched systems described by the differential equation: t ∈ +, (9) x(t) ˙ = Aσ (t) x(t) + Bu(t), where Ai , i ∈ P, are n × n Metzler matrices, while the n × m matrix B is nonnegative. Also in this case, we start by adapting the monomial reachability characterization provided in Proposition 1 to the specific class of systems we are considering. It turns out system (9) is monomially rechable if and only if there exist indices ji ∈ P, i ∈ n, such that A ji ei = αi ei , for some αi ≥ 0, and B has an n × n monomial submatrix. It is easily seen that if B has an n × n monomial submatrix and we are dealing with nonnegative input signals, Bu(t) is an arbitrary vector in m + at every time t. As a consequence, system (9) is reachable if and only if the following system is reachable. t ∈ +. (10) x(t) ˙ = Aσ (t) x(t) + u¯ (t), By making use of this necessary and sufficient condition, we provide a complete characterization of reachability property for the class of systems (9). Proposition 4. A continuous-time positive switched system (9), switching among |P| subsystems (Ai , B), i ∈ P, is reachable if and only if the following two conditions hold: a) B has an n × n monomial submatrix and b) ∀ S ⊆ n, IS = 0, / i.e. ∃ j(S ) ∈ P such that ZP(eA j(S ) eS ) = S . If any of the previous equivalent conditions holds, then every positive vector x f belongs to R|ZP(x f )| and hence, in particular, Rn = n+ . Proof. Suppose, first, that system (9) is reachable, then b) is true by Lemma 1. On the other hand, the system is a fortiori monomially reachable, and hence a) holds. Conversely, assume that a) and b) hold. Condition b) applied to all sets S of unitary cardinality ensures that for every i ∈ n there exists an index ji ∈ P such that A ji ei = αi ei , for some αi ≥ 0. So, under assumptions a) and b), monomial reachability is ensured, and hence, by the preliminary discussion, it remains to be shown that the associated switched system (10) is reachable. Given any positive vector x f ∈ n+ , set r := |ZP(x f )| and assume w.l.o.g. that ZP(x f ) = {i1 , i2 , . . . , ir }, with i1 < i2 < · · · < ir . For every h ∈ {1, 2, . . ., r}, set Sh := {i1 , i2 , . . . , ih } and let j(Sh ) be the index which makes assumption b) satisfied, and A hence ZP(e j(Sh ) eSh ) = Sh . Now, we show that by suitably choosing a final time t > 0, the values of the switching instants th , h = 0, 1, . . . , r − 1, with 0 = t0 < . . . < tr−1 < t, and a nonnegative input, which takes as a value a suitable monomial vector eih u¯h in every time interval [th−1 , th ), we may ensure that
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x f = eA j(Sr ) (t−tr−1 ) . . . e + ... +
t
tr−1
A j(S ) (t2 −t1 ) 2
t1 t0
e
A j(S ) (t1 −τ ) 1 dτ e
i1 u¯1
eA j(Sr ) (t−τ ) d τ eir u¯r
(11)
(which amounts to proving that every positive vector x f is reachable in |ZP(x f )| steps for system (10)). By the previous considerations, every term in (11) has a nonzero pattern included in Sr . Moreover, it is easy to conclude that, since every exponential matrix can be made, by choosing the time interval between two consecutive switching instants sufficiently small, as close as we want to the identity matrix (see Lemma 2 in [4]) A and since eih is an ih th monomial vector, with {ih } ⊆ ZP(e j(Sh ) eih ) ⊆ Sh ⊆ Sr , then each positive term vh := eA j(Sr ) (t−tr−1 ) . . . e
th
(t −t ) h+1 ) h+1 h
A j(S
th−1
e
A j(S ) (th −τ ) h dτ e
ih
can be made as close as we want to the monomial vector eih (and, of course, with nonzero pattern included in Sr ). Once the switching instants have been selected, in order to ensure that the aforementioned vectors vh are desired approximations of all the monomial vectors eih , ih ∈ Sr , and that ZP(vh ) ⊆ Sr , surely x f is an internal point of the cone generated by the vectors v1 , v2 , . . . , vr . So, nonnegative values u¯h can be found such that (11) holds. This part of the proof also ensures that the last part of the proposition statement holds true.
References 1. Farina, L., Rinaldi, S.: Positive linear systems: theory and applications. Series on Pure and Applied Mathematics. Wiley-Interscience, New York (2000) 2. Mason, O., Shorten, R.N.: Some results on the stability of positive switched linear systems. In: Proceedings of the 43rd Conference on Decision and Control, Paradise Island, Bahamas, pp. 4601–4606 (2004) 3. Mason, O., Shorten, R.N.: On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Transactions on Automatic Control 52, 1346–1349 (2007) 4. Santesso, P., Valcher, M.E.: An algebraic approach to the structural properties of continuous time positive switched systems. In: Commault, C., Marchand, N. (eds.) LNCIS, pp. 185–192. Springer, Heidelberg (2006) 5. Santesso, P., Valcher, M.E.: Controllability and reachability of switched positive systems. In: Proceedings of the Seventeenth International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan (2006); File MoP01.3.pdf
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6. Valcher, M.E.: On the k-switching reachability sets of single-input positive switched systems. Submitted to the 2009 American Control Conference, St. Louis (2009) 7. Valcher, M.E., Santesso, P.: On the reachability of single-input positive switched systems. In: Proceedings of the 47th Conference on Decision and Control, Cancun, Mexico, pp. 947–952 (2008)
Inverse-Positive Matrices with Checkerboard Pattern Manuel F. Abad, Mar´ıa T. Gass´o and Juan R. Torregrosa
Abstract. A nonsingular real matrix A is said to be inverse-positive if all the elements of its inverse are nonnegative. This class of matrices contains the M-matrices, from which inherit some of their properties and applications, especially in Economy. In this work we analyze the inverse-positive concept for a particular type of pattern: the checkerboard pattern. In addition, we study the Hadamard product of certain classes of inverse-positive matrices whose entries have a particular sign pattern.
1 Introduction In economics as well as other sciences, the inverse-positivity of real square matrices has been an important topic. A nonsingular real matrix A is said to be inversepositive if all the elements of its inverse are nonnegative. An inverse-positive matrix being also a Z-matrix is a nonsingular M-matrix, so the class of inverse-positive matrices contains the nonsingular M-matrices, which have been widely studied and whose applications, for example, in iterative methods, dynamic systems, economics, mathematical programming, etc, are well known. Of course, every inverse-positive matrix is not an M-matrix. For instance, −1 2 A= 3 −1 is an inverse-positive matrix that is not an M-matrix. The concept of inverse-positive is preserved by multiplication, left or right positive diagonal multiplication, positive diagonal similarity and permutation similarity. Manuel F. Abad, Mar´ıa T. Gass´o and Juan R. Torregrosa Instituto de Matem´atica Multidisciplinar, Universidad Polit´ecnica de Valencia, 46022 Valencia, Spain, e-mail:
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 185–194. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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So we may assume, without loss of generality, that all diagonal entries are equal to 1 when they are positive. Now, we present some examples of inverse-positive matrices that appear in different numerical processes. Example 1. The following n × n tridiagonal matrix ⎛ ⎞ a −1 0 . . . 0 0 1+ a−b ⎜ ⎟ ⎜ −1 2 −1 . . . 0 0 ⎟ ⎜ ⎟ ⎜ 0 −1 2 . . . 0 0 ⎟ ⎟, T =⎜ ⎜ .. .. .. .. .. ⎟ ⎜ . . . . .⎟ ⎜ ⎟ ⎝ 0 0 0 . . . 2 −1 ⎠ 0 0 0 . . . −1 1 with a > 0 and a > b is an inverse-positive matrix since T −1 = (1/a)C with C = (ci j ) where ci j = min{ai − b, a j − b}, i, j = 1, 2, . . . , n. Example 2. Let x = [x1 , x2 , . . . , xn ]T be a vector of Rn+ , that is, xi > 0, i = 1, 2, . . . , n. The lower bidiagonal matrix ⎛ ⎞ 1 0 0 ... 0 0 ⎟ ⎜ ⎜ x1 ⎟ ⎜ −1 1 ⎟ ⎜ ⎟ 0 ... 0 0 ⎟ ⎜ ⎜ x1 x2 ⎟ ⎜ ⎟ −1 1 ⎜ ⎟ ⎜ 0 ... 0 0 ⎟ ⎜ ⎟ x x 2 3 P(x, n) = ⎜ ⎟ .. .. .. .. ⎟ ⎜ .. ⎜ . ⎟ . . . . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 . . . −1 1 0 ⎟ ⎜ ⎟ xn−2 xn−1 ⎜ ⎟ ⎝ −1 1 ⎠ 0 0 ... 0 xn−1 xn is an inverse-positive matrix since ⎛
x1 x2 x3 .. .
0 x2 x3 .. .
⎜ ⎜ ⎜ ⎜ −1 P(x, n) = ⎜ ⎜ ⎜ ⎝ xn−1 xn−1 xn xn
0 ... 0 0 ... 0 x3 . . . 0 .. .. . . xn−1 . . . xn−1 xn . . . xn
0 0 0 .. .
⎞
⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ 0⎠ xn
Johnson in [3] studied the possible sign patterns of a matrix which are compatible with inverse-positiveness. Following his results we analyze the inverse-positive concept for a particular type of pattern: the checkerboard pattern.
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In addition, it is well known that the inverse of a nonsingular sign regular matrix is either a matrix with checkerboard pattern or is the opposite of a matrix with checkerboard pattern. In particular, the inverse of a nonsingular totally nonnegative matrix has checkerboard pattern. Definition 1. An n × n real matrix A = (ai j ) is said to have a checkerboard pattern if sign(ai j ) = (−1)i+ j , i, j = 1, 2, . . . , n. We study in this paper the inverse-positivity of bidiagonal, tridiagonal and lower (upper) triangular matrices with checkerboard pattern. We obtain characterizations of the inverse-positivity for each class of matrices. Definition 2. The Hadamard (or entry-wise) product of two n × n matrices A = (ai j ) and B = (bi j ) is A ◦ B = (ai j bi j ). Several authors have investigated about the Hadamard product of matrices. Johnson [2] showed that if the sign pattern is properly adjusted the Hadamard product of M-matrices is again an M-matrix and for any pair M, N of M-matrices the Hadamard product M ◦ N −1 is again an M-matrix. This result does not hold in general for inverse-positive matrices. We analyze when for any M, N checkerboard pattern inverse-positive matrices, the Hadamard product M ◦ N −1 is closed. The submatrix of a matrix A of size n × n, lying in rows α and columns β , in which α , β ⊆ N = {1, ..., n}, is denoted by A[α |β ], and the principal submatrix A[α |α ] is abbreviated to A[α ]. On the other hand, A(α |β ) is the submatrix obtained from A by deleting the rows indexed by α and columns indexed by β .
2 Checkerboard Inverse-Positive Matrices For matrices of size 2 × 2, it is easy to prove the following result. Proposition 1. Let A be a 2 × 2 nonsingular matrix. A is inverse-positive matrix if and only if, a) A has a checkerboard pattern and det(A) > 0, or b) −A has a checkerboard pattern and det(A) < 0. In general, this result does not hold for nonsingular matrices of size n × n, n ≥ 3, as we can see in the following example. Example 3. The nonsingular matrix ⎛
1 0 A = ⎝ −1 1 2 −1
⎞ 0 0 ⎠, 1
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has positive determinant and checkerboard pattern, but it is not an inverse-positive matrix. Now, we are going to analyze the inverse-positivity of bidiagonal, tridiagonal and lower (upper) triangular checkerboard matrices of size n × n, n ≥ 3. If A is a bidiagonal matrix with checkerboard pattern, then A is an M-matrix, so Proposition 2. If A is an n × n bidiagonal nonsingular matrix with checkerboard pattern, then A is an inverse-positive matrix. By other hand, we can observe that if A is a bidiagonal matrix and −A has checkerboard pattern, then A never is an inverse-positive matrix. Note that, in general, a tridiagonal matrix with checkerboard pattern it is not an inverse-positive matrix, as we can observe in the following example. Example 4. The tridiagonal matrix ⎛
⎞ 1 −1 0 A = ⎝ −2 1 −3 ⎠ 0 −4 1 is nonsingular with checkerboard pattern, but it is not an inverse-positive matrix. In the following results we present necessary and sufficient conditions for a tridiagonal matrix to be inverse-positive. Theorem 1. Let A = (ai j ) be an n × n tridiagonal nonsingular matrix with checkerboard pattern. Then, A is an inverse-positive matrix if and only if det A[α ] ≥ 0, α ⊆ {1, 2, . . . , n} |α | ≥ 2.
(1)
Proof. We are going to prove the sufficiency of condition (1). By hypothesis we have that all diagonal entries are positive, so we can assume that matrix A has the form ⎞ ⎛ 0 0 1 −a12 . . . ⎟ ⎜ −a21 1 . . . 0 0 ⎟ ⎜ ⎟ ⎜ .. . . . .. .. .. A=⎜ . ⎟. ⎟ ⎜ ⎝ 0 0 ... 1 −an−1,n ⎠ 0 0 . . . −an,n−1 1 1 (−1)i+ j det A( j|i). By using condition (1) it is Let A−1 = (bi j ) with bi j = det A easy to prove that bi j ≥ 0, for all i, j. The necessity of condition (1) follows a similar reasoning since if A is an inversepositive tridiagonal matrix, with checkerboard pattern, then aii = 0, i = 1, 2, . . . , n. When −A has a checkerboard pattern, we can establish the following result.
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Proposition 3. Let A be an 3 × 3 tridiagonal nonsingular matrix such that −A has a checkerboard pattern. Then, A is an inverse-positive matrix if and only if det A > 0, detA[{1, 2}] ≥ 0 and det A[{2, 3}] ≥ 0. We can observe that, if A = (ai j ) is an n × n, n > 3, tridiagonal nonsingular matrix with |ai j | > 0, |i − j| < 2, such that −A has a checkerboard pattern, then A is not an inverse-positive matrix. Finally, when A is a nonsingular lower (upper) triangular matrix, with checkerboard pattern, the nonnegativity of its inverse is not guaranteed. Example 5. Let us consider the lower triangular matrix ⎛ ⎞ 1 0 0 0 ⎜ −2 1 0 0 ⎟ ⎟ A=⎜ ⎝ 3 −1 1 0 ⎠ −4 5 −1 1 It is easy to check that A is not inverse-positive. We introduce a condition related with the associated graph of a matrix, which we call P-condition. Definition 3. Let A = (ai j ) be an n × n lower (upper) triangular matrix. A satisfies the P-condition if ai j ≤ aik ak j , i > k > j (i < k < j). We need the next lemma in order to get the main result for the inverse-positivity of this class of matrices. Lemma 1. Let A be an n × n nonsingular lower triangular matrix, with checkerboard pattern, that satisfies the P-condition. Then sign(detA[{i, i + 1, . . ., n}|{i − 1, i, . . ., n − 1}]) = (−1)n+i−1 , i = 2, 3, . . . , n. If A is an upper triangular matrix, the thesis of the above lemma is sign(detA[{i − 1, i, . . ., n − 1}|{i, i + 1, . . ., n}]) = (−1)n+i−1 , i = 2, 3, . . . , n. Theorem 2. Let A be an n × n nonsingular lower (upper) triangular matrix with checkerboard pattern, that satisfies the P-condition. Then, A is an inverse-positive matrix. Proof. Suppose that A is upper triangular. The proof is by induction on n. For n=3 ⎛ ⎛ ⎞ ⎞ 1 −a12 a13 1 a12 a12 a23 − a13 ⎝0 1 ⎠ ≥ 0, A3 = ⎝ 0 1 −a23 ⎠ and A−1 a23 3 = 0 0 1 0 0 1
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by using the P-condition. A matrix of size n × n can be partitioned as An−1 vn−1 An = , 0 1 where An−1 is the upper triangular submatrix A[{1, 2, . . ., n − 1}] and vn−1 is the submatrix A[{1, 2, . . ., n − 1}|{n}]. We can observe that −1 vn−1 An−1 −A−1 −1 n−1 An = , 0 1 where −A−1 vn−1 = n−1n+1 T (−1) detA[{1, . . . , n − 1}|{2, . . ., n}], . . . , (−1)2n−1 det A[{n − 1}|{n}] . The hypothesis of induction and Lemma 1 allow us to assure that A−1 n ≥ 0.
In general, the converse does not hold, as the following example shows. Example 6. Consider the lower triangular matrix ⎛ 1 0 0 ⎜ −7 1 0 A=⎜ ⎝ 0 −1 1 −1 1 −1
⎞ 0 0⎟ ⎟. 0⎠ 1
It is easy to check that A is inverse-positive, but a41 > a42 a21 . However, we can establish that Theorem 3. Let A be an 3 × 3 nonsingular lower (upper) triangular matrix with checkerboard pattern. Then A satisfies the P-condition if and only if A is an inversepositive matrix. In order to obtain a necessary condition in the general case, we introduce the following notation (see [4]). Given an n × n matrix A and the positive integers 1 ≤ m1 < m2 < · · · < mk ≤ n, we denote am1 ,m2 ,...,mk = (−1)k det A[{m1 , m2 , · · · , mk−1 }|{m2 , m3 , · · · , mk }]. We can establish the following result Theorem 4. Let A be an n × n nonsingular lower (upper) triangular matrix with checkerboard pattern. Then, A is an inverse-positive matrix if and only if for any 1 ≤ m1 < m2 < · · · < mk ≤ n, 1 ≤ k ≤ n, am1 ,m2 ,...,mk ≤ 0.
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Proof. Suppose that A is lower triangular. If A−1 = (bi j ) it is easy to observe that bi j = −a j, j+1,...,i . Analogously, if A is upper triangular we have bi j = −ai,i+1,..., j . We can extend the P-condition for general matrices in the following way: Given an n × n matrix A = (pi j ), A satisfies the P-condition if pi j ≤ pik pk j , i = j = k. Finally, by using the above condition we establish the following result. Theorem 5. Let A be an n × n, P- matrix with checkerboard pattern. If A satisfies the P-condition then, A is an inverse-positive matrix.
3 The Hadamard Product of Inverse-Positive Matrices Several authors have investigated about the Hadamard product of matrices. A celebrate and well-known theorem of Schur is that if A and B are positive semidefinite (nonnegative definite) matrices of the same size, then so is A ◦ B. For Mmatrices, though they have great many analogies to the positive definite matrices, the Hadamard product of two M-matrices is not M-matrix. Johnson [2] shown that the Hadamard product of an M-matrix and its own inverse is again M-matrix. Recently several authors have investigated the Hadamard product of inverse M-matrices. For example, Wang et al. in [4], proved that the class of inverse M-matrix is Hadamard -closed if and only if n ≤ 3. Our purpose here is to study the inverse-positivity of the Hadamard product A ◦ B−1 when A and B are inverse-positive matrices. In general, for matrices of size 2 × 2, it is easy to prove that Proposition 4. If A and B are 2 × 2 inverse-positive matrices and sign(detA) = sign(detB) then, A ◦ B−1 is an inverse-positive matrix. Now, we analyze when the class of lower (upper) triangular inverse-positive matrix, with checkerboard pattern, is closed under the Hadamard product. First, we consider the following technical result. Proposition 5. Let A = (ai j ) and B = (bi j ) be upper triangular matrices with −1 unit main diagonal. If (B)−1 i j denotes the (i, j) entry of B , we have (B)−1 i j = −bi,i+1,..., j
and
(B)−1 i,i+1 = −bi,i+1
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and, being C = A ◦ B−1 = (ci j ), ci j = −ai j bi,i+1,..., j ,
for all i < j.
(2)
We can establish a similar technical result for lower triangular matrices. Theorem 6. Let A and B be n × n, n ≥ 3, upper (lower) triangular matrices, with checkerboard pattern and unit main diagonal, satisfying the P-condition. Then A ◦ B−1 is an inverse-positive. Proof. Suppose that A and B are upper triangular matrices (the proof for lower triangular matrices is similar). The proof is by induction on n. For n = 3 ⎞ ⎛ ⎞ ⎛ 1 −b12 b13 1 −a12 a13 B = ⎝ 0 1 −b23 ⎠ A = ⎝ 0 1 −a23 ⎠ , 0 0 1 0 0 1 ⎞ 1 b12 b12 b23 − b13 ⎠ ≥ 0. b23 B−1 = ⎝ 0 1 0 0 1 ⎛
and
We observe that
⎞ 1 −a12 b12 a13 det B[{1, 2}|{2, 3}] ⎠ ≥ 0. 1 −a23 b23 C = A ◦ B−1 = ⎝ 0 0 0 1 ⎛
It is easy to prove that detC[{1, 2}|{2, 3}] ≥ 0, so C satisfies the P-condition and therefore C is inverse-positive. Now, let A and B be upper triangular matrices of size n × n, n > 3, A11 a¯12 B11 b¯ 12 A= and B = . 0 1 0 1 Note that, C = A◦B
and C
−1
=
−1
=
−1 ¯ A11 ◦ B−1 11 −a¯12 ◦ B11 b12 0 1
−1 [r , r , . . . , r T (A11 ◦ B−1 1n 2n n−1n ] 11 ) 0 1
.
By using the hypothesis of induction, we only need to prove that r jn ≥ 0, j = 1, 2, . . . , n − 1, where r jn = (C)−1 jn = −c j, j+1,...,n . By using (2), we have rn−1,n = −cn−1,n = −an−1,n(−bn−1,n) ≥ 0. Now,
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rn−2,n = −cn−2,n−1,n = detC[{n − 2, n − 1}|{n − 1, n}] = cn−2,n−1 cn−1,n − cn−2,n and by (2) rn−2,n = (−an−2,n−1bn−2,n−1)(−an−1,n bn−1,n) − (−an−2,nbn−2,n−1,n). Taking into account that A and B have checkerboard pattern and satisfy the Pcondition, we obtain rn−2,n = (an−2,n−1an−1,n − an−2,n)(bn−2,n−1bn−1,n ) + an−2,nbn−2,n ≥ 0. In a similar way, rn−3,n = −cn−3,n−2,n−1,n = − detC[{n − 3, n − 2, n − 1}|{n − 2, n − 1, n}] = = −cn−3,n−2rn−2,n + cn−3,n−1cn−1,n − cn−3,n. Since C has checkerboard pattern we have −cn−3,n−2rn−2,n ≥ 0 and, by using the P-condition and the checkerboard pattern of A and B, we obtain cn−3,n−1cn−1,n − cn−3,n = (an−3,n−1an−1,n − an−3,n)(bn−3,n−2,n−1bn−1,n ) + +an−3,n(−bn−3,n−2bn−2,n + bn−3,n) ≥ 0. Therefore, rn−3,n ≥ 0. In a similar way, we prove that r jn ≥ 0, for j = n − 4, n − 5, . . ., 2. Finally, we are going to prove that r1n ≥ 0. n+1 r1n = (C)−1 detC[{1, 2, . . . , n − 1}|{2, 3, . . ., n}] = 1n = (−1)
= (−1)n+1 [c12 detC[{2, 3, . . . , n − 1}|{3, 4, . . ., n}]− −c13 detC[{3, 4, . . ., n − 1}|{4, 5, . . ., n}] +
+ c14 detC[{4, . . . , n − 1}|{5, . . ., n}] + · · · + (−1)n−1c1,n−1 cn−1,n + (−1)nc1n .
By using a similar reasoning as rn−3,n , if n is even, and a similar reasoning as rn−2,n when n is odd we obtain that r1n ≥ 0. Acknowledgements. Supported by Ministerio de Ciencia y Tecnolog´ıa MTM2007-64477.
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References 1. Berman, A., Plemmons, R.J.: Nonnegative matrices in the Mathematical Sciences. SIAM, Philadelphia (1994) 2. Johnson, C.R.: A Hadamard Product Involving M-matrices. Linear Algebra and its Applications 4, 261–264 (1977) 3. Johnson, C.R.: Sign patterns of inverse nonnegative matrices. Linear Algebra and its Applications 55, 69–80 (1983) 4. Wang, B.Y., Zhang, X., Zhang, F.: On the Hadamard Product of Inverse M-matrices. Linear Algebra and its Applications 305, 23–31 (2000)
Some Remarks on Links between Positive Invariance, Monotonicity, Strong Lumpability and Coherency in Max-Plus Algebra Mourad Ahmane and Laurent Truffet
Abstract. In this paper, we make clearly appear links in Max-Plus algebra between four apparently different concepts encountered in various domains, such as Positive invariance, Monotonicity, Strong lumpability and Coherency. The first concept concerns Positive invariance of a particular set by a (linear) map. The second concept concerns Monotonicity of a given matrix. The third concept concerns Strong lumpability and the last concept concerns Coherency. To achieve these objectives, we begin first by recalling the idempotent version of Haar’s lemma, which gives necessary and sufficient conditions for the inclusion of ”one sided” idempotent polyhedra. Then, we generalize this result to the case of the inclusion of ”two sided” idempotent polyhedra. Finally, these results allow to formulate conditions for each concept and to give links between them. Recalling that Strong lumpability and Coherency are used for the aggregation (reduction) of systems. All the proofs are based on residuation theory, which play a central role in duality theory.
Basic Notations • All vectors are column vectors. ≤n denotes the component-wise or product on X n where (X , ≤) is a poset (i.e. x ≤n y ⇐⇒ ∀i, xi ≤ yi ). • Sm×k denotes the set of all m × k-dimensional matrices in a semifield S. • For all matrix A, ai, j , al,· and a·,k denote the entry (i, j), the lth row and the kth column of A, respectively. • If A, B ∈ Sn×m then A ≤ B denotes the entry-wise comparison of A and B. Mourad Ahmane SET Laboratory, University of Technology of Belfort-Montb´eliard, 90010 Belfort, France, e-mail:
[email protected] Laurent Truffet Ecole des Mines de Nantes, 4 rue Alfred Kastler, BP.20722, 44307 Nantes Cedex3, France e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 195–204. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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1 Introduction In our previous work [1], the following result formulated in Max-plus algebra is given to characterize in algebraic form the inclusion of ”one sided” idempotent polyhedra . Result 1 (Idempotent version of Haar’s Lemma) Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield. Given two polyhedra P(P, p) = {x ∈ Sd : P ⊗ x ≤m p} and P(Q, q) = {x ∈ Sd : Q ⊗ x ≤m q}, where P (resp. Q) be a m × d (resp. m × d) matrix and p (resp. q) be a m (resp. m ) dimensional column vector. Define that supp(v) = {l : vl = ε }, Tv = {l : vl = '} and ζ the complementary set of ζ . Assume the following hypotheses: / (i.e. P have non null columns); (H0 ) : ∀ j, supp(P·, j ) = 0, / (H1 ) : ∀ j, supp(q) ∩ TP·, j = 0. The assertion P(P, p) ⊆ P(Q, q) is true if and only if there exists a matrix H ∈ Sm ×m such that the following conditions hold: (i). Q ≤ H ⊗ P
and
(ii). H ⊗ p ≤m q.
(1)
It is almost classical that the following linear system over idempotent semiring or semifield (S, ⊕, ⊗, ε , e) defined by: x(0) ∈ Sd , A ∈ Sd×d (d, A) : (2) x(n) = A ⊗ x(n − 1), n ≥ 1, model linear systems on dioids of practical interest (e.g. some manufacturing systems, communication protocols (TCP), transmission of flows in network, etc.). For more details, see e.g. [2], [8]. Recall that the ith component (A⊗ x(n − 1))i of System (2) is expended as max (Ai, j + x j (n − 1)). j=1,...,d
The objectives of this paper are as follows. First, we generalize the idempotent version of Haar’s Lemma given by Result 1, which gives necessary and sufficient conditions for the inclusion of ”one sided” idempotent polyhedra, to the case of the inclusion of ”two sided” idempotent polyhedra (see Proposition 1). Then, according to Result 1 or Proposition 1, we give conditions for each property, such as Positive invariance, Monotonicity, Strong lumpability and Coherency. Finally, we show the links between these different properties. Recalling the following definitions for each property: • The first concept is Positive invariance of particular sets by a (linear) map. This concept is of particular importance because it leads to control strategies (see e.g. [6]). A set E ⊂ Sd is said to be positively invariant by the map f : Sd → Sd if f (E) ⊂ E. In this paper, f will be a linear map, i.e. f (x) = A⊗ x, where A ∈ Sd×d . In this case, we say that E is A-invariant. • The second concept concerns Monotonicity of a given matrix. Given two matrices W,W ∈ Sm×d , a matrix A is said to be (W,W )-monotone if ∀x, y ∈ Sd : x ≤W,W y =⇒ A ⊗ x ≤W,W A ⊗ y. The binary relation ≤W,W is defined by:
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∀x, y ∈ Sd : x ≤W,W y =⇒ W ⊗ x ≤m W ⊗ y. The links between Positive invariance and Monotonicity have already been noticed in the context of non linear dynamical systems (see e.g. [5]), and in the context of linear dynamical systems over idempotent semirings (see e.g. [15], [14]). • The two last concepts concern Strong lumpability and Coherency. These properties are used for the aggregation of systems (see e.g. [13, pp. 16,17]). A basic way to reduce the dimension of these linear systems over dioids is to lump or collapse some states into a single Mega-state. Thus we obtain a partition {C(1), ...,C(N)} of S = {1, ..., d} of System (2) into Σ = {1, ..., N} classes with N ≤ d. Given such a partition, we define a non-decreasing surjective map Φ from the state set S into the set Σ by ∀k ∈ S, ∀l ∈ Σ : Φ (k) = l ⇔ k ∈ C(l). The map Φ will be referred as a lumping map. We associate with the map Φ a lumping matrix V ∈ SN×d defined by ∀I ∈ Σ , ∀ j ∈ S : vI, j = δ{Φ ( j)=I} , where the {ε , e}-valued function δ{.} is e if the logical assertion is true, and ε otherwise. Then we deal with the system: ⎧ ⎨ x(0) ∈ Sd , (d, A,V ) : x(n) = A ⊗ x(n − 1), n ≥ 1, (I) . (3) ⎩ z(n) = V ⊗ x(n) In general, the series of vectors z(n) does not verify a difference equation of the 7 ∈ SN×N such form as (3,(I)). A condition under which there exist a matrix A 7 that z(n + 1) = A ⊗ z(n), is called a lumping condition (see e.g. [13]). These lumpability conditions are the counterparts of those existing for Markov chains ([12]). The paper is organized as follows. In Section 2, we introduce the main definitions of Max-Plus algebra (see e.g. ([2], [3], [11]). In Section 3, we give a generalization of the idempotent version of Haar’s Lemma given by Result 1, which gives necessary and sufficient conditions for the inclusion of ”one sided” idempotent polyhedra, to the case of the inclusion of ”two sided” idempotent polyhedra (see Proposition 1). In Section 4, we express different formulations of each concept: Positive invariance, Monotonicity, Strong lumpability and Coherency, under the form of Result 1 or Proposition 1. In Section 5, we give a conclusion where we express the different links between the various concepts mentioned above.
2 Basic Algebraic Structures For any set, (S, ⊕, ⊗, ε , e) is a semiring if (S, ⊕, ε ) is a commutative monoid, (S, ⊗, e) is a monoid, ⊗ distributes over ⊕, the neutral element ε for ⊕ is also absorbing for ⊗, i.e. ∀a ∈ S, ε ⊗ a = a ⊗ ε = ε , and e is the neutral element for (⊗). (S, ⊕, ⊗, ε , e) is an idempotent semiring (called also dioid) if (S, ⊕, ⊗, ε , e) is a semiring, ⊕ is idempotent, i.e. ∀a ∈ S, a ⊕ a = a. If (S, ⊗, e) is a commutative monoid, then the idempotent semiring (S, ⊕, ⊗, ε , e) is said commutative.
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(S, ⊕, ⊗, ε , e) is a an idempotent semifield if (S, ⊕, ⊗, ε , e) is an idempotent semiring and (S\{ε }, ⊗, e) is a group, i.e. (S\{ε }, ⊗, e) is a monoid such that all its elements are invertible (∀a ∈ S\{ε }, ∃a−1 : a ⊗ a−1 = a−1 ⊗ a = e). Also if (S\{ε }, ⊗, e) is a commutative monoid, then the idempotent semifield (S, ⊕, ⊗, ε , e) is said commutative. Let (S, ⊕, ⊗, ε , e) be an idempotent semiring. Each element of Sn is a n-dimensional column vector. We equip Sn with the two laws ⊕ and ⊗ as follows: not.
de f
∀x, y ∈ Sn , (x ⊕ y)i = xi ⊕ yi , ∀s ∈ S, (s.x)i = (s ⊗ x)i = s ⊗ xi , i = 1, . . . , n. The addition ⊕ and the multiplication ⊗ are naturally extended to matrices with compatible dimension. Any n × p matrix A is associated with a (⊕, ⊗)-linear map A : S p → Sn . The (i, j) entry, the lth row-vector and the kth column-vector of matrix A are denoted Ai, j , Al,· and A·,k , respectively. Let (S, ⊕, ⊗, ε , e) be an idempotent semiring or an idempotent semifield, then (S, ⊕, ε ) is an idempotent monoid, which can be equipped with the natural order relation ≤ defined by de f
∀a, b ∈ S : a ≤ b ⇔ a ⊕ b = b.
(4)
We say that (S, ⊕, ⊗, ε , e) is complete if it is complete as a naturally ordered set and if the respective left and right multiplications, λa , ρa : S → S, λa (x) = a ⊗ x, ρa (x) = x ⊗ a are continuous for all a ∈ S. We adopt the following notations ◦ = ∀a, b ∈ S : λa (b) = a\b not.
de f
de f not. ◦ = ρa (b) = b/a
8 8
{x ∈ S : x ⊗ a ≤ b},
{x ∈ S : a ⊗ x ≤ b}.
A typical example of complete dioid is the top completion of an idempotent semi◦ = a−1 ⊗ b and b/a ◦ = b ⊗ a−1 . field. Let us note that if a ∈ S is invertible then: a\b 8 not. Let us note also that as S is complete it possesses a top element S = ' = +∞. We have by convention the following identities:
ε ⊗ ' = ' ⊗ ε = ε , and ∀a ∈ S, a ⊕ ' = ', a ∧ ' = ' ∧ a = a. We suppose besides that (for a discussion, see e.g. ([2, p. 163-164]): ∀a = ε , a ⊗ ' = ' ⊗ a = '. By definition of /◦ (idem for \◦ ) and properties of ', ε and of (5) we have: ε if a = ' ε if a = ε ◦ ◦ ◦ ◦ . , ε/a = ∀a ∈ S : a/ε = ', '/a = ', a/' = ' if a = ' ' if a = ε
(5)
(6)
◦ ·\· ◦ are extended to matrices and vectors with compatible dimenThe operations ·/·, sions assuming that all the elements of these matrices and vectors are in a complete set S with ≤ denote the entry-wise comparison of matrices:
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8
◦ j ); (A\B) ◦ i = ∧ j (a j,i\y ◦ i, j = ( {X : A ⊗ X ≤ B})i, j = ∧k (ak,i\b ◦ k, j ); (A\y)
8
de f
◦ i, j = ( {X : X ⊗ C ≤ D})i, j = ∧ j (di,l/c ◦ j,l ). (D/C)
3 Generalization of Idempotent Version of Haar’s Lemma The aim of this section is to generalize the idempotent version of Haar’s Lemma given in Result 1, which provides necessary and sufficient conditions for the inclusion of ”one sided” idempotent polyhedra, to the case of the inclusion of ”two sided” idempotent polyhedra. Before that, we need the following result. Result 2 ([1]) Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield. Let A ∈ Sm×d be a matrix and b, g ∈ Sm be two vectors. Assume that Hypotheses (H0 ) and (H1 ) of Result 1 in this case are respected. Then the following equality holds: b/◦ (A\◦ g) = (b/◦ g) ⊗ A.
(7)
Proposition 1. Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield. Given ”two sided” idempotent polyhedra P(P, f (y)) = {x ∈ Sd , y ∈ Sd : P ⊗ x ≤m f (y)} and P(Q, g(y)) = {x ∈ Sd , y ∈ Sd : Q ⊗ x ≤m g(y)} where P ∈ Sm×d , Q ∈ Sm ×d and f : Sd → Sm , g : Sd → Sm be two functions. Assume that Hypotheses (H0 ) and (H1 ) of Result 1 are respected. The assertion ∀y ∈ Sd : P(P, f (y)) ⊆ P(Q, g(y)) is true if and only if there exist a matrix H ∈ Sm ×m such that the following condition hold: (8) ∀y ∈ Sd : (i). Q ≤ H ⊗ P and (ii). H ⊗ f (y) ≤m g(y).
Proof. The assertion ∀y ∈ Sd : P(P, f (y)) ⊆ P(Q, g(y)) can be rewritten as follows:
∀y ∈ Sd : {x ∈ Sd : P ⊗ x ≤m f (y)} ⇒ {x ∈ Sd : Q ⊗ x ≤m g(y)}, ◦ which is equivalent to (by definition of \):
∀y ∈ Sd : {x ∈ Sd : x ≤d P\◦ f (y)} ⇒ {x ∈ Sd : Q ⊗ x ≤m g(y)}. Since ⊗ is non-decreasing, this last assertion is true if and only if
∀y ∈ Sd : Q ⊗ (P\◦ f (y)) ≤m g(y),
◦ ∀y ∈ Sd : Q ≤ g(y)/(P ◦ \ ◦ f (y)). which is equivalent to (by definiton of /): Using Result 2, we obtain:
∀y ∈ Sd : Q ≤ (g(y)/◦ f (y)) ⊗ P.
(9)
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It is sufficient to take H = g(y)/◦ f (y). From the definition of the residuation, it is clear that the condition (ii) of Proposition 1 holds true, and from (9), we realize that (i) is satisfied too, which ends the proof.
4 Positive Invariance, Monotonicity, Strong Lumpability and Coherency In this section, we give the formulation of each property under the form of Result 1 or Proposition 1. Remark 1. In each case of the rest of the paper, Hypotheses (H0 ) and (H1 ) of Result 1 must be respected.
4.1 Positive Invariance A set E ⊂ Sd is said to be positively invariant by the map f : Sd → Sd if f (E) ⊂ E. In this paper, f will be a linear map, i.e. f (x) = A ⊗ x, where A ∈ Sd×d . In this case, we say that E is A-invariant. Proposition 2. Let (S, semifield. Let us 9⊕, ⊗, ε , e) be a complete idempotent : consider the set E := x ∈ Sd , y ∈ Sd : K ⊗ x ≤m K ⊗ y , with K ∈ Sm×d and x m×d have non null columns. Let z = ∈ Sd+d be a vector and M ∈ K ∈S y S(d+d )×(d+d ) be a matrix. The set E is M-invariant if there exists a matrix H ∈ S(m+m)×(m+m) such that the following conditions hold: (10) (i). T ⊗ M ≤ H ⊗ T and (ii). H ⊗ L ⊗ z ≤m+m L ⊗ M ⊗ z, E K K K (m+m)×(d+d ) , L = where T = ∈ S ∈ S(m+m)×(d+d ) and E is a E K K K m × d-matrix which its all entries are ε . Proof. From assertion (4), the inequaliy {K ⊗ x ≤m K ⊗ y} of the set E can be rewritten in Max-Plus algebra as {(K ⊗ x) ⊕ (K ⊗ y) = K ⊗ y}. This last assertion (K ⊗ x) ⊕ (K ⊗ y) ≤m K ⊗ y , which is equivalent to: can be written as follows: K ⊗ y ≤m (K ⊗ x) ⊕ (K ⊗ y) . K K E K x ⊗ z ≤ ⊗ z = {T ⊗ z ≤m+m L ⊗ z} . z= ∈ Sd+d : m+m E K K K y The set E is said to be M-invariant if z ∈ E ⇒ M ⊗ z ∈ E, which is equivalent to: {T ⊗ z ≤m+m L ⊗ z ⇒ T ⊗ M ⊗ z ≤m+m L ⊗ M ⊗ z} .
(11)
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From Proposition 1 and by taking Q = T ⊗ M, P = T, f (z) = L ⊗ z, g(z) = L ⊗ M ⊗ z, equation (1) becomes:
∀z ∈ Sd+d : P(T, L ⊗ z) ⊆ P(T ⊗ M, L ⊗ M ⊗ z), which reads
∀g ∈ Sd+d , ∀z ∈ Sd+d : T ⊗ g ≤ L ⊗ z ⇒ T ⊗ M ⊗ g ≤ L ⊗ M ⊗ z.
(12)
We remark that equation (12) implies equation (11), and the result is proved.
Result 3 ([1]) Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield and D, G ∈ Sm×d be two matrices. The following equality holds: ; y∈Sd
((D ⊗ y)/◦ (G ⊗ y)) = D/◦ G.
(13)
Proposition 3.9Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield. Let us con: d d sider the set E := x ∈ S , y ∈ S : K ⊗ x ≤m K ⊗ y with K ∈ Sm×d and K ∈ Sm×d x have non null columns. Let z = ∈ Sd+d be a vector and M ∈ S(d+d )×(d+d ) be y a matrix. The set E is M-invariant if there exists a matrix H ∈ S(m+m)×(m+m) such that the following conditions hold: (i). T ⊗ M ≤ H ⊗ T and (ii). H ⊗ L ≤ L ⊗ M. (14) K K E K where T = ∈ S(m+m)×(d+d ) , L = ∈ S(m+m)×(d+d ) and E is a E K K K m × d-matrix which its all entries are ε .
Proof. By residuation, Condition (ii) of assertion (1) becomes: ◦ H ≤ (L ⊗ M ⊗ z)/(L ⊗ z).
By using of Result 3, we obtain H ⊗ L ≤ L ⊗ M. Condition (i) is the same as Condition (i) of Proposition 2, which ends the proof.
4.2 Monotonicity Definition 1. Let A ∈ Sd×d and W,W ∈ Sm×d . Matrix A is said to be (W,W )monotone if ∀x, y ∈ Sd : x ≤W,W y ⇒ A ⊗ x ≤W,W A ⊗ y. (15) In the literature, this property is used to simplify some problems of performances assessement of linear systems on dioids. Here, we can characterize some monotonous
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operations in a more general case and therefore to hope to build more elaborate techniques of simplifications. Result 4 ([1]) Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield and W,W ∈ Sm×d be two matrices. Then a matrix A ∈ Sd×d is (W,W )-monotone if and only if W ⊗ A ≤ (W ⊗ A/◦W ) ⊗ W.
(16)
Proposition 4. Let (S, ⊕, ⊗, ε , e) be a complete idempotent semifield. Let us consider three matrices A ∈ Sd×d , W,W ∈ Sm×d . Assume that Hypotheses (H0 ) and (H1 ) of Result 1 are respected. Then, a matrix A ∈ Sd×d is (W,W )-monotone if and only if there exists a matrix H ∈ Sm×m such that the following conditions hold: (i). W ⊗ A ≤ H ⊗ W
and (ii). H ⊗ W ≤ W ⊗ A.
(17)
◦ and the proof is Proof. From equation (16), it suffice to take H = W ⊗ A/W achieved.
Proposition 5. When W = W in Proposition 4, we say that a matrix A is (W,W )monotone if and only if there exists a matrix H ∈ Sm×m such that the following condition holds: W ⊗ A = H ⊗ W. (18) Remark 2. Proposition 5 coincides with Theorem 4.2 of [15] obtained by a different proof.
4.3 Strong Lumpability Let Φ be a lumping map from S into Σ and V the corresponding lumping matrix (∀I ∈ Σ , ∀ j ∈ S : vI, j = δ{Φ ( j)=I} ). We can define a partition of S into N aggregates Φ −1 (J) = [mJ , MJ ] such that cardinal(Φ −1 (J)) = dJ , J ∈ Σ . Recall the system given by (3): ⎧ ⎨ x(0) ∈ Sd (d, A,V ) : x(n) = A ⊗ x(n − 1), n ≥ 1 . ⎩ z(n) = V ⊗ x(n) The series (x(n), n ≥ 0) of system (3) with a given initial data x(0) is said to be lumpable if the aggergated series (z(n), n ≥ 0) satisfy the reduced equation z(n + 7 ⊗ z(n), for some matrix A 7 ∈ SN×N . 1) = A Definition 2. Let V ∈ SN×d be a lumping matrix. The matrix A ∈ Sd×d is said to 7 ∈ SN×N such be strongly lumpable by V , or simply V -lumpable [13] if there exists A that 7 ⊗ V, V ⊗A = A (19) or equivalently: ∀I ∈ Σ , ∀J ∈ Σ , ∀ j ∈ Φ −1 (J),
0 means that its components are positive: Mi j > 0, and M ≥ 0 means that its components are nonnegative: Mi j ≥ 0. diag(λ ) is the diagonal matrix whose diagonal is formed by the components of the vector λ . M 0 , where M is a symmetric real matrix, means that M is definite positive.
2 Statements and Preliminaries This section provides some necessary preliminary statements and technical keys that are primordial for the characterization and the treatment of positive systems satisfying a differential delayed equation. The introduced facts and results will be essentially used in development and the derivation of our main stability result.
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The system under investigation is described by a general forced linear differential delayed equation. m dx = Ax + ∑ Ai x(t − τi (t)) + Bu(t), (1) dt i=1 the given matrices A, A1 , . . . , Am ∈ Ren×n and B ∈ Renu ×n are time-invariant and 0 ≤ τ1 (·), · · · , 0 ≤ τm (·) are time-varying delays that are supposed to be Lebesgue measurable. Throughout the paper we use the notation
τ := max sup τi (t). 1≤i≤m t≥0
The vector x(t) ∈ Ren is the instantaneous system state at time t and u(t) ∈ Renu represents an external input. The whole state at time t of system (1) is infinite dimensional which is given by the set {x(s)| − τ ≤ s ≤ t}. Following [14], it can be shown that the solution to the system’s equation (1) exists, unique and totally determined by any given initial Locally Lebesgue integrable vector function φ (·) such that x(s) = φ (s) for − τ ≤ s ≤ 0. Throughout this paper, the free system is assumed to satisfy a positivity constraint on its states as follows. Definition 1. For any nonnegative initial condition φ (t) ∈ Ren+ such that x(t) = φ (t) for − τ ≤ t ≤ 0, System (1) is said to be positive if the corresponding trajectory is nonnegative, that is x(t) ∈ Ren+ for all t ≥ 0. We stress out that that intrinsic properties of the delayed system’s positivity behavior are related to Metzlerian matrices and positive matrices. Definition 2. A real matrix M is called a Metzler matrix if its off-diagonal elements are nonnegative: Mi j ≥ 0, i = j. Definition 3. A real matrix M is called a positive matrix if all its elements are nonnegative: Mi j ≥ 0. Note that the following result shows how Metzlerian matrices are intrinsically connected to positivity. Lemma 1. Let M be a Metzler matrix then the following holds true. (a) M Metzler ⇔ etM ≥ 0, ∀t ≥ 0. (b) if v > 0, then etM v > 0, ∀t ≥ 0. Proof. Item (a) is well-know [22]. Item (b) is trivial.
We emphasize that the following result can be interpreted as an extension of the classical result on positive linear systems (see [22]) and its proof can be obtained in the same spirit of reasoning and then omitted. Also, this result offers an easy test for checking the positivity of the free system. Proposition 1. System (1) with u = 0 is positive if and only if A is a Metzler matrix and A1 , . . . , Am are positive matrices.
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3 Main Result In this section, the stability of autonomous linear positive systems subject to timevarying delays is studied. The relevant derived result involves necessary and sufficient conditions. Previous results and equivalent conditions for a Metzler matrix to be Hurwitz can be found in many places in the literature, see for example [4, 10, 18]. In the following, some well-known facts are presented and will be used in order to derive our main result. Lemma 2. Let M be a Metzler matrix. Then, the following conditions are equivalent i) M is Hurwitz (has eigenvalues with negative real part). ii) The inverse of M exists and all its components are negative: M −1 ≤ 0. iii)There exist a vector λ > 0 such that M λ < 0. iv) There exist a diagonal matrix D 0 such that M T D + DM ≺ 0. In the sequel, conditions for the asymptotic stability of the general positive linear time-delay system (1) are derived. But, before we need the following technical lemma. Lemma 3. Let A, A1 , . . . , Am ∈ Ren×n be constant matrices and assume that the matrix A is Metzler and A1 , . . . , Am are nonnegative. Consider the following delayed linear system with constant delay τ . m dz = Az(t) + ∑ Ai z(t − τ ), dt i=1 z(s) = λ , for − τ ≤ s ≤ 0.
(2)
m
Then, if (A + ∑ Ai )λ < 0 and λ > 0, we have that z(t) is strictly decreasing for i=1
t ≥ 0. Moreover, z(t) converges asymptotically to zero. That is z˙(t) < 0, ∀t > 0 and lim z(t) = 0.
t→0
Proof. Since A is Metzler and A1 , . . . , Am are nonnegative, the minus derivative −˙z of the solution to system (2) satisfies a linear delayed positive system equation m
and it can be easily shown that −˙z(t) > 0, ∀t ≥ 0. Also, because that (A + ∑ Ai )λ < i=1
0 and λ > 0 we have that z(t) converges asymptotically to zero (see for this fact [6, 13] and by this our claim is proved. Theorem 1. Assume that system (1) is positive, or equivalently that the matrix A is Metzler and A1 , . . . , Am are positive matrices. Then, the following statements are equivalent.
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i) There exist a constant-time delays τ1∗ , . . . , τm∗ and a nonnegative initial functional condition φ ∗ (·) with φ ∗ (0) > 0 for which the free system (1) (u=0) is asymptotically stable. ii) System (1) is asymptotically stable for every nonnegative initial condition φ (·) ≥ 0 and for any bounded arbitrary time-varying delays. iii)System (1) is asymptotically stable for every initial condition taking values in Ren (φ (·) has indefinite sign) and for any bounded arbitrary time-varying delays. iv) There exists λ ∈ Ren such that m
(A + ∑ Ai )λ < 0, λ > 0.
(3)
i=1
Proof. The implications (iii) ⇒ (ii) ⇒ (i) are obvious. The rest of the proof will be proceeded in 3 steps. We emphasize that the second step is the more delicate and subtle part of the proposed proof. Step 1 : (i) ⇒ (iv) By integrating System (1) we have x(t) − x(0) = A
T 0
m
T
i=1
0
x(t)dt + ∑ Ai
x(t − τi∗ )dt,
which by change of variable can be expressed as the following identity m
T
i=1
0
(A + ∑ Ai ) m
x(T ) + ∑ Ai i=1
x(t)dt =
T −τ ∗ i T
m
x(t)dt − ∑ Ai i=1
since x(T ) goes to zeros, then also
m
∑ Ai
i=1
m
T −τ ∗ i T
negative and x(0) is positive, the term ∑ Ai i=1
0 −τi∗
0 −τi∗
φ ∗ (t)dt − x(0),
x(t)dt. Moreover, since φ ∗ is non-
φ ∗ (t)dt + x(0) is constant and pos-
itive. Thus, regarding to these facts, it suffices to select a sufficiently large T to get m
(A + ∑ Ai )λ < 0, λ > 0,
T
i=1
where λ is defined as λ = 0 x(t)dt which is positive due to the fact that x(0) is positive and the trajectory x(t) is continuous. Step 2 : (iv) ⇒ (ii) Let φ (·) ≥ 0 be any initial functional condition and consider m
its associated trajectory x(·). Now, take any vector λ > 0 satisfying (A + ∑ Ai )λ < 0. Of course, there exist a positive constant scalar α > 0 such that
αφ (s) < λ , ∀s : −τ ≤ s ≤ 0.
i=1
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Note that by linearity the associated trajectory to αφ (·) is α x(·), so that the scaled trajectory is solution to
α x(t) = etA α x(0) +
t 0
m
e(t−s)A ∑ Ai α x(s − τi (s))ds.
(4)
i=1
Now, consider the following delayed linear system with constant delay τ , such that τ ≥ max sup τi (t). 1≤i≤m t≥0
m dz = Az(t) + ∑ Ai z(t − τ ), dt i=1 z(s) = λ , for − τ ≤ s ≤ 0.
Next, we claim that
(5)
α x(t) < z(t), ∀t ≥ 0.
If this fact does not hold, let t ∗ be the maximal time such that there exist at least a component xi (t ∗ ) of x(t ∗ ) such that α xi (t ∗ ) ≥ zi (t ∗ ) and α x(s) < z(s), ∀s : −τ ≤ s < t ∗. Based on the integral expression (4), we are going to perform a comparison at ∗ time t ∗ by using the fact that et A (z(0) − α x(0)) > 0 (since z(0) > α x(0)) and also ∗ e(t −s)A ≥ 0 if t ∗ ≥ s (apply Lemma 1). Thus, since the matrices A1 , . . . , Am are positive (and do not forget that α x(s) < z(s), ∀s : −τ ≤ s < t ∗ ), we obtain ∗
α x(t ∗ ) < et A z(0) +
t∗
e(t
∗ −s)A
0
m
∑ Ai z(s − τ )ds.
(6)
i=1
At this moment, one can wonder why this holds true? To give a positive answer we use Lemma 3 that asserts that z(t) is strictly decreasing, then from this fact, we can of course see that
α x(t − τi ) < z(t − τi (t)) ≤ z(t − τ ) ∀t : −τ ≤ t < t ∗ . Multiplying α x(t − τi (t)) − z(t − τ ) ≤ 0 by e(t from 1 up to m, we obtain t∗
e(t
∗ −s)A
0
∗ −t)A
m
t∗
i=1
0
∑ Ai x(s − τi (s))ds ≤
Ai ≥ 0 , integrating and summing
e(t
∗ −s)A
m
∑ Ai z(s − τ )ds,
i=1 ∗
∗
so that by keeping in mind the strict inequality et A z(0) > α et A x(0), now we can be sure that inequality (6) occurs at time t ∗ . At this stage, we give the crucial conclusion. The right hand side of the claimed inequality (6) is nothing else than z(t). Consequently, we got α x(t ∗ ) < z(t ∗ ), which turn out to contradict the fact that there is a component such that α xi (t ∗ ) ≥ zi (t ∗ ), and we are almost done. Because, we have now 0 ≤ α x(t) < z(t), ∀t and z(t) goes to m
zero (since A + ∑ Ai is Hurwitz see for this Lemma 3). Henceforth, we have shown i=1
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that system (1) is asymptotically stable for every initial functional condition φ (·) taking values in Ren+ . Step 3 : (ii) ⇒ (iii) This implication results from the linearity of the system and the fact that φ can be decomposed as φ = φ + − φ − where φ + ≥ 0, φ − ≥ 0. So that the proof is complete. Corollary 1. Assume that system (1) is positive, or equivalently that the matrix A is Metzler and A1 , . . . , Am are positive matrices. Then, the following statements are equivalent. i) There exist a constant-time delays τ1∗ , . . . , τm∗ and a nonnegative initial functional condition φ ∗ (·) with φ ∗ (0) > 0 for which the free system (1) (u=0) is asymptotically stable. ii) system (1) is asymptotically stable for every initial condition and for any arbitrary bounded time-varying delays. m
iii)The inverse of A + ∑ Ai exists and all its components are negative i=1
m
(A + ∑ Ai )−1 ≤ 0. i=1
m
iv) There exist a vector λ such that (A + ∑ Ai )λ < 0, λ > 0. i=1
v) There exist a diagonal matrix D such that m
m
i=1
i=1
(A + ∑ Ai )D + D(A + ∑ Ai )T ≺ 0, D
0.
m
v) A + ∑ Ai is a Hurwitz matrix. i=1
Proof. It suffices to apply Theorem 1 and Proposition 2.
4 Controllers Design The aim of this section is to show how our stability result can be applied in order to compute stabilizing feedback controllers. In particular, those control laws that take only nonnegative values will be considered owing to their importance in practice. The following result provides necessary and sufficient conditions for the existence of stabilizing control law that preserves the positivity of the system. It also provides an easy and efficient approach for checking the solvability of the stabilization problem and for computing any stabilizing state-feedback control by either using LP or LMI softwares.
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Theorem 2. Assume that A1 , . . . , Am are positive matrices. Then, the following statements are equivalent i) There exists a stabilizing memoryless state-feedback law u(t) = Kx(t) such that the resulting closed-loop system (1) is positive and asymptotically stable for arbitrary bounded time-varying delays. ii)There exists a matrix K ∈ Renu ×n such that A + BK is Metzler matrix and A + m
BK + ∑ Ai is a Hurwitz matrix. i=1
iii)The following LP problem in the variables λ ∈ Ren and Z ∈ Renu ×n is feasible ⎧ ⎡ ⎤ 1 ⎪ ⎪ m ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎨ (A + ∑ Ai )λ + BZ ⎣ . ⎦ < 0, i=1 (7) 1 ⎪ ⎪ ⎪ Adiag(λ ) + BZ + I ≥ 0, ⎪ ⎪ ⎩ λ > 0, Moreover, a gain matrix K satisfying the conditions (i) and (ii) can be computed as follows K = Zdiag(λ )−1 , where the vector λ and the matrix Z are any feasible solution to the above LP problem. iv) The following LMI problem in the variables D ∈ Ren×n and Y ∈ Renu ×n is feasible ⎧ m m ⎪ T T T T ⎪ ⎨ (A + ∑ Ai )D + D(A + ∑ Ai ) + BY + Y B ≺ 0, i=1 i=1 (8) ⎪ AD + BY + I ≥ 0, ⎪ ⎩ D 0. Moreover, a gain matrix K satisfying the conditions (i) and (ii) can be computed as follows K = Y D−1 , where the matrices D and Z are any feasible solution to the above LMI problem. Proof. The equivalence between i) and ii) is straightforward from Theorem 1. Now let us show that ii) and iii) are equivalent. First, consider the implication ii) → iii). Note that since A + BK is Metzler and A1 , . . . , Am are positive matrices then m
m
i=1
i=1
A + BK + ∑ Ai is Metzler. So that by using Corollary 1, we have A + BK + ∑ Ai is Hurwitz if and only if there exists a vector λ > 0 such that m
(A + BK + ∑ Ai )λ < 0. i=1
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Now, define K = Zdiag(λ )−1 . Thus, with this change of variable, the above inequality is effectively the first inequality in condition iii). The second inequality in the LP constraints, is obtained as follows. Note that A + BK is Metzler if and only if (A + BK)diag(λ ) is Metzler, or equivalently (by adding the identity matrix I) (A + BK)diag(λ ) + I ≥ 0, this holds true by choosing λ with sufficiently small components (since the stability condition is homogeneous in λ ). Thus, by recalling that K = Zdiag(λ )−1 , the above inequality is nothing else than the second inequality AT diag(λ ) + BZ + I ≥ 0, in the LP constraints. The reverse implication iii) → ii) can be trivially obtained by a simple matrix manipulation as shown above. Also, to show the equivalence between ii) and iv), it suffices to use the LMI condition given by Corollary 1, make the change of variable K = Y D−1 and follow the same line of argument as for the LP formulation. Thus, the proof is complete. Now, the following result provides necessary and sufficient conditions for the existence of a stabilizing nonnegative control law that preserves the positivity of the system. Theorem 3. The following statements are equivalent i) There exists a stabilizing nonnegative memoryless state-feedback law u(t) = Kx(t) ≥ 0 such that the resulting closed-loop system (1) is positive and asymptotically stable for arbitrary time varying delays. ii)There exists a matrix K ∈ Renu ×n such that K ≥ 0, A + BK is Metzler matrix and m
A + BK + ∑ Ai is a Hurwitz matrix. i=1
iii)The following LP problem in the variables λ ∈ Ren and Z ∈ Renu ×n is feasible ⎧ ⎡ ⎤ 1 ⎪ ⎪ m ⎪ ⎪ ⎢ .. ⎥ ⎪ (A + A ) λ + BZ ⎪ ⎣ . ⎦ < 0, ∑ i ⎪ ⎨ i=1 1 (9) ⎪ λ ) + BZ + I ≥ 0, Adiag( ⎪ ⎪ ⎪ ⎪ Z ≥ 0, ⎪ ⎪ ⎩ λ > 0. Moreover, a gain matrix K satisfying the conditions (i) and (ii) can be computed as follows K = Zdiag(λ )−1 , where the vector λ and the matrix Z are any feasible solution to the above LP problem.
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iv) The following LMI problem in the variables D ∈ Ren×n and Y ∈ Renu ×n is feasible ⎧ m m ⎪ T ⎪ (A + A )D + D(A + ATi ) + BY + Y T BT ≺ 0, ⎪ i ∑ ∑ ⎪ ⎨ i=1 i=1 (10) AD + BY + I ≥ 0, ⎪ ⎪ ⎪Y ≥ 0 ⎪ ⎩ D 0. Moreover, a gain matrix K satisfying the conditions (i) and (ii) can be computed as follows K = Y D−1 , where the matrices D and Z are any feasible solution to the above LMI problem. Proof. It is easy to see that u(t) ≥ 0 is equivalent to the positivity of its gain K ≥ 0. The rest of the proof mimics that one of Theorem 2, so that it is omitted.
5 Conclusions We have provided necessary and sufficient conditions for the asymptotic stability of linear positive systems subject to time-varying delays. We have introduced an original method for solving directly the proposed stability and stabilization problems. In addition to developing theoretical results, all the proposed conditions are necessary and sufficient, which turn out to be solvable in terms of LP or LMI. Acknowledgements. This work is funded by Ramon y Cajal grant, Spain.
References 1. Ait Rami, M., Tadeo, F.: Controller Synthesis for Linear Systems to Impose Positiveness in Closed-Loop States. In: Proceedings of the IFAC World Congress, Prague (2005) 2. Ait Rami, M., Tadeo, F.: Positive observation for positive discrete linear systems. In: IEEE CDC (2006) 3. Ait Rami, M., Tadeo, F.: Linear programming approach to impose positiveness in closedloop and estimated states. In: Proceedings Sixteenth International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan (2006) 4. Ait Rami, M., Tadeo, F.: Controller Synthesis for Positive Linear Systems with Bounded controls. IEEE Trans. on Circuits and Sys. II 54, 151–155 (2007) 5. Ait Rami, M., Cheng, C.H., de Prada, C.: Tight robust interval observers: an LP approach. In: IEEE CDC (2008) 6. Ait Rami, M., Helmke, U., Tadeo, F.: Positive observation problem for time-delays linear positive systems. In: Proceedings of the 15th IEEE Med. conf., Athens (2007) 7. Ait Rami, M., Helmke, U., Tadeo, F.: Positive observation problem for linear time-lag positive systems. In: The 3rd IFAC Symposium on System, Structure & Control, Foz do Iguau, Brazil (2007)
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8. Ait Rami, M., Hmamed, A., Alfidi, M.: L1 stability and stabilization of positive 2D contineous systems. Syst. and Control Letters (2008) (accepted) 9. Ait Rami, M., Tadeo, F., Benzaouia, A.: Control of constrained positive discrete systems. In: Proceedings of the American Control Conf., New York (2007) 10. Berman, A., Plemmon, R.J.: Nonnegative matrices in the mathematical sciences. SIAM Classics Appl. Maths (1994) 11. Berman, A., Neumann, M., Stern, R.J.: Nonnegative Matrices in Dynamic Systems. Wiley, New York (1989) 12. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000) 13. Haddad, W.M., Chellaboina, V.: Stability theory for nonegative and compartmental dynamical systems with time delay. Syst. and Control Letters 51, 355–361 (2004) 14. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977) 15. Hmamed, A., Ait Rami, M., Afidi, M.: Controller synthesis for positive 2D systems described by the Roesser model. In: IEEE CDC (2008) 16. Hmamed, A., Benzaouia, A., Ait Rami, M., Tadeo, F.: Positive stabilization of discretetime systems with unknown delay and bounded controls. In: European Control Conference, Kos, Greece (2007) 17. Hmamed, A., Benzaouia, A., Ait Rami, M., Tadeo, F.: Memoryless control to drive states of delayed continuous-time systems within the nonnegative orthant. In: Proceedings of the 17th World Congress, Seoul, Korea (2008) 18. Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge Univ. Press, Cambridge (1991) 19. Kaczorek, T.: Positive 1D and 2D Systems. Springer, UK (2001) 20. Lewis, R.M., Anderson, B.D.O.: Insensitivity of a class of Nonlinear Compartmental Systems to the Introduction of Arbitrary Time Delays. IEEE Trans. on Circuit and Sys. 27, 604–612 (1980) 21. Lewis, R.M., Anderson, B.D.O.: Necessary and Sufficient Condition for DelayIndependent Stability of Linear Autonomous Systems. IEEE Trans. Aut. Contr. 25, 735– 739 (1980) 22. Luenberger, D.G.: Introduction to Dynamic Systems. Wiley, New York (1979) 23. Ohta, Y.: Stability Criteria for Off-Diagonally Monotone Nonlinear Dynamical Systems. IEEE Trans. on Circuit and Sys. 27, 956–962 (1980)
Linear Programming Approach for 2-D Stabilization and Positivity Mohammed Alfidi, Abdelaziz Hmamed and Fernando Tadeo
Abstract. The problem of synthesizing stabilizing state-feedback controllers is solved when the closed-loop system is required to remain positive, for the class of 2-D linear systems described by the Fornasini-Marchesini second model. First, a constructive necessary and sufficient condition expressed as a Linear Programming problem is provided for stabilization of these systems when the states must be nonnegative (assuming that the boundary conditions are nonnegative). It is shown how it is simple to include additional constraints (such as positive controls). Moreover, this result is also extended to include uncertainty in the model, making possible to synthesize robust state-feedback controllers, solving Linear Programming problems. Some numerical examples are included to illustrate the proposed approach for different design problems. Keywords: positive 2-D systems, stabilization, Fornasini-Marchesini second model, linear programming.
1 Introduction During the last two decades, the two-dimensional (2-D) systems theory has been given a considerable attention by many researchers. These 2-D linear models were introduced in the seventies [7, 10] and have found many applications, in areas such as digital data filtering, image processing [18], partial differential equations [17], Mohammed Alfidi, Abdelaziz Hmamed LESSI, Department of Physics, Faculty of Sciences Dhar El Mehraz, B.P. 1796, 30000 Fes-Atlas, Morocco, e-mail:
[email protected],
[email protected] Fernando Tadeo Universidad de Valladolid, Dept. Ingenieria de Sistemas y Automatica, 47005 Valladolid, Spain, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 217–232. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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etc. In connection with the Roesser [18] and Fornasini-Marchesini [7] models, some important problems, such as realization, controllability or minimum energy control, have been extensively investigated (see for example [12]). On the other hand, the stabilization problem is not fully investigated and still not completely solved. In particular, it has not been completely solved for positive 2-D systems, that is, 2-D systems where the states always remain nonnegative. Recently, a growing interest in both theory and application of positive 2-D systems has been seen [2, 4, 13, 19, 21]. For a complete monograph on positive 2-D systems, see [13]. In the present paper, we first analyze the stability of linear positive 2-D models [7], following ideas borrowed from 1-D systems [1], already used by some of the authors for Roesser models [11], deriving a necessary and sufficient condition for 2D stability, based on simple linear inequalities. From this result, a simple numerical method is proposed for a complete treatment of the stabilization problem of these positive 2-D systems, when they can be described by a Fornasini-Marchesini second model (A parallel result for Roesser 2-D systems is presented in [2]). Moreover, based on this approach, necessary and sufficient conditions, expressed in terms of a Linear Programming problem, are proposed for the stabilization problem. We must point out that although it is known that the stability of linear digital 2-D systems can be reduced to checking the stability of a 2-D characteristic polynomial [3, 22], it seems difficult to apply this result in practice for the control synthesis problem. Thus, in the literature, various types of easily checkable but only sufficient conditions for asymptotic stability and stabilization problems for 2-D linear systems have been proposed [9, 15, 16, 23]. In contrast, the results in this paper are given as necessary and sufficient conditions. This paper is organized as follows: Section 2 presents the problem formulation and some preliminary results. Section 3 presents first the 2-D stability problem, which is then used to derive the main controller synthesis results, that are also extended to other related problems (bounded control and uncertainty). Finally, after presenting some illustrative examples in Section 4, some conclusions are given. Notations Rn+ denotes the non-negative orthant of the n-dimensional real space Rn . M T denotes the transpose of the real matrix M. For a real matrix M, M > 0 denotes a positive matrix, with all its components positive (mi j > 0), and M ≥ 0 denotes a nonnegative matrix, with none of its components negative (mi j ≥ 0). I denotes the identity matrix of appropriate order and N denotes the set of integer numbers. ρ (M) denotes the spectral radius of M ∈ Rn×n .
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2 Problem Formulation and Preliminaries Consider a linear homogeneous 2-D system described by the following FornasiniMarchesini second model [7]:
x(i + 1, j) u(i + 1, j) x(i + 1, j + 1) = A +B (1) x(i, j + 1) u(i, j + 1) where A = [A1 A2 ] ∈ Rn×2n and B = [B1 B2 ] ∈ Rn×2m are given real matrices, x(i, j) ∈ Rn is the state matrix and u(i, j) ∈ Rm is the input matrix. The boundary conditions for (1) are given by x(i, 0) = xi0 ∀ i ∈ N (2) x(0, j) = x j0 ∀ j ∈ N In the sequel, the following definition will be used. Definition 1. System (1)-(2) with zero input (u = 0) is said to be a positive 2D Fornasini-Marchesini system if for any given nonnegative boundary conditions x j0 ≥ 0 and xi0 ≥ 0, the resulting states are always nonnegative, that is, x(i, j) ≥ 0 for all i, j ∈ N. The following result shows how one can check the positiveness of System (1) (see [13]). Proposition 1. System (1)-(2) with zero input (u = 0) is positive if and only if A ∈ Rn×2n . + The asymptotic stability for general Fornasini-Marchesini second models [7] has been extensively studied in the literature. For example, a well-known necessary and sufficient frequency condition for asymptotic stability is stated in the following result. Lemma 1. Let A1 ∈ Rn×n and A2 ∈ Rn×n be given constant real matrices. Then, the 2-D system described by the Fornasini-Marchesini second model (1) with zero input, is asymptotically stable if and only if any of the following conditions holds (i) ρ (A1 + zA2 ) < 1,
|z| = 1 (3)
(ii) det(I − z1 A1 − z2 A2 ) = 0, ∀(z1 , z2 ) ∈ {(z1 , z2 ) : |z1 | ≤ 1, |z2 | ≤ 1} In the sequel, our purpose is to investigate the existence of state-feedback control laws of the form u(i, j) = Kx(i, j), (4) such that the resulting closed-loop system: x(i + 1, j + 1) = A¯
x(i + 1, j) x(i, j + 1)
(5)
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is positive and asymptotically stable, where K [ki j ] ∈ Rl×n is the controller gain to be determined and
K 0 ¯ ¯ ¯ A = A + BK, K = . (6) 0 K Of course, if one utilizes directly the results of Lemma 1 and Proposition 1, one can have the following necessary and sufficient condition for the closed-loop system to be positive and asymptotically stable: ⎧ • A1 + B1 K1 is a nonnegative matrix, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ • A2 + B2K is a nonnegative matrix, and (7) ⎪ ⎪ ⎪ ⎪ • det(In1 − z1 (A1 + B1 K1 ) − z2 (A2 + B2 K)) = 0, ⎪ ⎪ ⎩ ∀(z1 , z2 ) ∈ {(z1 , z2 ) : |z1 | ≤ 1, |z2 | ≤ 1}. However, this is a formulation which leads to a problem hard to solve, since we have a linear constraint (the positivity constraint) mixed with the very highly nonlinear infinite dimensional constraint (the asymptotic stabilizability constraint). A significant contribution of this paper is reflected by the simplicity and completeness of the solution provided. Effectively, all the provided main results involve easily checkable necessary and sufficient conditions. In fact, it will be shown how we can completely solve problem (7) in terms of a Linear Programming problem, which avoids unnecessary computational burdens. Previous results and equivalent conditions for a nonnegative matrix M to be Schur (or equivalently ρ (M) < 1) can be found in the literature [6]. In the following, some known results are presented and will be used in order to derive our main results. Proposition 2. [1] Let M be a nonnegative matrix. Then, the following conditions are equivalent: (i) The 1-D system x(k + 1) = Mx(k) is asymptotically stable (or equivalently, ρ (M) < 1). (ii)There exists a positive vector λ > 0 such that (M − I)λ < 0.
3 Stabilization with Positivity of 2-D Fornasini- Marchesini Systems 3.1 Stability Analysis for Positive 2-D Systems This section provides preliminary stability results for the free linear 2-D system described by the Fornasini-Marchesini second model (1) when A ≥ 0:
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x(i + 1, j + 1) = A
x(i + 1, j) x(i, j + 1)
(8)
In fact, it will be shown that the asymptotic stability of System (8) (under the positivity constraint) is equivalent to the stability and positivity of the following 1-D linear discrete-time system: x(k ˜ + 1) = (A1 + A2 )x(k), ˜
(9)
where system (9) is called positive if for any given nonnegative initial condition the resulting trajectories are also nonnegative. In fact, it has already been proved that if A1 ≥ 0, and A2 ≥ 0, the system (9) is positive [1]. Now, some results are needed in order to establish our main stability result. Theorem 1. [19, 20] Assume that the system (8) is positive (or equivalently that A1 and A2 are nonnegative). Then, the following statements are equivalent: i) det(In − z1 A1 − z2 A2 ) = 0, ∀(z1 , z2 ) ∈ {(z1 , z2 ) : |z1 | ≤ 1, |z2 | ≤ 1}. ii) ρ (A1 + A2 ) < 1. iii)The positive 2-D system (8) is asymptotically stable. iv) The positive 1-D system (9) is asymptotically stable. Now, we are in position to state the result that will be used in the rest of the paper. Corollary 1. Assume that the matrices A1 ≥ 0 and A2 ≥ 0, the following statements are equivalent: i) The 2-D system described by the Fornasini-Marchesini second model (8) is positive and asymptotically stable. ii) The 1-D system described by (9) is positive and asymptotically stable. iii)The matrices A1 and A2 are nonnegative and there exists a vector d ∈ Rn such that the following Linear Program condition is fulfilled: (A1 + A2 − I)d < 0, d > 0
(10)
Proof. ii) ⇔ iii) results from Proposition 2. i) ⇒ iii): Using Proposition 1 we have that A1 and A2 are nonnegative, so from Lemma 1 the asymptotic stability of the 2-D system (8) is equivalent to det(In − z1 A1 − z2 A2 ) = 0, ∀(z1 , z2 ) ∈ {(z1 , z2 ) : |z1 | ≤ 1, |z2 | ≤ 1}, which by Theorem 1 is equivalent to ρ (A1 + A2 ) < 1, and using Proposition 2 implies iii). Finally, iii) ⇒ i) follows from Proposition 1 combined with Proposition 2, Theorem 1 and Lemma 1.
3.2 Proposal for Synthesis of Stabilizing Controllers This section studies the stabilization problem of linear 2-D systems described by the Fornasini-Marchesini second model under state-feedback of the form u(i, j) = Kx(i, j). This control law is designed to ensure the positivity and the asymptotic
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stability of the resulting closed-loop system. As will be shown, our proposed approach does not impose any restriction on the dynamics of the governed system. For instance, the free Fornasini-Marchesini second model can possibly be nonpositive. In this case, our synthesis design can be interpreted as enforcing the system to be positive. Now, consider the closed-loop Fornasini-Marchesini second model:
x(i + 1, j) ¯ x(i + 1, j + 1) = (A + BK) (11) x(i, j + 1) where A = [A1 A2 ] ∈ Rn×2n and B = [B1 B2 ] ∈ Rn×2m are supposed to be any real matrices (not necessarily nonnegative). In what follows we provide the main result of this section. Theorem 2. The closed-loop system (11) is positive and asymptotically stable for any boundary conditions, if and only if there exist n + 1 vectors d = [d1 . . . dn ]T ∈ Rn and y1 . . . yn ∈ Rm such that ⎧ n ⎪ ⎪ (A + A − I )d + (B + B ) yi < 0, ⎪ n 1 2 1 2 ∑ ⎪ ⎨ i=1 (12) d > 0, ⎪ ⎪ ⎪ a d + b1i y j ≥ 0, 1 ≤ i, j ≤ n, ⎪ ⎩ 1i j j a2i j d j + b2i y j ≥ 0, 1 ≤ i, j ≤ n, with A1 = [a1i j ], A2 = [a2i j ], BT1 = [bT11 ... bT1n ] and BT2 = [bT21 ... bT2n ]. Moreover, the gain matrix K is given by: K = [d1−1 y1 . . . dn−1 yn ].
(13)
Proof. Assume that condition (12) is satisfied and define the appropriate matrix K = [k1 , . . . , kn ] with columns constructed as ki = di−1 yi , for i = 1, . . . , n, and partitioned as K = [K K]. By this construction, it is easy to see that A1 + B1 K and A2 + B2 K are nonnegative matrices. Effectively, from the last inequalities in condition (12) we have for i, j = 1, . . . , n: 0 ≤ (a1i j d j + b1iy j )d −1 j = a1i j + b1i k j = (A1 + B1 K)i j . 0 ≤ (a2i j d j + b2iy j )d −1 j = a2i j + b2i k j = (A2 + B2 K)i j . Next, we show the asymptotic stability under the feedback control u = Kx. Using n
the previous gain, we obtain by calculation B1 Kd + B2 Kd = (B1 + B2 )( ∑ yi ) which i=1
is used in condition (12) and leads to (A1 + B1K + A2 + B2 K − In)d < 0. Now, since d > 0, A1 + B1 K and A2 + B2 K are nonnegative, then by using Corollary 1, we conclude that the 2-D system described by the closed-loop Fornasini-Marchesini second model (11) is asymptotically stable. The rest of the proof follows the same line of argument, so it is omitted.
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Remark 1. We emphasize that the LP formulation proposed in Theorem 2 does not impose any restriction on the dynamics of the governed system. In fact, the matrix A may have negative components, or equivalently, the free system may be nonpositive. In this case the proposed synthesis methodology can be viewed as enforcing a nonpositive system to be positive. It must be pointed out that the requirement of positivity for the controls can also be handled by an LP approach similar to (12), that is now provided, and can be proved following the same ideas: Theorem 3. The following statements are equivalent: i) There exists a positive state-feedback law u(i, j) = Kx(i, j) ≥ 0 such that the closed-loop system (11) is positive and asymptotically stable for any initial boundary conditions. ii) There exists a matrix K ∈ Rm×n such that K ≥ 0, A1 + B1 K and A2 + B2 K are nonnegative matrices and A1 + B1 K + A2 + B2 K is a Schur matrix (that is, ρ (A1 + B1 K + A2 + B2K) < 1). iii)The following LP problem in the variables d = [d1 . . . dn ]T ∈ Rn and y1 , . . . , yn ∈ Rm is feasible. ⎧ n ⎪ ⎪ (A1 + A2 − In )d + (B1 + B2) ∑ yi < 0, ⎪ ⎪ ⎪ ⎪ i=1 ⎨ d > 0, (14) ⎪ yi ≥ 0, 1 ≤ i ≤ n, ⎪ ⎪ ⎪ ⎪ a d + b1iy j ≥ 0, 1 ≤ i, j ≤ n, ⎪ ⎩ 1i j j a2i j d j + b2iy j ≥ 0, 1 ≤ i, j ≤ n, with A1 = [a1i j ], A2 = [a2i j ], BT1 = [bT11 ...bT1n ] and BT2 = [bT21 ...bT2n ]. Moreover, the gain matrix in conditions i) and ii) can be chosen as: K=[d1−1 y1 . . . dn−1 yn ], where d and y1 , . . . , yn are given by any feasible solution to the above LP problem Remark 2. Note that if a negative state-feedback control law is to be considered it suffices to impose yi ≤ 0 instead of yi ≥ 0 in the previous LP formulation.
3.3 Synthesis with Uncertain Plant An important issue when designing control systems for practical applications is ensuring the robust stability, that is, the closed-loop system should remain stable under uncertainty in the plant model. In fact, the system matrices that describe a 2-D system are frequently uncertain [5, 8]. Thus, this section considers the robust stabilization of Fornasini-Marchesini second models, when the dynamics are not exactly known. More precisely, we assume
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that the uncertainties can be captured using a polytopic domain representation [8]. That is, we consider that our 2-D system can be represented by the following uncertain model:
x(i + 1, j) u(i + 1, j) + B(α ) , (15) x(i + 1, j + 1) = A(α ) x(i, j + 1) u(i, j + 1) where A(α ) = [A1 (α ) A2 (α )] ∈ Rn×2n and B(α ) = [B1 (α ) B2 (α )] ∈ Rn×2m are supposed to be not exactly known, but they belong to the following convex set: A1 (α ) A2 (α ) B1 (α ) B2 (α ) ∈ D, (16) 9 D :=
l
l
i=1
i=1
∑ αi [Ai1 Ai2 Bi1 Bi2], ∑ αi = 1, αi ≥ 0
: ,
(17)
where [A11 A12 B11 B12 ], . . . , [Al1 Al2 Bl1 Bl2 ] are known matrices. Thus, the robust synthesis design problem consists in finding a single constant gain matrix K for which the following closed-loop system is positive and asymptotically stable for every [A1 (α ) A2 (α ) B1 (α ) B2 (α )] ∈ D:
x(i + 1, j) ¯ x(i + 1, j + 1) = (A(α ) + B(α )K) . (18) x(i, j + 1) Using a simple convexity property, the LP formulation proposed in Theorem 2 can be easily extended to these systems with polytopic uncertainties, by just repeating the LP conditions for all the vertices in the polytope, using similar ideas as those proposed by some of the authors in [11], as stated in the following result. Theorem 4. There exists a robust state-feedback law u(i, j) = Kx(i, j) such that the resulting closed-loop system (18) is positive and asymptotically stable for any initial boundary conditions and for every [A1 (α ) A2 (α ) B1 (α ) B2 (α )] ∈ D, if the following LP problem in the variables d = [d1 . . . dn ]T ∈ Rn and y1 , . . . , yn ∈ Rm , is feasible. ⎧ n k + Ak − I )d + (Bk + Bk )( ⎪ ⎪ (A n ⎪ 1 2 1 2 ∑ yi ) < 0, f or k = 1, ..., l ⎪ ⎨ i=1 d > 0, ⎪ ⎪ ak j d j + bk1iy j ≥ 0, 1 ≤ i, j ≤ n, k = 1, ..., l, ⎪ ⎪ ⎩ 1i ak2i j d j + bk2iy j ≥ 0, 1 ≤ i, j ≤ n, k = 1, ..., l T
T
T
T
T
(19)
T
with Ak1 = [ak1i j ], Ak2 = [ak2i j ] Bk1 = [bk11 . . . bk1n ] and Bk2 = [bk21 . . . bk2n ] where n = n1 + n2 and k = 1, ..., l. Moreover, the gain matrix can be computed as K = [d1−1 y1 . . . dn−1 yn ] where d and y1 , . . . , yn are given by any feasible solution to the LP problem 19.
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Proof. If a solution {d1 , . . . , dn , y1 , . . . , yn } fulfills (19), then it also fulfills the following LP for any A∗1 A∗2 B∗1 B∗2 ∈ D: ⎧ n ⎪ ⎪ (A∗1 + A∗2 − In)d + (B∗1 + B∗2 ) ∑ yi < 0, f or k = 1, ..., l ⎪ ⎪ ⎨ i=1 d > 0, ⎪ ∗ ∗ ⎪ ⎪ ⎪ a1i j d j + b1i y j ≥ 0, 1 ≤ i, j ≤ n, k = 1, ..., l, ⎩ ∗ a2i j d j + b∗2i y j ≥ 0, 1 ≤ i, j ≤ n, k = 1, ..., l.
(20)
Thus, by convexity it is only necessary to check the LP (12) at the vertices of the polytope, which gives the result.
4 Numerical Examples 4.1 Stabilization Example As first illustration of the proposed design methodology for stabilization, we deal with the following system given by (1) and the following matrices: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −1.5 0.1 0 00 0 1 0 A1 = ⎣ 0.2 0.5 0.3 ⎦, A2 = ⎣ 0 0 0 ⎦, B1 = ⎣ 1 ⎦ and B2 = ⎣ 0 ⎦ . 0 0 0 1 1 0.2 0 0 Of course, as the matrix A1 is not nonnegative, the free system (i.e., when u = 0) is not positive. This fact is also illustrated by the evolution of the free system depicted in Figure 1 (starting from nonnegative boundary conditions). According to the result given in Lemma 1, the system in open-loop is also unstable, i.e., the spectral radius ρ (A1 + zA2 ) > 1, where z = e jω , for all ω ∈ [0, 2π ] (see Figure 2). Then, the objective is to design a state-feedback controller that stabilizes the system and enforces it to be positive. For this purpose, it suffices to use the result of Theorem 2 and find a solution fulfilling the inequalities (12). Since we have shown in Theorem 2 that the gain of a stabilizing control is given by K = [y1 d1−1 y2 d2−1 y3 d3−1 ], we have used the following feasible solution to the corresponding inequalities (12): ⎡ ⎤ ⎡ ⎤ d1 8.5299 ⎢ d2 ⎥ ⎢ 138.5801 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ d3 ⎥ ⎢ 202.3965 ⎥ ⎢ ⎥=⎢ ⎥ ⎢ y1 ⎥ ⎢ 14.5909 ⎥ , ⎢ ⎥ ⎢ ⎥ ⎣ y2 ⎦ ⎣ −11.4508 ⎦ y3 0.8102 to obtain the following gain of a stabilizing controller: K = [1.7106 − 0.0826 0.0040].
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Fig. 1 Open-loop evolution of x1 (i, j)
1.525
spectral radius
1.52 1.515 1.51 1.505 1.5
0
1
2
3
4
5
6
7
w Fig. 2 Spectral radius of the open-loop system
The corresponding system matrices in closed-loop are given by: ⎡ ⎡ ⎤ ⎤ 0.2106 0.0174 0.0040 0 0 0 A1 + B1K = ⎣ 1.9106 0.4174 0.3040 ⎦ and A2 + B2 K = ⎣ 0 0 0 ⎦ . 0 0 0 1 1 0.2 Hence, it suffices to look at the entries of the matrices A1 + B1 K and A2 + B2 K, to conclude that the closed-loop system is positive (according to Proposition 1). In addition, according to Corollary 1, the closed-loop system is asymptotically stable (it can be checked that the matrix A1 + B1 K + A2 + B2K has all the eigenvalues inside the unit circle, namely λ1 = 0.92, λ2 = 0.17 and λ3 = −0.26).
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Figure 3 shows the evolution of the closed-loop state x2 (i, j), using the proposed feedback law, from nonnegative boundary conditions: we can see that, effectively, all the states are nonnegative and converge to zero.
Fig. 3 Closed-loop evolution of x2 (i, j) under the designed state feedback
4.2 Positive Feedback Example Let us consider the non-positive 2D system described by (5) with the following system matrices:
0.2 0.7 0 0 0 0 A1 = , A2 = , B1 = and B2 = . 0 0 0.9 −0.8 0 1 Note that the system is open-loop unstable, since the spectral radius ρ (A1 + e jω A2 ) has values greater than 1.0 for some ω ∈ [0, 2π ] (see Figures 4 and 5). Here, our task is to use a positive state-feedback control, in order to stabilize the system and enforce the state to be positive. Based on the result provided by Theorem 3 the following inequalities must be fulfilled: ⎤ ⎡ −0.8 0.7 0 0 ⎡ ⎤ ⎢ 0.9 −1.8 1 1 ⎥ d1 ⎥ ⎢ ⎢ −1 0 0 0 ⎥ ⎢ d2 ⎥ ⎥⎢ ⎥ ⎢ ⎢ 0 −1 0 0 ⎥ ⎣ y1 ⎦ < 0, ⎥ ⎢ ⎣ 0 0 −1 0 ⎦ y2 0 0 0 −1
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Fig. 4 Uncertain plant example: Open-loop evolution of x1 (i, j)
1.3
spectral radius
1.2 1.1 1 0.9 0.8
0
1
2
3
4
5
6
w Fig. 5 Uncertain plant example: Open-loop spectral radius
⎡
−0.2 0 0 ⎢ 0 −0.7 0 ⎢ ⎣ −0.9 0 −1 0 0.8 0
⎤⎡ ⎤ 0 d1 ⎢ d2 ⎥ 0 ⎥ ⎥ ⎢ ⎥ ≤ 0. 0 ⎦ ⎣ y1 ⎦ y2 −1
Now, from the following feasible solution to the above LP problem: ⎡ ⎤ ⎡ ⎤ d1 97.3594 ⎢ d2 ⎥ ⎢ 106.6566 ⎥ ⎢ ⎥=⎢ ⎥ ⎣ y1 ⎦ ⎣ 6.6901 ⎦ , y2 89.5453
7
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we obtain (as stated in Theorem 3) that a stabilizing controller has the gain K = [0.0687 0.8396] (computed from K = [y1 d1−1 y2 d2−1 ]). Thus, the dynamic matrices of the closed-loop system are:
0.2 0.7 0 0 and A2 + B2 K = , A1 + B1 K = 0 0 0.9687 0.0396 It can be seen that all the states of system are convergent and asymptotically stable (see Figure 6). It can also be checked that the matrix A1 + B1 K + A2 + B2 K has all its eigenvalues inside the unit circle, namely at λ1 = 0.9471 and λ2 = −0.7076. From this, it follows that the system is asymptotically stable, by simple looking at the entries of the dynamic matrices A1 + B1 K and A2 + B2 K. Then, we can conclude that, according to Proposition 2.1, the governed system is positive.
Fig. 6 Positive feedback example: Closed-loop evolution of x3 (i, j)
4.3 Uncertain Plant Example In this example, we consider an uncertain system (15) subject to a parametric perturbation, described by the following matrices: ⎤ ⎡ ⎡ ⎡ ⎤ ⎤ −1.5 0.1 0 0 0 0 1 ⎦, B1 (α ) = ⎣ 1 − 0.01α ⎦ 0 A1 = ⎣ 0.2 0.5 0.3 ⎦ , A2 (α ) = ⎣ 0 0 0 0 0 0 1 1 0.2 − 0.01α ⎡ ⎤ 0 and B2 = ⎣ 0 ⎦ . 0
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The uncertain parameter is 0 ≤ α ≤ 1. We are looking for a robust statefeedback control which stabilizes and enforces the positivity of all the plants between the two extreme plants (α = 0 and α = 1). By applying Theorem 4, based on K = [y1 d1−1 y2 d2−1 y3 d3−1 ]), we obtain the following gain of a robust stabilizing controller: K = [1.7446 − 0.0836 0.0039].
Fig. 7 Uncertain plant example: Closed-loop state evolution for the extreme plant corresponding to α = 0
Hence, with this gain all the closed-loop systems between the two extreme plants (from α = 0 to α = 1) are positive and asymptotically stable. As illustration, the state evolution of the two extreme plants (α = 0 and α = 1), starting from initial positive boundaries, are depicted in Figures 7 and 8, respectively.
5 Conclusion An approach for solving the stability synthesis problem for 2-D systems described by the Fornasini-Marchesini second model under the requirement of positivity of the closed-loop system is presented. For this, necessary and sufficient conditions for the solvability of the stabilization problem have been proposed, including the presence of bounds on the control signal and uncertainty in the plant. Moreover, it has been shown that the proposed conditions are solvable in terms of Linear Programs, which are simple to solve using off-the-self software. Some numerical example illustrate the proposed approach. It must be pointed out that the proposed approach is quite general and can be extended to more involved problems. In fact, work is being
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Fig. 8 Uncertain plant example: Closed-loop state evolution for the extreme plant corresponding to α = 1
carried out to extend these ideas to more general 2-D systems, 2-D systems with delays, n-D systems, etc. Acknowledgements. This work is funded by MiCInn project DPI2007-66718-C04-02. The authors would like to thank Dr. Ait Rami for many helpful discussions.
References 1. Ait Rami, M., Tadeo, F.: Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits and Systems-II 54(2), 151–155 (2007) 2. Alfidi, M., Hmamed, A.: Control for stability and positivity of 2D linear discrete-time systems. WSEAS Transactions on Systems and Control 12(2), 546–556 (2007) 3. Anderson, B.D.O., Agathoklis, P., Jury, E.I., Mansour, M.: Stability and the Matrix Lyapunov Equation for Discrete 2-Dimensional Systems. IEEE Transactions on Circuits and Systems 33, 261–266 (1986) 4. Bailo, E., Bru, R., Gelonch, J., Romero, S.: On the Reachability Index of Positive 2-D Systems. IEEE Transactions on Circuits and Systems-II 53(19), 997–1001 (2006) 5. Dua, C., Xie, L., Zhang, C.: H∞ control and robust stabilization of two-dimensional systems in Roesser models. Automatica 37(2), 205–211 (2001) 6. Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Pure and Applied mathematics. John Wiley & Sons, Inc., New York (2000) 7. Fornasini, E., Marchesini, G.: State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control 21(4), 484–492 (1976) 8. Gao, H., Lam, J., Wang, C., Xu, S.: H∞ model reduction for uncertain two-dimensional discrete systems. Optimal Control Appl. and Methods 26(4), 199–227 (2005)
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9. Galkowski, K., Rogers, E., Xu, S., Lam, J., Owens, D.H.: LMIs-A Fundamental Tool in Analysis and Controller Design for Discrete Linear Repetitive Process. IEEE Transactions on Circuits and Systems I 49(6), 768–778 (2002) 10. Givone, D.D., Roesser, R.P.: Multidimensional linear iterative circuits - General properties. IEEE Transactions on Computers 21(10), 1067–1073 (1972) 11. Hmamed, A., Ait Rami, M., Alfidi, M.: Controller synthesis for positive 2D systems described by the Roesser model. In: Proceedings of the 47th Conference on Decision and Control, Cancun, Mexico, pp. 387–391 (2008) 12. Kaczorek, T.: Realization problem, reachability and minimum energy control of positive 2D Roesser model. In: Proceedings of the 6th Annual International Conference on Advances in Communication and Control, pp. 765–776 (1997) 13. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 14. Lancaster, P., Tismenetsky, M.: The Theory of Matrices. Academic Press, London (1985) 15. Lee, E.B., Lu, W.S.: Stabilization of Two-Dimensional Systems. IEEE Transactions on Automatic Control 30, 409–411 (1985) 16. Lu, W.S.: Some New Results on Stability Robustness of Two-Dimensional Discrete Systems. Multidimensional Systems and Signal Processing 5, 345–361 (1994) 17. Marszalek, W.: Two dimensional state-space discrete models for hyperbolic partial differential equations. Applied Mathematical Modelling 8, 11–14 (1984) 18. Roesser, R.: A discrete state-space model for linear image processing. IEEE Transactions on Automatic Control 20, 1–10 (1975) 19. Twardy, M.: An LMI approach to checking stability of 2D positive systems. Bulletin of the Polish Academy of Sciences: Technical Sciences 55(4), 385–395 (2007) 20. Valcher, M.E.: On the Internal Stability and Asymptotic Behavior of 2-D Positive Systems. IEEE Transactions on Circuits and Systems I 44(7), 602–613 (1997) 21. Valcher, M.E., Fornasini, E.: State models and asymptotic behaviour of 2D positive systems. IMA Journal of Mathematical Control and Information 12(1), 17–36 (1995) 22. Wu-Sheng, L., Lee, E.B.: Stability analysis for two-dimensional systems via a Lyapunov Approach. IEEE Transactions on Circuits and Systems 32, 61–68 (1985) 23. Yaz, E.: On State-feedback Stabilization of Two-Dimensional Digital Systems. IEEE Transactions on Circuits and Systems 32, 1069–1070 (1985)
An Algorithmic Approach to Orders of Magnitude in a Biochemical System Eric Benoˆıt and Jean-Luc Gouz´e
Abstract. We use orders of magnitudes of variables and parameters of a chemical system described by an ordinary differential equation, to obtain a partition of the state space in boxes (hyper-rectangles). From the fast system in each box, we derive rules of transition, and obtain a transition graph. This graph can be used for a qualitative simulation and validation of the system.
1 Introduction For biological models, the positivity of the variables is of primary importance, but the units (dimensions) and the scaling (order of magnitude) of the variables and the parameters also play a central role. Making the model without units permits to compare the variables between them, and to group parameters to reduce their number ([8]). The orders of magnitude are a good way of simplifying a big model, by keeping only the “large” or “fast” part and neglecting the “small” or “slow” part. To be more precise, consider a classical biochemical model, described by an ordinary differential equation. The well known method of “quasi-steady state approximation” ([6])(or, more mathematically, the singular perturbation method [7]) is related to the above kind of approximation: the order of magnitude of some groups of parameters leads to the fact that the rates of variation wrt. time of some variables are far greater than some others, and (under some assumptions) these “fast” variables can be put to their quasi-equilibria, leading to a simpler differential system with only “slow” variables. Eric Benoˆıt Laboratoire de Math´ematiques et Applications, Universit´e de la Rochelle, and COMORE, INRIA Sophia-Antipolis, France, e-mail:
[email protected] Jean-Luc Gouz´e COMORE, INRIA Sophia-Antipolis, France, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 233–241. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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But the order of magnitude of the variables is also important, although more rarely used [6]. For a biologist, it is very important to check that, given the parameters, the system will evolve in some domain where, for example, the concentration of some complex is very small with respect to the concentrations of the other reactants: this gives a mean to check qualitatively the model versus experimental measurements and a priori knowledge. Here our aim will be to combine the different approaches, using the order of magnitudes of the variables and of the parameters, and also the positivity of some terms and the structure of the system (chemical systems). Our goal will be to analyze large chemical systems, in an algorithmic way, with the help of a computer. This paper gives the first steps of this methodology. We define what we call orders of magnitude for the variables and parameters. We partition the space of the variables (without units) into rectangular hyper rectangles (called boxes), each box representing an order of magnitude. Within a box, we study the system and separate it into a “fast” and a “slow” part. Then we study the fast subsystem, and show that, with good hypotheses, we can obtain information on the full system. Roughly, we can distinguish two types of boxes: in the first type, any solution will exit the box in finite time and go to other boxes. The box is transitory, and our algorithm gives the possible transitions between the transitory boxes. We give sufficient conditions on the parameters so that the exit is possible or forbidden through a given face of the box. In the second type of box, a solution can remain during a “long” time, possibly until reaching the equilibrium. In this kind of box, it is possible to proceed to a singular or regular perturbation analysis to simplify the system ([7]). Finally, we obtain a qualitative simulation through the transition graph of boxes, and the biologist can compare this simulation to his experiments. In particular, he can check that the temporal order of the changes in the orders of magnitude is compatible with the graph. This approach is quite similar to qualitative simulation and hybrid system approaches, where the transition graph is an abstraction (in the hybrid system sense) of the continuous system. For example, this method was used to describe the qualitative behavior of large genetic networks [5]: in each box, a piecewise linear system is given by the model, but changes from box to box. The qualitative simulations can be compared with experiments and noisy data. It is also possible to check properties of the transition graph by model checking techniques [1, 2]. The notion of reachability (the set of boxes that can be attained from an initial set) plays an important role. Another related work, more oriented toward control aspects, concerns the multiaffine systems defined inside hyper rectangles ([3]). The authors derive sufficient conditions for driving all the solutions in the rectangle through desired faces of exit in finite time. The paper is organized as follows: we first define the system (of biochemical type), then the scaling and the boxes. We give some mathematical lemmas describing the behavior of the solutions inside a box or on a face of a box; then we describe the algorithm implemented with Maple. We take the classical example of the Michaelis-Menten mechanism for enzymes, which was the subject of numerous
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papers concerning the different quasi steady states approximations obtainable from this system ([9]). Here we give the transition graph obtained for given values of the parameters.
2 The System We consider a biochemical system with N nonnegative variables Xi , i = 1 . . . N (these variables are typically chemical concentrations), and P positive parameters K j , and we write the general system: X˙ = F(X, K). In general, the variables Xi could have different units, but we suppose here, for the sake of simplicity, that we have scaled the variables by defining Xi0 = Xi /Xie where Xie is the chosen unit for Xi and Xi0 is the new unitless variable. The new system can be written: X˙ 0 = F(X 0 , K 0 ) where the K 0j are the new unitless parameters defined from the parameters K j and from the units Xie . We keep the same notations. In the following, we will need the explicit form of the system. We suppose that it can be written in the following form: X˙i = Ai00 + ∑ Ai j0 X j + j=i
∑
j=i,k=i, j≤k
Ai jk X j Xk − Xi (Bi0 + ∑ Bi j X j )
(1)
j
In the expression, the positive terms represent the reactants of chemical reactions producing Xi . The first term Ai00 is a constant input, the second term Ai j0 X j describes the kinetics of the reaction X j → Xi , the third positive term Ai jk X j Xk describes the kinetics of the reaction X j + Xk → Xi . The negative terms represent the decay of the reactant Xi . The first term Bi0 Xi can be seen as a degradation rate toward the exterior of the system, or as a term due to a reaction Xi → X j + Xk or Xi → X j , and the second term describes the kinetics of the reaction X j + Xi → Xk . We have done some (reasonable) hypotheses: in particular we suppose that the reactions are at most of order two (bi-molecular). We have chosen to restrict our study to such a particular form to be able to write explicit formulas and give explicit lemmas in the following. Yet, the principle of our method remains valid for more complex systems. The main hypothesis is that we are able to separate the right-hand side into groups with very different orders of magnitude. These hypotheses on the structure of the system give us the nonnegativity of the solutions, by showing that the field is repulsive on the boundary (Xi = 0 ⇒ X˙i ≥ 0).
3 The Scaling We first define a hyper rectangular “box“ corresponding to an order of magnitude. Let ω be a real (large) number, that will define the scale (ω > 1). The box B in the space of the variables is described by the notation B = [n1 , . . . , nN ] (where the n j are integers) and means that, for each i = 1, . . . N, the variable Xi is such that
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ω ni −1/2 ≤ Xi ≤ ω ni +1/2 . This box is of (logarithmic) length ω and of center ω ni . Now we can scale the variables inside each box by the change of variables xi = Xi /(ω ni ) so that, in the box, the new variable verifies ω −1/2 ≤ xi ≤ ω +1/2 . The new parameters (i.e., the coefficients of the monomials in the equations) after the scaling of the variables are denoted by k. The new scaled system inside some particular box [n1 , . . . nN ] is now written as x˙ = f (x, k) and the explicit form is the same as equation (1) with scaled variables and new parameters: x˙i = ai00 + ∑ ai j0 x j + j=i
∑
j=i,k=i, j≤k
ai jk x j xk − xi (bi0 + ∑ bi j x j )
(2)
j
The numbers ai jk , −bi j will now be called the coefficients of fi and designated as Coe f ( fi ) (a vector). We remark that they are positive or negative with this definition. We will write x > 0 to say that the vector x has nonnegative coordinates, not all equal to zero. To be more explicit, easy computations give that ai jk = ω (n j +nk −ni ) Ai jk , bi j = ω n j Bi j . In a given box, because of the scaling, the coefficients ai jk and −bi j will often have different order of magnitude, and the right-hand side of the equation (2) will be the sum of a large (or fast) part and of a small (or slow) part. Let us remark that, until now, we have not scaled the time. Therefore we can choose a new time scale τ , with τ = Ct, such that the larger coefficients are of order greater than one, and are separated from the other coefficients by a gap of length G. Let us explain that more precisely. Definition 1. In a box, for the appropriate change of time, the system (2) can be written in the form: ddxτ = f f (x, k) + f s (x, k) where the non-zero coefficients of f f are greater than G in absolute value, and the non-zero coefficients of f s are smaller than 1 in absolute value. The number G is called the gap between the coefficients in the box. This gap is nothing but the interval between the larger coefficients and the other. The grouping of the parameters around the gap is not unique. We will show below that, if the gap G is large enough, then the full system can be well approximated by the fast subsystem ddxτ = f f (x, k) and we deduce several properties concerning the fact that the solutions in the box have to escape this box via some faces, and that the exit through some other faces is forbidden.
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4 The Behavior in a Given Box Suppose, as above, that the solution is in some box B = [n1 , . . . , nN ]. The following lemma gives a sufficient condition for one coordinate to be monotone in B. We first define four different disjoint sets of indexes, each being a subset of {1, . . ., N}. Let E0 = {i;Coe f ( fif ) = 0}, E+ = {i;Coe f ( fif ) > 0}, E− = {i;Coe f ( fif ) < 0}. The set E∗ will contain the rest of the indexes, i.e. the indexes such that the fast part fif has positive and negative coefficients in the box. Of course E+ ∪ E− ∪ E0 ∪ E∗ = {1, . . . , N}. i Lemma 1. Suppose that i ∈ E+ . Then, if G > (N + 1)ω 2, then dx d τ > 0, i.e. the i-th coordinate is increasing in B. The box is exited in finite time. The face xi = ω +1/2 is a possible face of exit. The solution cannot exit through the face xi = ω −1/2 .
f
Proof. Because i ∈ E+ , all the coefficients of ( fi ) are zero or positive and greater than G, and at least one is positive. The coefficients of fis are lower than 1, and of any sign. In the box B, the following inequalities are valid for any j:
ω −1/2 ≤ x j ≤ ω +1/2 Now, given the explicit form (2), we can minorate x˙i with the above inequalities. The first part of the right-hand side ai00 + ∑( j=i) ai j0 x j + ∑( j=i,k=i) ai jk x j xk is (because of the positivity of the elements) minorated by G or Gω −1/2 or G/ω ; we keep G/ω because ω > 1 . The second (negative) part −xi (bi0 + ∑( j) bi j x j ) is minorated by −ω +1/2(1 + N ω +1/2 ), more roughly by (N + 1)ω because ω > 1. Finally i we obtain that, if G > (N + 1)ω 2, then dx d τ > 0. Now we give a lemma about the possible exit on a face. This lemma is used by the algorithm to compute if the exit by some face is possible. Lemma 2. Suppose that i ∈ E+ . Then, if G > (N + 1)ω , a solution in the box cannot exit through the face ω −1/2 . The proof is similar to the preceding one, except that xi has the value ω −1/2 . The third lemma is similar to Lemma 1, but for negative coefficients in the fast system. i Lemma 3. Suppose that i ∈ E− . Then, if G > N(N + 1)ω 2 /2, then dx d τ < 0, i.e. the i-th coordinate is decreasing in B. The box is exited in finite time. The face xi = ω −1/2 is a possible face of exit. The solution cannot exit through the face xi = ω 1/2 .
Lemma 4. Suppose that i ∈ E− . Then, if G > N(N + 1)ω /2, a solution in the box cannot exit through the face ω 1/2 . The next lemma is technical and gives an upper bound for the time needed for a solution to escape the box when the assumptions of Lemma 1 are fulfilled.
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Lemma 5. Suppose that i ∈ E+ . Then, if G > (N + 1)ω q , with real q > 2, then a solution x(τ ) is such that xi (τ ) cannot stay between ω −1/2 and ω 1/2 longer than 1 time T = N+1 ω 3/2−q (1 + O(1/ω )). Suppose that i ∈ E− . Then, if G > (N + 1)ω q , with real q > 2, then a solution x(τ ) is such that xi (τ ) cannot stay between ω −1/2 and ω 1/2 longer than time T = 1 1/2−q ln ω + O(ω 5/2−2q ). N+1 ω Next lemma makes use of the above estimates of the maximal time during which a coordinate xi stays in the box. We suppose now that the velocities of some coordinate i of the fast system are equal to zero (for these coordinates, the coefficients are all slow), and that some other coordinates j of the fast system have a positive or negative velocity. Then we show that, roughly, (up to some approximation depending on the gap) the solutions will escape the box through the faces corresponding to x j , and not xi . / Then if xi (0) < Bω 1/2 , with Lemma 6. Suppose that i ∈ E0 , and that E+ ∪E− = 0. N (2−q) (2−q) B = 1− 2 ω + o(ω ), then x(τ ) cannot escape through the face xi = ω 1/2 . Similarly, suppose that i ∈ E0 , and that E+ ∪ E− = 0. / Then if xi (0) > B ω −1/2 ,with (2−q) (2−q) B = 1+ω + o(ω ), then x(τ ) cannot escape through the face xi = ω −1/2 . Remark that, if the gap G is large, the bounds B and B in the lemma are close to 1. In the algorithm below, the faces xi will be called ”almost forbidden“ to express the fact that the algorithm will consider that no solution escapes through the faces xi = ω ±1/2 . Due to the lack of space, we do not give the proofs (see [4]). Remark 1 (Stoichiometric invariants). Very often, a chemical or biochemical system has invariants that are linear first integrals of the system. Indeed, the system is written x˙ = Sv(x) where the v(x) are monomials of degree at most two. The invariants are the elements of the left kernel of stoichiometric matrix S, and they are given by (here X is not scaled) I(X) = ∑i ki Xi . They are used in the algorithm to reduce the possible number of faces of exit.
5 The Algorithm We wrote a Maple program to perform this study. The structure is briefly described below: • Specify the data : – The stoichiometric matrix S and the rates of the reactions v(X) as in (1). – The numerical values of the parameters K, and an initial condition. – Choose the values of ω and q that are control parameters for the algorithm. Then G = (N + 1)ω q .
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• Preliminaries : – Write the equation in the form (1). – Determination of the left kernel of S to define the stoichiometric invariants. – Determination of the initial box and of the value of the invariants in this box. • The main loop : – – – –
Scale the equation to obtain (2). Sort the coefficients of f , and find (if possible) a gap greater than G. Write the ”fast” vector field f f . When E+ or E− is non empty, using lemmas, we can write a list of forbidden (lemmas 2 and 4), or almost forbidden exit faces (lemma 6). – Eliminate from the list of the possible transitions the boxes that are not compatible with the value of the stoichiometric invariants. – Return to the loop with a new box. • The program ends when all the encountered boxes are studied or when the number of boxes is greater than a given large number. • The result of the algorithm is a graph with boxes as edges and possible transitions as vertices.
6 Example: Michaelis-Menten Equation We now test the Maple algorithm with the classical Michaelis Menten equation. Of course, our goal is to apply this algorithm to systems of higher dimensions, but this simple example will illustrate our approach. ⎧ S˙ = −k+ SE + k− C ⎪ ⎪ ⎨˙ E = −k+ SE + k− C + k2 C ⎪ C˙ = k+ SE − k− C − k2 C ⎪ ⎩ ˙ P = k2 C We choose k+ = 1, k− = 2 and k2 = 3, all of the same order of magnitude. We choose ω = 102 and G = 106 , an initial box [0, −2, −25, −15]. Therefore the concentrations of C and of P are almost zero. The invariants are S + C + P and E + C. If we only consider the orders of magnitude of the variables, these invariants can be computed as max(n1 , n3 , n4 ) and max(n2 , n3 ). Their values are 0 and −2. Because C is always lower than ω −2 , the first invariant can be simplified to max(n1 , n4 ) = 0. For this simple example, we have a planar representation of the graph of boxes: because max(n1 , n4 ) = 0, the value of n1 − n4 characterizes n1 and n4 . So we can draw a box as the point (n1 − n4 , n2 − n3 ). For technical reasons, an edge, oriented from the left to the right is drawn as : . Therefore, a thick segment corresponds to two opposite edges. A vertex surrounded by a small gray disk (left of the figure) is a ! vertex such that E+ E− = 0/ : a more precise study could be done in this box using the singular perturbation theory ([9]), but it is not our aim here.
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On the picture, the trajectory of boxes starts at point n = [0, −2, −25, −15], corresponding to the point (15, 23). Then the graph of transition shows that n2 is decreasing and n1 , n3 , n4 stay constant until n2 is between −7 and −10. Up to here, the pathway is unique. After that, n4 and n3 are increasing, with a mean slope around 1/2. In a third phase, n3 and n1 are decreasing. The trajectory enters the region where singular perturbation analysis is possible. A biologist can therefore compare this qualitative trajectory throughout orders of magnitude with his experiments. In fact, the number of paths found by the algorithm is excessive, because we did not give enough constraints. For example, some trajectories stay on a vertical line n1 − n4 = 5, with n2 − n3 moving between −6 and 4, which is not possible. Improving the lemmas will drastically decrease the number of trajectories.
7 Conclusion The paper gives the first steps of the method; room is left for many improvements. In particular, it is possible to obtain stronger results on the possibility of exiting via a face of the box, using the invariants and the structure of the chemical system. The
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transition graph that is obtained at the end of the procedure can be very big. Existing computer tools, like model checking, could be adapted to be able to manage large networks and verify given properties, in a spirit similar to [2]. Acknowledgements. The authors acknowledge the support of the French ANR Biosys Metagenoreg program.
References 1. Batt, G., Belta, C., Weiss, R.: Model checking genetic regulatory networks with parameter uncertainty. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 61–75. Springer, Heidelberg (2007) 2. Batt, G., Ropers, D., de Jong, H., Geiselmann, J., Mateescu, R., Page, M., Schneider, D.: Validation of qualitative models of genetic regulatory networks by model checking: analysis of the nutritional stress response in Escherichia coli. Bioinformatics 21(supp. 1), i19–i28 (2005) 3. Belta, C., Habets, L.: Controlling a Class of Nonlinear Systems on Rectangles. IEEE Transactions on Automatic Control 51(11), 1749 (2006) 4. Benoˆıt, E., Gouz´e, J.-L.: A mathematical and algorithmic approach to orders of magnitude in a biochemical system. Research report, INRIA (2009) 5. de Jong, H., Gouz´e, J.-L., Hernandez, C., Page, M., Sari, T., Geiselmann, J.: Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull. Math. Biol. 6(2), 301–340 (2004) 6. Heinrich, R., Schuster, S.: The Regulation of Cellular Systems. Chapman & Hall, Boca Raton (1996) 7. Khalil, H.: Nonlinear systems. Prentice Hall, Upper Saddle River (2002) 8. Lin, C., Segel, L.: Mathematics Applied to Deterministic Problems in the Natural Sciences. Society for Industrial Mathematics (1988) 9. Murray, J.: Mathematical Biology: I. An Introduction. Springer, Heidelberg (2002)
Structural Identifiability of Linear Singular Dynamic Systems Bego˜na Cant´o, Carmen Coll and Elena S´anchez
Abstract. Structured singular systems depending on a parametric vector are considered. The identification of the parameters is analyzed in terms of the input-output behavior of the system. The role of the reachability and observability properties in this analysis is studied and a characterization of the structural identifiability property is given. Finally, the structural identifiability of a positive reachable system is studied.
1 Introduction Structured systems are used in the modeled of mechanical, electrical, biological and economics models. Normally, the mathematical model incorporates parameters that symbolize empirical relations among variables. Given a parametrized state space model, structural identifiability is concerned with whether the unknown parameters within the model can be identified from the experiment considered. A structural global identifiability analysis of the model is important in the modeling process and is necessary for the system identification or the parameter estimation. It means uniqueness of the parametric structure in the model. If the structural global identifiability property holds, it is possible to determine the values of the parameters uniquely in terms of known quantities. Several results on structured and global identifiability linear system have been published. For instance, a survey on linear structured systems is given in [6] and references therein. For references of global identifiability see [1] and [8]. In the singular case, few results have been given, see for instance [2] and [7]. In this paper a characterization for the structural identifiability of linear singular systems is treated. Bego˜na Cant´o, Carmen Coll and Elena S´anchez Institut de Matem`atica Multidisciplinar, Universitat Polit`ecnica de Val`encia, 46071, Val`encia, Spain, e-mail:
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 243–249. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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The motivation for the study of structural identifiability of positive systems is the used of positive structured linear systems in economics, chemistry and other research areas. In these systems the structure of coefficients matrices plays an important role because there are the positive restrictions on the behavior of the system. A powerful tool used in the solution of the singular systems is the Drazin inverse. Let M ∈ n×n a square matrix. The matrix M D is the Drazin inverse of M if it satisfies M D MM D = M D , MM D = M D M and M k+1 M D = M k , where k = ind(M), the least nonnegative integer such that rank(M k ) = rank(M k+1 ), is the index of matrix M. Consider a structured singular system E(p)x(k + 1) = A(p)x(k) + B(p)u(k),
(1)
y(k) = C(p)x(k) where the matrices E(p), A(p) ∈ n×n , B(p) ∈ n×m and C(p) ∈ m×n have a fixed structure with p belongs to a subset P ⊆ ∇ . Matrix E(p) is singular with n1 = rank(E D (p)E(p)) and l = ind(E(p)). This structured singular system is denoted by S(p) = (E(p), A(p), B(p),C(p), P). If there exists λ ∈ such that det[λ E(p) − A(p)] = 0 the system S(p) has solution. The output of the system is given by k y (k) = C(p)(E D (p)A(p) E D (p)E(p)x (0) k−1 k−i−1 + ∑ C(p)E D (p) E D (p)A(p) B(p)u (i) i=0
l−1 i −C(p) I − E D (p)E(p) ∑ E(p)AD (p) AD (p)B(p)u (k + i), i=0
if E(p)A(p) = A(p)E(p) and ker(E(p)) ∩ ker(A(p)) = {0}. The admissible initial conditions set, X0 , is given by X0 = Im E D (p)E(p) H0 . . . Hq−1 i where Hi = I − E D (p)E(p) E(p)AD (p) AD (p), i = 0, . . . , l − 1. Before solving the identifiability problem we will remind the definition of reachable, observable and similar properties. A system S(p) is called structurally reachable if, for all p ∈ P the reachability matrices n−1 B(p) R f (S(p)) = E D (p)B(p) . . . E D (p) E D (p)A(p) Rb (S(p)) =
I − E D (p)E(p) AD (p)B(p) . . . l−1 D A (p)B(p) , . . . I − E D (p)E(p) E(p)AD (p)
have full rank, that is rank(R f (S(p))) = n1 and rank(Rb (S(p))) = n − n1 . And, a system S(p) is called structurally observable if, for all p ∈ P the observability matrices
Structural Identifiability of Singular Systems
n−1 T O f (S(p)) = (C(p)E D (p))T . . . (C(p)E D (p) E D (p)A(p) ) Ob (S(p)) = (C(p) I − E(p)D E(p) AD (p))T . . . l−1 D A (p))T . . . (C(p) I − E D (p)E(p) E(p)AD (p)
245 T
T
have full rank, that is rank(O f (S(p))) = n1 and rank(Ob (S(p))) = n − n1. Two systems S(p) and S(q), p, q ∈ P are structurally similar if there exists an invertible matrix T such that E(p) = T E(q)T −1 , A(p) = TA(q)T −1 , B(p) = T B(q) and C(p) = C(q)T −1 .
2 Structural Identifiability Problem The identifiability of the parameters of the system is concerned with the determination of these from the external behavior of the system. The response of the system from an input given can be analyzed in the z-domain or using the input-output application. That is, to determine the input-output behavior (io) of a model S(p) we can use the transfer matrix G(z, p) = C(p)(zE(p) − A(p))−1 B(p) or the Markov parameters associated to the system S(p). These parameters are given by j V ( j, p) = C(p)E D (p) E D (p)A(p) B(p), j ≥ 0 j (2) H( j, p) = C(p)(I − E(p)E D (p)) E(p)AD (p) AD (p)B(p) j = 0, . . . , l − 1. The concept of structural identifiability is given in the following definition. Definition 1. The system S(p) is structurally identifiable if and only if, for almost any two candidates parameter vector values p, q ∈ P, io(p) = io(q) implies p = q, where io(·) denotes the input-output behavior of the system S(p). This concept is also named in the literature global identifiability (see [2], [8] to continuous-time case). The structural reachability and observability are related to the structural identifiability property. However, these properties are neither necessary nor sufficient conditions for global identifiability. For example, consider the system S(p) with the following structure ⎞ ⎛ ⎛ ⎞ 01 p1 p2 0 0 ⎜1 0⎟ ⎜ 0 p3 0 0 ⎟ ⎟ ⎜ ⎟ E(p) = ⎜ (3) ⎝ 0 0 0 p4 ⎠ , A(p) = I + E(p), B = ⎝ 0 0 ⎠ , 0 00 0 11 and
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C=
1010 0101
where the parametric vector p ∈ P = {(p1 , p2 , p3 , p4 ) ∈ 4+ , pi > 0}. Constructing the reachability and observability matrices it is easy to see that they have full rank, then the system is structurally reachable and observable but the system is unidentifiable since its transfer matrix is ⎞ ⎛ −p2 (z − 1) p4 (1 − z) ⎟ ⎜ G(z, p) = ⎝ (zp1 − 1 − p1)(zp3 − 11− p3) ⎠ −1 zp3 − 1 − p3 and the parameters are undetermined. Then they cannot be estimated using inputoutput data, even using well-design experiments. On the other hand there exist structurally globally identifiable systems which are not structurally reachable. For example, the structured system (A(p), B) given by 0 1 1−α , B= , A(p) = 0 1−α 0 with the parametric vector p ∈ {α ∈ / α = 0}. Since the reachability matrix satisfies 1 1−α = 1, rank 0 0 the system is not reachable but as its transfer function is 1 1 G(z, p) = , z+α −1 0 the parameter α is uniquely determined and the system is globally identifiable. In the following result we have a characterization of the structural global identifiability property when the system is structurally reachable and observable. Theorem 1. Consider the structured system S(p) given in (1) structurally reachable and structurally observable. Then, S(p) is structurally identifiable if and only if S(p) and S(q) structurally similar implies that p = q and T = I. Proof. Consider S(p) and S(q) structurally similar, then they have the same Markov parameters. That is, the input-output behavior is the same io(p) = io(q) and by hypothesis p = q. Reciprocally, consider S(p) and S(q) such that they have the same input-output behavior. It is known that each one of them is equivalent to (see [5]) B1 (p) I O A1 (p) O ¯ , (C1 (p) C2 (p)) , S(p) = , O I B2 (p) O N(p)
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¯ S(q) =
247
I O A1 (q) O B1 (q) , , , (C1 (q) C2 (q)) O N(q) O I B2 (q)
and they satisfy io(p) = io(q). Using the definition (2) of the Markov parameters and the definition of the reachability and observability matrices we show that O f (S(p))B1 (p) = O f (S(q))B1 (q) C1 (p)R f (S(p)) = C1 (q)R f (S(q)) O f (S(p))A1 (p)R f (S(p)) = O f (S(q))A1 (q)R f (S(q))
(4)
and
Ob (S(p))B2 (p) = Ob (S(q))B2 (q) C2 (p)Rb (S(p)) = C2 (q)Rb (S(q)) Ob (S(p))N(p)Rb (S(p)) = Ob (S(q))N(q)Rb (S(q)). Tf O Constructing T = with O Tb
(5)
T T f = R f (S(p))R Tf (S(q)) R f (S(q))R Tf (S(q)) T Tb = Rb (S(p))RbT (S(q)) Rb (S(q))RbT (S(q)) ¯ and using (4)-(5) it is easy to show that T is nonsingular and the systems S(p) and ¯ S(q) are similar under the transformation matrix T . By hypothesis, this implies that T = I and p = q, and hence, the structured system is identifiable. Before we have indicated that the structured system S(p) given by (3) is unidentifiable, now we can observe that it neither satisfies the condition of the theorem 1. Consider two structured systems S(p) and S(q) of type (3) such that ⎛ ⎞ ⎞ ⎛ p1 p2 0 0 p1 p2 0 0 ⎜ 0 p3 0 0 ⎟ ⎜ 0 p3 0 0 ⎟ ⎟ ⎟ ⎜ E(p) = ⎜ ⎝ 0 0 0 p4 ⎠ and E(q) = ⎝ 0 0 0 q4 ⎠ . 0 00 0 0 00 0 We can prove that these systems are similar by means of the transformation matrix ⎞ ⎛ 1000 ⎜ 0 1 0 0⎟ ⎟ ⎜ L = ⎜ p4 0 1 0 ⎟ . ⎠ ⎝q 4 0001 Hence, if we choose p4 = q4 then p = (p1 , p2 , p3 , p4 ) = (p1 , q2 , p3 , q4 ) = q and L = I.
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3 Structural Identifiability of Positive Systems When positive restrictions are considered a concept of reachability of positive states by means of positive controls is used. This concept is named positive reachability (see [3]). In this context, we are interested in the study of the structural identifiability problem when the system satisfies this property. Thus, we say that a structured system S(p) is (i) structurally identifiable positive system if it is positive for all p ∈ P and it is structurally identifiable, and (ii) structurally reachable positive system if it is positive and positively reachable for all p ∈ P. Consider the system S(p) where the matrices are given by B1 (p) DO A1 (p) O B(p) = , E= ≥ 0, A(p) = O I B2 ON
(6)
where D is a nonsingular diagonal matrix, N is a nilpotent matrix, B2 ≤ 0, and ⎞ ⎛ ⎛ ⎞ 0 p1 0 0 0 ⎟ ⎜ ⎜ .. ⎟ ⎜ 0 0 ... 0 ⎟ ⎟ ⎟ , and B1 (p) = ⎜ A1 (p) = ⎜ ⎜ . ⎟, ⎟ ⎜ . . ⎝ ⎠ . . 0 ⎝ 0 . . pn1 −1 ⎠ b pn 0 0 0 1
being P = {p = (p1 , . . . , pn1 , b) ∈ n1 +1 / pi , b > 0}. This structured system is positive since it holds the conditions established in [4] in order to be a positive system, i that is, E D E ≥ 0, E D A(p) ≥ 0, E D B(p) ≥ 0, and I − E D E EAD (p) AD (p)B(p) ≤ 0, i = 0, 1, ..., l − 1, where l is the index of E. Moreover, the reachability matrices R f (S(p)) and R f (S(p)) have a monomial matrix of size n1 and n − n1 , respectively, then the positive structural reachability property holds (see [3]). In the following result we prove that it is also structurally globally identifiable. Theorem 2. Consider the positive structured system (6). This system is structurally globally identifiable. Proof. We consider two positive structured systems S(p) and S(q) of type (6) with p, q ∈ P such that they have the same input-output behavior (io) V ( j, p) = V ( j, q), j ≥ 0 H( j, p) = H( j, q), j = 0, . . . , l − 1, ¯ by and we shall prove that p = q. If p = (p1 , . . . , pn1 , b) and q = (q1 , . . . , qn1 , b), definition of the Markov parameters in (2) and by the structure of the matrices of the system (6) we have
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E D B(p) = E D B(q) ⇒ b = b E D A(p)B(p) = E D A(q)B(q) ⇒ pn1 −1 = qn1 −1 .. . D n1 −1 (E A(p)) B(p) = (E D A(q))n1 −1 B(q) ⇒ p1 = q1 (E D A(p))n1 B(p) = (E D A(q))n1 B(q) ⇒ pn1 = qn1 Hence, p = q.
4 Conclusions The problem of structural global identifiability of the model consists in determinate the uniqueness of the parameter vector when an input-output response is considered. This problem has been studied for structured singular systems. The relation between the structural properties and the structural identifiability has been analyzed for these systems and a characterization of this property has been given. Finally, the structural identifiability of a class of positive reachable system has been treated. Acknowledgements. Supported by Spanish DGI grant MMT2007-64477.
References 1. Audoly, S., D’Angi´o, L., Saccomani, M.P., Cobelli, C.: Global identifiability of linear compartmental models. IEEE Trans. Biomed. Eng. 45, 36–47 (1998) 2. Ben-Zvi, A., McLellan, P.J., McAuley, K.B.: Identifiability of linear time-invariant differential-algebraic systems. I. The generalized Markov parameter approach. Ind. Eng. Chem. Res. 42, 6607–6618 (2003) 3. Bru, R., Coll, C., Romero, S., S´anchez, E.: Some problems about structural properties of positive descriptor systems. LNCIS, vol. 294, pp. 233–240. Springer, Heidelberg (2003) 4. Bru, R., Coll, C., S´anchez, E.: Structural properties of positive linear time-invariant difference-algebraic equations. Linear Algebra and its Applications 349, 1–10 (2002) 5. Dai, L.: Singular Control Systems. LNCIS, vol. 118. Springer, Heidelberg (1989) 6. Dion, J.M., Commault, C., Van der Woude, J.: Generic properties and control of linear structured systems: a survey. Automatica 39, 1125–1144 (2003) 7. Miyamura, A., Kazuyuki, K.: Identifiability of delayed singular systems. In: Proceedings 5th Asian Control Conference, Melbourne, Australia, pp. 789–797 (2004) 8. Tayakout-Fayolle, M., Jolimaitre, E., Jallut, C.: Consequence of strutural identifiability properties on state model formulation for linear inverse chromatography. Chemical Eng. Science 55, 2945–2956 (2000)
On Positivity of Discrete-Time Singular Systems and the Realization Problem Rafael Cant´o, Beatriz Ricarte and Ana M. Urbano
Abstract. In this work we introduce different positivity concepts for singular systems. Discrete-time regular singular systems are considered and the minimal realization problem is discussed for the case of weakly positive systems and internally positive systems.
1 Introduction Positive singular systems are widely applied in different fields like engineering problems such as electrical circuits network, power systems, aerospace engineering or chemical processing, and social, economic or biological systems among others. In the literature singular systems have been called by different names as for example descriptor variable systems or the generalized state-space systems [2, 13]. Nevertheless, we can find different concepts of positivity in systems theory such us weak positivity, internal positivity or simple positivity [2, 7, 10, 11, 13]. In this work we consider those definitions and their properties for singular systems. Then discrete-time regular singular systems are considered and the realization problem is discussed for the case of weakly positive systems and internally positive systems. Moreover, we try to give conditions to obtain the minimal positive realization when it is possible. r×s [z], matrices E, A ∈ n×n , We recall that given a transfer matrix T (z) = M(z) d(z) ∈ n×s r×n B∈ and C ∈ such that T (z) = C [zE − A]−1 B Rafael Cant´o, Beatriz Ricarte and Ana M. Urbano Institut de Matem`atica Multidisciplinar, Universitat Polit`ecnica de Val`encia, 46071 Val`encia, Spain, e-mail:
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 251–258. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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are called a realization of T (z). It is denote by (E, A, B,C). The size of A is called the dimension of the realization. The realization is minimal if it has minimum dimension.
2 Preliminary Results In this paper we consider the discrete-time singular system E¯ x(k ¯ + 1) = A¯ x(k) ¯ + B¯u(k) ¯ = C¯ x(k) ¯
y(k) ¯
(1)
where x(k) ¯ ∈ n is the state vector, u(k) ¯ ∈ s is the input vector, y(k) ¯ ∈ r is the n×n n×s r×n ¯ ¯ ¯ ¯ ¯ ,B∈ and C ∈ . If E = I, the system is called output vector and E, A ∈ standard, if E¯ is a singular matrix, then the system is called singular. System (1) is ¯ B, ¯ ¯ A, ¯ C). denoted by (E, It is well known that if the system (1) satisfies the regularity condition, i.e. there exists an scalar λ ∈ such that ¯ = 0 det[λ E¯ − A] then (1) is equivalent to the canonical forward-backward form (see [7]) given by Ex(k + 1) = Ax(k) + Bu(k) y(k)
= Cx(k)
(2)
(3)
with E = diag(In1 , N), A = diag(A1 , In2 ), n1 + n2 = n, where n1 is the degree of polynomial det[zE − A], A1 ∈ n1 ×n1 and N ∈ n2 ×n2 is a nilpotent matrix with index q, i.e., N q−1 = 0 and N q = 0. B1 Consider B = with B1 ∈ n1 ×s and B2 ∈ n2 ×s , and let C = [C1 C2 ] with B2 C1 ∈ r×n1 and C2 ∈ r×n2 , then the system (3) can be broken down into the following subsystems: • the standard (or forward) subsystem x1 (k + 1) = A1 x1 (k) + B1 u(k) y1 (k)
= C1 x1 (k)
• the complete singular (or backward) subsystem Nx2 (k + 1) = x2 (k) + B2 u(k) y2 (k)
= C2 x2 (k)
(4)
(5)
Positive Realization Problem
where x(k) =
253
x1 (k) and y(k) = y1 (k) + y2 (k). x2 (k)
Lemma 1. [10, Lemma 4.2] The transfer matrix of the system (3) is equal to the sum T (z) = Tsp (z) + W (z) of the strictly proper transfer matrix of the subsystem (4) Tsp (z) = C1 [zIn1 − A1 ]−1 B1 and the polynomial transfer matrix of the subsystem (5) W (z) = C2 (zN − In2 )−1 B2 Definition 1. Matrices A1 ∈ n1 ×n1 , B1 ∈ n1 ×s , B2 ∈ n2 ×s , N ∈ n2 ×n2 , C1 ∈ C2 ∈ r×n2 are called the realization in Weierstrass canonical form (WCF realization) of the matrix T (z) ∈ r×s (z). r×n1 ,
From Lemma 1 we deduce that the realization problem in regular singular systems can really be dealt as two realization subproblems, a realization problem in a standard system and a realization problem of a complete singular system. Positive realization problem has been wide studied for standard systems, see for instance [1, 3, 4, 8, 12] for SISO systems or [5, 6] for MIMO systems. With respect to polynomial matrices of complete singular systems, some algorithms have been developed to obtain minimal realizations (see, for instance, [6, 7]). From now on, we denote by Jt,α ∈ t α ×t α the following nilpotent matrix of nilpotent index t ⎤ ⎡ O O ... O O ⎢ Iα O . . . O O ⎥ ⎥ ⎢ ⎥ ⎢ Jt,α = ⎢ O Iα . . . O O ⎥ . ⎢ .. .. .. .. ⎥ ⎣ . . . .⎦ O O . . . Iα O In this work we base on the following result we can find in [6]. Proposition 1. Consider W (z) = Wt−1 zt−1 + · · · + W1 z + W0 ∈
r×s [z].
(1) If rankWt−1 = s then a minimal realization (N, Its , B,C) of W (z) with a nilpotent matrix N ∈ ts×ts , B ∈ ts×s and C ∈ r×ts is given by ⎡ ⎤ Ip ⎢O⎥ ⎢ ⎥ N = Jt,s , B = ⎢ . ⎥ and C = [−W0 − W1 · · · − Wt−2 − Wt−1 ]. ⎣ .. ⎦ O
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(2) If rankWt−1 = r then a minimal realization (N, Itr , B,C) of W (z) with a nilpotent matrix N ∈ tr×tr , B ∈ tr×s and C ∈ r×tr is given by ⎤ ⎡ −W0 ⎢ −W1 ⎥ ⎥ ⎢ ⎥ ⎢ T N = Jt,r , B = ⎢ ... ⎥ and C = [Ir O · · · O O]. ⎥ ⎢ ⎣ −Wt−2 ⎦ −Wt−1 As we comment in the Introduction, in the literature there are many different concepts of positivity. In next sections we consider those definitions and the realization problem.
3 Weakly Positive System Definition 2. [10, p. 92] The system described by Equation (1) is called weakly positive if and only if E¯ ∈
n×n + ,
A¯ ∈
n×n + ,
B¯ ∈
n×s + ,
and C¯ ∈
r×n + .
From now on, we consider regular singular systems in canonical forward-backward form. Therefore, a regular singular system will be weakly positive if and only if 1 2 and C2 ∈ r×n A1 ∈ n+1 ×n1 , B1 ∈ n+1 ×s , B2 ∈ n+2 ×s , N ∈ n+2 ×n2 , C1 ∈ r×n + + , i.e., we obtain a positive realization in Weierstrass canonical form. Recall that by Lemma 1 we can broken down the problem to obtain a positive realization in Weierstrass canonical form into two subproblems, the computation of a positive realization (A1 , B1 ,C1 ) of the strictly proper matrix Tsp (z) and the computation of the positive realization (N, I, B2 ,C2 ) of the polynomial matrix W (z). Let us see this second subproblem. Corollary 1. Let W (z) = Wt−1 zt−1 + · · · + W1 z + W0 ∈ r×s [z] be a transfer matrix. If rankWt−1 = s or rankWt−1 = r and −Wi ∈ r×s + , for i = 0, 1, . . . ,t − 1, then there exists a minimal realization (N, I, B,C) of W (z) = C(zN − I)−1 B with N ≥ O, B ≥ O and C ≥ O. Proof. By Proposition 1, there exist a minimal realization (N, I, B,C) of W (z) where N, B and C are nonnegative matrices if −Wi ≥ O, for i = 0, 1, . . . ,t − 1. Taking B2 = B and C2 = C we obtain the desired realization.
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4 Internally Positive System Definition 3. [2, 11] The singular system (1) is called internally positive (or simply positive) if for any admissible initial condition x(0) ¯ ∈ n+ and for every input s n ¯ ∈ + and y(k) ¯ ∈ r+ for k ∈ + . control sequence u(k) ¯ ∈ + , k ∈ + , we have x(k) The following algebraic characterization of positive singular systems is given in [2, Proposition 1]. We denote by M D the Drazin inverse of a matrix M, and by ¯ the index of E, ¯ that is, q is the smallest nonnegative integer such that q = ind(E) rank(E¯ q ) = rank(E¯ q+1 ). ¯ B). ¯ A, ¯ Suppose that E¯ E¯ D ≥ O and E¯ A¯ = Proposition 2. Consider the system (E, ¯ B) ¯ The system (E, ¯ A, ¯ is positive if and only if E¯ D A¯ ≥ O, E¯ D B¯ ≥ O and A¯ E. D D i D ¯ E¯ A¯ ) A¯ B¯ ≤ O, i = 0, 1, . . . , q − 1, where q = ind(E). ¯ (I − E¯ E)( ¯ B) ¯ A, ¯ satisfies the regularity condition (2), then it is equivalent If the system (E, to the canonical forward-backward form given by Ex(k + 1) = Ax(k) + Bu(k). The authors [2] prove that in this case E D E = diag(In1 , O), E D A = diag(A1 , O), T T E D B = BT1 O and (I − E D E)(EAD )i AD B = O (N i B2 )T . Hence, a forwardbackward system is positive if and only if A1 ≥ 0, B1 ≥ 0 and N i B2 ≤ 0, i = 0, 1, . . . , q − 1. ¯ B, ¯ A, ¯ and Next result gives necessary and sufficient conditions on the matrices E, ¯ ¯ ¯ ¯ A, B, ¯ C) is positive [9, Theorem 2.1]. C such that the system (E, ¯ B, ¯ Suppose that E¯ E¯ D ≥ O, E¯ A¯ = A¯ E¯ ¯ A, ¯ C). Corollary 2. Consider the system (E, ¯ = {0}. The system (E, ¯ B, ¯ is internally positive if and only ¯ ∩ ker(A) ¯ A, ¯ C) and ker(E) ¯ − 1: if the following conditions hold, for i = 0, 1, . . . , ind(E) 1. E¯ D A¯ ≥ O, 2. E¯ D B¯ ≥ O, 3. C¯ E¯ D E¯ ≥ O, ¯ E¯ A¯ D )i A¯ D B¯ ≤ O, 4. (I − E¯ D E)( ¯ ¯ 5. C(I − E E¯ D )(E¯ A¯ D )i A¯ D B¯ ≤ O. ¯ B, ¯ satisfies the regularity condition (2), then it is equiva¯ A, ¯ C) If the system (E, lent to the canonical forward-backward form given by (3). Now, we introduce the following algebraic characterization for this system can be internally positive. Proposition 3. A singular system given in the canonical forward-backward form is internally positive if and only if the following conditions hold, for i = 0, 1, . . ., ind(E) − 1: 1. A1 ≥ O, 2. B1 ≥ O, 3. C1 ≥ O,
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4. N i B2 ≤ O, 5. C2 N i B2 ≤ O. Proof. By Corollary 2 and taking into account that
In1 0 D = [C1 0 ] ≥ 0, then C1 ≥ O CE E = [C1 C2 ] 0 0
O D D i D = C2 N i B2 ≤ 0. C(I − EE )(EA ) A B = [C1 C2 ] N i B2 Directly from the previous result we obtain the following Corollary. Corollary 3. Consider a singular system in canonical forward-backward form expressed by (4) and (5). If A1 ≥ O, B1 ≥ O, C1 ≥ O, N ≥ O, B2 ≤ O, C2 ≥ O then this system is internally positive. Remark 1. Note that in general, neither a singular weakly positive system (1) is internally positive, nor an internally positive system is weakly positive except for the particular case when B2 = O. Kaczorek [11] observed a similar conclusion. By Corollary 3 and Lemma 1 the problem to obtain an internally positive realization of a singular system given in the canonical forward-backward (4) and (5) can be separated into two subproblems, the computation of a positive realization (A1 , B1 ,C1 ) of the strictly proper matrix Tsp (z) and the computation of a realization (N, I, B2 ,C2 ) of the polynomial matrix W (z) with N ≥ O, B2 ≤ O, and C2 ≥ O. Let us see this second subproblem. Corollary 4. Let W (z) = Wt−1 zt−1 + · · · + W1 z + W0 ∈ r×s [z] be a transfer matrix. If rankWt−1 = s or rankWt−1 = r and Wi ∈ r×s + , for i = 0, 1, . . . ,t − 1, then there exists a minimal realization (N, I, B,C) of W (z) = C(zN − I)−1 B with N ≥ O, B ≤ O and C ≥ O. Proof. By Proposition 1, we can obtain a minimal realization (N, I, B,C) of W (z) where N ≥ O, B ≤ O and C ≥ O if Wi ≥ O, for i = 0, 1, . . . ,t − 1. Taking B2 = B and C2 = C we obtain the desired realization.
5 Conclusions Taking into account the different kinds of positivity existing in system theory, if specific conditions are satisfied in each particular case then positive realizations of a transfer matrix T (z) ∈ r×s (z) can be computed by the use of the following procedure.
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Step 1. Given a transfer matrix T (z) , express it as the sum of an strictly proper rational matrix Tsp (z) and a polynomial matrix W (z). Step 2. Compute a positive realization (A1 , B1 ,C1 ) for Tsp (z). Step 3. Compute a minimal realization (N, I, B2 ,C2 ) for W (z). Step 4. Compose the positive realization (E, A, B,C) for T (z) from the realizations given in Steps 2 and 3. Note that if the realizations obtained in Step 2 is minimal, then the global realization in Step 4 is also minimal.
6 Example Obtain an internally positive realization of the transfer matrix T (z):
1 2z − 1.7 4z − 3.4 z+1 1 T (z) = + 0.6 1.2 0 z (z − 1)(z − 0.7) Step 1. T (z) = Tsp (z) + W (z) with
1 M(z) 2z − 1.7 4z − 3.4 = 0.6 1.2 d(z) (z − 1)(z − 0.7)
z+1 1 1 0 1 1 W (z) = = z+ = W1 z + W0 0 z 0 1 0 0
Tsp (z) =
where W0 ≥ 0, W1 ≥ 0 and detW1 = 0 (Corollary 4). Step 2. By [5] we obtain the minimal positive realization (A1 , B1 ,C1 ) of Tsp (z):
0.85 0.15 0 0 0 2 B1 = C1 = . A1 = 0.15 0.85 1 2 4 0 Step 3. We obtain a minimal realization (N, I4 , B2 ,C2 ) by Proposition 1: ⎡ ⎡ ⎤ ⎤ 0 0 0 0 −1 0
⎢0 0 0 0⎥ ⎢ 0 −1 ⎥ 1 1 1 0 ⎢ ⎢ ⎥ ⎥ N=⎣ . B2 = ⎣ C2 = 0 0 0 1 1 0 0 0⎦ 0 0⎦ 0 1 0 0 0 0 Step 4. We compose the minimal internally positive realization (E, A, B,C):
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⎡
1 ⎢0 ⎢ ⎢0 E =⎢ ⎢0 ⎢ ⎣0 0
0 1 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0
⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0
⎡
⎤ 0.85 0.15 0 0 0 0 ⎢ 0.15 0.85 0 0 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 1 0 0 0⎥ ⎢ ⎥ A=⎢ 0 0 1 0 0⎥ ⎢ 0 ⎥ ⎣ 0 0 0 0 1 0⎦ 0 0 0 0 0 1
⎡
⎤ 0 0 ⎢ 1 2⎥ ⎢ ⎥ ⎢ −1 ⎥ 0 ⎢ ⎥ B=⎢ ⎥ 0 −1 ⎢ ⎥ ⎣ 0 0⎦ 0 0
0 2 1 1 1 0 C= . 4 0 0 0 0 1
Acknowledgements. Supported by the Spanish DGI grant MTM2007-64477 and by the UPV under its research program.
References 1. Benvenuti, L., Farina, L.: A Tutorial on the Positive Realization Problem. IEEE Transactions on Automatic Control 49(5), 651–664 (2004) 2. Bru, R., Coll, C., S´anchez, E.: Structural properties of positive linear time-invariant difference-algebraic equations. Linear Algebra and its Applications 349, 1–10 (2002) 3. Bru, R., Cant´o, R., Ricarte, B., Rumchev, V.: A Basic Canonical Form of DiscreteTime Compartmental Systems. International Journal of Contemporary Mathematical Sciences 2(6), 261–273 (2007) 4. Cant´o, R., Ricarte, B., Urbano, A.M.: On Positive Realizations of Irreducible Transfer Matrices. In: Commault, C., Marchand, N. (eds.). LNCIS, vol. 341, pp. 41–48. Springer, Heidelberg (2006) 5. Cant´o, R., Ricarte, B., Urbano, A.M.: Positive Realizations of Transfer Matrices with real poles. IEEE Trans. Circuits Syst. II, Expr. Briefs 54(6), 517–521 (2007) 6. Cant´o, R., Ricarte, B., Urbano, A.M.: Computation of realizaciones of complete singular systems (submitted) 7. Dai, L.: Singular Control Systems. LNCIS, vol. 118. Springer, Heidelberg (1989) 8. Halmschlager, A., Matolcsi, M.: Minimal Positive Realizations for a Class of Transfer Functions. IEEE Trans. Circuits Syst. II, Expr. Briefs 52(4), 177–180 (2005) 9. Herrero, A., Ram´ırez, A., Thome, N.: An algorithm to check the nonnegativity of singular systems. Applied Mathematics and Computation 189, 355–365 (2007) 10. Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002) 11. Kaczorek, T.: Externally and Internally Positive Singular Discrete-Time Linear Systems. International Journal of Appl. Math. Comput. Sci. 12(2), 197–202 (2002) 12. Nagy, B., Matolcsi, M.: Minimal Positive Realizations of Transfer Functions with Nonnegative Multiple Poles. IEEE Transactions on Automatic Control 50(9), 1447–1450 (2005) 13. Virnik, E.: Stability analysis of positive descriptor systems. Linear Algebra Appl. 429(10), 2640–2659 (2008)
Multi-Point Iterative Methods for Systems of Nonlinear Equations Alicia Cordero, Jos´e L. Hueso, Eulalia Mart´ınez and Juan R. Torregrosa
Abstract. A family of multi-point iterative methods for solving systems of nonlinear equations is described. Some classical methods are included in the mentioned family. Under certain conditions, convergence order is proved to be 2d + 1, where d is the order of the partial derivatives required to be zero in the solution. Moreover, different numerical tests confirm the theoretical results and allow us to compare these variants with Newton’s method.
1 Introduction Let us consider the problem of finding a real zero of a function F : D ⊆ Rn −→ Rn , that is, a solution α ∈ D of the nonlinear system F(x) = 0. This solution can be obtained by means of the fixed point iteration method. The best known fixed point method is the classical Newton’s method. In different types of applications it is required to find a solution of a nonlinear system. For example, many physical systems can be described by a system of differential equations Alicia Cordero, Jos´e L. Hueso, Juan R. Torregrosa Instituto de Matem´atica Multidisciplinar, Universidad Polit´ecnica de Valencia, 46022 Valencia, Spain, e-mail:
[email protected],
[email protected],
[email protected] Eulalia Mart´ınez Instituto de Matem´atica Pura y Aplicada, Universidad Polit´ecnica de Valencia, 46071 Valencia, Spain, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 259–267. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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⎫ ⎪ dx1 ⎪ = f1 (x1 , x2 , . . . , xn ) ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dx2 ⎬ = f2 (x1 , x2 , . . . , xn ) ⎪ dt . (1) .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dxn ⎭ = fn (x1 , x2 , . . . , xn ) ⎪ dt Equations of this type arise quite often in biological and physical applications, economical models, etc. The existence of equilibrium values x¯ = (x¯1 , x¯2 , . . . , x¯n )T , for which x(t) ≡ x¯ are solutions of (1), is an important problem in the qualitative theory of differential equations. An equilibrium value x¯ is actually a solution of the nonlinear system F(x) = 0, where F is a vectorial function with f1 , f2 , . . . , fn as its coordinate functions. Also bifurcations can be analyzed by solving the corresponding nonlinear system of equations with different values of a parameter as, for example, in Lorentz equations: ⎫ ⎪ dx ⎪ = σ (x + y) ⎪ ⎪ ⎪ dt ⎪ ⎬ dy = −xz + μ x − y ⎪ . dt ⎪ ⎪ ⎪ ⎪ dz ⎭ = xy + β z ⎪ dt These equations have an equilibrium point at the=origin for μ ≤ 1 and two new equilibrium points appear for μ > 1, at x = y = ± β (μ − 1), z = μ − 1. The construction of numerical methods for the approximation to the solution α of a nonlinear system is an interesting task in numerical mathematics and applied scientific branches. There is a collection of papers concerned with multi-point iterative methods; for example, for a nonlinear system F(x) = 0, the authors suggest in [1] the extension of the application of quadrature formulas in the development of new adjustments of Newton’s method. In [2] a family of modified Newton’s methods is obtained whose general expression is * x
(k+1)
=x
(k)
−
m
∑ AhJF (ηh (x
+−1 (k)
))
F(x(k) ),
(2)
h=1
with ηh (x(k) ) = x(k) − τh JF (x(k) )−1 F(x(k) ), where JF (x) is the jacobian matrix of F. In this paper we analyze a collection of multi-point iterative methods obtained from Newton’s method by replacing F(x(k) ) by a linear combination of values of F(x) in different points. Specifically, the general method is % $ x(k+1) = x(k) − JF (x(k) )−1
m
∑ Ah F(ηh (x(k) ))
h=1
,
(3)
Variants of Newton’s Method for Nonlinear Systems
261
where τh and Ah are parameters to be chosen in [0, 1] and R, respectively, and m is a positive integer. As we will see, the value of these parameters plays an important role in the order of convergence of the method. We consider the definition of efficiency index (see [3]) as p1/d , where p is the order of convergence and d is the total number of new function evaluations by step. Since (3) can be considered as an iterative fixed point formula, we study the convergence of the different methods by using the following result. Theorem 1. ([4]) Let G(x) be a fixed point function with continuous partial derivatives of order p with respect to all components of x. The iterative method x(k+1) = G(x(k) ) is of order p if G(α ) = α ; ∂ k gi (α ) = 0, for all 1 ≤ k ≤ p − 1, 1 ≤ i, j1 , . . . , jk ≤ n; ∂ x j1 ∂ x j2 . . . ∂ x jk ∂ p gi (α ) = 0, for at least one value of i, j1 , . . . , j p ∂ x j1 ∂ x j2 . . . ∂ x j p where gi are the component functions of G. The rest of the paper is structured as follows. In Section 2 we show some technical results that we need to prove the convergence of the methods. In Section 3, we analyze the general iterative formula (3) and we study the conditions that parameters τh and Ah must satisfy in order to obtain a method with a particular order of convergence. Finally, the last section is dedicated to the numerical results obtained by applying some of the described methods to several nonlinear systems.
2 Preliminary Results We consider x ∈ Rn , n > 1, and denote by Ji j (x) the (i, j)-entry of the jacobian matrix of F, and by Hi j (x) the respective entry of its inverse, so n
∑ Hi j (x)J jk (x) = δik .
(4)
j=1
If f j (x), j = 1, 2, . . . , n denote the coordinate functions of F(x), it is easy to prove that: n n ∂ H ji (x) ∂ fi (x) ∂ 2 fi (x) (5) ∑ ∂ xl ∂ xr = − ∑ H ji (x) ∂ xl ∂ xr , i=1 i=1 n n ∂ 2 H ji (x) ∂ fi (x) ∂ H ji (x) ∂ 2 fi (x) ∂ H ji (x) ∂ 2 fi (x) = −∑ −∑ − ∂ xr ∂ xl ∂ xs ∂ xr i=1 ∂ xs ∂ xr ∂ xl i=1 ∂ xs ∂ xl i=1 n
∑
∂ 3 fi (x) . − ∑ H ji (x) ∂ xs ∂ xr ∂ xl i=1 n
(6)
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The following results are useful in the proof of the main theorem. Lemma 1. Let λ (x) be the iteration function of classical Newton’s method, n
whose coordinates are λ j (x) = x j − ∑ H ji (x) fi (x), j = 1, . . . , n. Then, i=1
∂ 3 λ j (α ) = ∂ xs ∂ xr ∂ xl
n
∑
i=1
∂ λ j (α ) = 0, ∂ xl
(7)
n ∂ 2 λ j (α ) ∂ 2 fi (α ) = ∑ H ji (α ) , ∂ xr ∂ xl ∂ xr ∂ xl i=1
(8)
∂ H ji (α ) ∂ 2 fi (α ) ∂ H ji (α ) ∂ 2 fi (α ) ∂ H ji (α ) ∂ 2 fi (α ) + + + ∂ xr ∂ xs ∂ xl ∂ xs ∂ xr ∂ xl ∂ xl ∂ xs ∂ xr
n
+2 ∑ H ji (α ) i=1
∂ 3 fi (α ) , ∂ xs ∂ xr ∂ xl
(9)
for j, l, r, s ∈ {1, 2, . . ., n}. Let us note that the convergence order of Newton’s method, can be obtained by applying Theorem 1 and using expressions (7) and (8). Lemma 2. Let ηk (x) be the iteration functions ηk (x) = x − τk JF−1 (x)F(x), where τk ∈ [0, 1], for k = 1, . . . , m. Then,
∂ (ηk (x))q = (1 − τk )δql , ∂ xl |x=α
(10)
n ∂ 2 (ηk (x))q ∂ 2 fi (α ) = ∑ τk Hqi (α ) , ∂ xr ∂ xl |x=α i=1 ∂ xr ∂ xl
(11)
and n ∂ 3 (ηk (x))q ∂ Hqi (α ) ∂ 2 fi (α ) ∂ Hqi (α ) ∂ 2 fi (α ) = τk ∑ + ∂ xs ∂ xr ∂ xl |x=α ∂ xr ∂ xs ∂ xl ∂ xs ∂ xr ∂ xl i=1 n ∂ Hqi (α ) ∂ 2 fi (α ) ∂ 3 fi (α ) + , + 2τk ∑ Hqi (α ) ∂ xl ∂ xs ∂ xr ∂ xs ∂ xr ∂ xl i=1
(12)
for q, l, r, s ∈ {1, 2, . . ., n}. Lemma 3. Let ηk (x), k = 1, . . . , n, be the functions used in Lemma 2 and fi (x), i = 1, . . . , n, the coordinate functions of F(x). Then,
∂ fi (ηk (x)) = (1 − τk )Jil (α ), ∂ xl |x=α
(13)
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∂ 2 fi (ηk (x)) ∂ 2 fi (α ) = (1 + τk2 − τk ) , ∂ xr ∂ xl |x=α ∂ xr ∂ xl
(14)
and
∂ 3 fi (ηk (x)) ∂ 3 fi (α ) = (1 − τk3 + 3τk2 − τk ) − (15) ∂ xs ∂ xr ∂ xl |x=α ∂ xs ∂ xr ∂ xl 2 n ∂ fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) −τk2 ∑ Hqi (α ) − − , ∂ xq ∂ xl ∂ xs ∂ xr ∂ xr ∂ xq ∂ xs ∂ xl ∂ xs ∂ xq ∂ xr ∂ xl q=1 for i, l, r, s ∈ {1, 2, . . . , n}.
3 Description and Convergence Analysis of the Methods Let F : D ⊆ Rn −→ Rn be a sufficiently differentiable function and α ∈ D a zero of the nonlinear system F(x) = 0. Let G be the fixed point function that allows us to describe (3), % $ G(x) = λ (x) + JF (x)−1 F(x) − JF (x)−1
m
∑ AhF(ηh (x))
,
h=1
where ηh (x) = x − τh JF (x)−1 F(x), τh ∈ [0, 1] and λ (x) = x − JF (x)−1 F(x). The ith component of this function can be rewritten as $ % n
∑ Ji j (x) g j (x) − λ j (x) −
j=1
n
m
p=1
h=1
∑ H j p(x) f p (x) + ∑ Ah fi (ηh (x)) = 0.
(16)
By direct differentiation of (16), being i and l arbitrary and fixed, % $ n n ∂ Ji j (x) g j (x) − λ j (x) − ∑ H j p (x) f p (x) + ∑ j=1 ∂ xl p=1 $ % n n n ∂ g j (x) ∂ λ j (x) ∂ H j p (x) ∂ f p (x) + ∑ Ji j (x) − −∑ f p (x) − ∑ H j p (x) + ∂ xl ∂ xl ∂ xl ∂ xl j=1 p=1 p=1 m n ∂ fi (ηh (x)) ∂ (ηh (x))q + ∑ Ah ∑ = 0. (17) ∂ (ηh (x))q ∂ xl q=1 h=1 When x = α , by applying (4), (7), (10) and taking into account that g j (α ) = α , λ j (α ) = α and fi (α ) = 0, we have $ % n m ∂ g j (α ) ∑ Ji j (α ) ∂ xl + Jil (α ) −1 + ∑ Ah(1 − τh) = 0. j=1 h=1
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Therefore, we can conclude the following result. Proposition 1. If parameters Ah and τh satisfy
m
∑ Ah(1 − τh ) = 1 and matrix
h=1
JF (x) is continuous and nonsingular in x = α , then the iterative method (3) has at least order of convergence 2. If we take A1 and τ1 such that A1 (1 − τ1 ) = 1, we obtain methods whose order of convergence is at least 2. For example, by using A1 = 1 and τ1 = 0, we have Newton’s method. Now, by direct differentiation of (17), being r arbitrary and fixed, by substituting x = α and applying (4), (5), (7),(8) and (14) and taking into account that
m
∑ Ah (1 −
h=1
τh ) = 1, we obtain ∂ 2 g j (α ) ∂ 2 fi (α ) J ( α ) + i j ∑ ∂ xr ∂ xl ∂ xr ∂ xl j=1 n
$
m
−1 + ∑
% Ah τh2
= 0.
h=1
So, we can affirm: Proposition 2. If parameters Ah and τh satisfy m
∑ Ah (1 − τh) = 1
h=1
m
and
∑ Ah τh2 = 1,
(18)
h=1
and matrix JF (x) is continuous and nonsingular in x = α , then the iterative method (3) has at least order of convergence 3. In this case, for m = 1 we obtain the iterative method √ 3+ 5 (k+1) (k) JF (x(k) )−1 F(η1 (x(k) )), =x − (19) x 2 √ 5−1 (k) (k) where η1 (x ) = x − JF (x(k) )−1 F(x(k) ). The order of this method is 3 and 2 2 its efficiency index 31/(n +2n) . We note that this index is greater than the efficiency 2 index of Newton’s method, whose value is 21/(n +n) . For m = 2, parameters A j and τ j must satisfy a system that has infinite solutions. One of them, A1 = A2 = 1, τ1 = 0 and τ2 = 1 gives us the method described by Traub in [4]. In an analogous way, by a new direct differentiation of (17), being s arbitrary and fixed, by taking x = α and by applying (5), (6), (8), (9), (15) and conditions (18), we have
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$
%
265
∂ 3 g j (α ) − ∂ xs ∂ xr ∂ xl j=1 h=1 n n ∂ 2 fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) ∂ 2 fi (α ) − ∑ Hqi (α ) − ∑ Hqi (α ) + ∂ xq ∂ xl ∂ xs ∂ xr q=1 ∂ xq ∂ xr ∂ xs ∂ xl q=1 n ∂ 2 fi (α ) ∂ 2 fi (α ) + ∑ Hqi (α ) = 0. ∂ xs ∂ xq ∂ xr ∂ xl q=1 ∂ 3 fi (α ) ∂ xs ∂ xr ∂ xl
m
1− ∑
n
+ ∑ Ji j (α )
Ah τh3
Therefore, we can establish the following proposition: Proposition 3. If parameters Ah and τh satisfy m
m
∑ Ah(1 − τh) = 1, ∑ Ah τh2 = 1
h=1
and
m
and
h=1
∑ Ahτh3 = 1,
(20)
h=1
∂ 2 fi (α ) = 0, ∀a, b, i ∈ {1, 2, . . . , n}, then (3) has at least order 4. ∂ xa ∂ xb
For m = 3 one of the infinite solutions allows us to obtain the method x(k+1) = x(k) − JF (x(k) )−1 4F(η1 (x(k) )) − 6F(η2 (x(k) )) + 4F(η3 (x(k) )) ,
(21)
where τ1 = 1/4, τ2 = 1/2andτ3 = 3/4. This method has order 4 and efficiency index 2 41/(n +4n) . Again, being u arbitrary and fixed, using conditions of Proposition 3 and the results of the previous section, it can be proved that $ % m n ∂ 4 g j (α ) ∂ 4 fi (α ) 4 −1 + ∑ Ah τh + ∑ Ji j (α ) = 0. ∂ xu ∂ xs ∂ xr ∂ xl ∂ xu ∂ xs ∂ xr ∂ xl j=1 h=1 Therefore, we can establish a similar result to the previous proposition. Proposition 4. If parameters Ah and τh satisfy m
m
m
∑ Ah (1 − τh) = 1, ∑ Ah τh2 = 1, ∑ Ahτh3 = 1
h=1
h=1
h=1
m
and
∑ Ahτh4 = 1
(22)
h=1
∂ 2 fi (α ) = 0, ∀a, b, i ∈ {1, 2, . . ., n}, then the iterative method (3) has at least ∂ xa ∂ xb order 5. and
In general, we can establish the following result: Theorem 2. Let F : D ⊆ Rn −→ Rn be sufficiently differentiable at each point of an open neighborhood D of α ∈ Rn , that is a solution of the system F(x) = 0. Let us suppose that JF (x) is continuous and nonsingular in α . Then the sequence {x(k) }k≥0 obtained using the iterative expression (3) converges to α with convergence order:
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m
h=1
h=1
∑ Ah (1 − τh) = 1, ∑ Ahτhp = 1,
p = 2, 3, . . . , 2d − 1 and
∂ j fi (α ) = 0, 1 ≤ i, a1 , . . . , a j ≤ n, j = 2, 3, . . . , d. ∂ xa 1 ∂ xa 2 . . . ∂ xa j • 2d + 1, if
m
m
h=1
h=1
∑ Ah(1 − τh) = 1, ∑ Ah τhp = 1,
p = 2, 3, . . . , 2d and
∂ j fi (α ) = 0, 1 ≤ i, a1 , . . . , a j ≤ n, j = 2, 3, . . . , d. ∂ xa 1 ∂ xa 2 . . . ∂ xa j
4 Numerical Examples In this section we apply classical Newton’s method (CN), Traub’s method (TM) and the new methods described by (19) and (21) (denoted by NM1 and NM2, respectively) in order to estimate the zeros of the following nonlinear functions. T (a)F(x1 , x2 ) = (sin(x1 ) + x2 cos(x √ 1 ), x1 − x2 ), α = (0, 0) .T 2 (b)F(x1 , x2 ) = (exp(x1 ) − exp( 2x1 ), x1 − x2), α = (0, 0) . x2 (c)F(x1 , x2 ) = (− 2 + exp(x2 ) + x1 − 2, x2 − 2x1 + 2), α = (1, 0)T . 2 (d)F (x) = ( f1 (x) , f2 (x) , . . . , fn (x)), where x = (x1 , x2 , . . . , xn )T and fi : Rn → R, i = 1, 2, . . . , n, such that fi (x) = xi xi+1 − 1, i = 1, 2, . . . , n − 1 and fn (x) = xn x1 − 1. The exact zeros of F(x) are α1 = (1, 1, . . . , 1) and α2 = (−1, −1, . . . , −1) when n is odd. Results appearing in Table 1 are obtained for n = 999 and all the methods converge to α1 . (e)F(x1 , x2 ) = (x1 + exp(x2 ) − cos(x2 ), 3x1 − x2 − sin(x2 )), α = (0, 0)T . 1 (f) F(x1 , x2 , x3 ) = (− sin(x1 ) + cos(x2 ), xx31 − , exp(x1 ) − x23 ), x2 α = (0.9095695, 0.6612268, 1.575834)T .
These nonlinear functions have been chosen in order to have different points of view: the second partial derivatives of functions from (a) to (c) are null in their respective solutions, so that the convergence order of the methods increase; the case of function (d) is that of a big-sized system; functions (e) and (f) have singular, respectively bad conditioned, jacobian matrix. done using MATLAB. The stopping criterion used is - - were- All computations - - (k+1) (k) (k) − x - + -F x - < 10−12. For every method, we analyze the number -x of iterations needed to converge to the solution and the order of convergence estimated by - - ln(-x(k+1) − α - / -x(k) − α -) - - . (23) p≈ ln(-x(k) − α - / -x(k−1) − α -)
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Table 1 Numerical results for nonlinear systems
F(x)
x(0)
Solution
(a)
(0.4, 0.4)T (1, −2)T (−0.5, 0.5)T (−1, −0.5)T (−1, −2)T (2.5, 1.5)T (2, . . . , 2)T (3, −3, . . . , 3)T (2, −2)T (1, 1)T (1, 1, 1)T (0.5, 0.1, 0.7)T
α α α α α α α1 α1 α α α α
(b) (c) (d) (e) (f)
Iterations p CN NM1 TM NM2 CN NM1 TM NM2 5 4 4 4 6 5 4 4 5 4 4 4 6 5 5 5 5 5 4 4 5 5 4 4 6 5 5 5 7 5 5 5 n.c. 43 n.c. 75 6 5 5 5 8 8 n.c. n.c. 10 7 7 7
3.0 3.0 3.0 3.1 2.8 3.0 2.0 2.2 2.0 2.0
3.0 3.0 3.0 3.0 3.0 3.0 3.1 3.1 2.9 -
5.0 4.9 5.1 5.1 3.6 4.4 2.9 3.2 2.9 -
5.0 4.8 4.8 4.7 3.7 3.9 3.0 2.9 2.9 2.9 -
In Table 1 we can observe several results obtained using the previously described methods in order to estimate the zeros of functions from (a) to (f). For every function, the following items are specified: the initial estimation x(0) and, for each method, the approximate solution found, the number of iterations needed (n.c. denotes that the method does not converge) and the estimated computational order of convergence p. The value of p that appears in Table 1 is the last coordinate of vector p when the variation between its coordinates is small. When this does not happen, the value of p is said to be not conclusive, and is denoted by ” - ” in the mentioned table. Acknowledgements. Supported by Ministerio de Ciencia y Tecnolog´ıa MTM2007-64477.
References 1. Cordero, A., Torregrosa, J.R.: Variants of Newton’s method for functions of several variables. Applied Mathematics and Computation 183, 199–208 (2006) 2. Frontini, M., Sormani, E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation 149, 771–782 (2004) 3. Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, New York (1966) 4. Traub, J.F.: Iterative methods for the solution of equations. Chelsea Publishing Company, New York (1982)
Identifiability of Nonaccessible Nonlinear Systems Leontina D’Angi`o, Maria Pia Saccomani, Stefania Audoly and Giuseppina Bellu
Abstract. Identifiability is a fundamental prerequisite for model identification. Differential algebra tools have been applied to study identifiability of dynamic systems described by nonlinear polynomial equations. In a previous paper a differential algebra method for testing identifiability for locally and globally non accessible systems has been proposed. In this paper we describe a strategy to simplify the above differential algebra method to test identifiability of systems which are non accessible from everywhere. In particular we make the method more efficient and thus of more general applicability. A strategy for testing identifiability also of nonlinear models described by non polynomial equations is proposed.
1 Introduction Global identifiability concerns uniqueness of the model parameters determined from input-output data, under ideal conditions of noise-free observations and error-free model structure. There different methods which have been proposed to check global identifiability of nonlinear systems [2, 3, 7–9, 11, 13, 14]. In particular the differential algebra approach has been recently generalized [11] to deal with both locally and globally non accessible systems. Unfortunately in this last case, i.e. systems nonaccessible from everywhere, the proposed method requires the calculation of Leontina D’Angi`o and Giuseppina Bellu Department of Mathematics University of Cagliari, 09100 Cagliari Italy, e-mail:
[email protected],
[email protected] Maria Pia Saccomani Department of Information Engineering University of Padova, 35131 Padova Italy, e-mail:
[email protected] Stefania Audoly Department of Structural Engineering University of Cagliari, 09100 Cagliari Italy, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 269–277. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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a closed form expression for the integral of a suitable differential form, which of course is a difficult task. To the best of our knowledge, a general procedure to do this integration is not known. The principal goal of this paper is to show that, to check global identifiability of globally nonaccessible systems, the above integration is not necessary. In this paper we also provide the theoretical basis for correctly testing global identifiability of systems involving non polynomial, for example exponential or logarithmic, functions. As an elementary example consider a (non-polynomial) system like x˙1 = a exp(−x2 ) + u x˙2 = −bx1 where a and b are the unknown parameters, u the input, x1 and x2 the state variables. This system can be rendered polynomial by introducing a new state x3 = a exp(−x2 ) and by differentiating it the following additional equation x˙3 = −x˙2 x3 is provided. This differential equation will turn it into a third order system of the following form x˙1 = x3 + u x˙2 = −bx1 x˙3 = bx1 x3
which is indeed polynomial (and time-invariant). Furthermore, in many biological and physiological applications, very often in the differential equations describing the phenomena, time-varying coefficients appear with a known functional form but depending on some unknown parameters. Consider for example a system like x˙1 = a exp(−bt)x1 + u This system also can be rendered polynomial by introducing a new state x2 = exp(−bt) and an additional equation x˙2 = −bx2 , which will turn it into the following second order polynomial (and time-invariant) system x˙1 = ax1 x2 + u x˙2 = −bx2 With this technique one can handle many classical situations where time-varying coefficients of known functional form appear, and in fact even non algebraic nonlinearities arising to an augmented model which is trivially globally nonaccessible since the evolution of the system obtained by adding the new state variable is constrained to take place in some invariant submanifold.
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2 Background 2.1 A priori Identifiability This section provides the reader with the definitions which are necessary to set the notations used in the paper. For a formal treatment of differential algebra, see [10]. Consider a nonlinear dynamic system described in state space form x˙ (t) = f(x(t), p) + ∑m i=1 gi (x(t), p)ui (t) (1) y(t) = h(u(t), x(t), p) where the state variable x evolves in an open set X of the n-dimensional space n ; u is the m-dimensional input ranging on some vector space of piecewise smooth (infinitely differentiable) functions and y is the r-dimensional output. The constant unknown p-dimensional parameter vector p belongs to some open subset P of the p-dimensional Euclidean space p . Whenever initial conditions are specified, the relevant equation x(0) = x0 is added to the system. The essential assumptions here are that there is no feedback, there is the affine structure in u and f, g1 , . . . , gm and h are vectors of rational functions in x. The dependence on p may be rational. We adopt the identifiability definitions used in [11] and recall here only that of global identifiability. Let y = ψx0 (p, u) be the input-output map of the system (1) started at the initial state x0 (we assume that this map exists). Definition 1. The system (1) is a priori globally (or uniquely) identifiable from input-output data if, for at least a generic set of points p∗ ∈ P, there exists (at least) one input function u such that the equation
ψx0 (p, u) = ψx0 (p∗ , u) has only one solution p = p∗ for almost all initial states x0 ∈ X ⊆
(2) n.
2.2 Accessibility Here we recall a concept of geometric nonlinear control theory [4, 5, 12] called accessibility. In particular the accessibility can be view as a weak counterpart of the concept of reachability (from an arbitrary initial state). Definition 2. The system (1) is accessible from x0 if the set of states reachable from x0 (at any finite time) has a nonempty interior, i.e. contains an open ball in n . To study accessibility one looks at the Control Lie Algebra, i.e. the smallest Lie algebra C containing the vector fields f, g1 , ..., gm of (1) and invariant under Lie bracketing with f, g1 , ..., gm . To the Lie algebra C we associate the distribution ΔC mapping each x ∈ n into the vector space
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ΔC (x) = span{τ (x) : τ ∈ C } We recall from the literature [4, 5, 12] the so called accessibility rank condition. Theorem 1. For analytic, in particular polynomial, systems, a necessary and sufficient condition for accessibility from x0 is that dimΔC (x0 ) = n
3 Identifiability of Globally Nonaccessible Systems Our goal here is to investigate the differential algebra algorithm, to see if it correctly tests identifiability of globally nonaccessible systems. The starting point will be the results in [11] holding for local nonaccessible systems. We know that the characteristic set of the ideal generated by the polynomials defining the dynamical system is independent of the system initial conditions; in fact the pseudodivision algorithm for calculating the characteristic set does not take into account the initial conditions. If the system is accessible, whatever the initial point, the whole space where the solutions evolve is correctly described by the ideal generated by the differential polynomials describing the system [11]. In case of systems nonaccessible from initial points belonging to a thin set, the above ideal does not any more describe the space where the system solution evolves. In this case the ideal to be considered in the identifiability test should include also the invariant submanifold where the solution of the system starting from that particular point, evolves. This submanifold can be obtained by Frobenius Theorem, calculating the Control Lie algebra as indicated at the end of the previous section. Since our system is polynomial (and hence analytic), there exists a unique maximal submanifold Mx0 of n through x0 which carries all the trajectories of the control system (1) started at x0 . In particular, if the dimension of ΔC (x0 ) is n then dim(Mx0 ) = n [4]. Thus the ideal which correctly describes the whole space where the solutions evolve is the one generated by the polynomials defining the dynamical system plus the polynomials defining the invariant submanifold. The case of a globally nonaccessible system is different. In this case many invariant submanifolds exist, one for each initial point. We would like to stress that, in this case, the initial state does not belong to a thin set, but rather to a generic set of points in the state space. This implies that, when the polynomials defining the invariant submanifold are added to the original polynomials, no additional conditions are added to the original system. We have shown this in many examples. We have also observed that the polynomials defining the invariant submanifold could be calculated by integrating some suitable combination of the polynomials describing the dynamic system. By following this line of reasoning, we can conclude that, in case of globally nonaccessible model, the system obtained by adding the polynomials defining the invariant submanifold to the original ones, is equivalent to the original system itself where an integration in closed form has been performed.
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Note that this integration in closed form can be performed in both the accessible and nonaccessible models, providing the equation of the invariant submanifold. In fact, by adding this equation to those defining the dynamical system and by suitably reducing them, the system order always decreases, i.e. the maximum order derivative variable disappears (the only difference in case of accessible model being that the invariant submanifold depends on the system input). Thus, we conclude that to test identifiability of a nonlinear dynamical model, one has first to test the accessibility of the model from its initial conditions. If the model is globally non accessible, the identifiability test on the original system gives the correct result, independently from the knowledge of the invariant submanifold where the solution of the system evolves.
3.1 Two Examples To give evidence to the method discussed in the previous section, we present two examples of globally nonaccessible systems. Example 1. Consider the following model ⎧ x˙ = p1 ux3 ⎪ ⎪ ⎨ 1 x˙2 = p2 x1 x˙3 = p3 x1 x2 ⎪ ⎪ ⎩ y = p 4 x2
(3)
It is easy to see that by combining the second and the third differential equations we obtain x˙3 /x˙2 − (p3 /p2 )x2 = 0. By integrating it, one can see that, no matter of which input is chosen, the system evolves in the algebraic submanifold of 3 described by
φ (x) = −p3 x22 /2 + p2x3 + p3x220 /2 − p2x30 = 0
(4)
which clearly shows that we have a system which is not accessible from any point in 3 . To check this formally, one computes the matrix made with the vector fields f , g and the Lie brackets [ f , g], [ f , [ f , g]], [ f , [ f , [ f , g]]], . . .. Since all Lie brackets in the above sequence are zero after the fifth one, the distribution is involutive and all covectors μ orthogonal to the distribution ΔC have the form:
μ = α (x)[0, −p3 x2 , p2 ]T
(5)
where α (x) is an arbitrary non-zero smooth function. Therefore dim ΔC < 3 for all x ∈ 3 . By applying Frobenius Theorem in a relatively open neighborhood of any x0 for which dim ΔC (x0 ) = 2, it must follow that for some suitable α , the covector μ generates a closed differential form. In fact, it is easy to check that φ has differential d φ (x) = −p3 x2 dx2 + p2dx3 which, by integration provides just eq. (4).
(6)
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To test the identifiability of system (3), consider first the dynamic system regardless of accessibility. In this case an easy computation provides the following characteristic set: ... ¨ 4 + y up4 − yp ˙ 1 p3 u2 y A1 ≡ −u˙yp A2 ≡ y˙ − p2 p4 x1 (7) A 3 ≡ y − p 4 x2 A4 ≡ y¨ − p1 p2 p4 ux3 hence the exhaustive summary is p1 p3 /p4 and the system (3) is nonidentifiable. Now assume that the system is started from a generic initial condition x0 = [x10 , x20 , x30 ]T with all the components different from zero so that dim ΔC (x0 ) = 2. In this case the solution of system (3) evolves in the slice described by φ (x) = 0. Note that dim ΔC = 2 for all x ∈ { 3 − T } where T is the “thin” set of equilibrium points where dim ΔC = 0: T := {x : x1 = 0,
x2 = x20 ,
x3 = 0}
(8)
By following the same line of reasoning applied to the generically accessible systems, we add the equation φ (x) = 0 to the system equations (3) and the characteristic set turns out to be ¨ 4 + p1(−2p2 p24 ux30 + p3 (p24 ux220 − uy2)) Aˆ 1 ≡ 2yp ˆ A2 ≡ y˙ − p2 p4 x1 Aˆ 3 ≡ y − p4x2 Aˆ 4 ≡ p2 p24 (2x3 − 2x30) + p3 (p24 x220 − y2 )
(9)
By comparing this with the previous characteristic set (7), one can see that the system order is decreased. However, the identifiability test based on this characteristic set gives exactly the same identifiability results, i.e. (p1 p3 /p4 ), obtained regardless of initial conditions. Thus it is not necessary to calculate a closed form expression for the integral of the differential form (6). Example 2. It may well happen that the invariant submanifold is not algebraic. In this case one can not enlarge the ideal in the ring of differential polynomials. Consider the system ⎧ x˙1 = p1 ux3 + x2 ⎪ ⎪ ⎨ x˙2 = p2 x1 (10) ⎪ x˙3 = p3 x1 x3 ⎪ ⎩ y = x3 By integrating x˙3 /x˙2 − (p3 /p2 )x3 = 0, one can see that, no matter of which input is chosen, the system evolves in the submanifold described by
φ (x) = p2 (log(x3 ) − log(x30)) − p3 (x2 − x20 ) = 0
(11)
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where x0 = [x10 , x20 , x30 ]T . Hence the system is not accessible from any point in 3 . Given the form of eq. (11), there is no algebraic polynomial vanishing at the solutions of the system started at any x0 , i.e. Mx0 is not algebraic, see section 2.2. Formally, by computing the matrix with the vector fields f , g and the Lie brackets [ f , g], [ f , [ f , g]], [ f , [ f , [ f , g]]], . . ., it can be shown that all covectors μ orthogonal to the distribution ΔC must be of the form:
μ = α (x)[0, −p3 x3 , p2 ]T
(12)
where α (x) is an arbitrary non-zero smooth function. Therefore dim ΔC < 3 for all x ∈ 3 . By applying Frobenius Theorem it follows that the covector μ generates the following closed differential form: d φ (x) = −p3 x3 dx2 + p2dx3
(13)
which, by integration provides just eq. (11). First, identifiability of system (10) regardless of initial conditions is tested. The characteristic set is the following ... A1 ≡ uy ˙ 4 p1 p3 − y y2 + 3y¨yy ˙ − 2y˙3 + yuy ˙ 3 p1 p3 + yy ˙ 2 p2 A2 ≡ y˙ − p3x1 y (14) A3 ≡ −yy ¨ + y˙2 + uy3 p1 p3 + x2 y2 p3 A4 ≡ −x3 + y hence the exhaustive summary is (p2 , p1 p3 ) showing that p2 has one solution while p1 and p3 have an infinite number of solutions. Note that if the initial condition is known, the system becomes globally identifiable. Now assume that the system is started from an unknown initial condition x0 = [x10 , x20 , x30 ]T with all components different from zero so that dim ΔC (x0 ) = 2. The solution of system (10) evolves in the slice described by the nonalgebraic equation φ (x) = 0. This obviously can not be added to the system equations (10). However, by suitably manipulating the polynomials defining the characteristic set (14) together with the transcendent function defining the invariant submanifold (11), we obtain the following new set of functions Aˆ 1 ≡ −yy ¨ + y˙2 + (log(y) − log(x30))p2 y2 + p1 p3 uy3 + p3 x20 y2 Aˆ 2 ≡ y˙ − p3 x1 y Aˆ 3 ≡ (log(y) − log(x30))p2 − (x2 − x20)p3 Aˆ 4 ≡ −x3 + y
(15)
It is easy to see that Aˆ 1 , which now depends also from the initial condition, provides exactly the same identifiability result of that obtained regardless of initial conditions, i.e. the system (10) is nonidentifiable. Again, if the initial condition is known, the system becomes globally identifiable. Note that in this example also it is shown that it is not necessary to calculate a closed form expression for the integral of the differential form (13).
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4 A Biological Model Consider the following biological model. It is a twelve-compartments model recently proposed in [6] to describe the nitrogen metabolism in humans. ⎧ g˙ = −k21 g + u ⎪ ⎪ ⎪ ⎪ il˙ = k21 g − (k32 + k42 )il + k24sa ⎪ ⎪ ⎪ ⎪ e˙ = k32 il ⎪ ⎪ ⎪ ⎪ sa ˙ = k42 il + k45 scp + k47 pl + k410 bu − (k24 + k54 + k64 + k74 + k104 + k124 )sa ⎪ ⎪ ⎪ ⎪ scp ˙ = k54 sa − k45scp ⎪ ⎪ ⎨ sep ˙ = k64 sa ˙ = k74 sa + k78 pa − (k47 + k87)pl pl ⎪ ⎪ ⎪ ⎪ pa ˙ = k87 pl + k89 pp − (k78 + k98 + k128)pa ⎪ ⎪ ⎪ ⎪ pp ˙ = k98 pa − k89 pp ⎪ ⎪ ⎪ ⎪ ˙ = k104 sa − (k1110 + k410)bu bu ⎪ ⎪ ⎪ ⎪ uu ˙ = k1110 bu ⎪ ⎪ ⎩ ua ˙ = k124 sa + k128 pa (16) The measurement equations are: y1 = e,
y2 = sep,
y3 = pl,
y4 = bu,
y5 = uu,
y6 = ua
(17)
The unknown parameters are: p = [k21 , k32 , k24 , k42 , k45 , k47 , k54 , k64 , k74 , k104 , k124 , k78 , k87 , k89 , k98 , k128 , k410 , k1110 ] To study the accessibility of this model, one should apply the Frobenius theorem and calculate dim ΔC (x0 ). Given the high dimension of the model this calculation is very complex. However, it is easy to see that the model is globally nonaccessible simply by manipulating some of its equations. For example one can calculate: bu = uu/k ˙ 1110
sa = sep/k ˙ 64
(18)
which substituted in the 10th equation of system (16) provides: ˙ + k64 uu(k k1110 k64 bu ˙ 1110 + k410) − k1110k104 sep ˙
(19)
This equation can be finitely integrated
φˆ = k64 k1110 bu + k64uu(k1110 + k410) − k1110k104 sep + constant
(20)
The solution of the model evolves thus in this submanifold showing that the model is globally nonaccessible (φˆ is not necessarily the equation of the whole invariant submanifold φ which should be calculated with the Frobenius theorem). As discussed in the previous section, one can correctly perform the identifiability test of
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this model by calculating the ideal generated only by the polynomials defining the system. The model is globally identifiable [6].
5 Conclusions In this paper we show that the differential algebra method based on the characteristic set of the ideal generated only by the polynomials defining the system can be successfully used in testing the identifiability of globally nonaccessible systems. This result is particular relevant since it allows to correctly test the identifiability of many models described by nonpolynomial equations. A software tool DAISY [1] to test identifiability of nonlinear systems under the conditions described in this paper is available in htt p : //www.dei.unipd.it/ ∼ pia/.
References 1. Bellu, G., Saccomani, M.P., Audoly, S., D’Angi`o, L.: DAISY: a new software tool to test global identifiability of biological and physiological systems. Computer Methods and Programs in Biomedicine 88, 52–61 (2007) 2. Chapman, M.J., Godfrey, K.R., Chappell, M.J., Evans, N.D.: Structural identifiability of non-linear systems using linear/non-linear splitting. Int. J. Control 76(3), 209–216 (2003) 3. Chappell, M.J., Godfrey, K.R.: Structural identifiability of the parameters of a nonlinear batch reactor model. Math. Biosci. 108, 245–251 (1992) 4. Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Automatic Control AC-22(5), 728–740 (1977) 5. Isidori, A.: Nonlinear control systems, 3rd edn. Springer, London (1995) 6. Juillet, B., Saccomani, M.P., Bos, C., Gaudichon, C., Tom´e, D., Fouillet, E.: Conceptual, methodological and computational issues in compartmental modeling of a complex biological system: the postprandial inter-organ metabolism of dietary nitrogen in humans. Math. Biosci. 204, 282–309 (2006) 7. Ljung, L., Glad, S.T.: On global identifiability for arbitrary model parameterizations. Automatica 30(2), 265–276 (1994) 8. Margaria, G., Riccomagno, E., Chappell, M.J., Wynn, H.P.: Differential algebra methods for the study of the structural identifiability of rational function state-space models in the biosciences. Math. Biosci. 174, 1–26 (2001) 9. Ollivier, F.: Le probl`eme de l’identifiabilit´e structurelle globale: e´ tude th´eorique, ´ m´ethodes effectives et bornes de complexit´e. Th`ese de Doctorat en Science, Ecole Polyt´echnique, Paris, France (1990) 10. Ritt, J.F.: Differential Algebra. RI: American Mathematical Society, Providence (1950) 11. Saccomani, M.P., Audoly, S., D’Angi`o, L.: Parameter identifiability of nonlinear systems: the role of initial conditions. Automatica 39, 619–632 (2003) 12. Sontag, E.D.: Mathematical control theory, 2nd edn. Springer, Berlin (1998) 13. Denis-Vidal, L., Joly-Blanchard, G.: Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems. Automatica 40, 287–292 (2004) 14. Walter, E., Lecourtier, Y.: Global approaches to identifiability testing for linear and nonlinear state space models. Math. Biosci. 24, 472–482 (1982)
Trajectory Tracking Control of a Timed Event Graph with Specifications Defined by a P-time Event Graph Philippe Declerck and Abdelhak Guezzi
Abstract. The aim of this paper is a trajectory tracking control of Timed Event Graphs with specifications defined by a P-time Event Graph. Two problems are solved on a fixed horizon knowing the current state: The optimal control for favorable past evolution; The prediction of the earliest future evolution of the process. These two parts make up an on-line control which is used on a sliding horizon. Completely defined in (max, +) algebra, the proposed approach is a Model Predictive Control using the componentwise order relation.
1 Introduction In this paper, we focus on the trajectory tracking control of Timed Event Graphs with reference model defined by a P-time Event Graph. The P-time Event Graph describes the desired behavior of the interconnections of all the internal transitions. Some events are stated as controllable, meaning that the corresponding transitions (input) may be delayed from firing until some arbitrary time provided by a supervisor. We wish to determine the greatest input in order to obtain the desired behavior defined by the desired output and the specifications. This problem is denoted problem 1 in the document. Moreover, the aim of this paper is also the extension of problem 1 to Predictive Control on infinite horizon. This extension is denoted problem 2. Using a receding horizon principle, Model Predictive Control is a form of control in which the current control is obtained by solving on-line, a finite open-loop optimal control problem at each sampling instant. The current state of the process is considered as the initial Philippe Declerck and Abdelhak Guezzi LISA EA4014, University of Angers, 62 avenue Notre-Dame du Lac, 49000 Angers, France, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 279–290. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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state. The optimization yields an optimal control sequence but only the first control in this sequence is applied to the plant. This procedure can be repeated infinitely. In this paper, we complete the approaches developed in [4] and [6] by introducing specifications defined by a P-time Event Graph as in the preliminary study [5] which is generalized to a sliding horizon. The framework of this proposed study can be found in [6] where a comparison with [15] is given in the standard algebra. The approach is based on the concept of earliest desired output which was introduced in [4] to the best of our knowledge. A similar concept was also considered in [11]: as this last approach uses the past control and not the current state, we can prove that the relevant updated desired output (called reference input in [11]) can be lower. Let us recall that a simple forward technique gives the earliest desired output while the control is given by the classical backward approach. However, this simple technique does not hold if some specifications are introduced in the problem as shown in parts 2 and 3: the structure of matrix Dh in part 2.2 shows the forward and backward connections of inequality X ≥ Dh ⊗ X for instance. In this paper, we consider that each transition is observable: the event date of each transition firing is assumed to be available. Let us note that we have developed software written in Scilab composed of estimation, prediction and control. No hypothesis is taken on the structure of the Event Graphs which does not need to be strongly connected. The initial marking should only satisfy the classical liveness condition and the usual hypothesis that places should be First In First Out (FIFO) is taken. In the context of the trajectory tracking control (problem 2), we consider different structures of matrix B. Defined in part 3.3.2, the case of fully controlled transitions can be found in the modeling of railway system where each departure of train must be controlled [2, 14]. This structure is also considered in urban bus networks where the timetable must be respected at each stop [9]. The paper is structured as follows: The optimal control on a fixed horizon (problem 1) and its extension to a sliding horizon (problem 2) are successively considered. The resolution of problem 2 is based on the prediction of the earliest desired output. By reason on the lack of place, we cannot give a complete presentation of the preliminary remarks but the reader can easily find more information in [1] and [8]. The presentation of the model of the P-time Event Graph is also omitted: the reader can find the preliminaries and the presentation of the models in [5]. Maximization and addition operations are denoted respectively ⊕ and ⊗. The set of n.n matrices with entries in dioid D = max = ( ∪{−∞} ∪{+∞}, ⊕, ⊗) including the two operations ⊕ and ⊗ is a dioid, which is denoted Dn.n . Mapping f is said to be residuated if for all y ∈ D, the least upper bound of subset {x ∈ D | f (x) ≤ y} exists and lies in this subset. Mapping x ∈ ( max )n → A ⊗ x, defined over max is residuated (see [1]) and the left ⊗−residuation of B by A is denoted by: A\B = max{x ∈ ( max )n such that A ⊗ x B}. The following Theorem uses the Kleene star < i defined by: A∗ = +∞ i=0 A . Theorem 1. [1, Theorem 4.75 part 1] Consider equation x = A ⊗ x ⊕ B and inequality x ≥ A ⊗ x ⊕ B with A and B in complete dioid D. Then, A∗ B is the least solution to these two relations.
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Variable xi (k) is below the date of the kth firing of transition xi .
2 Control on a Fixed Horizon (Problem 1) Let us consider the objective of problem 1.
2.1 Objective The problem of this paper is the determination of the greatest control of a plant described by a Timed Event Graph when the state and control trajectories are constrained by additional specifications defined by a P-time Event graph. Applications of P-time Event Graphs can be found in production systems, microcircuit design, transportation systems, real-time systems and food industry. The objective is to calculate the greatest control u on horizon [ks + 1, k f ] such that its application to the Timed Event Graph defined by x(k + 1)=A ⊗ x(k) ⊕ B ⊗ u(k + 1) (1) y(k) = C ⊗ x(k) satisfies the following conditions: a) y ≤ z knowing the trajectory of the desired output z on a fixed horizon [ks + 1, k f ] with h = k f − ks ∈ ; b) The state trajectory follows the model of the autonomous P-time Event Graph defined by ε A+ x(k) x(k) ⊗ (2) ≥ A− A= x(k + 1) x(k + 1) for k ≥ ks ; c) The first state vector of the state trajectory x(k) for k ≥ ks is finite and is known vector x(ks ) . This “ non-canonical ” initial condition can be the result of a past evolution of a process. Underlined symbols like x(ks ), z(k) correspond to known data of the problem and x(k) and y(k) are estimated in the following resolutions based on the information available at number of events ks . A simple example of this problem is a production system composed of two tasks which are the cooking of a product and its packaging with an additional constraint: the cooking time must not be too excessive , otherwise, the product would be damaged. In the following part 2.2, we present the relations which describe a trajectory of a Timed Event Graph satisfying the specifications defined by a P-time Event Graph (constraint b)). The introduction of the ”Just-in-time” objective (constraint a)) in part 2.3 allows the resolution of the control problem on a fixed horizon.
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2.2 Trajectory Description From (1) and (2), we deduce a system which describes the trajectories on horizon [ks , k f ] . Let us introduce the following notations. Let X = t x(k + 1)t x(k + 2)t · · · x(k − 1)t x(k )t t ) x(k s s s f f ⎞ (t: transposed) and Dh = ⎛ + ε A ε ··· ε ε ε ⎜ A ⊕ A− A= A+ · · · ε ε ε ⎟ ⎟ ⎜ ⎜ε A ⊕ A− A= · · · ε ε ε ⎟ ⎟ ⎜ ⎜ ··· ··· ··· ··· ··· ··· ··· ⎟ ⎟ . Matrix Dh presents an original ⎜ ⎜ε ε ε · · · A= A+ ε ⎟ ⎟ ⎜ ⎝ε ε ε · · · A ⊕ A− A= A+ ⎠ ε ε ε ··· ε A ⊕ A− A= block tridiagonal structure: this is a square matrix, composed of a lower diagonal (square submatrices A ⊕ A− ), a main diagonal (square submatrices A= except the first element) and an upper diagonal (square submatrices A+ ), with all other blocks being zero matrices (ε ). As n is the dimension of x, Dh is a n.(h + 1)) x n.(h + 1)) matrix. Theorem 2. The state trajectories of a Timed Event Graph (1) starting from x(ks ) and following the specifications defined by a P-time Event Graph (2) on horizon [ks , k f ] satisfy the following system ⎧ X ≥ Dh ⊗ X ⎪ ⎪ ⎨ x(k) ≥ B ⊗ u(k) for k ∈ [ks + 1, k f ] (3) x(k) ≤ A ⊗ x(k − 1) ⊕ B ⊗ u(k) for k ∈ [ks + 1, k f ] ⎪ ⎪ ⎩ x(ks ) = x(ks ) Proof. System (3) is directly deduced from the models of the Timed Event Graph (1)and the P-time Event Graph (2). For instance, equality (1) is equivalent to A ⊗ x(k − 1) ⊕ B ⊗ u(k) ≤ x(k) for k ≥ ks . x(k) ≤ A ⊗ x(k − 1) ⊕ B ⊗ u(k)
2.3 Greatest Trajectory We now introduce the ”Just-in-time” objective defined by constraint a). Using the previous description of the state and control trajectories (3), the problem is rewritten under a general fixed point formulation x ≤ f (x) which allows the resolution of control problem 1. The greatest estimated state trajectory X and its relevant state x(k) are denoted X + and x+ (k) , respectively. Theorem 3. The greatest state and control trajectory of a Timed Event Graph (1) starting from x(ks ) and following specifications defined by a P-time Event Graph (2) on horizon [ks , k f ] is the greatest solution of the following fixed point inequality system
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⎧ X ≤ Dh \X ⎪ ⎪ ⎨ u(k) ≤ B\x(k) for k ∈ [ks + 1, k f ] ⎪ x(k) ≤ [A ⊗ x(k − 1) ⊕ B ⊗ u(k)] ∧C\z(k) for k ∈ [ks + 1, k f ] ⎪ ⎩ x(ks ) ≤ x(ks )
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(4)
with condition x(ks ) ≤ x+ (ks ). Proof. From Dh ⊗ X ≤ X and B.u(k) ≤ x(k) , we deduce X ≤ Dh \X and u(k) ≤ B\x(k) on horizon [ks + 1, k f ]. The constraints of the desired output y ≤ z and y(k) = Cx(k) can be introduced in the fixed-point formulation with x(k) ≤ C\z(k) . So, x(k) ≤ [Ax(k − 1) ⊕ B.u(k)] ∧ C\z(k) on horizon [ks + 1, k f ] . The constraint x(ks ) = x(ks ) can be written x(ks ) ≤ x(ks ) and x(ks ) ≤ x(ks ). Therefore, a condition is x(ks ) ≤ x(ks ). If condition x(ks ) ≤ x+ (ks ) is satisfied, then x(ks ) = x+ (ks ) and condition c) are satisfied. Therefore, the calculated state trajectory for k ≥ ks is consistent with the past evolution k ≤ ks : In other words, the Timed Event Graph can follow calculated trajectory X + after ks which obeys the specifications defined by the P-time Event Graph. System (4) leads to a fixed-point formulation whose general form is such that x ≤ f (x). Containing (min, max, +) term [A ⊗ x(k − 1) ⊕ B ⊗ u(k)] ∧ C\z(k) , f is also a (min, max, +) function. It can be defined by the following grammar: f = b, x1 , x2 , . . . , xn | f ⊗ a | f ∧ f | f ⊕ f where a, b are arbitrary real numbers (a, b ∈ ). The effective calculation of the greatest control can be made by a classical iterative algorithm of Mc Millan and Dill [12] which particularizes the algorithm of Kleene to (min, max, +) expressions. The general resolution of x ≤ f (x) is given by the iterations of xi ← xi−1 ∧ f (xi−1 ) if the finite starting point is greater than the final solution. Here, number i represents the number of iterations and not the number of components of vector x. The general algorithm of Mc Millan and Dill [12] is known to be pseudo-polynomial in practice. The aim of the following part is the extension of problem 1 to predictive control.
3 Predictive Control (Problem 2) We present below the principle of the sliding horizon in predictive control and the general technique of the proposed approach. Another description can be found in [6] where the control of a Timed Event Graph without specification is described in standard algebra. We assume that each event date of transition firing is available for current number of event k: at step k = ks , uks and xks are known. A future control sequence u(k) for k ∈ [ks + 1, ks + h] is determined such that this control is the optimal solution of the problem. The first element of the optimal sequence (here u(ks + 1)) is applied to the process. At the next number of event ks + 1, the horizon is shifted: at step ks + 1, the
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problem is updated with new information uks +1 and xks+1 and a new optimization is performed.
3.1 Principle of the Proposed Approach After the calculation of state trajectory x+ and control u at step ks , condition c) xks = x+ (ks ) must be checked in order to guarantee the coherence of the state trajectory between each iteration: this verification shows that future trajectory k ≥ ks + 1 is the extension of the past trajectory (k ≤ ks ). The on-line comparison of the two vectors xks and x+ (ks ), is similar to the comparator of the closed-loop of classical continuous control which compares a desired trajectory and its measure: when the two data are equal, the objective is obtained. In our context, an optimal control is similarly found. Let us consider the different cases. • If condition xks = x+ (ks ) is satisfied, we can conclude that control problem 1 has a solution for data z and xks : there is an optimal control such that, starting from the current state xks , the Timed Event Graph can follow a trajectory obeying the specifications defined by a P-time Event Graph with a Just-in-time criteria. • If xks = x+ (ks ), we can conclude that control problem 1 has no solution for data z and xks : the process presents some delays produced by a disruption of the process activity for instance. The Timed Event Graph cannot (provisionally) follow a trajectory obeying the constraints of the problem, i.e. the three conditions a), b) and c). Consequently, at least a specification and/or the Just-in-time criteria, is not satisfied if we directly apply the calculated control of part 2.3 to the Timed Event Graph starting from the initial condition xks . Therefore, the problem must be modified such that condition c) xks = x+ (ks ) is satisfied. In this paper, we consider that the model of the Timed Event Graph cannot be modified. If we assume that the initial condition is the result of a past evolution, xks is a datum of the problem and only condition a) and/or condition b) can be changed.
3.2 Predictive Control Objective Suppose that the fulfillment of the specifications (condition b)) is essential. In consequence, the only possibility is to modify the just in time criteria of condition a) and to put the desired output back such that problem 1 presents a solution. Therefore, an aim is the determination of a desired output such that control problem 1 presents a solution. Particularly, the state trajectory must start from current state xks . As a minimal desired output allows the limitation of the delays, the problem is to find the earliest desired output denoted z− such that
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• there is control such that its application to the Timed Event Graph generates a state trajectory which starts from the current state xks (condition c)) • this state trajectory follows the additional specifications defined by the P-time Event Graph on horizon [ks + 1, ks + h] (condition b)). This earliest desired output is a limit such that the Timed Event Graph cannot follow a lower trajectory satisfying the different constraints of the problem. Knowing earliest desired output z− , the optimal approach of part 2.3 can be applied to modified desired output trajectory zm (k) = z(k) ⊕ z− (k) for k ∈ [ks + 1, ks + h] such that this procedure yields a control which can be applied to the process. Therefore, condition a) is satisfied for the modified desired output zm and the relevant calculated control is optimal for zm . Below, we characterize an arbitrary state trajectory obeying the specifications (condition b)). System (3) will be rewritten under a fixed point formulation f (x) ≤ x allowing the prediction problem of the earliest desired output z− .
3.3 Prediction of the Earliest Desired Output z− An arbitrary state trajectory obeying the specifications is now described with a fixed point form. From system (3), we deduce the following system ⎧ ⎨ X ≥ Dh ⊗ X (5) x(k) ≥ B ⊗ u(k) for k ∈ [ks + 1, k f ] ⎩ x(ks ) = x(ks ) which allows the determination of an interesting desired output. Indeed, this system is a fixed-point form f (X) ≤ X where f is a (max,+) function (if we assume that control u is known). Therefore, we can apply the concept of componentwise order to the desired output as follows: The resolution makes the prediction of the earliest state trajectory x− (k) for k ∈ [ks + 1, ks + h] and so, of the earliest output trajectory z− (k) = C ⊗ x− (k). The modified desired output zm is consequently obtained: zm (k) = z(k) ⊕ z− (k) for k ∈ [ks + 1, ks + h]. We now characterize the set of trajectories of systems (3) and (5). Property 1. Each trajectory of system (3) satisfies (5). Proof. Immediate: As system (3) contains an additional constraint, any trajectory of this system satisfies relaxed system (5).
3.3.1
Earliest Firing Rule
As x(k) ≥ A ⊗ x(k − 1) ⊕ B ⊗ u(k) is already satisfied in (5), constraint x(k) ≤ A ⊗ x(k − 1) ⊕ B ⊗ u(k) guarantees the earliest firing rule. In this part, we determine the
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conditions such that this last relation can be disregarded in the determination of the trajectory. If we now consider only inequality x(k) ≥ B ⊗ u(k) of system (5), the greatest control is obviously u(k) = B\x(k). This control law is considered below. We assume that no row of B is null. Theorem 4. A trajectory of (5) x satisfies (3) if this state trajectory x also satisfies condition B ⊗ (B\x(k)) = x(k) for k ∈ [ks + 1, k f ]. Proof. Let us prove that inequality x(k) ≤ A ⊗ x(k − 1) ⊕ B ⊗ u(k) of system (3) is also satisfied in (5). The relaxation of x(k) = A ⊗ x(k − 1) ⊕ B ⊗ u(k) for k ∈ [ks + 1, k f ] gives x(k) ≥ A ⊗ x(k − 1) ⊕ B ⊗ u(k) or, x(k) ≥ B ⊗ u(k) and x(k) ≥ A ⊗ x(k − 1). This last inequality is expressed in (5) with X ≥ Dh ⊗ X. Let us suppose that an arbitrary trajectory denoted X satisfies system (5). Particularly, X satisfies X ≥ Dh ⊗ X and so inequality x (k) ≥ A⊗ x (k − 1) is satisfied. We want the Timed Event Graph defined by its state equation to follow given trajectory X (neither earlier, nor later) by applying a specific control. For given x (k), a possible control is u(k) = B\x (k) which is the greatest control satisfying inequality x (k) ≥ B ⊗ u(k). As B ⊗ u(k) = B⊗(B\x (k)) = x (k) and x (k) ≥ A⊗x (k −1), we can deduce that A⊗x (k − 1) ⊕ B ⊗ u(k) is equal to x (k). Particularly, equality x (k) = A ⊗ x (k − 1) ⊕ B ⊗ u(k) implies inequality x (k) ≤ A ⊗ x (k − 1) ⊕ B ⊗ u(k) . This control guarantees the values of trajectory X and consequently, the consistency of X ≥ Dh ⊗ X . Therefore, condition on state trajectory B ⊗ (B\x(k)) = x(k) leads to a control satisfying x(k) = B ⊗ u(k) (and not only x(k) ≥ B ⊗ u(k) ). The relation expressing the earliest firing rule x(k) ≤ A ⊗ x(k − 1) ⊕ B ⊗ u(k) can be disregarded in the determination of the trajectory. 3.3.2
Structures 1 and 2
As above, we assume that no row of B is null. Moreover, we assume that each column of B contains a non-null element at the most (but a row can contain more than one element). With this structure of matrix B (denoted structure 1), there is a control such that B ⊗ u(k) = x(k) and condition B ⊗ (B\x(k)) = x(k) is satisfied. Indeed, as a general result of residuation is (A\b)i =
m >
j=1
A ji \b j where A is an m × n
matrix, we obtain ui = (B\x(k))i = B j i \x j (k) for a specific row j and equality B j i ⊗ ui = x j (k) is satisfied. As no row of B is null, B ⊗ u(k) = x(k) . A more restrictive condition (structure 2) is as follows. We can also assume that each column and each row of B contain a non-null element at the most . This last assumption also corresponds to the hypothesis of ”fully controlled” transitions i.e B = I. Therefore, the firing of each transition can be delayed in a control way and all the transitions are said to be controllable. Modeling of transportation network with timetable often leads to this assumption [2, 3, 9, 14]. Consequently, B ⊗ (B\x(k)) = x(k) is always satisfied for any state trajectory and the control law is obviously u(k) = x(k) .
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Using the Kleene star, a simple resolution of relaxed fixed-point form (5) in (max, +) algebra can now give the earliest state trajectory denoted X − and so, the earliest output trajectory z− (k) = C ⊗ x− (k) where x− (k) is the earliest state vector for k ∈ [ks + 1, k f ]. Let us⎛now determine the earliest state trajectory X − of the prediction problem. ⎞ x(ks ) ⎟ ⎜ε ⎟ ⎜ ⎟ ⎜ε ⎟ Let E = ⎜ ⎜ · · · ⎟ As constraint x(ks ) = x(ks ) can be written x(ks ) ≤ x(ks ) and ⎟ ⎜ ⎠ ⎝ε ε x(ks ) ≤ x(ks ), the earliest state trajectory X − is given by the resolution of X ≥ Dh ⊗ X ⊕ E with condition x(ks ) ≥ x− (ks ). The application of Kleene star by Theorem 1 gives the lowest solution X − = (Dh )∗ ⊗ E with condition x(ks ) ≥ x− (ks ). The control is given by u− (k) = B\x− (k). 3.3.3
Generalization
Condition on state trajectory B ⊗ (B\x(k)) = x(k) leads to control u(k) = B\x(k) which produces the exact calculated state trajectory. The same result can be obtained with assumptions on the structure of matrix B (structure 1 or 2). In fact, this technique can be generalized as we can only consider only transitions whose dates obey the additional constraints and neglect the other ones. Using the previously calculated state trajectory, the application of control u(k) = B\x(k) must lead to the exact firing dates of the first class but can minimize the firing dates of the second class. The structure of B is defined as follows: Divide the set of transitions T R into Tc defined below, and its complement Tnc with T R = Tc ∪ Tnc ; Set Tc is the set of + = transitions xi such that there is a non-null coefficient A− i j or Ai j or Ai j . Recall that + = xi (k + 1) ≥ A− i j ⊗ x j (k) , xi (k + 1) ≥ Ai j ⊗ x j (k + 1) and xi (k) ≥ Ai j ⊗ x j (k + 1) , for k ≥ ks . After reorganization of the rows and columns, matrix B is as follows: vector xc (respectively xnc ) expresses the firing dates of transitions xi ∈ Tc (respectively xi ∈Tnc ); With no conditions B22 , onB12 and B11 B12 u1 (k) xc (k) = where B11 follows structure 1 and B21 = ε . ⊗ B21 B22 xnc (k) u2 (k) So, the control can satisfy xc (k) = B11 ⊗ u1 (k) with xc (k) ≥ B12 ⊗ u2 (k) and xnc (k) ≥ B22 ⊗ u2(k) .
3.4 Causality Approaches based on a feedback defined by a Petri net are limited by the condition that the temporisation and initial marking of each added place are non-negative. The
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existence of a linear state feedback is discussed in [10]: this problem is reminiscent of difficulties of the theory of linear dynamical systems over rings [7]. Similarly, Model Predictive Control is limited by the following behavior. Let um (respectively, u− ) be the calculated control corresponding to the modified desired output zm (respectively, earliest desired output z− ). In general, not all components of x(k) are known at the same time and some of the components of x(ks + j) for some j > 0 might be known when the control um is calculated. In this part, we only consider the usual procedure used in model predictive control. Consequently, the application of control um (ks + 1) must be made after the dates of x(ks ) which are data of the problem. So, each component (um )i (ks + 1) must be greater than the date of the possible application which is the addition (in standard algebra) of the maximum of components of x(ks ) and the CPU time TCPU . More formally, the causality condition is um (ks + 1) ≥ Fu ⊗ x(ks ) where Fu is the ⊗−product of TCPU and a full matrix of zeros with appropriate dimensions. Moreover, each calculated date xi (ks + 1) is the result of the application of the control and we can similarly write x(ks + 1) ≥ Fx ⊗ x(ks ) where Fx is defined as Fu with appropriate dimensions. As the complete analysis of these conditions needs an extensive study (see part 7.2 ”Directions for future research” in [13]), we only give the following results. Remark 1. If matrix B has no null row, then the first causality relation implies the second one. Indeed, x(ks + 1) ≥ B ⊗ um (ks + 1) ≥ B ⊗ Fu ⊗ x(ks ) ≥ Fx ⊗ x(ks ) . Different authors give examples following this assumption on B (see [15], chapter 3 and 4 in [13] for instance). The following result assumes that the predictive control approach gives x− (ks ) = x(ks ) . Property 2. Suppose that the control procedure gives a control u− such that B ⊗ u− (k) = x− (k) and x− (ks ) = x(ks ) . The causality conditions x(ks + 1) ≥ Fx ⊗ x(ks ) and um (ks + 1) ≥ Fu ⊗ x(ks ) are satisfied for any x(ks ) if I ⊕ A ⊕ A− ≥ Fx and B\[(I ⊕ A ⊕ A−)] ≥ Fu , respectively. Proof. Let us consider the causality condition on state x. So, x(ks + 1) ≥ x− (ks + 1) ≥ (A ⊕ A−) ⊗ x− (ks ) ⊕ B ⊗ u−(ks + 1) = (A ⊕ A− ) ⊗ x−(ks ) ⊕ x− (ks + 1) = (I ⊕ A ⊕ A−) ⊗ x− (ks ) = (I ⊕ A ⊕ A−) ⊗ x(ks ) As relation x(ks + 1) ≥ (I ⊕ A ⊕ A−) ⊗ x(ks ) is always satisfied and assumption I ⊕ A ⊕ A− ≥ Fx is taken, we can deduce that x(ks + 1) ≥ Fx ⊗ x(ks ) . Let us consider the causality condition on control um . So, um (ks + 1) ≥ u− (ks + 1) = B\x− (ks + 1) ≥ B\[(I ⊕ A ⊕ A−) ⊗ x(ks )] ≥ B\[(I ⊕ A ⊕ A−)] ⊗ x(ks ) (property f12 in [1]). If assumption B\[(I ⊕ A⊕ A−)] ≥ Fu is taken, we can deduce that um (ks + 1) ≥ Fu ⊗ x(ks ) .
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4 Conclusion In this paper, we present a trajectory tracking control of Timed Event Graphs with specifications defined by a P-time Event Graph. The proposed approach presents the following characteristics. The approach is completely defined in (max, +) algebra and does not use standard algebra. Except the algorithm of Kleene star, every used mathematical tool is present in the document which gives a complete description of the approach. The two parts of the trajectory tracking control are: a) the optimal control; b) the updating of a desired output based on a prediction of the earliest possible desired output trajectory. These two parts use a special block tridiagonal matrix. This type of matrix is often encountered in numerical solutions of engineering problems (e.g. computational fluid dynamics, finite element method). In the general case, a pseudo-polynomial algorithms gives the control and proposes an initial condition which must satisfy a condition of coherence of the state trajectory. This technique is sufficient when the control system can apply the calculated initial condition to the process. For different structures of matrix B, the proposed trajectory tracking control is composed of two polynomial algorithms. Trials show that the approach can be applied on-line for relatively important sizes of Event Graphs and horizon of calculation. It can offset unfavorable initial situations while the specifications are met.
References 1. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity. In: An Algebra for Discrete Event Systems. Wiley, New York (1992), http://maxplus.org 2. Braker, J.G.: Algorithms and Applications in Timed Discrete Event Systems, PhD thesis, Department of Technical Mathematics and Informatics, Delft University of Technology, Delft, the Netherlands (1993) 3. Cofer, D.D., Garg, V.K.: A max-algebra solution to the supervisory control problem for real-time discrete event systems. In: 11th International Conference on Analysis and Optimization of Systems Discrete Event Systems, Sophia-Antipolis (June 15-17, 1994) 4. Declerck, P.: Control synthesis using the state equations and the “ARMA” model in Timed Event Graphs. In: 5th IEEE Mediterranean Conference on Control and Systems, CD-ROM, invited session, Paphos Chypre (July 1997) 5. Declerck, P., Didi Alaoui, M.K.: Extremal trajectories in P-time Event Graphs: application to control synthesis with specifications. In: Proc. 44th IEEE Conference on Decision and Control and European Control Conference ECC, CDC-ECC 2005, Seville, Spain, pp. 7621–7626 (2005), http://www.istia.univ-angers.fr/˜declerck/ 6. Guezzi, A., Declerck, P., Boimond, J.-L.: From monotone inequalities to Model Predictive Control. In: ETFA 2008, Hamburg, Germany(September 15-18, 2008), http://www.istia.univ-angers.fr/˜declerck/ 7. Hautus, M.L.J.: Controlled invariance in systems over rings. In: Feedback Control of Linear and Nonlinear Systems, vol. 39, pp. 107–122. Springer, Berlin (1982)
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8. Heidergott, B., Olsder, G.J., van der Woude, J.: Max Plus at Work. Princeton University Press, Princeton (2006) 9. Houssin, L., Lahaye, S., Boimond, J.-L.: Just in Time Control of constrained (max,+)Linear Systems. Discrete Event Dynamic Systems 17(2), 59–178 (2007) 10. Katz, R.D.: Max-Plus (A,B)-Invariant Spaces and Control of Timed Discrete-Event Systems. IEEE Transactions on Automatic Control 52(2), 229–241 (2007) 11. Menguy, E., Boimond, J.-L., Hardouin, L.: Optimal control of discrete event systems in case of updated reference input. In: Proceedings of the IFAC conference on system structure and control, Nantes, France, July 1998, pp. 601–607 (1998) 12. Mc Millan, K., Dill, D.: Algorithms for interface timing verification. In: Proceedings of the IEEE, International Conference on Computer Design: VLSI in Computers and Processors (1992) 13. Necoara, I.: Model Predictive Control for Max-Plus-Linear and Piecewise Affine Systems, PhD Thesis, Delft Center for Systems and Control, Delft University of Technology, The Netherlands (October 2006) 14. Olsder, G.J., Subiono, S., Mc Gettrik, M.: On time tables and allocation of trains, Wodes 1998, Cagliary, Italy (1998) 15. De Schutter, B., van den Boom, T.: Model predictive control for max-plus-linear discrete event systems. Automatica 37(7), 1049–1056 (2001)
Tropical Scaling of Polynomial Matrices St´ephane Gaubert and Meisam Sharify
Abstract. The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.
1 Introduction A classical problem is to compute the eigenvalues of a matrix polynomial P(λ ) = A0 + A1 λ + · · · + Ad λ d where Al ∈ n×n , l = 0 . . . d are given. The eigenvalues are defined as the solutions of det(P(λ )) = 0. If λ is an eigenvalue, the associated right and left eigenvectors x and y ∈ n are the non-zero solutions of the systems P(λ )x = 0 and y∗ P(λ ) = 0, St´ephane Gaubert and Meisam Sharify INRIA Saclay – ˆIle-de-France & Centre de Math´ematiques appliqu´ees, Ecole Polytechnique, 91128 Palaiseau, France, e-mail:
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 291–303. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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respectively. A common way to solve this problem, is to convert P into a “linearized” matrix pencil L(λ ) = λ X + Y, X,Y ∈ nd×nd with the same spectrum as P and solve the eigenproblem for L, by standard numerical algorithms like the QZ method [16]. If D and D are invertible diagonal matrices, and if α is a non-zero scalar, we may consider equivalently the scaled pencil DL(αλ )D . The problem of finding the good linearizations and the good scalings has received a considerable attention. The backward error and conditioning of the matrix pencil problem and of its linearizations have been investigated in particular in works of Tisseur, Li, Higham, and Mackey, see [11, 12, 17]. A scaling on the eigenvalue parameter to improve the normwise backward error of a quadratic polynomial matrix was proposed by Fan, Lin, and Van Dooren [8]. This scaling only relies on the norms γl := Al , l = 0, 1, 2. In this paper, we introduce a new family of scalings which also rely on these norms. The degree d is now arbitrary. These scalings originate from the work of Akian, Bapat, and Gaubert [1, 2], in which the entries of the matrices Al are functions, for instance Puiseux series, of a (perturbation) parameter t. The valuations (leading exponents) of the Puiseux series representing the different eigenvalues were shown to coincide, under some genericity conditions, with the points of non-differentiability of the value function of a parametric optimal assignment problem (the tropical eigenvalues), a result which can be interpreted in terms of amoebas [13]. Indeed, the definition of the tropical eigenvalues in [1, 2] makes sense in any field with valuation. In particular, when the coefficients belong to , we can take the map z → log |z| from to ∪ {−∞} as the valuation. Then, the tropical eigenvalues are expected to give, again under some non degeneracy conditions, the correct order of magnitude of the different eigenvalues. The tropical roots used in the present paper are an approximation of the tropical eigenvalues, relying only on the norms γl = Al . A better scaling may be achieved by considering the tropical eigenvalues, but computing these eigenvalues requires O(nd) calls to an optimal assignment algorithm, whereas the tropical roots considered here can be computed in O(d) time, see Remark 3 below for more information. We examine such extensions in a further work. As an illustration, consider the following quadratic polynomial matrix −3 10 2 −18 1 2 −18 12 15 +λ + 10 P(λ ) = λ 10 34 16 45 34 28 By applying the QZ algorithm on the first companion form of P(λ ) we get the eigenvalues -Inf,- 7.731e-19 , Inf, 3.588e-19, by using the scaling proposed in [8] we get -Inf, -3.250e-19, Inf, 3.588e-19. However by using the tropical scaling we can find the four eigenvalues properly: - 7.250e-18 ± 9.744e-18i, - 2.102e+17 ± 7.387e+17i. The result was shown to be correct (actually, up to a 14 digits precision) with PARI, in which an arbitrarily large precision can be set. The above computations were performed in Matlab (version 7.3.0).
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The paper is organized as follows. In Section 2, we recall some classical facts of max-plus or tropical algebra, and show that the tropical roots of a tropical polynomial can be computed in linear time, using a convex hull algorithm. Section 3 states preliminary results concerning matrix pencils, linearization and normwise backward error. In Section 4, we describe our scaling method. In Section 5, we give a theorem locating the eigenvalues of a quadratic polynomial matrix, which provides some theoretical justification of the method. Finally in Section 6, we present the experimental results showing that the tropical scaling can highly reduce the normwise backward error of an eigenpair. We consider the quadratic case in Section 6.1 and the general case in Section 6.2. For the quadratic case, we compare our results with the scaling proposed in [8].
2 Tropical Polynomials The max-plus semiring max , is the set ∪ {−∞}, equipped with max as addition, and the usual addition as multiplication. It is traditional to use the notation ⊕ for max (so 2 ⊕ 3 = 3), and ⊗ for + (so 1 ⊗ 1 = 2). We denote by the zero element of the semiring, which is such that ⊕ a = a, here = −∞, and by the unit element of the semiring, which is such that ⊗ a = a ⊗ = a, here = 0. We refer the reader to [3, 4, 14] for more background. A variant of this semiring is the max-times semiring max,× , which is the set of nonnegative real numbers + , equipped with max as addition, and × as multiplication. This semiring is isomorphic to max by the map x → log x. So, every notion defined over max has an max,× analogue that we shall not redefine explicitly. In the sequel, the word “tropical” will refer indifferently to any of these algebraic structures. Consider a max-plus (formal) polynomial of degree n in one variable, i.e., a < formal expression P = 0≤k≤n Pk X k in which the coefficients Pk belong to max , and the associated numerical polynomial, which, with the notation of the classical algebra, can be written as p(x) = max0≤k≤n Pk + kx. Cuninghame-Green and Meijer showed [7] that the analogue of the fundamental theorem of algebra holds in the max-plus setting, i.e., that p(x) can be written uniquely as p(x) = Pn + ∑1≤k≤n max(x, ck ), where c1 , . . . , cn ∈ max are the roots, i.e., the points at which the maximum attained at least twice. This is a special case of more general notions which have arisen recently in tropical geometry [13]. The multiplicity of the root c is the cardinality of the set {k ∈ {1, . . . , n} | ck = c}. Define the Newton polygon Δ (P) of P to be the upper boundary of the convex hull of the set of points (k, Pk ), k = 0, . . . , n. This boundary consists of a number of linear segments. An application of Legendre-Fenchel duality (see [2, Proposition 2.10]) shows that the opposite of the slopes of these segments are precisely the tropical roots, and that the multiplicity of a root coincides with the horizontal width of the corresponding segment. (Actually, min-plus polynomials are considered in [2], but the max-plus case reduces to
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the min-plus case by an obvious change of variable). Since the Graham scan algorithm [10] allows us to compute the convex hull of a finite set of points by making O(n) arithmetical operations and comparisons, provided that the given set of points is already sorted by abscissa, we get the following result. Proposition 1. The roots of a max-plus polynomial in one variable can be computed in linear time. The case of a max-times polynomial reduces to the max-plus case by replacing every coefficient by its logarithm. The exponentials of the roots of the transformed polynomial are the roots of the original polynomial.
3 Matrix Pencil and Normwise Backward Error Let us come back to the eigenvalue problem for the matrix pencil P(λ ) = A0 + A1 λ + · · · + Ad λ d . There are many ways to construct a “linearized” matrix pencil L(λ ) = λ X + Y, X,Y ∈ nd×nd with the same spectrum as P(λ ), see [15] for a general discussion. In particular, the first companion form λ X1 + Y1 is defined by ⎛ ⎞ Ak−1 Ak−2 . . . A0 ⎜ −In 0 . . . 0 ⎟ ⎜ ⎟ Y1 = ⎜ . X1 = diag(Ak , I(k−1)n ), .. .. . . ⎟ . ⎝ .. ⎠ . . . 0
. . . −In 0
In the experimental part of this work, we are using this linearization. To estimate the accuracy of a numerical algorithm computing an eigenpair, we shall consider, as in [17], the normwise backward error. The latter arises when considering a perturbation
Δ P = Δ A0 + Δ A1 λ + · · · + Δ Ad λ d . The backward error of an approximate eigenpair (x, ˜ λ˜ ) of P is defined by
η (x, ˜ λ˜ ) = min{ε : (P(λ˜ ) + Δ P(λ˜ ))x˜ = 0, Δ Al 2 ≤ ε El 2 , l = 0, . . . m} . The matrices El representing tolerances. The following computable expression for η (x, ˜ λ˜ ) is given in the same reference, r2 η (x, ˜ λ˜ ) = α˜ x ˜ 2 where r = P(λ˜ )x˜ and α˜ = ∑ |λ˜ |l El 2 . In the sequel, we shall take El = Al . Our aim is to reduce the normwise backward error, by a scaling of the eigenvalue λ = α μ , where α is the scaling parameter. This kind of scaling for quadratic
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polynomial matrix was proposed by Fan, Lin and Van Dooren [8]. We next introduce a new scaling, based on the tropical roots.
4 Construction of the Tropical Scaling Consider the matrix pencil modified by the substitution λ = α μ ˜ μ ) = A˜ 0 + A˜ 1 μ + · · · + A˜ d μ d P( where A˜ i = β α i Ai . The tropical scaling which we next introduce is characterized by the property ˜ μ ) has at least two matrices A˜ i with an (induced) that α and β are such that P( Euclidean norm equal to one, whereas the Euclidean norm of the other matrices are all bounded by one. This scaling is inspired by the work of M. Akian and R. Bapat and S. Gaubert [1], which concerns the perturbation of the eigenvalues of a matrix pencil. The theorem on the location of the eigenvalues which is stated in the next section provides some justification for the present scaling. We associate to the original pencil the max-times polynomial tp(x) = max(γ0 , γ1 λ , · · · , γd λ d ) , where
γi := Ai
(the symbol t stands for “tropical”). Let α1 ≤ α2 ≤ . . . ≤ αd be the tropical roots of tp(x) counted with multiplicities. For each αi , the maximum is attained by at least two monomials. Subsequently, the transformed polynomial q(x) := βi tp(αi x), with βi := (tp(αi ))−1 has two coefficients of modulus one, and all the other coefficients have modulus less than or equal to one. Thus α = αi and β = βi will satisfy the goal. The idea is to apply this scaling for all the tropical roots of tp(x) and each time, to compute n out of nd eigenvalues of the corresponding scaled matrix pencil, because replacing P(λ ) by P(αi μ ) is expected to decrease the backward error for the eigenvalues of order αi , while possibly increasing the backward error for the other ones. More precisely, let α1 ≤ α1 ≤ . . . ≤ αd denote the tropical roots of tp(x). Also let μ , . . . , μn , μ , . . . , μ , . . . , μ(d−1)n+1, . . . , μnd ? 1 @A B ? n+1 @A 2nB ? @A B ˜ μ ) sorted by increasing modulus, computed by setting be the eigenvalues of P( −1 α = αi and β = tp(αi ) and partitioned in d different groups. Now, we choose the ith group of n eigenvalues, multiply by αi and put in the list of computed eigenvalues. By applying this iteration for all i = 1 . . . d, we will get the list of the eigenvalues of P(λ ). Taking into account this description, we arrive at Algorithm 1. It should
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be understood here that in the sequence μ1 , . . . , μnd of eigenvalues above, only the eigenvalues of order αi are hoped to be computed accurately. Indeed, in some extreme cases in which the tropical roots have very different orders of magnitude (as in the example shown in the introduction), the eigenvalues of order αi turn out to be accurate whereas the groups of higher orders have some eigenvalues Inf or Nan. So, Algorithm 1 merges into a single picture several snapshots of the spectrum, each of them being accurate on a different part of the spectrum. Algorithm 1 Computing the eigenvalues using the tropical scaling INPUT: Matrix pencil P(λ ) OUTPUT: List of eigenvalues of P(λ ) 1. Compute the corresponding tropical polynomial tp(x) 2. Find the tropical roots of tp(x) 3. For each tropical root such as αi do 3.1 Compute the tropical scaling based on αi 3.2 Compute the eigenvalues using the QZ algorithm and sort them by increasing modulus 3.3 Choose the ith group of the eigenvalues To illustrate the algorithm, let P(λ ) = A0 + A1 λ + A2 λ 2 be a quadratic polynomial matrix and let tp(λ ) = max(γ0 , γ1 λ , γ2 λ 2 ) be the tropical polynomial corresponding to this quadratic polynomial matrix. + − + − We refer to the tropical roots C of tp(x) by α ≥ α . If α = α which happens γ when γ12 ≤ γ0 γ2 then, α = γ02 and β = tp(α )−1 = γ0−1 . This case coincides with C the scaling of [8] in which α ∗ = γγ02 .
When α + = α − , we will have two different scalings based on α + = and two different β corresponding to the two tropical roots:
β + = tp(α + )−1 =
γ2 , γ12
β − = tp(α − )−1 =
γ1 γ2 ,
α− =
γ0 γ1
1 . γ0
To compute the eigenvalues of P(λ ) by using the first companion form linearization, we apply the scaling based on α + , which yields % 1 $1 γ2 A A A 1 0 2 2 γ γ1 , λ γ2 + 1 I −I 0 to compute the n biggest eigenvalues. We apply the scaling based on α − , which yields $γ % 0 1 A1 γ12 A0 2 A2 γ λ 1 + γ1 , −I 0 I to compute the n smallest eigenvalues.
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In general, let α1 ≤ α1 ≤ . . . ≤ αd be the tropical roots of tp(x) counted with multiplicities. To compute the ith biggest group of eigenvalues, we perform the scaling for αi , which yields the following linearization: ⎞ ⎞ ⎛β α d−1 A ⎛ d d−1 . . . β αi A1 β A0 i β αi Ad −I 0 ... 0 ⎟ ⎟ ⎜ ⎜ I ⎟ ⎟ ⎜ ⎜ ⎜ .. ⎟ . ⎟ ⎜ . . ⎜ . .. ⎟+ ⎜ λ⎜ 0 −I . ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ .. .. ⎝ I ⎠ ⎝ . . 0 ⎠ I 0 . . . −I 0 where β = tp(αi )−1 . Doing the same for all the distinct tropical roots, we can compute all the eigenvalues. Remark 1. The interest of Algorithm 1 lies in the accuracy (since it allows us to solve instances in which the data have various order of magnitudes). Its inconvenient is to call several times (once for each distinct tropical root, and so, at most d times) the QZ algorithm. However, we may partition the different tropical roots in groups consisting each of eigenvalues of the same order of magnitude, and then, the speed factor we would loose would be reduced to the number of different groups.
5 Splitting of the Eigenvalues in Tropical Groups In this section we state a simple theorem concerning the location of the eigenvalues of a quadratic polynomial matrix, showing that under a non degeneracy condition, the two tropical roots do provide the correct estimate of the modulus of the eigenvalues. We shall need to compare spectra, which may be thought of as unordered sets, therefore, we define the following metric (eigenvalue variation), which appeared in [9]. We shall use the notation spec for the spectrum of a matrix or a pencil. Definition 1. Let λ1 , . . . λn and μ1 . . . μn denote two sequences of complex numbers. The variation between λ and μ is defined by v(λ , μ ) := min {max |μπ (i) − λi |} , π ∈Sn
i
where Sn is the set of permutations of {1, 2, . . ., n}. If A, B ∈ n×n , the eigenvalue variation of A and B is defined by v(A, B) := v(spec A, specB). Recall that the quantity v(λ , μ ) can be computed in polynomial time by solving a bottleneck assignment problem. We shall need the following theorem of Bathia, Elsner, and Krause [5]. Theorem 1. [5] Let A, B ∈ n×n . Then v(A, B) ≤ 4 × 2−1/n(A + B)1−1/nA− B . 1/n
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The following result shows that when the parameter δ measuring the separation between the two tropical roots is sufficiently large, and when the matrices A2 , A1 are well conditioned, then, there are precisely n eigenvalues of the order of the maximal tropical root. By applying the same result to the reciprocal pencil, we deduce, under the same separation condition, that when A1 , A0 are well conditioned, there are precisely n eigenvalues of the order of the minimal tropical root. So, under such conditions, the tropical roots provide accurate a priori estimates of the order of the eigenvalues of the pencil. Theorem 2 (Tropical splitting of eigenvalues). Let P(λ ) = λ 2 A2 + λ A1 + A0 where Ai ∈ n×n , and γi := Ai , i = 0, 1, 2. Assume that the max-times polynomial p(λ ) = max(λ 2 γ2 , λ γ1 , γ0 ) has two distinct tropical roots, α + := γ1 /γ2 and α − = γ0 /γ1 , and let δ := α + /α − . Assume that A2 is invertible. Let ξ1 , . . . , ξn denote the eigenvalues of the pencil λ A2 + A1, and let us set ξn+1 = · · · = ξ2n = 0. Then, v(spec P, ξ ) ≤
Cα + , δ 1/2n
where 1/2n condA2 1−1/2n C := 4 × 2−1/2n 2 + 2 condA2 + condA2 , δ and
α + (cond A1 )−1 ≤ |ξi | ≤ α + condA2 ,
1≤i≤n .
(1)
Proof. Let us make the scaling corresponding to the maximal tropical root α + = γ1 /γ2 , with β + = γ2 /γ12 , which amounts to considering the new polynomial matrix Q(μ ) = β + P(α + μ ) = A¯ 2 μ 2 + A¯ 1 μ + A¯ 0 where A¯ 2 = γ2−1 A2 ,
A¯ 1 = γ1−1 A1 ,
γ2 A¯ 0 = 2 A0 . γ1
Since A2 is invertible, λ is an eigenvalue of the pencil P if and only if λ = α + μ where μ is an eigenvalue of the matrix: −1 ¯ −A¯ 2 A¯ 1 −A¯ −1 2 A0 X= I 0 Let μi , i = 1, . . . , 2n denote the eigenvalues of this matrix. Consider −1 −A¯ 2 A¯ 1 0 Y= I 0 Observe that A¯ 1 = 1 and A¯ 0 = γ2 γ0 /γ12 = 1/δ . Since the induced Euclidean norm · is an algebra norm, we get
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−1 −1 ¯ −1 ¯ ¯ ¯ X ≤ I + A¯ −1 2 A1 + A2 A0 ≤ 1 + A2 A2 + A2 A2 A0 = 1 + condA2 (1 + 1/δ ) .
Moreover,
X − Y = (cond A2 )/δ .
Y ≤ 1 + condA2 , Using Theorem 1, we deduce that
v(spec X, specY ) ≤ C/δ 1/2n . Since the family of eigenvalues of P coincide with α + (spec X), and since the family of numbers ξi coincides with α + (specY ), the first part of the result is proved. If ξ is an eigenvalue of A2 λ + A1 , then, we can write ξ = α + ζ , where ζ is ¯ an eigenvalue of A¯ 2 μ + A¯ 1 . We deduce that |ζ | ≤ A¯ −1 2 A1 = condA2 , which establishes the second inequality in (1). The first inequality is established along the same lines, by considering the reciprocal pencil of A¯ 2 μ + A¯ 1 . Remark 2. Theorem 2 is a typical, but special instance of a general class of results that we discuss in a further work. In particular, this theorem can be extended to matrix polynomials of an arbitrary degree, with a different proof technique. Indeed, the idea of the proof above works only for the two “extreme” groups of eigenvalues, whereas in the degree d case, the eigenvalues are split in d groups (still under nondegeneracy conditions). Note also that the exponent in δ 1/2n is suboptimal Remark 3. In [1, 2], the tropical eigenvalues are defined as follows. The permanent of a n × n matrix B = (bi j ) with entries in max is defined by per B := max
∑
σ ∈Sn 1≤i≤n
biσ (i) .
This is nothing than the value of the optimal assignment problem with weights (bi j ). The characteristic polynomial of a matrix C = (ci j ) is defined as the map from max to itself, x → PC (x) := per(C ⊕ xI) , where I is the max-plus identity matrix, with diagonal entries equal to 0 and offdiagonal entries equal to −∞. The sum C ⊕ xI is interpreted in the max-plus sense, so if i = j ci j (C ⊕ xI)i j = max(cii , x) if i = j. The tropical eigenvalues are defined as the roots of the characteristic polynomial. The previous definition has an obvious generalization to the case of tropical matrix polynomials: if C0 , . . . ,Cd are n × n matrices with entries in max , the eigenvalues of the matrix polynomial C(x) := C0 ⊕ C1 x ⊕ · · · ⊕ Cd xd are defined as the roots of the polynomial function x → per(C(x)). The roots of this function can be computed in polynomial time by O(nd) calls to an optimal assignment solver (the case in which C(x) = C0 ⊕ xI was solved by Burkard and Butkoviˇc [6]; the generalization to the
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degree d case was pointed out in [1]). When the matrices A0 , . . . , Ad are scalars, the logarithms of the tropical roots considered in the present paper are readily seen to coincide with the tropical eigenvalues of the pencil in which Ck is the logarithm of the modulus of Ak , for 0 ≤ k ≤ d. When these matrices are not scalars, in view of the asymptotic results of [1], the exponentials of the tropical eigenvalues are expected to provide more accurate estimates of the moduli of the complex roots. This alternative approach is the object of a further work, however, the comparative interest of the tropical roots considered here lies in their simplicity: they only depend on the norms of A0 , . . . , Ad , and can be computed in linear time from these norms. They can also be used as a measure of ill-posedness of the problem (when the tropical roots have different orders of magnitude, the standard methods in general fail).
6 Experimental Results 6.1 Quadratic Polynomial Matrices Consider first P(λ ) = A0 + A1 λ + A2 λ 2 and its linearization L = λ X + Y . Let z be the eigenvector computed by applying the QZ algorithm to this linearization. Both ζ1 = z(1 : n) and ζ2 = z(n + 1 : 2n) are eigenvectors of P(λ ). We present our results for both of these eigenvectors; ηs denotes the normwise backward error for the scaling of [8], and ηt denotes the same quantity for the tropical scaling. Our first example coincides with Example 3 of [8] where A2 2 ≈ 5.54 × 10−5 , A1 2 ≈ 4.73 × 103, A0 2 ≈ 6.01 × 10−3 and Ai ∈ 10×10 . We used 100 randomly generated pencils normalized to get the mentioned norms and we computed the average of the quantities mentioned in the following table for these pencils. Here we present the results for the 5 smallest eigenvalues, however for all the eigenvalues, the backward error computed by using the tropical scaling is of order 10−16 which is the precision of the computation. The computations were carried out in SCILAB 4.1.2. |λ | 2.98E-07 5.18E-07 7.38E-07 9.53E-07 1.24E-06
η (ζ1 , λ ) 1.01E-06 1.37E-07 5.81E-08 3.79E-08 3.26E-08
η (ζ2 , λ ) 4.13E-08 3.84E-08 2.92E-08 2.31E-08 2.64E-08
ηs (ζ1 , λ ) 5.66E-09 8.48E-10 4.59E-10 3.47E-10 3.00E-10
ηs (ζ2 , λ ) 5.27E-10 4.59E-10 3.91E-10 3.36E-10 3.23E-10
ηt (ζ1 , λ ) 6.99E-16 2.72E-16 2.31E-16 2.08E-16 1.98E-16
ηt (ζ2 , λ ) 1.90E-16 1.83E-16 1.71E-16 1.63E-16 1.74E-16
In the second example, we consider a matrix pencil with A2 2 ≈ 10−6, A1 2 ≈ 103 , A0 2 ≈ 105 and Ai ∈ 40×40 . Again, we use 100 randomly generated pencils with the mentioned norms and we compute the average of all the quantities presented in the next table. We present the results for the 5 smallest eigenvalues. This time, the computations shown are from MATLAB 7.3.0, actually, the results are insensitive to this choice, since the versions of MATLAB and SCILAB we used both rely on the QZ algorithm of Lapack library (version 3.0).
Tropical Scaling of Polynomial Matrices
|λ | 1.08E+01 1.75E+01 2.35E+01 2.93E+01 3.33E+01
η (ζ1 , λ ) 2.13E-13 5.20E-14 4.56E-14 4.18E-14 3.77E-14
η (ζ2 , λ ) 4.97E-15 4.85E-15 5.25E-15 5.99E-15 5.28E-15
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ηs (ζ1 , λ ) 8.98E-12 7.71E-13 6.02E-13 5.03E-13 4.52E-13
ηs (ζ2 , λ ) 4.19E-13 4.09E-13 4.01E-13 3.97E-13 3.84E-13
ηT (ζ1 , λ ) 5.37E-15 6.76E-16 5.54E-16 4.80E-16 4.67E-16
ηT (ζ2 , λ ) 3.99E-16 3.95E-16 3.66E-16 3.47E-16 3.53E-16
6.2 Polynomial Matrices of Degree d Consider now the polynomial matrix P(λ ) = A0 + A1 λ + · · · + Ad λ d , and let L = λ X +Y be the first companion form linearization of this pencil. If z is an eigenvector for L then ζ1 = z(1 : n) is an eigenvector for P(λ ). In the following computations, we use ζ1 to compute the normwise backward error of Matrix pencil, however this is possible to use any z(kn + 1 : n(k + 1)) for k = 0 . . . d − 1. To illustrate our results, we apply the algorithm for 20 different randomly generated matrix pencils and then compute the backward error for a specific eigenvalue of these matrix pencils. The 20 values x-axis, in Fig. 1 and 2, identify the random instance while the y-axis shows the log10 of backward error for a specific eigenvalue. Also we sort the eigenvalues in a decreasing order of their absolute value. We firstly consider the randomly generated matrix pencils of degree 5 where the order of magnitude of the Euclidean norm of Ai is as follows: A1 A2 A3 A4 A5 A0 O(10−3) O(102 ) O(102 ) O(10−1) O(10−4 ) O(105 ) Fig. 1 shows the results for this case where the dotted line shows the backward error without scaling and the solid line shows the backward error using the tropical scaling. We show the results for the minimum eigenvalue, the “central” 50th eigenvalue and the maximum one from top to down. In particular, the picture at the top shows a dramatic improvement since the smallest of the eigenvalues is not computed accurately (backward error almost of order one) without the scaling, whereas for the biggest of the eigenvalues, the scaling typically improves the backward error by a factor 10. For the central eigenvalue, the improvement we get is intermediate. The second example concerns the randomly generated matrix pencil with degree 10 while the order of the norm of the coefficient matrices are as follows: A1 A2 A3 A4 A5 A0 −5 −2 −3 −4 O(10 ) O(10 ) O(10 ) O(10 ) O(102 ) O(1) A8 A9 A10 A6 A7 O(103) O(10−3 ) O(104 ) O(102 ) O(105 ) In this example, the order of the norms differ from 10−5 to 105 and the space dimension of Ai is 8. Figure 2 shows the results for this case where the dotted line shows the backward error without scaling and the solid line shows the backward error using tropical scaling. Again we show the results for the minimum eigenvalue, the 40th eigenvalue and the maximum one from top to down.
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Fig. 1 Backward error for randomly generated Fig. 2 Backward error for randomly genmatrix pencils with n = 20, d = 5. erated matrix pencils with n = 8, d = 10.
References 1. Akian, M., Bapat, R., Gaubert, S.: Perturbation of eigenvalues of matrix pencils and optimal assignment problem. C. R. Acad. Sci. Paris, S´erie I 339, 103–108 (2004) 2. Akian, M., Bapat, R., Gaubert, S.: Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem (2005), arxiv:math.SP/0402090 3. Akian, M., Bapat, R., Gaubert, S.: Max-plus algebras. In: Hogben, L. (ed.) Handbook of Linear Algebra, Discrete Mathematics and Its Applications, ch. 25, vol. 39, Chapman & Hall/CRC (2006) 4. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity. Wiley, Chichester (1992) 5. Bhatia, R., Elsner, L., Krause, G.: Bounds for the variation of the roots of a polynomial and the eigenvalues of a matrix. Linear Algebra Appl. 142, 195–209 (1990) 6. Burkard, R.E., Butkoviˇc, P.: Finding all essential terms of a characteristic maxpolynomial. Discrete Appl. Math. 130(3), 367–380 (2003) 7. Cuninghame-Green, R.A., Meijer, P.F.J.: An algebra for piecewise-linear minimax problems. Discrete Appl. Math. 2(4), 267–294 (1980) 8. Fan, H.-Y., Lin, W.-W., Van Dooren, P.: Normwise scaling of second order polynomial matrices. SIAM J. Matrix Anal. Appl. 26(1), 252–256 (2004) 9. Gal´antai, A., Heged˝us, C.J.: Perturbation bounds for polynomials. Numer. Math. 109(1), 77–100 (2008) 10. Graham, R.L.: An efficient algorithm for determining the convex hull of a finite planar set. Inf. Proc. Lett. 1(4), 132–133 (1972)
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11. Higham, N.J., Li, R.-C., Tisseur, F.: Backward error of polynomial eigenproblems solved by linearization. SIAM J. Matrix Anal. Appl. 29(4), 1218–1241 (2007) 12. Higham, N.J., Mackey, D.S., Tisseur, F.: The conditioning of linearizations of matrix polynomials. SIAM J. Matrix Anal. Appl. 28(4), 1005–1028 (2006) 13. Itenberg, I., Mikhalkin, G., Shustin, E.: Tropical algebraic geometry. Oberwolfach seminars, Birkh¨auser (2007) 14. Kolokoltsov, V.N., Maslov, V.P.: Idempotent analysis and its applications. In: Mathematics and its Applications, vol. 401. Kluwer Academic Publishers Group, Dordrecht (1997) 15. Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28(4), 971–1004 (2006) 16. Moler, C.B., Stewart, G.W.: An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256 (1973) 17. Tisseur, F.: Backward error and condition of polynomial eigenvalue problems. Linear Algebra Appl. 309(1-3), 339–361 (2000); Proceedings of the International Workshop on Accurate Solution of Eigenvalue Problems, University Park, PA (1998)
Scrutinizing Changes in the Water Demand Behavior Manuel Herrera, Rafael P´erez-Garc´ıa, Joaqu´ın Izquierdo and Idel Montalvo
Abstract. Time series novelty or anomaly detection refers to automatic identification of novel or abnormal events embedded in normal time series points. In the case of water demand, these anomalies may be originated by external influences (such as climate factors, for example) or by internal causes (bad telemetry lectures, pipe bursts, etc.). This paper will focus on the development of markers of different possible types of anomalies in water demand time series. The goal is to obtain early warning methods to identify, prevent, and mitigate likely damages in the water supply network, and to improve the current prediction model through adaptive processes. Besides, these methods may be used to explain the effects of different dysfunctions of the water network elements and to identify zones especially sensitive to leakage and other problematic areas, with the aim to include them in reliability plans. In this paper, we use a classical Support Vector Machine (SVM) algorithm to discriminate between nominal and anomalous data. SVM algorithms for classification project low-dimensional training data into a higher dimensional feature space, where data separation is easier. Next, we adapt a causal learning algorithm, based on the reproduction of kernel Hilbert spaces (RKHS), to look for possible causes of the detected anomalies. This last algorithm and the SVM’s projection are achieved by using kernel functions, which are necessarily symmetric and positive definite functions.
1 Introduction The anomaly detection of water demand time series aims to correct likely data errors in measures from telemetry systems. These systems are used by most water Manuel Herrera, Rafael P´erez-Garc´ıa, Joaqu´ın Izquierdo and Idel Montalvo CMMF, Universidad Polit´ecnica de Valencia 46022 Valencia, Spain, e-mail:
[email protected],
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 305–313. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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companies in big cities for control and operation purposes. This will allow more accurate estimations that can be used to immediately detect severe anomalies, such as service disruption. Simultaneously, it can identify more rapidly light anomalies which can develop insidious and progressively [7]. If no errors are found in data, the novelties significance is the occurrence of some physical change in the water supply network or in the demand behavior caused by external influences, such as climate factors. Changes in time series behavior may exhibit permanent or transitional effects. The causes of these are diverse, but could be divided into external and internal causes. Examples of the first class are weather or calendar factors. For the internal causes one can have wrong telemetry readings, water leakage or failure of one or more valves. There is a need to divide the problem into two phases: anomaly detection and action taking. In this way, one can obtain early warning methods to identify and mitigate likely damages in the water supply network or to improve the current prediction model through some adaptive process. To distinguish between normal and abnormal deviations, novelties will be sought in three specific cases: when data loggers identify a disruption of service, when the discrepancies between the last observations and their prediction are significant, and when the last observations lack of the expected random characteristics. Here, we consider working with sliding time windows to include all possible cases. The sliding window method is based on a window size W ; only the latest W observations are used for detection. As an observation arrives, the oldest observation in the sliding window expires. An alert processing method based on Support Vector Machines will be proposed to extract trends and to highlight punctual discrepancies between observed and predicted data [14]. To look for possible causes of the detected anomalies, we propose using the recently developed causal learning algorithm based on the reproduction of kernel Hilbert spaces [20]. By using this methodology the statistical dependences can always be detected by correlations after the data are mapped into an appropriate feature space. The algorithm is an improvement of the inductive causation (IC) algorithm [18], which generalizes in several ways. The control of the consequences of novelties in the earlier stages can avoid, among other things, economic and water losses, which are of great importance from the point of view of water as a scarce resource. This paper will focus on the development of markers of the different possible types of anomalies in water demand time series, explaining their causes, and proposing a feasible integration mechanism in the prediction system. Figure 1 summarizes the process. The paper is organized as follows. Section 2 shows a brief literature review about detection of anomalies in water distribution systems. In section 3 we present a methodology to detect the possible types of novelties within time series data, to develop a way of classifying them and to discuss their causes. Section 4 gives some application results and summarizes the conclusions.
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Fig. 1 Scheme of the methodology proposed
2 Brief Literature Review Detecting novel events is an important ability of any signal classification scheme. This is the main reason for several models of novelty detection, which have proved to perform well on different data, to exist. It is clear that there is no single best model for novelty detection, and success depends not only on the type of method used but also on the statistical properties of the data handled. Thus, several applications have been published in the literature with the goal of detecting and classifying possible outliers or abnormal data. In the last years, Neural Networks approaches [1] have been replaced by Support Vector Machines applications in this regard [10, 14]. As different alternatives, other methods include the control charts, proposed by Nong [12], and the techniques based on fuzzy rough clustering, tested by Chimphlee et al. [2], to increase the detection rates and reduce false positive rates in the intrusion detection system. Herrera et al. [5] have proposed hybrid nonlinear models for interpolation in the case of having problems with telemetry lectures of water consumption. Izquierdo et al. [7] have presented a neuro-fuzzy approach to fault detection in water supply systems. In addition to novelty detection, we also aim to identify its causes. Pearl [13] has shown that, under reasonable assumptions, it is possible to get hints about causal relationships from non-experimental data. Sch¨olkopf and Smola [14] have proposed the idea of measuring dependences by reproducing kernel Hilbert spaces. Sun et al. [20] and Fukumizu et al. [3] have worked on an algorithm describing the causal learning method, which we will follow in this paper. Being able to establish the cause-effect relationships in a water supply environment in the presence of anomalies, would certainly produce better understanding of the demand behavior.
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3 Methodology To obtain abnormal data (anomaly events) in an easy way and to train correctly the Machine Learning procedures, we propose working with our real system replicated by its EPANET [15] model. This way, we can run the water demand simulations under different novelty scenarios and check the response of our methodology. The next step will be to detect the abnormal data. Then, by using a kernel-based causal algorithm we will try to establish the causes of the observed anomalies.
3.1 EPANET Simulation The above methodology is tested on the simulated consumption of water by using EPANET. The first premise is to work with a correct pattern demand curve. We propose generating curves by using the current model for prediction. For the sake of simplicity we use a simple and novel weighted pattern-based model for water forecasting, which has been tested by the authors with very good results [6]: this method is based on the pattern of the demand, which considers its seasonal properties. This proposal contains two components: a first part that reflects the seasonal pattern of the water demand; and a second part that corrects/adjusts this initial forecast to account for the specificities of the day for which a prediction is being obtained. Both parts use exponentially decreasing weights, which give more importance to more recent values of the water demand. Equation (1) gives the formal definition of the model. L
yˆk = ∑ α (1 − α )l−1 yk−24l + β l−1 (1 − β )l−2Δk−24l ,
(1)
l=1
where k = 25, 26, ..., L is the number of items to include in the predictor, Δh = yh − yˆh and α and β are the exponential coefficients of the weights. These weights are independent but, seeking the stability of the model, usually the seasonality pattern part weight, α , will be higher than error part weight, β . The model can discriminate between the different days of the week. All the characteristics of this model are reproduced as a generator system of the EPANET’s pattern demand curve. The different anomalies are also simulated in EPANET at randomized points of time: valves failures are schematized straightforward (with a programmed change of their characteristics as a function of the loss ratio) and, for example, leaks may be modeled as shown in Figure 2. From the hydraulic point of view, a leak can be simulated by a model consisting of a valve and a node with zero manometric pressure. The loss ratio of the valve will be proportional to the effective section of the fault, and depends on the coefficients of contraction and the velocity.
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Fig. 2 Leakage simulated under EPANET
3.2 Detecting Anomalies Support Vector Machines provide a novel approach to the classification problem, learning to perform the classification task through a supervised learning procedure. Vapnik [21, 22] and Shaw-Taylor and Cristianini [17] are two of the essential references for SVM. These are complemented with the works by Karatzouglou [8, 9], implementing SVM and kernel methods environment in R Language. In this work, we propose a classical SVM to discriminate the nominal and anomalous data obtained in the last sliding time window. The basis for the SVM algorithm for classification is the projection of the low-dimensional training data in a higher dimensional feature space, since it is easier to separate the input data in this higher dimensional feature space. This projection is achieved by using kernel functions. According to Mercer’s theorem [11, 19], kernel functions necessarily are symmetric and positive definite functions. The proposed working-line keeps the next flowing scheme: 1. measure the distance between the predicted and observed data within the last W -long performed array; 2. use the SVM algorithm to classify this array; 3. if an anomaly is detected then reclassify it as: outlier, trend or service disruption These are the necessary steps to complete the anomaly detection phase.
3.3 Kernel-Based Causal Algorithm As stated, the identification of the cause-effect relationships in a water supply environment in the presence of anomalies will produce better understanding of the demand behavior. To achieve this we propose the application of the kernel-based
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causal learning algorithm (KCL) developed by Sun et al. [20]. This approach assumes that a variable Z is likely to be a common effect of X and Y , if conditioning on Z increases the dependence between X and Y . Based on this assumption, the algorithm collect “votes” for hypothetical causal directions and orient the edges by the majority principle. The algorithm is an improvement of inductive causation (IC) algorithm [18], generalizing it in several ways: First, it handles both discrete and continuous variables. Next, it does not need the assumption of special kinds of distributions. Let (X , BX ) and (Y , BY ) be measurable spaces and let (HX , KX ) and (HY , KY ) be reproducing kernel Hilbert spaces of funtions on X and Y , with positive definite kernels KX , KY . We consider random vector (X,Y ) on X × Y such that expectations EX [KX (X, X)] and EY [KY (Y,Y )] are finite. We define ΣXY as the cross-covariance operator and ΣXY |Z as the conditional cross-covariance operator. We have that ΣXY = 0 ⇐⇒ X ⊥ Y . The strength of the marginal and conditional dependence can be defined by
XY := ||ΣXY ||2 HS ,
(2)
XY |Z := βZ ||ΣXY ||2 HS ,
(3)
with βZ := 1/||TZ ||2HS and TZ is defined by h2 , TZ h1 = E[h1 (Z)h2 (Z)] for arbitrary h1 , h2 ∈ HZ . Gretton et al. [4] obtained consistent estimators of these dependences. The algorithm is based on the next heuristics: conditioning on a common effect has the tendency to generate dependence between the causes. This is true when the unconditional dependences between the causes are small. Based on this, a votinglike procedure for orientation of edges is introduced: for any triple (X,Y, Z), one gets a vote for Z being a common effect of X and Y , if and only if XY |Z > λ XY , with appropriate λ > 0. By continuing with these votes we may direct most edges in the direction. We choose λ1 very large in the first run and set λ2 := 9 majority : ZX|Y ZY |X max ZX , ZY in the second run. If the result is balanced, leave the edge undirected. See [20] for further details.
3.4 Action Taking Phase To take suitable actions when there is evidence of anomaly data detection we propose following the next steps: 1. Repairing the data with interpolation to continue working with the current prediction model 2. Analyzing the anomaly origin a. External cause i. Detect the cause type ii. Adapt the prediction model to these novelties
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b. Internal cause i. Detect the cause type ii. Repair it iii. Prevent it, explaining characteristics and checking the common points with other anomalies in the water supply network
4 Conclusions and Results We have tested this methodology in a hourly EPANET simulation of the conditions of a water supply network zone along 100 days. This is a real-world case study exhibiting high intensity in the presence of different dysfunctions what makes it suitable for training the learning algorithms involved. Working with the demand variable, in a time window of 12 hours, the SVM algorithm is able to detect all the anomalies in the validation (40 days) and the testing data (10 days). In this case, we not only managed to detect the anomalies in the water demand behavior, but also we have been able to find justifiable cause-effect relationships in the water demand environment. The causal model includes the continuous variables: pressure, valve position (that represents the water entrance to our zone from other parts of the water supply network), diameter of the leakage and also the discrete inputs from the previous classification stage. This model offers deeper knowledge of the water consumption behavior and the supply network and their elements, as seen in our working example of the Figure 3. This graph shows some effects (for example, valves are related to outliers but are only a cause regarding trend novelties) that are obtained with the KCL algorithm. Future work will aim to obtain more information of the factors, to improve the identification of the anomaly causes, and to take actions to prevent them or mitigate their effects. The kernel-based independence measures benefit from the power of detecting nonlinear dependence and can keep, for example, type II errors (deciding independence when there is dependence) at a very low level. The methodology we have shown is a perfect supplement to improve the current prediction models of water demand, since it can be easily adapted to various anomaly scenarios. In the future, methods to screening the water network to find specially sensible or vulnerable zones, where abnormal events may have more important consequences, should be explored. Acknowledgements. Supported by grants BES-2005-9708 and MAEC-AECI 0000202066, awarded to two of the authors.
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Fig. 3 Final step of the KCL causal-effect structure
References 1. Augusteijn, M.F., Folkert, B.A.: Neural network classification and novelty detection. International Journal of Remote Sensing 23(14), 2891–2902 (2002) 2. Chimphlee, W., Abdullah, A.H., Sap, M.N., Srinoy, S., Chimphlee, S.: Anomaly-based intrusion detection using fuzzy rough clustering. In: 2006 International Conference on Hybrid Information Technology (ICHI 2006), vol. 1, pp. 329–334 (2006) 3. Fukumizu, K., Bach, F., Gretton, A.: Statistical consistency of kernel canonical correlation analysis. Journal of Machine Learning Research 8, 361–383 (2007) 4. Gretton, A., Bousquet, O., Smola, A.J., Sch¨olkopf, B.: Measuring Statistical Dependence with Hilbert-Schmidt Norms. In: Jain, S., Simon, H.U., Tomita, E. (eds.) ALT 2005. LNCS, vol. 3734, pp. 63–77. Springer, Heidelberg (2005) 5. Herrera, M., Garc´ıa–D´ıaz, J.C., P´erez, R., Mart´ınez, J.F., L´opez, P.A.: Interpolaci´on con redes neuronales artificiales en series temporales intervenidas para la predicci´on de la demanda urbana de agua. In: Proceedings NOLINEAL 2007. Ciudad Real, Spain (2007) 6. Herrera, M., Torgo, L., Izquierdo, J., P´erez, R.: Predictive models for forecasting hourly urban water demand (submitted, 2009) 7. Izquierdo, J., L´opez, P.A., Mart´ınez, F.J., P´erez, R.: Fault detection in water supply systems using hybrid (theory and data–driven) modelling. Mathematical and Computing Modelling 46, 341–350 (2007) 8. Karatzouglou, A.: Kernel methods software, algorithms and applications. PhD. dissertation, Technischen Universitat Wien, Austria (2006) 9. Karatzouglou, A., Meyer, D., Hornik, K.: Support Vector Machines. R. Journal of Statistical Software 15(9) (2006), http://www.jstatsoft.org/v15/i09 (accessed on January 2009) 10. Ma, J., Perkins, S.: Time-series novelty detection using one-class support vector machines. In: Proceedings of the International Joint Conference on Neural Networks, vol. 3, pp. 1741–1745 (2003) 11. Mercer, J.: Functions of positive and negative and their connection with the theory of integral equations. Philos. Trans Royal Soc. 209, 415–446 (1909) 12. Nong, Y., Qian, C.: Computer intrusion detection through EWMA for autocorrelated and uncorrelated data. IEEE Transactions on Realibility 52(1), 75–82 (2003) 13. Pearl, J.: Causality: Models, reasoning, and inference. Cambridge University Press, Cambridge (2000)
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14. Rocco, M.C., Zio, E.: A support vector machine integrated system for the classification of operation anomalies in nuclear components and systems. Reliability Eng. & System Safety 92, 593–600 (2007) 15. Rossman, L.: EPANET-User’s Manual. United States Environmental Protection Agency (EPA), Cincinnati, OH (2000) 16. Sch¨olkopf, B., Smola, A.: Learning with kernels. MIT Press, Cambridge (2002) 17. Shawe-Taylor, J., Cristianini, N.: An Introduction to Support Vector Machines. Cambridge University Press, Cambridge (2000) 18. Spirtes, P., Gylmour, C.: An algorithm for fast recovery of sparse causal graphs. Social Science Computer Review 9, 67–72 (1991) 19. Sun, H.: Mercer theorem for RKHS on noncompact sets. Journal of Complexity 21(3), 337–349 (2005) 20. Sun, X., Janzig, D., Sch¨olkopf, B., Fukumizu, K.: A kernel-based causal learning algorithm. In: Proc. 24th Annual International Conference on Machine Learning (ICML 2007), pp. 855–862 (2007) 21. Vapnik, V.: The Nature of Statistical Learning Theory. Springer, Heidelberg (1995) 22. Vapnik, V.: Statistical Learning Theory. John Wiley and Sons, Chichester (1998)
Characterization of Matrices with Nonnegative Group-Projector Alicia Herrero, Francisco J. Ram´ırez and N´estor Thome
Abstract. In [Jain, Tynan, Linear Algebra and its Applications 379, 381–394, 2004], the authors shown that a nonnegative square matrix A satisfies that AA# ≥ O, being A# the group inverse of A, if and only if A is permutationally similar to a matrix with a special structure. In this paper, a similar structure for this kind of matrices, slightly simplified, is presented, where the restriction of the nonnegativity of the matrix A is omitted. In addition, this result to characterize the {k}-group involutory matrices is applied.
1 Introduction For a given matrix F ∈ n×n , a matrix G ∈ n×n is called its group inverse if the properties FGF = F, GFG = G, and FG = GF hold. When this matrix exists, it will be denoted by F # and it is unique [1]. Through this paper we will assume that the involved matrices have group inverse. We will stand A ≥ O for a matrix A with nonnegative entries and AT for the transpose of A. We remember that a square matrix A is called {k}-group periodic if it satisfies A# = Ak−1 where k belongs to {2, 3, . . .}. It is well-known that AA# is a projector on the range of A along the null space of A# . In order to distinguish this projector among others defined using other generalized inverses, we will call group-projector to AA# . The group inverse has been widely studied in the literature and applied to solve real problems. For instance, it is applied in model electric networks, Markov chains, symmetric singular control systems, numerical methods, etc. [1, 4–9, 11, 14, 15]. In Alicia Herrero and N´estor Thome Instituto de Matem´atica Multidisciplinar, Universidad Polit´ecnica de Valencia, 46022 Valencia, Spain, e-mail:
[email protected],
[email protected] Francisco J. Ram´ırez Instituto Tecnol´ogico de Santo Domingo, Av. Los Pr´oceres, Gal´a, Santo Domingo, Rep´ublica Dominicana, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 315–320. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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particular, in [12] a similar form to the one studied in this paper has been used to obtain conditions to guarantee the nonnegativity of control linear systems. Moreover, the problem of characterizing group involutory matrices (that is, {2}-group periodic matrices) has been studied in [3]. The purpose of this paper is to give a characterization of square matrices A such that AA# ≥ O without any restriction on the matrix A. In addition we apply the result in order to characterize the {k}-group periodic matrices.
2 Main Results We start this section with a result partially given in [10] where the converse is also included here. We recall that we assume that all involved matrices have group inverse. Lemma 1. Let A ∈ n×n a nonzero matrix with rank(A) = r. Then A is an idempotent nonnegative matrix if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XY XY M O O O ⎦ PT (1) A = P⎣ O NXY NXY M O where M, N are arbitrary nonnegative matrices of appropriate sizes and X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ) being xi and y j positive column vectors with i, j ∈ {1, . . . , r} such that Y X = I. Proof. The sufficiency is given in the proof of the Lemma 2.1 in [10]. Since P, M, N, X, and Y are nonnegative matrices, from (1) we get that A ≥ O. Again from (1) and making a simple block product we get A2 = A. The necessity is then proved. An important particular case is shown in the next corollary. Corollary 1. Let A ∈ n×n with rank(A) = r such that AA# is a nonzero matrix. Then AA# ≥ O if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XY XY M O O O ⎦ PT (2) AA# = P ⎣ O NXY NXY M O where M, N are arbitrary nonnegative matrices of appropriate size and X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ) being xi and y j positive column vectors with i, j ∈ {1, . . . , r} such that Y X = I. Proof. The group inverse A# of the matrix A satisfies that (AA# )2 = AA# AA# = AA# , that is AA# is an idempotent matrix. Then, the results follow directly by applying the Lemma 1 to the matrix AA# when its nonnegativity is assumed since rank(AA# ) = rank(A).
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Now, we obtain the main result of this paper. The importance of this result is that the condition A ≥ O is suppressed and besides the form of the matrix A is simplified with respect to that given in Theorem 1 in [13]. More precisely, taking into account that for a given matrix A ∈ n×n it follows that AA# is idempotent, the nonnegativity of AA# allows to factorize this product in a special form by the Corollary 1. This factorization leads to the following result on the matrix A. Theorem 1. Let A ∈ n×n with rank(A) = r and AA# = O. Then AA# ≥ O if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XTY XTY M O O O ⎦ PT A = P⎣ O (3) NXTY NXTY M O where M, N are arbitrary nonnegative matrices of appropriate size, T ∈ r×r is nonsingular and X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ) being xi and y j positive column vectors with i, j ∈ {1, . . . , r} such that Y X = I. In this case, ⎡ ⎤ XT −1Y XT −1Y M O A# = P ⎣ O O O ⎦ PT . (4) −1 −1 NXT Y NXT Y M O Proof. From the Corollary 1 we can write AA# in the next form ⎡ ⎤ XY XY M O O O ⎦ PT AA# = P ⎣ O NXY NXY M O
(5)
being P, M, N, X, and Y as there. Now, we partition the matrix A in a 3 × 3 block matrix as in (5) as follows ⎤ ⎡ A1 A2 A3 A = P ⎣ A4 A5 A6 ⎦ PT , (6) A7 A8 A9 and we apply the property (AA# )A = A. The form of AA# and the partition of A lead to: A4 , A5 , and A6 are null matrices, Ai = XYAi for i = 1, 2, 3, A7 = NA1 , A8 = NA2 , and A9 = NA3 . Since A# A = AA# , the same property A(A# A) = A also yields to A3 = O, A1 = A1 XY , and A2 = A1 M. Summarizing, ⎡ ⎤ A1 A1 M O O O ⎦ PT , A = P⎣ O (7) NA1 NA1 M O where A1 = XYA1 = A1 XY . Then A1 = XTY being T = YA1 X. Note that rank(A1 ) = rank(A) = r. Thus, r = rank(XTY ) ≤ rank(T ) ≤ r because T ∈ r×r . The converse is evident.
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In the next result the nonnegativity of the matrix A is added in order to compare it with the one presented by S. Friedland and E. Virnik in [10]. Corollary 2. Let A ∈ n×n with rank(A) = r, A ≥ O, and AA# = O. Then AA# ≥ O if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XTY XTY M O A = P⎣ O O O ⎦ PT (8) NXTY NXTY M O where M, N are arbitrary nonnegative matrices of appropriate size, T ∈ r×r is a nonnegative and nonsingular matrix, X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ) being xi and y j positive column vectors with i, j ∈ {1, . . . , r} such that Y X = I. Proof. Applying the Theorem 1 one only has to show that T ≥ O. From A ≥ O and P ≥ O we have that PT AP ≥ O and thus, by (8), in particular we get XTY ≥ O. Since X ≥ O, Y ≥ O and Y X = I, premultiplying and postmultiplying the inequality XTY ≥ O by Y and X, respectively, we obtain T ≥ O. Note that, in general, for a nonnegative matrix A the condition A2 = A implies that AA# ≥ O, which corresponds to T = I in Corollary 2. However, the converse is not always true. In fact, the following nonnegative matrix ⎡ ⎤ 010 A = ⎣1 0 0⎦ 000 satisfies that AA# ≥ O but A2 = A. This implies that the Corollary 2 is a more general version of the Lemma 2.1 in [10]. Now, we applied the Theorem 1 in order to characterize the {k}-group periodic matrices. Theorem 2. Let A ∈ n×n with rank(A) = r, ind(A) = 1, and Ak ≥ O with Ak = O being k ∈ {1, 2, . . . }. Then Ak+1 = A if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XTY XTY M O O O ⎦ PT (9) A = P⎣ O NXTY NXTY M O where M, N are nonnegative matrices of appropriate size, X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ), xi and yi are positive column vectors with i, j ∈ {1, . . . , r} such that yTi xi = 1, and T ∈ r×r is a nonsingular matrix such that T k = I. Proof. Let k > 1. Following a similar reasoning as in [3], it is easy to see that whenever k > 1, the condition Ak+1 = A is equivalent to the condition A# = Ak−1 . This implies that AA# = Ak ≥ O and by hypothesis AA# = O. Then, from Theorem 1, we have that A and A# have the form (3) and (4) respectively. Moreover, using these expressions, we can write the condition A# = Ak−1 as the equivalent one
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⎤ ⎤ ⎡ XT −1Y XT −1Y M O XT k−1Y XT k−1Y M O ⎣ O O O⎦ = ⎣ O O O ⎦. −1 −1 k−1 k−1 NXT Y NXT Y M O NXT Y NXT Y M O ⎡
Then, it follows that XT −1Y = XT k−1Y . Premultiplying by Y and postmultipliying by X and using that Y X = I we obtain T −1 = T k−1 , that is T k = I. To prove the converse it is enough to compute Ak+1 using the expression (9) and take into account that T k = I. Note that the case k = 1 corresponds to Lemma 1 (where A2 = A and T = I). From this result, we can also keep the form of A in the case A ≥ O, taking into account that the matrix T must be nonnegative in this case. Corollary 3. Let A ∈ n×n with rank(A) = r, ind(A) = 1, A ≥ O and Ak = O being k ∈ {1, 2, . . . }. Then Ak+1 = A if and only if there exists a permutation matrix P ∈ n×n such that ⎡ ⎤ XTY XTY M O O O ⎦ PT (10) A = P⎣ O NXTY NXTY M O where M, N are arbitrary nonnegative matrices of appropriate size, and X = diag(x1 , . . . , xr ), Y = diag(yT1 , . . . , yTr ) being xi and y j positive column vectors with i, j ∈ {1, . . . , r} such that Y X = I and T ∈ r×r is a nonnegative and nonsingular matrix satisfying T k = I. Acknowledgements. The authors would like to thank the referees for their valuable suggestions and comments, which resulted in a great improvement of the original manuscript. This paper has been partially supported by the DGI project with number MTM200764477.
References 1. Ben-Israel, A., Greville, T.: Generalized inverses: Theory and applications. Wiley, New York (1974) 2. Berman, A., Plemmons, R.J.: Nonnegative matrices in Mathematical Sciences. SIAM Academic Press, New York (1979) 3. Bru, R., Thome, N.: Group inverse and group involutory matrices. Linear and Multilinear Algebra 45(2-3), 207–218 (1998) 4. Campbell, S.L.: Singular systems of differential equations. Pitman, London (1980) 5. Campbell, S.L., Meyer Jr., C.D.: Generalized inverses of linear transformations. Dover, London (1979) 6. Chen, J., Xu, Z., Wei, Y.: Representations for the Drazin inverse of the sum P+ Q + R + S and its applications. Linear Algebra and its Applications 430, 438–454 (2009) 7. Coll, C., Herrero, A., S´anchez, E., Thome, N.: Output feedback stabilization for symmetric control systems. Journal of the Franklin Institute 342, 814–823 (2005) 8. Deng, C.Y.: The Drazin inverses of products and differences of orthogonal projections. J. Math. Anal. Appl. 335, 64–71 (2007)
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9. Deng, C.Y.: The Drazin inverses of sum and difference of idempotents. Linear Algebra and its Applications 430, 1282–1291 (2009) 10. Friedland, S., Virnik, E.: Nonnegative of Schur complements of nonnegative idempotent matrices. Electronic Journal of Linear Algebra 17, 426–435 (2008) 11. Groß J.: Nonsingularity of the difference of two oblique projectors. SIAM Journal on Matrix Analysis and Application 21(2), 390–395 (1999) 12. Herrero, A., Ram´ırez, A., Thome, N.: An algorithm to check the nonnegativity of singular systems. Applied Mathematics and Computation 189, 355–365 (2007) 13. Jain, S.K., Tynan, J.: Nonnegative matrices A with AA# ≥ O. Linear Algebra and its Applications 379, 381–394 (2004) 14. Wei, Y.: Index splitting for the Drazin inverse and the singular linear systems. Applied Mathematics and Computation 95(2-3), 115–124 (1998) 15. Wei, Y.: On the perturbation of the group inverse and oblique projection. Applied Mathematics and Computation 98(1), 29–42 (1999)
Robust Design of Water Supply Systems through Evolutionary Optimization Joaqu´ın Izquierdo, Idel Montalvo, Rafael P´erez-Garc´ıa and Manuel Herrera
Abstract. Water Supply Systems (WSS) are clearly dynamical systems. Processes associated with WSS include design, planning, maintenance, control, management, rehabilitation, enlargement, etc. Modeling and simulation of these processes can be performed by using a number of variables and constraints that are non-negative in nature. Demands, diameters of pipes, flowrates, minimum pressure at demand nodes, volume of reservoirs, are only a few examples, taken from the purely technical context. In this paper we will focus on the design of WSS. This a mixed discrete-continuous constrained optimization problem that is addressed here by the use of an evolutionary technique based on swarm intelligence. Robustness is enforced by adding reliability to the system both to cope with abnormal conditions and by considering the likelihood of different state and load conditions. Application to a real-world problem is also provided.
1 Introduction Water Supply Systems are live beings. They born, grow, age and deteriorate, need care (preventive care but also surgery is sometimes needed), are expected to work properly, have to meet basic requirements even under adverse circumstances, and so on. The aim is quality long-lasting life. As a consequence, the design of WSS cannot be thought as single, material and static design. This is one reason for WSS design optimization to be one of the most heavily researched areas in Hydraulics (see [5, 11, 22] for detailed review). The objective of the optimal design of a WSS is to determine the values of all involved variables so that all the demands are satisfied, even under certain failure Joaqu´ın Izquierdo, Idel Montalvo, Rafael P´erez-Garc´ıa and Manuel Herrera Centro Multidisciplinar de Modelaci´on de Fluidos, Universidad Polit´ecnica de Valencia, 46071 Valencia, Spain, e-mail:
[email protected],
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 321–330. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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conditions, while the investment and maintenance costs are minimal [8]. This optimal design problem involves minimizing a fitness function that includes costs for layout and sizing using new components, reusing or substituting existing components, creating a working system configuration that fulfils all water demands -including water quality-, adhering to the design constraints, and guaranteeing a certain degree of reliability for the system [6, 7]. Reliability refers to the ability of the network to provide consumers with adequate and high quality supply under normal and abnormal conditions. Both hydraulic and mechanical reliability are considered. The former refers to uncertainty coming mainly from nodal demand and pipe roughness. The latter usually refers to failures of system components, such as pipe breakage. There is no universal agreement about what would be the best measure of reliability, redundancy or resilience and what an acceptable level of these concepts is [19]. The approach considered here is twofold. Firstly, it considers hydraulic aspects. Working conditions can change due to change in demand, change in pipe roughness, technical failures, and so on. Robust optimization-based designs must consider these scenarios. Secondly, it also faces mechanical problems. Early work by Alperovits and Shamir [1] already showed that designs obtained by purely minimizing the cost of the pipes produce branched networks. But branched networks cannot guarantee the service to consumers downstream of a broken pipe. To get looped networks, able to cope with such abnormal situations, redundancy must be added to the system. Enforcing minimum diameters seems unnatural, and current trends point towards adding certain economic costs. In this paper we considered a proposal recently raised in [14]. We claim that it enforces certain level of reliability by considering some costs incurred by the lack of satisfaction of the supply. Interestingly, the system improvement implies only moderate increase regarding the initial investment costs. Due to the sundry aspects considered into the fitness function and to the nature of the involved variables a general purpose global optimization technique must be used. Recently, evolutionary algorithms (EA) have turned to be the preferred -and more suitable- water system design optimization techniques for many researchers [3, 4, 12, 13, 15, 20, 23–25]. They use full network model simulation to evaluate solution quality and, as a consequence, may require substantial computing time when real networks are considered. But, on the other hand, they manage to get rid of the drawbacks associated to classical optimization methods. In this paper, we apply a derivative of Particle Swarm Optimization (PSO), recently introduced by the authors [16, 17], to the design of WSS by using reliability ideas from [14]. This derivative is able to consider mixed discrete-continuous optimization, since the problem we tackle here involves the use of both types of variables. Also, it is able to find optimum or near-optimum solutions much more efficiently and with considerably less computational effort because of the richer population diversity it introduces. Finally, the cumbersome aspect, which is common to all metaheuristics, of parameters’ selection is tackled through self-adaptive dynamic parameter control.
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2 The Optimization Problem The optimization model considers a number, s, of scenarios each with probability of occurrence Pi and ∑si=1 Pi = 1. We now describe the fitness function and the constraints for a given scenario. Although the diameters of the pipes are the main decision variables, storage volumes, pump heads, kind of rehabilitation to be performed, etc., are also frequently required. These variables, being discrete, continuous or binary, share one characteristic: they are non-negative. The estimation of individual costs depends on these variables. The correct approach to assess the costs for each element becomes important when defining the fitness function, which has to be fully adapted to the problem under consideration: design, enlargement, rehabilitation, operation design, etc. For the sake of simplicity, we include here only the cost of the pipes. Other costs can be included in a straightforward manner. For a given scenario, this cost is represented by L
F1 (D) = ∑ ci (Di )li ,
(1)
i=1
where one sums over all, L, individual pipes. D = (D1 , ..., DL )t is the vector of pipes’ diameters. The cost per meter, depending on the diameter of pipe i, is given by ci (Di ) and its corresponding length by li . Note that Di is chosen from a discrete set of available diameters and ci is a non linear function of diameter. A number of constraints can be considered. Again, we restrict ourselves here to the provision of quality water in terms of minimum supply pressure. Accordingly, piezometric head at demand nodes, H, must be bigger than certain positive value: Hmin . These problem constraints are, obviously, positive. They are included as penalty costs in the fitness function, such that the violation of one of the imposed constraints provokes an increase in its value: L
N
i=1
j=1
F2 (D) = ∑ ci (Di )li + ∑ H(Hmin − H j ) · p · (Hmin − H j ),
(2)
where penalties are added over all, N, demand nodes. H(·) is the Heaviside function, and the factor p, which multiplies with the head difference, represents a fixed value which becomes effective whenever the minimal head requirement is not met. Note that in this model the individual penalties grow linearly with this difference. Penalty is high enough to render the corresponding solution unfeasible. In addition, the distribution of flowrates through the network and the piezometric head values must satisfy the classical equations of continuity and energy enforced into the hydraulic model. The complete set of equations may be written, by using block matrix notation [9], as q −A10 H f A11 (q) A12 = , (3) At12 0 H Q
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where A12 is the so-called connectivity matrix describing the way demand nodes are connected through the lines; its size is L × N p , N p being the number of demand nodes; q is the vector of the flowrates through the lines; H the vector of unknown heads at demand nodes; A10 is an L × N f matrix, N f being the number of fixed head nodes with known head H f and Q is the N p -dimensional vector of demands. Finally, A11 (q) is an L × L diagonal matrix. System (3) is a non-linear problem, whose solution is the state vector x = (q, H)t of the system. Continuity and energy equations are enforced by the use of EPANET2 [18], which is the benchmark hydraulic analysis tool used worldwide. Following [14], reliability is added here from an economic point of view, by considering the costs of the water not delivered due to problems in the system. Finally, the fitness function adds up this additional cost: L
N
L
i=1
j=1
i=1
F3 (D) = ∑ ci (Di )li + ∑ H(Hmin − H j ) · p · (Hmin − H j ) + ∑ wi · li · D−u i .
(4)
Here, wi is a coefficient associated to each pipe, of the form a · t f · (c f + ca ·V f ); a · l · D−u gives the number of expected failures per year of one pipe, as a function of diameter, Di , and length, li , (a and u are known constants); t f is the average number of days required to repair the pipe; c f is the daily repairing average cost; ca is the average cost of the water supplied to affected consumers, in monetary units per unit volume; and V f = 86400 · Qbreak is the daily volume of water that should be supplied to the affected consumers due to the loss of water of Qbreak in cubic meters per second. The scenarios considered here follow the approach of “breaking” by turn all the pipes of a specific design to check if all the constraints are fulfilled subjected to this circumstance. If the test is negative the design is suitably penalized. This way, designs will develop increasing reliability. To undergo those tests, the system must be analyzed for any of those specific “breakages”.
3 Description of the PSO Variant Used A swarm consists of an integer number, M, of particles, Xi , moving in the search space, S ⊂ Rd , each representing a potential solution of the problem: Find minX ε S F(X), subject to appropriate constraints, where F is the fitness function associated with the problem that, without loss of generality, we consider a minimization problem. In each cycle of the evolution, t, each particle, i, with position vector Xi (t) = (xi1 , ..., xid ) has an associated velocity vector, Vi (t) = (vi1 , ..., vid ), and an associated best personal position at which the best fitness was encountered by the particle, Yi (t) = (yi1 , ..., yid ) = arg min(F(Xi (t)), F(Xi (t − 1))). Also, the position of the best particle of the swarm,
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Y ∗ = arg min{F(Xi (t)), i = 1, ..., M}, is identified for every t. In each generation, the velocity of each particle is updated: Vi = ω Vi + c1rand()(Yi − Xi ) + c2 rand()(Y ∗ − Xi ).
(5)
On each dimension, particle velocities are restricted to minimum and maximum velocities, which are user-defined parameters, Vmin ≤ V j ≤ Vmax ,
(6)
to control excessive roaming of particles outside the search space. The position of each particle is also updated every generation: Xi = Xi + Vi .
(7)
The parameters are as follows: ω is a factor of inertia [21] that controls the impact of the velocity history into the new velocity. Acceleration parameters c1 and c2 are typically two positive constants, called the cognitive and social parameters, respectively. rand( ) is a function that creates random numbers between 0 and 1, used to maintain the population diversity. The discussion so far has considered the standard PSO algorithm, which is applicable to continuous systems and cannot be used for mixed discrete-continuous problems. To tackle discrete variables, this algorithm takes the integer parts of the flying velocity vector’s discrete components into account: Vi = f ix(ω Vi + ci rand()(Yi − Xi ) + c2rand()(Y ∗ − Xi )),
(8)
where fix(·) implies that we only take the integer part of the result. The role of the inertia, ω , in (5) is considered critical for the PSO algorithm’s convergence behavior. As it facilitates the balancing of global and local searches, it has been suggested to allow to adaptively decrease linearly with time, usually in a way that at first emphasizes global search and then, with each cycle of the iteration, increasingly prioritizes local search [21]. A significant improvement in the performance of PSO is achieved by using [10]
ω = 0.5 +
1 . 2(ln(t) + 1)
(9)
The acceleration coefficients and the clamping velocity, however, are neither set to a constant value, as in standard PSO, nor set as a time-varying function, as in adaptive PSO variants [2]. Here, instead they are incorporated into the optimization problem. Each particle is allowed to self-adaptively set its own parameters by using the same process used by PSO and given by equations (5) or (8) and (7). To this end, these three parameters are considered as three new variables that are incorporated into position vectors Xi .
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Obviously, these new variables do not enter the fitness function, but rather they are manipulated by using the same mixed individual-social learning paradigm used in PSO. Note that also Vi and Yi increase their dimension, correspondingly. By using equations (5) or (8) and (7), each particle is additionally endowed with the ability to adjust its parameters by taking into account both the parameters it had at its best position in the past as well as the parameters of the leader, which facilitated this best particle’s move to its privileged position. As a consequence, particles use their cognition of individual thinking and social cooperation not only to improve their positions but also to improve the way they improve their position by accommodating themselves to the best known conditions, namely, their conditions and their leader’s conditions when they achieved the thus-far best position. Finally, in [16], PSO was endowed with a re-generation-on-collision formulation, which further improves the performance of standard discrete PSO. The random regeneration of the many birds that tended to collide with the best birds was shown to avoid premature convergence, as it prevented clone populations from dominating the search. The inclusion of this procedure into the discrete PSO produces greatly increased diversity and improved convergence characteristics and yields higherquality final solutions. In this study, a population size of M = 100 particles has been used. Also, if there is no improvement after 800 iterations, the process is stopped. The performance of the approach herein introduced can be observed from the results reported in the next section for a real-world problem.
4 Case Study In this case study the minimum pressure allowed is 15m and the available commercial diameters are given in Table 1, also including the Hazen-Williams coefficient, C, used in the hydraulic model, and the unit cost of the pipes. Table 1 Commercially available diameters
Diameter(mm) C Cost($ units) 100 150 200 250 300
140 140 140 140 140
117.14 145.16 191.42 241.09 333.16
The problem is solved by using the two fitness functions F2 and F3 defined in (2) and (4). The same penalty factor was used in both cases. The layout of the network can be seen in Figure 1. For the understanding of the results a code for colors has
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been used. Regarding pipes, blue, green, yellow and red colors represent 100, 150, 200 and 250mm pipes, respectively. Regarding nodes, dark blue means pressure above 15m; light blue, between 14 and 15m; green, between 12 and 14m; yellow, between 10 and 12m; and, finally, nodes with a pressure under 10m are represented in red. This network, which is fed by a tank, has 294 lines amounting to 18.337km of pipes and 240 nodes consuming 81.53l/s in total. Figure 1 (left) presents the solution obtained by using F3 (including reliability). This solution is only a mere 3.65% more expensive than the one obtained by using F2 (no reliability consideration), whose diameters can be seen in Figure 1 (right). Table 2 presents a comparison between the initial investment costs for both solutions. Table 2 Comparison between costs for both solutions
Diameter Without reliability With reliability (mm) Length (m) Cost($ units) Length(m) Cost($ units) 100 17731.10 2077021.41 15822.31 1853425.63 150 606.39 88023.28 2077.69 301597.04 0.00 0.00 328.79 62937.56 200 250 0.00 0.00 108.70 26206.24 300 0.00 0.00 0.00 0.00 Total cost ($ units) 2165044.69 2244166.47
The effect of closing the pipe pointed by the arrow can be observed in Figure 1(right) for the solution without reliability. It shows the great impact produced by a closed pipe. It does not happen for the more reliable design obtained from F3 (left), no matter which pipe is out of service.
Fig. 1 Solutions with (left) and without (right) reliability considerations for the case-study
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Designs with and without reliability perform in a completely different way under the event of a broken or closed pipe. In Figure 1 (right) only some of the nodes close to the tank are able to maintain the minimal pressure of 15 m. On the other hand, in Figure 1 (left) the solution obtained considering reliability was able to re-distribute the flowrates and to guarantee the demand at the required pressure at all the demand nodes of the network. This represents a great advantage from the operating and management point of view. In addition, as said before, this is achieved with only a small increase in the initial investment. Finally, if consideration is made not only of the initial investment costs, but also of the costs derived from breakages, the solution with reliability reveals itself as much more advantageous from an economical point of view. The following table shows the value of pressure at the most critical nodes when pipe indicated in Figure 1 is closed. Table 3 Pressure at most critical nodes
Node Without reliability With reliability Pressure (m) Pressure (m) ID 1111345 18.81 3.21 1102108 18.77 3.24 18.85 3.24 1112395 1106799 18.90 3.33 1098891 18.75 3.35 19.19 3.59 1103578 1113234 19.21 3.59 19.26 3.64 1107987 1100151 19.04 3.65 1099662 19.14 3.75 19.33 4.02 1094132 1062222 19.23 4.52 1049416 19.30 4.89 19.32 4.97 1047213
5 Conclusions Most processes on WSS fall clearly under the category of positive systems. We have tackled here the robust design of such a WSS. The solution cannot ignore the evaluation of aspects related to different scenarios and certain failure conditions. Consideration of only the initial investment costs will produce designs that will be cheaper but that will experiment serious difficulties to cope with abnormal situations. In this work, we have shown, through a case study, that more reliable designs do not necessarily must involve immoderate investment increase. Interestingly, the same case
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study shows, nonetheless, the much better performance of reliable designs in the case that failure events are represented by pipes being out of service. The concept of reliability we have used here takes into account the economical impact of the water not delivered due to this kind of failure events during the life of the network. Optimization has been carried out by using a variant of PSO devised by the authors that considers both discrete and continuous variables, has increased population diversity and manages self-adaptively its parameters. Having at one’s disposal an optimization tool like the one used here is of paramount importance, since other terms (load or service conditions, rehabilitation costs, life-long costs, and so on) can be added to the fitness function without rendering the problem conceptually more complex. In addition, this tool can be combined easily with hydraulic network simulation modules, thus allowing great versatility in the analysis of candidate solutions. Finally, the multi-agent approach that permeates the optimization algorithm constitutes and open-door environment for multi-objective formulations regarding the design of WSS. Acknowledgements. Supported by grants MAEC-AECI 0000202066 and BES-2005-9708, awarded to two of the authors.
References 1. Alperovits, E., Shamir, U.: Design of optimal water distribution systems. Water Resour. Res. 13(6), 885–900 (1977) 2. Arumugam, M.S., Rao, M.V.C.: On the improved performances of the particle swarm optimization algorithms with adaptive parameters, cross-over operators and root mean square (RMS) variants for computing optimal control of a class of hybrid systems. Appl. Soft Comput. 8(1), 324–336 (2008) 3. Cunha, M.C., Sousa, J.: Water distribution network design optimization: simulated annealing approach. J. Wat. Res. Plann. Mgmt. 125(4), 215–221 (1999) 4. Geem, Z.W.: Optimal cost design of water distribution networks using harmony search. Eng. Optm. 38(3), 259–280 (2006) 5. Goulter, I.C.: Systems analysis in water-distribution network design: From theory to practice. J. Wat. Res. Plann. Mgmt. 118(3), 238–248 (1992) 6. Goulter, I.C., Bouchart, F.: Reliability-Constrained Pipe Network Model. J. Hydraul. Eng. 116(2), 211–229 (1990) 7. Goulter, I.C., Coals, A.V.: Quantitative approaches to reliability assessment in pipe networks. J. Transp. Eng. 112(3), 287–301 (1986) 8. Izquierdo, J., P´erez, R., Iglesias, P.L.: Mathematical Models and Methods in the Water Industry. Math. Comput. Modelling (39), 1353–1374 (2004) 9. Izquierdo, J., Tung, M.M., P´erez, R., Mart´ınez, F.J.: Estimation of fuzzy anomalies in Water Distribution Systems. In: Progress in Industrial Mathematics at ECMI 2006, vol. 12, pp. 801–805. Springer, Berlin (2008) 10. Jin, Y.X., Cheng, H.Z., Yan, J.Y., Zhang, L.: New discrete method for particle swarm optimization and its application in transmission network expansion planning. Electri. Power Syst. Res. 77(3-4), 227–233 (2007)
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11. Lansey, K.E.: Optimal Design of Water Distribution Systems. In: Mays, L.W. (ed.) Water Distribution System Handbook. McGraw-Hill, New York (2000) 12. Liong, S.Y., Atiquzzama, M.: Optimal design of water distribution network using shuffled complex evolution. J. Inst. Eng. 44(1), 93–107 (2004) 13. Maier, H.R., Simpson, A.R., Zecchin, A.C., Foong, W.K., Phang, K.Y., Seah, H.Y., Tan, C.L.: Ant-colony optimization for design of water distribution systems. J. Wat. Res. Plann. Mgmt. 129(3), 200–209 (2003) 14. Mart´ınez, J.B.: Quantifying the economy of water supply looped networks. J. Hydraul. Eng. 133(1), 88–97 (2007) 15. Mat´ıas, A.S.: Dise˜no de redes de distribuci´on de agua contemplando la fiabilidad, mediante Algoritmos Gen´eticos. Departamento de Ingenier´ıa Hidr´aulica y Medio Ambiente. Universidad Polit´ecnica de Valencia. Doctoral dissertation (2003) 16. Montalvo, I., Izquierdo, J., P´erez, R., Iglesias, P.L.: A diversity-enriched variant of discrete PSO applied to the design of Water Distribution Networks. Engineering Optimization 40(7), 655–668 (2008) 17. Montalvo, I., Izquierdo, J., P´erez, R., Tung, M.M.: Particle Swarm Optimization applied to the design of water supply systems. Comput. Math. Appl. 56(3), 769–776 (2008) 18. Rossman, L.A.: EPANET, users manual, U.S. EPA, Cincinnati (2000) 19. Savic, D.A.: Coping with risk and uncertainty in urban water infrastructure rehabilitation planning. In: Acqua e citt`a - i convegno nazionale di idraulica urbana, S’Agnello (NA), pp. 28–30 (2005) 20. Savic, D.A., Walters, G.A.: Genetic algorithms for least-cost design of water distribution networks. J. Wat. Res. Plann. Mgmt. 123(2), 67–77 (1997) 21. Shi, Y., Eberhart, R.C.: A modified particle swarm optimizer. In: Proceedings of the IEEE Congress on Evolutionary Computation, Piscataway, NJ, pp. 69–73 (1998) 22. Walski, T.M.: State of the Art: Pipe Network Optimization, Computer Applications in Water Resources, ASCE (1985) 23. Wu, Z.Y., Simpson, A.R.: Competent genetic-evolutionary optimization of water distribution systems. J. Comput. Civ. Eng. 15(2), 89–101 (2001) 24. Wu, Z.Y., Walski, T.: Self-Adaptive Penalty Approach Compared with Other ConstraintHandling Techniques for Pipeline Optimization. J. Wat. Res. Plann. Mgmt. 131(3), 181– 192 (2005) 25. Zecchin, A.C., Simpson, A.R., Maier, H.R., Leonard, M., Roberts, A.J., Berrisford, M.J.: Application of two ant-colony optimisation algorithms to water distribution system optimisation. Math. Comput. Modelling 44(5-6), 451–468 (2006)
Applications of Linear Co-positive Lyapunov Functions for Switched Linear Positive Systems Florian Knorn, Oliver Mason and Robert Shorten
Abstract. In this paper we review necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for switched linear positive systems. Both the state dependent and arbitrary switching cases are considered and a number of applications are presented.
1 Introduction Positive systems, that is systems in which each state can only take positive values, play a key role in many and diverse areas such as economics [11, 16], biology [1, 9], communication networks [4, 17], decentralised control [21] or synchronisation / consensus problems [10]. Although these as well as switched systems have been the focus of many recent studies in the control engineering and mathematics literature — to name but a few [2, 3, 12, 20] — there are still many open questions relating to the stability of systems that fall into both categories: switched positive systems. Proving stability for switched systems involves determining a Lyapunov function that is common to all constituent subsystems, [18]. In that context, work discussed in [14, 15] provides necessary and sufficient conditions for the existence of a particular type of Lyapunov function, namely a linear co-positive Lyapunov function (LCLF). It is the aim of this paper to review these results and provide examples of their use. Our brief paper is structured as follows. In Section 2 we present a number of examples from various applications to motivate the problem. We then summarise conditions for the existence of a common LCLF for switched systems evolving in the entire positive orthant, as well as when the positive orthant is partitioned into Florian Knorn, Oliver Mason and Robert Shorten Hamilton Institute, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland, e-mail:
[email protected],
[email protected],
[email protected]@nuim.ie
R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 331–338. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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cones. Finally, in Section 4 we apply these results to the examples given at the beginning.
Notation and Mathematical Preliminaries Throughout, (resp. + ) denotes the field of real (resp. positive) numbers, n is the n-dimensional Euclidean space and n×n the space of n × n matrices with real entries. A closed, pointed convex cone C is a subset of n if and only if α x + β y ∈ C for any x, y ∈ C and non-negative scalars α , β . Matrices or vectors are said to be positive (resp. non-negative) if all of their entries are positive (resp. non-negative); this is written as A 0 (resp. A ! 0), where 0 is the zero-matrix of appropriate dimension. A matrix A is said to be Hurwitz if all its eigenvalues lie in the open left half of the complex plane. A matrix is said to be Metzler if all its off-diagonal entries are non-negative. We use ΣA to denote the linear time-invariant (LTI) system x˙ = Ax. Such a system is called positive if, for a positive initial condition, all its states remain in the positive orthant throughout time. A classic result shows that this will be the case if and only if A is a Metzler matrix, [3]. Similarly, a switched linear positive system is a dynamical system of the form x˙ = As(t) x, for x(0) = x0 where s : → {1, . . . , N} is the so-called switching signal and {A1 , . . . , AN } are the system matrices of the constituent systems, which are Metzler matrices. See [5, 18] for more details on systems of this type. Below we will just write x˙ = A(t)x for such a system. Finally, the function V (x) = vT x is said to be a linear co-positive Lyapunov function (LCLF) for the positive LTI system ΣA if and only if V (x) > 0 and V˙ (x) < 0 for all x ! 0 and x = 0, or, equivalently, v 0 and vTA ≺ 0.
2 Motivating Examples To motivate our results we shall first present a few situations to which they can be applied. 1) Classes of switched time-delay systems Consider the class of n-dimensional linear positive systems with time-delay τ ≥ 0, similar to those considered by Haddad et al. in [6], but where both the system and the delay matrices may be switching over time: x˙ (t) = A(t)x(t) + Ad (t)x(t − τ ),
x(θ ) = φ (θ ), −τ ≤ θ ≤ 0 (1) 0 / where we assume 0 matrix A(t) ∈ A1 , . . . , AN is Metzler, the delay / that the system matrix Ad (t) ∈ Ad1 , . . . , AdM is non-negative, A(t) + Ad (t) is Metzler and Hurwitz for all t ≥ 0, and where φ : [−τ , 0] → n is a continuous, vector valued function
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specifying the initial condition of the system. How can stability of the system for arbitrary switching and delays be shown? 2) Switched positive systems with multiplicative noise Consider the class of switched positive systems with feedback quantisation or where the states experience resets. In this type of system, the states on the right hand side are scaled by a (usually time-varying) diagonal matrix: 0 / x˙ = A(t)D(t)x, A(t) ∈ A1 , . . . , AN where we assume that A(t) is Metzler and Hurwitz for all t, and the diagonal matrix D(t) has strictly positive and bounded diagonal entries for all t. Under which conditions would such a system be stable? 3) Robustness of switched positive systems with channel dependent multiplicative noise An important class of positive systems is the class that arises in certain networked control problems. Here, the system of interest has the form: x˙ = A(t)x + C[1] (t) + · · · + C[n] (t) x where A(t) is Metzler and where C[i] (t) ! 0 is an n × n matrix that describes the communication path from the network states to the ith state; namely it is a matrix of unit rank with only one non-zero row. Usually, the network interconnection / [i]between[i] 0N different configurations, so that / structure0 varies with time A(t) ∈ A1 , . . . , AN and C[i] (t) ∈ C1 , . . . , CN for i = 1, . . . , n. Again, we assume that A(t) + C[1] (t) + · · · + C[n] (t) is also Metzler and Hurwitz for all t. What can be said regarding asymptotic stability here? 4) Numerical example Finally, to provide a more concrete example, assume we are given a switched linear positive system with the following three Metzler and Hurwitz matrices ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ −16 6 6 −10 4 0 −9 2 8 A = ⎣ 1 −18 2 ⎦ , B = ⎣ 8 −10 9 ⎦ , C = ⎣ 6 −10 4 ⎦ (2) 5 3 −20 4 3 −13 8 0 −16 Can we prove that it is stable under arbitrary switching?
3 Common Linear Co-positive Lyapunov Functions As mentioned in the introduction, work reported in [14] discusses conditions for the existence of a common LCLF for switched linear positive systems comprised of sets of LTI systems, where each of the constituent systems is assumed to be associated with a convex region of the positive orthant of the n .
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Let us briefly present two results. The first, more general result concerns situations where the state space (the positive orthant) is partitioned into smaller regions, and where only certain subsystems may be active in certain regions (this may be interpreted as state dependent switching). The other result focuses on the special case where each of those regions is the entire positive orthant itself, that is the system can switch to any subsystem in any given point in the state space.
3.1 Switching in Partitioned Positive Orthant Assume there are N closed pointed convex cones C j such that the closed positive orthant can be written as n+ = ∪Nj=1 C j . Moreover, assume that we are given stable positive LTI systems ΣA j for j = 1, . . . , N such that the jth system can only be active for states within C j . The following theorem then gives a necessary and sufficient condition for the existence of a common LCLF in this set-up. Theorem 1. Given N Metzler and Hurwitz matrices A1 , . . . , AN ∈ n×n and N closed, convex pointed cones C1 , . . . , CN such that n+ = ∪Nj=1 C j , precisely one of the following statements is true: 1. There is a vector v ∈ n+ such that vTA j x j < 0 for all non-zero x j ∈ C j and j = 1, . . . , N. 2. There are vectors x j ∈ C j not all zero such that ∑Nj=1 A j x j ! 0. Proof. 2 ⇒ ¬1:1 Assume 2 holds. Then, for any positive vector v 0 we have T 1 x1 + · · · + v AN xN ≥ 0 which implies that 1 cannot hold. ¬2 ⇒ 1: Assume 2 does not hold, i. e. there are no vectors x j ∈ C j not all zero such that ∑Nj=1 A j x j ! 0. This means that the following intersection of convex cones is empty: 9 : 9 : / ∑Nj=1 A j x j : x j ∈ C j , not all zero ∩ x ! 0 = 0. ? @A B ? @A B vTA
O1
O2
By scaling appropriately we can see that this is equivalent to: 9 : 9 : ∑Nj=1 A j x j : x j ∈ C j , ∑Nj=1 x j 1 = 1 ∩ x ! 0 = 0/ ? @A B ? @A B
(3)
O2
O¯1
where · 1 denotes the L1 -norm. Now, O¯1 and O2 are disjoint non-empty closed convex sets and additionally O¯1 is bounded. Thus, we can apply Corollary 4.1.3 from [8] which guarantees the existence of a vector v ∈ n such that max vTy < inf vTy
y∈O¯1 1
y∈O 2
That is, we show that if 2 is true, then 1 cannot hold.
(4)
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As the zero vector is in O2 , it follows that infy∈O 2 vTy ≤ 0. However, as O2 is the cone {x ! 0} it also follows that infy∈O 2 vTy ≥ 0. Thus, infy∈O 2 vTy = 0. Hence, vTy ≥ 0 for all y ! 0 and thus v ! 0. Moreover, from (4), we can conclude that for/any j = 1, . . . , N 0 and any x j ∈ C j with x j 1 = 1 we have vTA j x j < 0. As C j ∩ x ! 0 : x1 = 1 is compact, it follows from continuity that by choosing ε > 0 sufficiently small, we can guarantee / 0 that vε := v + ε 1 0 satisfies vT ε A j x j < 0 for all x j ∈ C j ∩ x ! 0 : x1 = 1 and all j = 1, . . . , N, where 1 is the vector of all ones. Finally, it is easy to see that vT ε A j x j < 0 is true even without the norm require ment on x j . This completes the proof of the theorem. A very practical way of partitioning the state space would be to partition it using simplicial cones C j . These are cones generated by non-negative, non-singular generating matrices Q j ∈ n×n : 9 : (i) C j := x x = ∑ni=1 αi q j , αi ≥ 0, i = 1, . . . , n (5) (i)
where j = 1, . . . , N and q j denotes the ith column of Q j . In that case, we may include the cone generating matrices into the second statement of Theorem 1 to reword it slightly to: [...] 2. There are vectors w j ! 0 not all zero s. t. ∑Nj=1 B j w j ! 0, with B j := A j Q j . This new statement 2 can now be easily tested by running a feasibility check on a suitably defined linear program, see [14] for more details.
3.2 Switching in Entire Positive Orthant An important special case of the previous results is when the Q j matrices are the identity matrix, namely when we seek a common linear co-positive Lyapunov function for a finite set of linear positive systems. For that, some additional notation is required: Let the set containing all possible mappings σ : {1, . . . , n} → {1, . . . , N} be called Sn,N , for positive integers n and N. Given N matrices A j , these mappings will then be used to construct matrices Aσ (A1 , . . . , AN ) in the following way: (1) (2) (n) (6) Aσ A1 , . . . , AN := aσ (1) aσ (2) . . . aσ (n) (i)
(i)
where a j denotes the ith column of A j . In other words, the ith column aσ of Aσ is the ith column of one of the A1 , . . . , AN matrices, depending on the mapping σ ∈ Sn,N chosen. We then have the following condition: Theorem 2. Given N Hurwitz and Metzler matrices A1 , . . . , AN ∈ lowing statements are equivalent:
n×n ,
the fol-
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1. There is a vector v ∈ n+ such that vTA j ≺ 0 for all j = 1, . . . , N. 2. Aσ (A1 , . . . , AN ) is Hurwitz for all σ ∈ Sn,N .
Proof. Given in [14].
Remark 1. Since the submission of [14] it has come to our attention that this result may also be deduced from the more general results on P-matrix sets given in [19]. Theorem 2 states that N positive LTI systems have a common linear co-positive Lyapunov function V (x) = vT x if and only if the Aσ (A1 , . . . , AN ) matrices are Hurwitz matrices, for all σ ∈ Sn,N . In that case, the switched system formed by these subsystems is uniformly asymptotically stable under arbitrary switching. Finally, note that when the A j Q j in Theorem 1 (or its reworded version) are Metzler and Hurwitz, then the Hurwitz condition of Theorem 2 can also be used to give a solution to the state dependent switching problem.
4 Solution to Motivating Examples We shall now use these results to answer the problems posed in Section 2. 1) Classes of switched time-delay systems We can show stability under arbitrary switching and delays if two conditions are ˜ d # 0 for all t, i. e. there is a ˜ d such that Ad (t) − A met: (a) there is a matrix A ˜ d that is entry-wise greater or equal than Adi for all i = 1, . . . , M; (b) for all matrix A ˜ d , . . . , AN + A ˜ d are Hurwitz. This can be seen by σ ∈ Sn,N the matrices Aσ A1 +A noting that (b) guarantees (by applying Theorem 2) the existence of a vector v 0 ˜ d ≺ 0. Then, consider the following Lyapunov-Krasovskii such that vT A(t) + A functional, [7, 13]. ˜d V (ψ ) = vT ψ (0) + vT A
0 −τ
ψ (θ )dθ
for some v 0. Clearly V (ψ ) ≥ vT ψ (0) ≥ aψ (0)∞ with a = mini {vi } > 0 and · ∞ being the maximum modulus norm. Next, define xt := { x(t + θ ) | θ ∈ [−τ , 0] } as the trajectory segment of the states in the interval [t − τ , t ]. Then, if condition (a) is met, the directional derivative of the above functional along the solutions of (1) will be ˜ d x(t) − x(t − τ ) V˙ (xt ) = vT x˙ (t) + vT A ˜ d x(t) − x(t − τ ) = vT A(t)x(t) + Ad (t)x(t − τ ) + vT A ˜ d x(t − τ ) ˜ d x(t) + vT Ad (t) − A = vT A(t) + A ? ? @A B @A B #0 ≤ −pT x(t) ≤
−β x(t)∞
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where β = mini {pi } > 0. It then follows (see for instance [7]) that the switched system is uniformly asymptotically stable. 2) Switched positive systems with multiplicative noise Through Theorem 2 we know that if Aσ (A1 , . . . , AN ) is a Hurwitz matrix for all σ ∈ Sn,N , then there exists a common LCLF for the system. In that case, since D(t)x 0, the system will be stable for any D(t). 3) Robustness of switched pos. systems with channel dep. multiplicative noise Again, our principal result can be used to give conditions such that this system is stable. A sufficient requirement for asymptotic stability here would be ∈ Sn,q , where that Aσ B1 , . . . , Bq is a Metzler and Hurwitz matrix for all σ [1] [n] q = N (n+1) and B1 , . . . , Bq are all the matrices of the form Ai0 + Ci1 + · · · + CiN with i0 , . . . , iN ∈ {1, . . . , N}. Further, by exploiting simple properties of Metzler matrices, we will also get the robust stability of the related system: x˙ = A(t)x + C[1] (t)D[1] (t) + · · · + C[n] (t)D[n] (t) x where the D[i] (t) are non-negative diagonal matrices whose diagonal entries are strictly positive, but with entries bounded less than one. This latter result is important as it can be used to model uncertain communication channel characteristics. 4) Numerical example With A, B, C given as in (2), it turns out that all Aσ (A, B, C) are Hurwitz matrices, for any σ ∈ S3,3 ; hence a switched linear positive system with these matrices will be uniformly asymptotically stable under arbitrary switching. If, however, the (3,1)element of C is changed from 8 to 14 — note that after the change C is still a Metzler and Hurwitz matrix — then the matrix A(3,2,3) = c(1) b(2) c(3) will have an eigenvalue λ * 1.7 which violates the Hurwitz condition.
5 Conclusion In this paper, after presenting a few motivating examples, we have reviewed necessary and sufficient conditions for the existence of a certain type of Lyapunov function for switched linear positive systems. We then illustrated and commented on the implications of our results. Future work will consider switched positive systems with time delay, and we suspect that the results reviewed here will be of great value in this future study.
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References 1. Arcak, M., Sontag, E.D.: Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica 42(9), 1531–1537 (2006) 2. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. In: Computer science and applied mathematics. Academic Press, New York (1979) 3. Farina, L., Rinaldi, S.: Positive Linear Systems. Wiley-Interscience Series. John Wiley & Sons, Inc., New York (2000) 4. Foschini, G.J., Miljanic, Z.: A simple distributed autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology 42(4), 641–646 (1993) 5. Gurvits, L., Shorten, R., Mason, O.: On the stability of switched positive linear systems. IEEE Transactions on Automatic Control 52(6), 1099–1103 (2007) 6. Haddad, W.M., Chellaboina, V.: Stability theory for nonnegative and compartmental dynamical systems with time delay. Systems & Control Letters 51(5), 355–361 (2004) 7. Hale, J., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. In: Applied Mathematical Sciences, vol. 99. Springer, New York (1993) 8. Hiriart-Urruty, J.B., Lemar´echal, C.: Fundamentals of convex analysis. Grundlehren Text Editions. Springer, Heidelberg (2001) 9. Jacquez, J.A., Simon, C.P.: Qualitative theory of compartmental systems. SIAM Review 35(1), 43–79 (1993) 10. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control 48(6), 988–1001 (2003) 11. Johnson, C.R.: Sufficient conditions for D-stability. Journal of Economic Theory 9(1), 53–62 (1974) 12. Johnson, C.R., Mehrmann, V., Olesky, D.D.: Sign controllability of a nonnegative matrix and a positive vector. SIAM Journal on Matrix Analysis and Applications 14(2), 398–407 (1993) 13. Kharitonov, V.L.: Robust stability analysis of time delay systems: A survey. Annual Reviews in Control 23, 185–196 (1999) 14. Knorn, F., Mason, O., Shorten, R.: On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica (2008) (to appear) 15. Mason, O., Shorten, R.: On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Transactions on Automatic Control 52(7), 1346– 1349 (2007) 16. Meyn, S.P.: Control Techniques for Complex Networks. Cambridge University Press, New York (2008) 17. Shorten, R., Wirth, F., Leith, D.J.: A positive systems model of tcp-like congestion control: asymptotic results. IEEE/ACM Transactions on Networking 14(3), 616–629 (2006) 18. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Review 49(4), 545–592 (2007) 19. Song, Y., Gowda, M.S., Ravindran, G.: On some properties of P-matrix sets. Linear Algebra and its Applications 290(1-3), 237–246 (1999) 20. Virnik, E.: Analysis of positive descriptor systems. Ph.D. thesis, Technische Universit¨at Berlin, Germany (2008) ˇ 21. Siljak, D.D.: Large-Scale Dynamic Systems: Stability and Structure. North-Holland Series in System Science and Engineering, vol. 3. North-Holland Publishing Co., New York (1979)
A Problem in Positive Systems Stability Arising in Topology Control Florian Knorn, Rade Stanojevic, Martin Corless and Robert Shorten
Abstract. We present a problem in the stability of switched positive systems that arises in network topology control. Preliminary results are given that guarantee stability of a network topology control problem under certain assumptions. Roughly speaking, these assumptions reduce the underlying stability problem to a nonlinear consensus problem with a driving term, that eventually becomes a Lur’e problem. Simulation results are given to illustrate our algorithm. While these results indicate that our assumptions can be removed, a proof of the general stability problem remains open.
1 Introduction Recent years have witnessed a growing interest in the control community in problems that arise when dynamic systems evolve over graphs. While the most high profile of these applications are in consensus applications such as formation flying and synchronisation problems, [4, 8, 11], many other applications have arisen where the manner in which network topologies change affect the performance of algorithms that are run over these networks. In such applications, an essential requirement is that the topology of the graph be such that some properties required to support communication and control are satisfied, the most basic of these being that the network is connected. Considerations of this kind have given rise to the emerging field of network topology control. Clearly, graph connectivity is an essential component in Florian Knorn, Rade Stanojevic, Robert Shorten Hamilton Institute, National University of Ireland Maynooth, Maynooth, Co. Kildare, Ireland, e-mail:
[email protected],
[email protected],
[email protected] Martin Corless School of Aeronautics & Astronautics, Purdue University, West Lafayette, IN 47907, USA, e-mail:
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 339–347. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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situations where a group of network nodes must work together, in a decentralised manner, to achieve some global task. This issue of graph connectivity is therefore very important and has achieved much attention in various contexts recently. In this paper we describe a recently proposed decentralised topology control algorithm [6] and suggest a simple way of adding weights to the states. This algorithm was posed to overcome some common assumptions in topology control (namely, that the underlying graph is symmetric). It exploits the fact that the rate of convergence of certain algorithms evolving over a graph is a good proxy for graph connectivity. Furthermore, this rate of convergence can be estimated in a decentralised manner, and can therefore be used to regulate graph connectivity. Under the assumption that the estimation problem and the control problem operate on different time-scales, stability can be demonstrated using elementary arguments. In particular, we show that the feedback system reduces to a consensus problem with an input, and it eventually becomes a scalar nonlinear system that can be analysed in a Lur’e problem framework. Simulations are presented to illustrate the validity of our results. These results also indicate that the feedback system is stable even when a separation of time-scales is not present. This latter problem in positive systems remains open and is posed in the concluding remarks of the paper.
2 Main Results In the context of this paper and the topology control problem discussed in [6], we are interested in the following type of n-dimensional positive systems with an input term: x(k + 1) = f x(k), k + u x(k), k 1, x(0) = x0 , k = 0, 1, 2, . . . where, x(k) ∈ n+ are the states, f : n+ × → n+ is a continuous vector-valued function, and u(x(k), k ∈ is an input term, that, through 1 = ( 1 . . . 1 )T , is thus equally added to all states and that we assume to be such that the system’s states do not leave the positive orthant. We would then like to investigate under which conditions the system’s states approach each other over time and, in the limit, eventually all take the same value — which may be time-varying, depending on the input term. More formally, we are looking for conditions such that limk→∞ xi (k)−x j (k) = 0 for all i, j ∈ {1, . . . , n}.
2.1 Affine Case Before stating our more general result we would like to present a more easily established result which can be obtained when the function f takes a particular linear form: f x, k = P(k)x where P(k) ∈ n×n is a sequence of primitive, row-stochastic
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(and thus non-negative) matrices with strictly positive main diagonal entries. This special form is often encountered in distributed averaging or consensus applications, for instance. Theorem 1. Let P(k) ∈ n×n be a sequence of matrices taken from a finite set of primitive, row-stochastic matrices with strictly positive main diagonal entries, and u(x(k), k a sequence of real, non-negative numbers. If x(k) ∈ n+ evolves for some x(0) = x0 ∈ n+ according to x(k + 1) = P(k)x(k) + u x(k), k 1 (1) where 1 = ( 1 . . . 1 )T , the elements of x(k) will approach each other over time, that is limk→∞ xi (k)−x j (k) = 0 for all i, j ∈ {1, . . . , n}. Proof. Given in [6].
2.2 General Case The previous result can be extended to classes of nonlinear consensus operators using the recent results of [7]. Borrowing its notation, we get: graphs1, Theorem 2. Let G (k) = V , A (k) be a sequence of strongly connected n u x(k), k a sequence of finite real numbers and f a map on × satisfying the following. Associated to each directed graph G = V , A with node set V = {1, . . . , n}, each node i ∈ V and each state x ∈ n+ , there is a compact set Ei (A )(x) ⊂ satisfying: ∀k ∈ ∀x ∈ n+ , 1. fi (x, k) ∈ Ei A (k) (x) 2. Ei (A )(x) = {xi } whenever the states of node i and its neighbouring nodes j are all equal, 3. Ei (A )(x) is contained in the relative interior of the convex hull of the states of node i and its neighbouring nodes j whenever the states of node i and its neighbouring nodes j are not all equal, 4. Ei (A )(x) depends continuously on x, that is, the set-valued function Ei (A ) : n ⇒ + is continuous. + Then, if the states x(k) ∈ n+ evolve for some initial condition x(0) = x0 ∈ according to x(k + 1) = f x(k), k + u x(k), k 1
n +
where 1= ( 1 . . . 1 )T, the elements of x(k) will approach each other over time, i. e. limk→∞ xi (k)−x j (k) = 0 for all i, j ∈ {1, . . . , n}. Proof. Given in [6].
Remark 1. Put simply, the theorem’s four conditions require that the updated state of each node must be a strict convex combination of its own and its neighbours’ states, and that the update function must be continuous. 1
That is, there is a directed path connecting any two nodes in the network.
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3 Application of Main Result 3.1 Distributed Averaging and Topology Control Consensus or distributed averaging algorithms have been the subject of an inordinate amount of attention in the past decade, as they arise in applications such as distributed sensing, clock synchronisation, flocking, or fusion of Kalman filter data; see for instance [2, 4, 9, 10]. Since the rate at which these algorithms converge strongly depends on structural properties of the network of nodes they are run on, it is an interesting problem to try to somehow regulate the topology of the graph in order to ultimately control the speed at which consensus will be achieved. But as control usually requires some form of measurement and feedback of the quantity of interest, in this case one would need to be able to determine the level of connectivity. While the primary focus of the present paper is neither on the properties nor the dynamics of consensus algorithms, we recall that the second eigenvalue in magnitude of the averaging matrix2 determines the rate at which the nodes in a network achieve consensus. Classically, the second smallest eigenvalue of the Laplacian (or transition Laplacian) matrix of a graph has been used as an algebraic measure for connectivity, [1, 3]. However, Laplacians are usually only defined for symmetric graphs, a restriction that we would like to avoid. In that regard, the second eigenvalue of an averaging matrix is also an excellent candidate measurement to indicate the degree of connectivity of an entire graph (whether the underlying graph is directed or not). It also has the added benefit of being able to be estimated locally. In [6] we describe several methods of estimating this important, global quantity in a distributed way. With the algorithms provided therein, each node in a network is able to estimate the second eigenvalue using only local, readily available information. In wireless networks (or, on a more abstract level, geometric graphs), this would offer the abovementioned possibility to control or maintain a certain level of interconnectedness: Each node could reduce or expand its communication radius if the connectivity is estimated to be larger or smaller than required (as decreasing or increasing this radius will lead to reducing or increasing the number of neighbours, hence changing connectivity). That such a strategy is well posed is evident and follows from the basic observation that if all nodes increase their communication radii sufficiently, then the graph will eventually achieved the desired level of connectedness. Let us investigate this control application more concretely in the following. 2
Many distributed averaging algorithms can be written as x(k +1) = Px(k) where x(k) is the vector containing the states of all the nodes in the network, at time k. The row-stochastic, so-called averaging matrix P describes how each node averages its own value with that of its neighbors.
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3.2 Control Strategy Given a wireless network, we wish to adjust the communication radius of each node in the network r1 , . . . , rn > 0 using the estimates of the second eigenvalue in magnitude of the averaging matrix, λ , with the ultimate objective of regulating λ to some neighbourhood of a target value; namely so that λ − λ∗ < ε for some λ∗ ∈ ( 0 , 1 ) and ε > 0. Since there will always be more than one set of communication radii { r1 , . . . , rn } that will guarantee this objective, we shall propose a control law that guarantees that the closed loop algorithm converges to a unique, single radius used by all nodes. Although this additional requirement is made to facilitate analytical tractability (that is, uniqueness of the solution), it can also be motivated from a practical standpoint: Having all nodes use the same broadcast radius helps to achieve similar battery lifetimes of the nodes, which is desirable in many applications. However, our framework is sufficiently general to allow other quantities of interest to be included in the control law design (but the convergence proofs will change accordingly). For instance, relaxing the requirements on the communication radii, one may require all nodes to have an equal number of neighbours (which would also yield a unique radius distribution). To achieve this, we propose updating the individual node radii using a convex combination of their neighbours’ radii, plus an input term that depends on the estimated second largest eigenvalue, that is we feed back of the current level of connectivity.3 Specifically, we suggest the following decentralised control law r(k + 1) = Pc (k)r(k) + η λ (k) − λ∗ 1 (2) for some initial, strictly positive radius distribution r(0) = r0 which guarantees a strongly connected graph. Here Pc (k) is a sequence of primitive, row-stochastic averaging matrices on the graphs induced by r(k), λ (k) is the magnitude of the second largest eigenvalue of the averaging matrix P as in Footnote 2 for the graph topology at time k, and η > 0 is a suitable control gain. To be fully precise, we could write λ r(k) to highlight that the second eigenvalue is ultimately a function of the topology of the graph, which in turn is dependent on r(k). Unfortunately both dependencies are rather complex in nature and hard to determine or express analytically. However, it can be shown that λ (k) may be treated as a sector bounded nonlinearity so that it can be treated in a Lur’e framework, [5]: Both Theorems 1 and 2 will guarantee that the control law forces all the radii, over time, to a common value. In other words, (2) will eventually become a scalar relation, so that the stability and convergence properties of the controlled system will eventually be governed by the scalar, positive system x(k + 1) = x(k) + u x(k), k 3
Note that we assume a certain separation of time scales between the estimation and the control scheme, i. e. we assume that the estimators have successfully converged to an exact estimate that is hence common to all nodes in the network.
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Since the properties of such systems are well understood, the above theorems offer interesting possibilities for the design of further control laws. This also allows us to determine how the control gain η must be chosen so that the closed loop system is stable, which is reported in depth in [6]. Note again, that any other consensus scheme (to which Theorem 2 can be applied) may be used as well. Also, we would like to stress that the proposed controller is decentralised in that each node only requires the radius information of its neighbours — information that can easily be broadcast along the communication that is necessary to run the algorithm used for estimating λ (k) in the first place. Remark 2. Let us comment on possible connectivity issues when using the above control law. When λ∗ is chosen very close to one, it may be possible that in some iteration the control law would adjust (reduce) the communication radii so much that network becomes disconnected. This can either be prevented by using a much smaller control gain than necessary for stability (which guarantees that λ∗ is approached without overshoot), or by introducing a “minimum radius” that the nodes’ radii are not allowed to fall below and that is large enough to guarantee that the graph always remains strongly connected.
3.3 Weighting In the scheme presented above, all nodes eventually reach a common radius. Now, imagine a setting where some nodes (say, nodes 17 and 25) are equipped with a longer-lasting power supply than others. In that case, these “special” nodes should be allowed to use a larger broadcast radius relative to the consensus value of the other, “ordinary” nodes. This would be an example of a situation were a certain “weighting” is applied to the states of each node. We shall now see that this can easily be incorporated in our set-up, without changing the convergence proofs. Let W := diag{wi } with wi > 0 for i = 1, . . . , n be the n × n diagonal matrix with positive entries wi along its main diagonal, and let r˜ := W−1 r. Then, run the control strategy for the “auxiliary” states r˜ — which will converge to a common value — but recover and utilise the “weighted” radii using r = W˜r. In the example above, this would mean that setting w17 = w25 = 2, and wi = 1 for all other nodes. Remark 3. The proposed weighting could also be used in a slightly more elaborate way. For instance, a node’s weight could be made a function of the remaining battery power, such that it is decreased over time as its battery is getting emptied. This way, nodes with little remaining battery power are allowed to use smaller radii than others so that they can “survive” a little bit longer: As their weight decreases, so will their communications radius, relative to the other nodes in the network. However, one will need to assure that the radius is not decreased too much so that the node disconnects from the network (see the remark at the end of the previous subsection).
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4 Simulation Results
−→
λ −→
1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0
ri
ri
−→
λ
−→
Let us now present some simulations that demonstrate our results. They all show experiments on networks with 200 nodes and randomly distributed initial radii in the [ 0.05 , 0.95 ] interval,4 in which the second largest eigenvalue in magnitude was regulated to some desired value. Depicted are the evolution over time of the second eigenvalue together with the nodes’ radii.
5
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1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0
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15 20 Time step k
25 −→
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Fig. 1 Evolution of the second eigenvalue λ (k) of the averaging matrix (upper subplots) and the individual nodes’ communication radii ri (k) (lower plots) in two networks of 200 nodes for (a) λ∗ = 0.9 and (b) λ∗ = 0.2.
Both Figures 1(a) resp. (b) show situations where the nodes were to achieve λ∗ = 0.9 resp. λ∗ = 0.2 using a common communication radius. It can be seen that at first the radii converge to a common value and then, on a slower time scale, change such that the second eigenvalue reaches the desired value. Figure 2 shows two examples where the nodes’ states were weighted. In (a), we simulate the setting where two nodes are equipped with different power supplies than the others. This presents an application of weighting the states as mentioned in Subsection 3.3. We picked two nodes which we wanted to use twice resp. half the radius as the other nodes in the network. This was achieved by setting the corresponding weights to 2 resp. 0.5. As can be seen in the plot, the second largest eigenvalue of the network converges quickly to its desired value of λ∗ = 0.9, and the nodes’ radii all converge to a common value but for the two special nodes of different weighting. An example for the remark at the end of Subsection 3.3 is given in Figure 2(b). Whilst again regulating the second largest eigenvalue in magnitude, we started to successively reduce one node’s weight starting at time k = 40. The plots show that the desired level of connectivity is, again, quickly achieved and maintained throughout. However, after k = 40 one node’s radius decreases bit by bit whereas the other 4
We deliberately chose different initial radii to show that consensus is achieved on these values.
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Fig. 2 Evolution of λ (k) and the individual nodes’ radii ri (k) in two networks of 200 resp. 50 nodes as the second eigenvalue is regulated to λ∗ = 0.9. In (a) two nodes where to have twice resp. half the radius as their peers; in (b) one node’s weight was successively reduced starting at k = 40.
nodes’ radii all commonly increase, slightly, to counter the effect of the reduction in radius of the special node.
5 Conclusion and Future Directions In the context of consensus algorithms, we presented two theorems that provide conditions for the convergence of the states to a common value, even when there are inputs to the system. We also suggested a simple modification that allows for different weightings to be applied to the states. We then used these results to control the topology of wireless networks or geometric graphs in general. The proposed decentralised control law, which adjusts the communication radii of the nodes so that the overall network achieves a predefined level of connectivity, poses such a consensus problem with inputs, and possible weighting of the states. This leads us to the more general, open problem of finding consensus conditions for systems of the type x(k + 1) = f x(k) , k + g x(k) , λ˜ (k) , k where f is some convex function of the system’s states, and g is a function of the (local) states and the (local) estimates λ˜ i (k) of the second largest eigenvalue of the averaging matrix of the graph. These systems are encountered when we drop the assumption of separation of time scales of estimation and control scheme, or in certain distributed optimisation problems, when g represents the derivative of some convex cost function.
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Acknowledgements. This work was supported by Science Foundation Ireland PI Award 07/IN.1/1901.
References 1. Chung, F.R.K. (ed.): Spectral Graph Theory. CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society, Province (1997) 2. Estrin, D., Girod, L.D., Pottie, G.J., Srivastava, M.: Instrumenting the world with wireless sensor networks. In: Proc. of the Int. Conf. on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, vol. 4, pp. 2033–2036 (2001) 3. Fiedler, M.: Algebraic connectivity of graphs. Czech. Math. J. 23(98), 298–305 (1973) 4. Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003) 5. Khalil, H.K.: Nonlinear Systems. Macmillan Publishing Co., New York (1992) 6. Knorn, F., Stanojevic, R., Corless, M., Shorten, R.: A framework for decentralised feedback connectivity control with application to sensor networks. Int. J. Control (to appear, 2009) 7. Moreau, L.: Stability of multiagent systems with time-dependent communication links. IEEE Trans. Autom. Control 50(2), 169–182 (2005) 8. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Autom. Control 49(9), 1520–1533 (2004) 9. Reynolds, C.W.: Flocks, herds, and schools: A distributed behavioral model. In: Proc. of the 14th Annual Conf. on Computer Graphics and Interactive Techniques, Anaheim, CA, USA, pp. 25–34 (1987) 10. Vicsek, T., Czir´ok, A., Ben-Jacob, E., Cohen, I., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995) 11. Zavlanos, M.M., Pappas, G.J.: Controlling connectivity of dynamic graphs. In: Proc. of the Joint 44th IEEE Conf. on Decision and Control, and the European Control Conf., Seville, Spain, pp. 6388–6393 (2005)
Control of Uncertain (min,+)-Linear Systems Euriell Le Corronc, Bertrand Cottenceau and Laurent Hardouin
Abstract. This paper deals with the control of uncertain (min,+)-linear systems which belong to an interval. Thanks to the residuation theory, a precompensator controller placed upstream of the studied system is given in such a way that even if the system’s behavior is not perfectly known, it has the property to delay the input as much as possible while keeping the input/output behavior unchanged. This precompensator is called neutral.
1 Introduction Discrete Event Dynamic Systems (DEDS) such as production systems, computing networks and transportation systems which are characterized by delay and synchronization phenomena can be described by linear models. Thanks to the particular algebraic structure called idempotent semiring (or dioid), this translation into a linear model is possible through for instance the (min,+)-algebra. This approach, detailed in [1] and [4], has numerous analogies with the classical automatic theory and in particular, the control of these systems can be considered. For instance, some model matching problems are solved by the way of different control structures (open-loop or close-loop structures) as presented in [2], [6] and [8]. These results rely on the residuation theory and assume that the model is perfectly known. This paper puts forward a control synthesis problem when the system is modeled with some parametric uncertainties. More precisely, the following conditions are assumed: • the system has a (min,+)-linear input/output behavior denoted h, Euriell Le Corronc, Bertrand Cottenceau and Laurent Hardouin Laboratoire d’Ing´enierie des Syst`emes Automatis´es, Universit´e d’Angers, 62, Avenue Notre Dame du Lac, 49000 Angers, France, e-mail:
[email protected],
[email protected],
[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 349–357. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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• because of uncertainties, h is unknown but belongs to an interval [ h , h ], the bounds of which are known. Under these assumptions, a precompensator controller p for the unknown system h is computed in order to achieve two goals: • the precompensator p is the greatest as possible, i.e. the one which delays the input as much as possible, • the input/output transfer1 is unchanged, i.e. h ∗ p = h. In a manufacturing context, such a controller allows the work-in process to be reduced while keeping the same process output. This enables to preserve input/output stream while decreasing internal congestions. It is important to note that our approach is different from the one presented in [7]. Indeed, in [7], the system also belongs to an interval (h ∈ [ h , h ]) but is subject to fluctuation2 within the interval limits and admits a precompensator p ∈ [ p , p ] such that h ∗ p ∈ [ h , h ]. In this paper, a precompensator p is computed such that the equality h ∗ p = h is satisfied, provided that h is a stationary (min,+)-linear system. In order to introduce this work, the paper is organized as follows. The second section recalls some algebraic tools required to the DEDS study through idempotent semiring and residuation theory. In the third section, models and controls of (min,+)-linear systems are presented. Finally, in the fourth section, the neutral precompensator controller p is proposed and an example is given.
2 Algebraic Preliminaries 2.1 Dioid Theory An idempotent semiring D is a set endowed with two inner operations denoted ⊕ and ⊗ (see [1, §4.2]). The sum ⊕ is associative, commutative, idempotent (i.e. ∀a ∈ D, a ⊕ a = a) and admits a neutral element denoted ε . The product ⊗ is associative, distributes over the sum and accepts e as neutral element. An idempotent semiring is said to be complete if it is closed for infinite sums and if the product distributes over infinite sums too. Moreover, the greatest element of D is denoted T (for Top) and represents the sum of all its elements. Due to the sum idempotency, an order relation can be associated with D by the following equivalences: ∀a, b ∈ D, a ! b ⇐⇒ a = a ⊕ b and b = a ∧ b. Because of the lattice properties of a complete idempotent semiring, a ⊕ b is the least upper bound of D whereas a ∧ b is its greatest lower bound. 1 2
where ∗ is the convolution product. h is not necessarily (min,+)-linear.
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Example 1. (min ). The set min = ( ∪ {−∞, +∞}), endowed with the min operator as sum ⊕ and the classical sum as product ⊗, is a complete idempotent semiring where ε = +∞, e = 0 and T = −∞. On min , the greatest lower bound ∧ takes the sense of the max operator.
2.2 Residuation Theory Residuation is a general notion in lattice theory which allows to define “pseudoinverse” of some isotone maps (see [1]). In particular, the residuation theory provides optimal solutions to inequalities such as f (x) # b (respectively f (x) ! b), where f is an order-preserving mapping defined over ordered sets. A mapping f defined over ordered sets is isotone, respectively antitone, if a # b ⇒ f (a) # f (b), respectively f (a) ! f (b). Now, let f : E → F be an isotone mapping, where (E , #) and (F , #) are ordered sets. Mapping f is said residuated if ∀b ∈ F , the greatest element denoted f (b) of subset {x ∈ E | f (x) # b} exists and belongs to this subset. Mapping f is called the residual of f . When f is residuated, f is the unique isotone mapping such that f ◦ f # IdF and f ◦ f ! IdE , where IdF (respectively IdE ) is the identity mapping on F (respectively on E ). Example 2. (Left product). Mapping La : x → a ⊗ x defined over a complete idempotent semiring D is residuated. Its residual represents the optimal solution to inequality a ⊗ x # b and is usually denoted La : x → a \◦ x (left quotient). Remark 1. (Isotony and antitony). ∀x, y, a ∈ D, an ordered set, these properties are given: a \◦ x # a \◦ y (x → a \◦ x is isotone), x#y ⇒ (1) x \◦ a ! y \◦ a (x → x \◦ a is antitone).
3 Models and Control of (min,+)-Linear Systems 3.1 Counter Functions Some idempotent semiring algebras enable to model DEDS which involve synchronization and delay phenomena. The behavior of such systems can be represented by discrete functions called “counter” functions. More precisely, a discrete variable x(t) is associated to an event labeled x and represents the occurrence number x at time t (the numbering conventionally beginning at 0). For negative values of t, these variables are defined as constant so they can be manipulated as mappings from to min . Thanks to these counter functions, the studied DEDS can be modeled on the idempotent semiring min by the following linear state representation:
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x(t) = Ax(t − 1) ⊕ Bu(t), y(t) = Cx(t),
(2)
where A ∈ min , B ∈ min and C ∈ min while n, p and q refer respectively to the state vector (x) size, the input vector (u) size and the output one (y). In the SISO3 case (p = 1 and q = 1), the state equation leads to the following input/output relation: D CAτ Bu(t − τ ). (3) y(t) = n×n
n×p
q×n
τ ≥0
CAτ B,
Moreover, setting h(τ ) = and defining the inf-convolution (or (min,+)convolution) as follows (see [9] and [5]), ∀ f , g ∈ min : ( f ∗ g)(t)
D
[ f (τ ) ⊗ g(t − τ )] = min [ f (τ ) + g(t − τ )], τ ≥0
τ ≥0
relation (3) can be rewritten as y(t) = (h ∗ u)(t), which is actually the transfer relation of the considered system, with h(t) the transfer function4. According to [1, Theorem 5.39] and [3], a (min,+)-linear system defined as (2) is necessarily such that h(t) is periodic and causal i.e.: ∃T0 , N, T ∈ | ∀t ≥ T0 , h(t + T ) = N ⊗ h(t) [periodicity],
h(t) = h(0) h(t) ≥ 0
for t < 0 for t ≥ 0
[causality].
(4)
(5)
Let us note that the set of nondecreasing5 mappings from to min endowed with the two inner operations ⊕ as pointwise addition and ∗ as inf-convolution is also an idempotent semiring denoted (min , ⊕, ∗) where ε and e are defined by: 0 for t < 0, (6) ∀t, ε : ε (t) → +∞ and e : e(t) → +∞ for t ≥ 0. In the MIMO6 case, the input/output relation becomes Y (t) = (H ∗ U)(t), where U ∈ (min ) p , Y ∈ (min )q and H ∈ (min )q×p is such that Hi j is periodic. The inf
t2 ⇒ h(t1 ) ≥ h(t2 ). Multiple Inputs Multiple Outputs.
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3.2 Precompensator Control A specific (min,+)-linear controller, called precompensator p, can be placed upstream of process h so that u(t) = (p ∗ v)(t) and where v is the controller input. In (min , ⊕, ∗), the output of the controlled system becomes y(t) = (h ∗ p ∗ v)(t). With this configuration, the controller p aims at slowing down the system input. Moreover, the residuation theory shows (see [8]) that there exists an optimal neutral precompensator given by: p(t) ˆ = sup{p(t) | (h ∗ p)(t) = h(t)} = (h \◦ h)(t), where the mapping x → a \◦ x is in that case the residual of the inf-convolution product. This optimal controller is said to be neutral since it lets the input/output behavior unchanged. Nevertheless, it delays the process input u as much as possible in order to avoid useless accumulations into h. The computation of pˆ requires thus the use of the residual of the inf-convolution product (see [9]): p(t) ˆ = (h \◦ h)(t) =
; τ ∈
[h(τ − t) \◦ h(τ )] = max [h(τ ) − h(τ − t)]. τ ∈
(7)
Remark 2. (Periodicity). If function h(t) is periodic, (h \◦ h)(t) is periodic too. Remark 3. (Argument of the maximum). Let us note that if h is periodic, there exists at least a τ0 (not necessarily unique) such that maxτ ∈ [h(τ ) − h(τ − t)] = h(τ0 ) − h(τ0 − t) and defined by:
τ0 ∈ argmax[h(τ ) − h(τ − t)].
(8)
τ ∈
4 Neutral Precompensator for Unknown Systems Usually, this optimal neutral precompensator is given for (min,+)-linear system the transfer function h of which is perfectly known. This section deals with the problem of finding such a precompensator when h presents some parametric uncertainties ◦ is the and belongs to an interval [ h , h ]. In such a case, we will see that pˆ = e ⊕ h\h greatest precompensator which is neutral for all systems, i.e. ∀h ∈ [ h , h ], h ∗ pˆ = h.
4.1 SISO Case
Proposition 1. Let [ h , h ] be an interval with h, h ∈ min : ∀hi ∈ [ h , h ], hi \◦ hi ! h \◦ h.
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◦ i! Proof. According to the left quotient isotony and antitony properties (1), hi \h ◦ i ! h\ ◦ h. h\h
Proposition 2. Let [ h , h ] be an interval with h, h ∈ min , two periodic and causal functions (see (4) and (5)): ∀ti > 0, ∃hi ∈ [ h , h ] such that (hi \◦ hi )(ti ) = (h \◦ h)(ti ). Proof. Thanks to the residual of the inf-convolution product definition > (see (7) and (8)): (h \◦ h)(ti ) = τ ∈ h(τ − ti ) \◦ h(τ ) = h(τi − ti ) \◦ h(τi ) with τi ∈ arg maxτ ∈ [h(τ ) − h(τ − ti )]. This τi leads to the hi following definition7: h(t), for t < τi , hi (t) (9) h(t), for t ≥ τi . On Fig. 1 is illustrated an example of the hi function for which ti = 1 and τi = 6. In ◦ = h(6) − h(5) = 4. As specified in remark 3, τi is that case (hi \◦ hi )(1) = (h\h)(1) not unique for ti = 1 and belongs to the set {3, 4, 6, 8, 10, . . .}.
Fig. 1 Example of a hi function where ti = 1 and τi = 6. The arrow represents the distance (= 4) between h and h for these values.
>
◦ i )(ti ) = τ ∈ hi (τ − ti ) \ ◦ hi (τ ). This latter exThen, equation (7) shows that: (hi \h pression can be factorized, since ∀τ , h(τ ) # h(τ ) and according to (1) we obtain:
; τ < τi
7
h(τ − ti ) \◦ h(τ ) !
; τ < τi
h(τ − ti ) \◦ h(τ ) ! h(τi − ti ) \◦ h(τi ),
It is important to recall that the order # of min is the opposite to the natural order ≥ of functions. Moreover, as illustrated in Fig. 1, hi (t) is still a nondecreasing function.
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h(τ − ti ) \◦ h(τ ) !
τi +ti ≤τ
;
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h(τ − ti ) \◦ h(τ ) ! h(τi − ti ) \◦ h(τi ).
τi +ti ≤τ
Moreover: h(τi − ti ) \◦ h(τi ) ∧
; τi 0, a subset of systems H ⊂ [ h , h ] (h\h)(t i ). So, thanks > ◦ i )(t) = (h \h)(t). ◦ exists such that hi ∈H (hi \h Moreover, for t = 0 and according > ◦ To conclude, ∀h ∈ [ h , h ], to remark 4, ∀t ≥ 0, hi ∈H (hi \◦ hi )(t) = e ⊕ (h\h)(t). > > ◦ ◦ ◦ h \h ! hi ∈H (hi \hi )(t) and finally e ⊕ h\h = h∈[ h , h ] h \◦ h.
Proposition 3 must be interpreted as follows: the precompensator pˆ = e ⊕ h\◦ h is the greatest precompensator which is neutral for all systems h ∈ [ h , h ] i.e. h ∗ pˆ = h.
4.2 MIMO Extension Proposition 3 given for all uncertain SISO systems belonging to an interval can be extended to MIMO systems. For an uncertain (min,+)-linear system H ∈ (min )q×p in an interval [ H , H ], the greatest neutral precompensator is now defined by ◦ where I is the identity matrix of (min ) p×p. Pˆ = I ⊕ H \H,
Proposition 4. Let ha , hb ∈ min : hb \◦ ha =
; ha ∈ [ ha , ha ] hb ∈ [ hb , hb ]
hb \◦ ha .
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Proof. Thanks to (1).
Proposition 5. Let [ H , H ] be a matrix interval with H, H ∈ (min )q×p and which represents the behavior of an uncertain p-input q-output system: I ⊕ H \◦ H =
;
H \◦ H.
H∈[ H , H ]
>
n ◦ ◦ k j. Proof. Thanks to [1, Equation (4.82)], ∀H ∈ [ H , H ], (H \H) i j = k=1 Hki \H >n ◦ ◦ Hki with Hki ∈ [ H ki , H ki ]. Thanks On the one hand, if>i = j, (H \H) H \ ii = k=1 ki >n ◦ ◦ ki )ii . On the other hand, to proposition 3, ( H∈[ H , H ] H \H) ii = k=1 (e ⊕ H ki \H >n ◦ ◦ for i = j, (H \H) = H \H with H ∈ [ H , H ki ], Hk j ∈ [ H k j , H k j ] ij kj ki ki k=1 ki > >n ◦ ◦ k j. and thanks to proposition 4, ( H∈[ H , H ] H \H)i j = k=1 (Hki \◦ Hk j )i j = H ki \H > Finally, ∀i, j, ( H∈[ H , H ] H \◦ H)i j = (I ⊕ H \◦ H)i j .
4.3 Example of Neutral Precompensator for MIMO Systems
A MIMO system with H ∈ (min )1×2 (two inputs, one output) the transfer function of which belongs to an interval ([ H 11 , H 11 ] [ H 12 , H 12 ]) is considered. The bounds are the periodic and causal functions given in table 1. Table 1 Bounds of H
t
2
3
4
t≥5
0
1
H 11 0
3
H 11 0
0
H 12 0
2
t ≥ 2, H 12 (t) = 2 ⊗ H 12 (t − 2)
H 12 0
0
t ≥ 2, H 12 (t) = 1 ⊗ H 12 (t − 1)
t ≥ 2, H 11 (t) = 3 ⊗ H 11 (t − 2) 0
1
2
t ≥ 5, H 11 (t) = 3 ⊗ H 11 (t − 2)
As previously said, (min,+)-linear systems are always characterized by periodic functions ([1, Theorem 5.39]) and according to remark 2, residuals of the infconvolution are periodic functions too. Thus, for this system and thanks to propo sition 5, the computation of the neutral precompensator Pˆ ∈ (min )2×2 given by ◦ Pˆ = I ⊕ H \H, is described in table 2. Let us note that for this example, Pˆ21 = ε with ε defined by (6).
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Table 2 Neutral precompensator Pˆ
t
0
1
2
t≥3
Pˆ11 Pˆ12
0
5
7
t ≥ 3, Pˆ11 (t) = 3 ⊗ P11 (t − 2) t ≥ 1, Pˆ12 (t) = 1 ⊗ P12 (t − 1)
Pˆ22
0
2 3
t ≥ 2, Pˆ22 (t) = 1 ⊗ P22 (t − 1)
5 Conclusion This paper has introduced the control of unknown (min,+)-linear systems belonging to an interval the bounds of which are known. A neutral precompensator controller placed upstream of these systems has been given without changing the input/output behavior while delaying the process input as much as possible. This precompensator is enabled both for SISO and MIMO systems and an example has been given in order to illustrate these propositions.
References 1. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronisation and linearity: an algebra for discrete event systems. Wiley and sons, Chichester (1992) 2. Cottenceau, B., Hardouin, L., Boimond, J.L., Ferrier, J.L.: Model reference control for timed event graphs in dioids. Automatica 37(9), 1451–1458 (2001) 3. Gaubert, S.: Th´eorie des syst`emes lin´eaires dans les dio¨ıdes. PhD Thesis. Ecole Nationale Sup´erieure des Mines de Paris (1992) 4. Heidergott, B., Olsder, G.J., Woude, J.: Max plus at work, modeling and analysis of synchronized systems. In: A course on max-plus algebra and its applications. Princeton University Press, Princeton (2006) 5. Le Boudec, J.Y., Thiran, P.: Network calculus: a theory of deterministic queuing systems for the internet. Springer, Heidelberg (2001) 6. Lhommeau, M., Hardouin, L., Cottenceau, B.: Optimal control for (max,+)-linear systems in the presence of disturbances. In: International Symposium on Positive Systems: Theory and Applications, Roma. POSTA 2003 (2003) 7. Lhommeau, M., Hardouin, L., Ferrier, J.L., Ouerghi, I.: Interval analysis in dioid: application to robust open-loop control for timed event graphs. In: 44th IEEE Conference on Decision and Control and European Control Conference, Seville. CDC-ECC 2005, pp. 7744–7749 (2005) 8. Maia, C.A., Hardouin, L., Santos-Mendes, R., Cottenceau, B.: Optimal closed-loop control of timed event graphs in dioids. IEEE Transactions on Automatic Control 48(12), 2284–2287 (2003) 9. Max Plus Second order theory of min-linear systems and its application to discrete event systems. In: Proceedings of the 30th IEEE Conference on Decision and Control, Brighton. CDC 1991(1991)
On a Class of Stochastic Models of Cell Biology: Periodicity and Controllability Ivo Marek
Abstract. This contribution is a natural continuation of a series of papers devoted to analysis of models utilized by specialists in Cell Biology around E. Bohl and W. Boos. Our novelty may be seen in enriching the models in direction of controllability in the spirit of biology engineering. Besides the standard properties of the models such as existence of appropriate solutions and their uniqueness the following issues are of interest: Asymptotic behavior (e.g. steady states and pseudo-steady states), controllability and also special features such as various types of symmetries, periodicity etc. Our aim is focused on periodicity of solutions of models whose state objects share the properties of concentrations, i.e. probabilities.
1 Definitions and Notation Let X be a generally infinite dimensional partially ordered Banach space, generated by a closed normal cone K e.g. space X = L p (Ω ) is consisting of classes of functions defined on (0, +∞) whose representatives possess convergent integral Ω
|x(t)| p dt < +∞, x ∈ {x} ⊂ L p (Ω ), 1 ≤ p < +∞.
In applications the role of the cone K is usually played by some version of the collection of classes of functions whose representatives assume nonnegative real values almost everywhere, i.e. K = L+p (Ω ) = {{x} ⊂ X : x(t) ≥ 0 a.e.within [0, +∞)} . Ivo Marek Czech University of Technology, School of Civil Engineering, Thakurova 7, 166 29 Praha 6, Czech Republic, e-mail: [email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 359–367. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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Assume X is a Banach space. Then symbol X denotes the dual of X , i.e. the space of all bounded linear functionals mapping X into reals R. Space X is equipped with the norm / 0 x X = sup |x (x)| : xX ≤ 1 . If the norm . is defined via an inner product [., .] the space X is then a Hilbert space and the norm of x ∈ X reads xX = [x, x]1/2 in this case. Symbol ρ (H) is exploited for denoting the spectral radius of H, i.e. ρ (H) = Max{|λ | : λ ∈ σ (H)}, where σ (H) is the spectrum of linear operator H.
2 Periodicity and Controllability Below we formulate a general problem that can be frequently met when modeling concrete situations of various research experiments, here we focus to research in Biology. We are going to show that the methods and results obtained in [4], [5], [3] are suitable tools to solve problems just mentioned. In particular, the properties concerned with asymptotic behavior of the required solutions will appear as decisive for establishing a theory adequate for the experiments studied. Problem 1. Let u ∈ U, where U is the Banach space of classes of scalar functions on (0, +∞) equipped with the L1 -norm uU =
+∞ 0
|u(t)|dt.
Let L {X } denote the space of bounded linear operators mapping X into X and G = G(u) : U → L {X } be an operator function each element of which is a bounded linear operator acting on X and G(x) = H(x) − ρ (H(x))I, H(x)K ⊂ K . Further, let B be the densely defined infinitesimal generator of a semigroup of linear operators of class C0 such that each of the operators of the semigroup T (t; B) leaves invariant the cone K . Furthermore, there is a vector xˆ ∈ K such that relations xˆ (x) > 0 hold for all 0 = x ∈ K and B xˆ = 0 = [G(x)] xˆ , where B denotes the dual of B and similarly G(u) is the dual with respect to G(u). Find a solution x = x(t, u) ∈ X such that dx(t) = Bx(t) + G(x(t))x(t)u(t), x(0) = x0 , dt where u ∈ U is assumed to satisfy umin ≤ u(t) ≤ umax , umin , umax ∈ R + .
(1)
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Remark 1. There is a variety of possible applications behind the fact that Problem 1 as formulated above is free of nonhomogeneous term. However, its possible presence produces no difficulties in the investigation. Problem 1 formulated above forms a basis for applications. Some results have been published already, see [3], [4] showing existence, uniqueness and asymptotic properties of Bohl’s models on an abstract level. A realization of a version of the model based on a system of ODE’s devoted to multi-level time aspects (MichaelisMenten kinetics) is presented in [5]. A natural prolongation of the results contained in the mentioned papers is presented in this contribution. It concerns existence of periodic solutions to Problem 1 and some properties of controllability with piecewise constant input. As the applications in Biology concerns, the earlier papers are related to Cell Biology and the present one to some issues related to microalgal ˇ Pap´acˇ ek [13] and [14], see also [7]. growth in the spirit of Thesis of S.
2.1 Periodic Solutions of Models based on ODE’s With no loss of generality, let t0 = 0. Further we let ⎧ ⎨ ua for 0 ≤ t < t1 u(t) = ⎩ ub for t1 ≤ t < t2 ,
(2)
where ua and ub are positive real numbers, and u(s) = u(t), for any real positive s such that s − t = k[t1 + t2 ],
(3)
where k ranges within the set of positive integers N = {1, 2, ...}. A piecewise constant function u defined in (2) periodically prolonged onto [0, +∞) is called intermittent control. Assume that X = R n equipped with some norm . and let A(t) = A(u(t)) = B + C(u(t)),
(4)
where B is a constant matrix and C = C(u) = (c jk ), c jk = c jk (u(t)), j, k = 1, ..., N,
(5)
with u defined in (3). Moreover, let B and C(u) be negatives of generally singular M-matrices [1, p.133]. The following collection of matrices will appear useful in approximating the matrix of system (4) under periodicity hypothesis. Let ε > 0 be arbitrary. Define
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Cε (t) =
⎧ C(ua ) − [C(ub) − C(ua)] t−ε ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ C(ua )
0≤t ≤ε for t ∈ [ε ,t1 )
⎪ 1 +ε ) ⎪ C(ub ) + tt−(t [C(ua ) − C(ub )] for t ∈ [t1 ,t1 + ε ] ⎪ ⎪ 1 −(t1 −ε ) ⎪ ⎪ ⎪ ⎪ ⎩ for t ∈ [t1 + ε ,t1 + t2 ) C(ub )
1 as above in defining C(t). and periodically for any t ∈ R+ Since t v(t) = exp{Bt} + exp{B(t − s)}C(u(s))v(s)ds 0
and similarly, vε (t) = exp{Bt} +
t 0
exp{B(t − s)}Cε (u(s))vε (s)ds,
we get estimate Max {v(s) − vε (s) : 0 ≤ s ≤ t1 + t2 } -9 :
≤ - 0+ε + tt11 +ε exp{B(t − s)} [C(u(s))v(s) − Cε (uε (s))vε (s)] ds≤ κε , where κ is a constant independent of ε and t ∈ [0, +∞). According to this relation, v is the uniform limit of sequence {vε } on any finite interval of the semiaxis [0, +∞). Our next aim is to prove Theorem 1. Under the hypotheses of this section Problem 1 possess a solution periodic in time variable. The number of linear independent periodic solutions is bounded above by the dimension of the kernel of the matrix system A(u) = B + C(u). Proof. We consider first the situation when matrix Cε is in place of matrix C(t). Since, according to [5], there exists a constant κ < +∞ such that expA(u(t)) ≤ κ we see that any fundamental matrix Φ of the investigated system satisfies Φ (t) ≤ κ . From the well known fact [8, pp.90-93] that fundamental matrix of a homogeneous periodic system whose matrix A = A(t) is continuous can be represented in form
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Φ (t) = P(t)exp{tR}, P(t + ν ) = P(t), ν = t1 + t2 , we deduce that spectrum of the approximation Rε , where Φε (t) = Pε (t)Rε denotes the fundamental matrix of Cε constructed in this section, has the following structure:
σ (Rε ) = {0} ∪ {ρ1 , ..., ρs } , R ρ j < 0, and due to the Markov property of our model, the block corresponding to eigenvalue zero is diagonalizable. Thus, taking one of the eigenvectors of Aε corresponding to eigenvalue 0, and denoting it by v0 , we check easily that vε satisfying vε (t) = exp{t}v0 +
t 0
exp{B(t − τ )}Cε (u(τ ))vε (s)dτ
is, by construction, periodic and thus, it represents a periodic solution to the Cauchy problem d X(t) = Aε (u(t))X(t), X(0) = v0 . dt 1 , where xˆ = eT = (1, ..., 1) ∈ R n , imply Relations BT xˆ = Cε (t)T xˆ = 0 for all t ∈ R+ that t v(t) = lim vε (t) = exp{Bt}v0 + exp{B(t − τ )}C(u(τ ))v(s)dτ
ε →0
0
is a solution to
d X(t) = A(u(t))X(t), X(0) = v0 . dt Moreover, since v0 is (t1 + t2 )-periodic, so is v. Since the part of the theorem concerned with the number of periodic solutions is obvious, the proof is complete. It is interesting that an analog of the previous theorem remains valid for a broader class of problems; actually, we have Theorem 2. In the following, let . denote some norm in R N . Assume a system of operators A(t) is defined via matrices B and G(t) possessing the following properties: (i) B is a constant N × N minus M-matrix; N , u ∈ U , where U is a set of scalar real valued functions (ii) G = G(x; u), x ∈ R+ on the reals, is a minus M-matrix;
(iii) There is a positive constant κ < +∞ independent of x ∈ R N and u ∈ U such that G(x; u) ≤ κ (iv) There is a positive constant quantity ν < +∞ independent of x ∈ R N and u ∈ U such that G(x; u) − G(y; u) ≤ ν x − y.
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Besides the hypotheses (i) − (iv) let u = u(t), u ∈ U be a periodic within the time interval [0, +∞) with period τ . Then there exist τ -periodic solutions to Problem 1. The number of linearly independent such periodic solutions coincides with the number of linearly independent N: solutions to the following nonlinear equation belonging to R+ Bw + G(w, u)w = 0.
(6)
Proof. First, according to [5], there exist solutions to (6). Let w be one of such solutions Then we can construct matrix G(w, u(t)). Since w is independent of time we see that this matrix and hence also Problem 1 with this data is periodic and satisfies conditions of Theorem 1. It follows that the solution obtained according to the Theorem just mentioned is periodic. Since the proof of the remaining part of Theorem 2 is obvious and is thus omitted. The proof is complete.
2.2 Models based on PDE’s This subsection is devoted to investigations generalizing the previous results and methods in order to cover by mathematical models situations without limitations to both sides of groups of researchers, the specialists in experimental as well as in theoretical area. Our approach gives freedom to both sides to consider very general situations on the one hand and broadens our experience for making realistic visions for a progress in the near future. Typical in this direction is appearance of a semigroup of operators in place of the exponential of a matrix. We are going to keep the generality in realistic bounds however by assuming that the infinite-dimensional objects can be relatively accurately approximated by their finite-dimensional counterparts. The class of problems we are going to consider in this subsection is characterized as follows: (α ) X is a Banach space equipped with norm . and is generated by a closed normal cone K [11]; (β ) Operator B is densely defined on D(B) ⊂ X and generates a semigroup of bounded linear operators of class C0 [10]; (γ ) each operator of the class of operators forming the semigroup mentioned in (β ) leaves invariant the cone K [11]; (δ ) For every x ∈ X linear operator G(x) is such that uniformly with respect to K G(x) ≤ κ < +∞; (ε ) Relations G(x) = H(x) − ρ (H(x))I, H(x)K ⊂ K hold for every x ∈ K ; (η ) There is a constant ν , 0 ≤ ν < +∞ such that relations G(x) − G(y) ≤ ν x − y hold for any x, y ∈ K .
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Without proof we formulate an abstract result whose validity is based on our results established by solving some concrete problems, some of them presented in the present contribution. The proofs exploit the ”finite-dimensional” techniques and convergence properties of some suitable approximations in order to establish appropriate generalizations in abstract infinite-dimensional spaces. Two examples of major interest are shown in the Appendix. Theorem 3. Hypotheses (α )-(η ) and periodicity of data imply that the initial value problem introduced as Problem 1 possesses solutions periodic in the time variable. A crucial step in proving Theorem 3 consists of showing that there is a suitable discretization of data appearing in Problem 1 such that the corresponding approximate problems are determined by data for which Theorem 2 applies. For operators exposed in the Appendix the step just mentioned is based on the discrete maximum principle valid for piecewise linear finite element discretizations of the spatial variables and convergence of the corresponding approximate solutions to the exact ones. This is the case of Example 1 (see [9, pp.285-286]). For the case of Example 2 a method of V.S. Vladimirov is appropriate [12, pp.73-76].
3 Concluding Remarks Our stochastic models introduced for analyzing actual problems of Biology and Chemistry have been further generalized and completed by enriching further structures such as more general partial order in order to examine formulation of more complex mathematical problems. Two areas of results are established: Existence of periodic solutions to some nonlinear boundary value problems in direction of theory belongs to the first category and an explanation of certain type of experiments concerned with growth of microalgae via intermittent control processes as application belongs to the second category.
4 Appendix Example 1. Diffusion operator Let Ω ⊂ R d , d ≤ 3, be a bounded domain with the Lifschitz boundary ∂ Ω . Diffusion operator L is defined as the following elliptic differential operator by formulas
∂ k=1 ∂ xk d
Lw(r) ≡ − ∑
∂w ∂ w(r) = 0 for r ∈ ∂ Ω . D(r) , r ∈ Ω and ∂ xk ∂n
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Example 2. Transport operator Let Ω ⊂ R d , d ≤ 3, be a bounded convex domain n denote the direction of the outer with smooth /boundary ∂ Ω . Let / 00 normal. Further, let V = [0, +∞) × ω , ω = v ∈ R 3 : |v|2 = v2x + v2y + v2z = 1 be the velocity space. By w we denote a density of particles. Transport operator L is then defined by relations Lw ≡ vgrad w + vΣ (r, v)w + μ w, v = |v|, r ∈ Ω , and w(r, v) = 0 for r ∈ ∂ Ω and v ∈ V whenever (n, v) < 0. The data in the definition of the transport operator is the total cross-section Σ (r, v) and a quantity possessing various meanings depending on the matter of research. In our treatment, μ makes the transport operator to satisfy our basic requirement: The semigroup of operators generated by corresponding infinitesimal generators should leave invariant some cone generating the appropriate space and preserve the total concentrations. Acknowledgements. Supported by grant from the Grant Agency of the Czech Republic under contract Nr. 201/09/1544 and by the grant from the Ministry of Education, Sports and Youth under contract Nr. MSM 6840770010.
References 1. Berman, A., Plemmons, R.: Non-negative Matrices in the Mathematical Sciences. Academic Press, New York (1979) 2. Bohl, E., Boos, W.: Quantitative analysis of binding protein-mediated ABC transport system. J. Theor. Biology 186, 65–74 (1997) 3. Bohl, E., Marek, I.: A stability theorem for a class of linear evolution problems. Integral equations Operator Theory 34, 251–269 (1999) 4. Bohl, E., Marek, I.: Existence and uniquness results for nonlinear cooperative systems. OperatorTheory: Advances and Applications 130, 153–170 (2001) 5. Bohl, E., Marek, I.: Input-output systems in Biology nad Chemistry and a class of mathematical models characterizing them. Appl. Math. 50, 219–245 (2005) 6. Brenner, P., Thom´ee, V.: On rational approximations of semigroups. SIAM J. Numer. Anal. 16, 683–694 (1979) ˇ 7. Celikovsk´ y, S.: On the Lipschitzian dependence of trajectories of bilinear systems of multi-input time dependent bilinear systems of control. Problems Control Information Theory 17, 231–238 (1988) 8. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential equations. McGrawHill Book Company, New York (1955); Russian translation Izdat. Innostrannoi Literatury, Moscow (1958) 9. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Element Methods. Springer, Heidelberg (2005) 10. Hille, E., Phillips, R.S.: Functional Analysis and Semigroups. In: Amer. Math. Socitey Coll. Publ., vol. XXXI. Third printing of Revised edition Providence, Rhode Island (1968) 11. Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Uspekhi mat. nauk III Nr. 1, 3–95 (1948) (in Russian); Eenglish translation in AMS Translations 26 (1950)
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12. Marchouk, G.I.: Methods of Nuclear Reactor Computation, Gosatomizdat, Moscow (1961) (in Russian) ˇ Photobioreactors for Cultivation of Microalgae Under Strong Irradiances 13. Pap´acˇ ek, S.: Modelling: Simulation and Design. Ph.D. Thesis, Technical University Liberec (2005) ˇ Celikovsk´ ˇ ˇ 14. Pap´acˇ ek, S., y, S., Stys, D., Ruiz-Le´on, J.: Bilinear system as a modelling framework for analysis of microalgal growth. Kybernetika 43, 1–20 (2007) ˇ y, 15. Sultangazin, U.M., Smelov, V.V., Akishev, A.S., Sabekov, A., Marek, I., M´ıka, S., Zitn´ K.: Some Mathematical Problems of Kinetic Tranport Theory, Nauka, Alma-Ata (1986) (in Russian)
Implementation of 2D Strongly Autonomous Behaviors by Full and Partial Interconnections Diego Napp Avelli and Paula Rocha
Abstract. In this paper we study linear discrete two-dimensional systems in the behavioral context where control is viewed as interconnection. Within the behavioral framework a natural concept of interconnection has been introduced by J.C.Willems, called regular interconnection. We investigate regular interconnections that yield finite dimensional behaviors, and prove that when a finite dimensional behavior can be achieved from a given behavior B by regular full/partial interconnection then the controllable part/manifest part of B is rectifiable.
1 Introduction As is well known, the central idea in the behavioral approach to control is the one of interconnection. This consists in the interconnection of a given behavior to be controlled B (the plant) with a suitable behavior (the controller), in order to obtain a desired behavior Bd . If this is possible, we say that Bd is implementable from B. In this context, there are two main situations to be considered: either all the system variables are available for control (i.e., are control variables) or only some of the variables are control variables. In order to distinguish these two cases, we respectively refer to full and partial control or interconnection. In this paper we focus on a particular kind of interconnection that is called regular interconnection. In such interconnection, the restrictions imposed on the plant by the Diego Napp Avelli Research Unit of Mathematics and Applications, Department of Mathematics, University of Aveiro, e-mail: [email protected] Paula Rocha Department of Electrical and Computer Engineering Faculty of Engineering University of Oporto, e-mail: [email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 369–378. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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controller are independent of the restrictions already present in the plant, as happens, for instance, in a feedback interconnection (see [7, 15]). More concretely, we are interested in studying regular interconnections that yield finite dimensional behaviors, i.e., we wish to characterize the behaviors from which a finite dimensional behavior is implementable by regular interconnection. This can be seen as a relaxation of the control objective of implementing the zero behavior by regular interconnection from a given behavior B, a problem that has already been addressed in [10, 15]. In this sense, regular implementation of a finite dimensional behavior can be regarded as almost regular implementation of zero, see [4, 5]. Using a notion of stability defined with respect to a specified stability region by adapting the ideas in [6] to the discrete case, it was recently proven, in [9], that the stable behaviors considered there have the property of being finite dimensional 2 linear subspaces of ( q ) . Thus, the possibility of stabilizing a behavior B is strictly connected with the regular implementation of a finite dimensional behavior from B. In the context of full interconnection, a complete characterization of the stabilization property was given in [9] under the assumption that the controllable part of the 2 given behavior B, denoted by Bc , is rectifiable, i.e., is a direct summand of ( q ) . This is a very strong property and allows to derive several results that are in general only valid for the one dimensional case (1D), such as, for instance the existence of a decomposition of the behavior into the direct sum of its controllable part and an autonomous part. However, in this paper we prove that if a finite dimensional behavior is implementable by regular interconnection from a given behavior B, then Bc is rectifiable. As a consequence of this result we conclude that the assumption about the rectifiability of Bc , used in [9] in order to obtain several results on stabilization, is indeed not restrictive since it is a necessary condition for stabilization. In contrast with the full interconnection case, we also treat the partial interconnection case and show that, under one condition, analog results can be obtained for this new situation. We begin by introducing some necessary background from the field of 2D discrete behavioral theory. Section 3 is devoted to an exposition of regular interconnection and finite dimensional behaviors. In Section 4 we move from the full interconnection context to the partial one.
2 Preliminaries In order to state more precisely the questions to be considered, we introduce some preliminary notions and results. We consider 2D behaviors B defined over 2 that can be described by a set of linear partial difference equations, i.e., B = ker R(σ , σ −1 ) := {w ∈ U | R(σ , σ −1 )w ≡ 0},
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where U is the trajectory universe, here taken to be ( q ) , σ = (σ1 , σ2 ), σ −1 = (σ1−1 , σ2−1 ), the σi ’s are the elementary 2D shift operators (defined by σi w(k) = w(k + ei ), for k ∈ 2 , where ei is the ith element of the canonical basis of 2 ) and R(s, s−1 ) is an 2D Laurent-polynomial matrix known as representation of B. Instead of characterizing B by means of a representation matrix R, it is also possible to characterize it by means of its orthogonal module Mod(B), which consists of all the 2D Laurent-polynomial rows r(s, s−1 ) ∈ 1×q [s, s−1 ] such that B ⊂ ker r(σ , σ −1 ), and can be shown to coincide with the [s, s−1 ]-module RM(R) generated by the rows of R, i.e., Mod(B) = RM(R(s, s−1 )). 2
Theorem 1. [15, pag.1074] Let B1 and B2 be two 2D behaviors. Then, B1 + B2 and B1 ∩ B2 are also 2D behaviors. Moreover, one has that Mod(B1 + B2 ) = Mod(B1 ) ∩ Mod(B2 ) and Mod(B1 ∩ B2 ) = Mod(B1 ) + Mod(B2 ). Definition 1. A behavior B ⊂ ( q ) is said to be controllable if for all w1 , w2 ∈ B there exits δ > 0 such that for all subsets U1 , U2 ⊂ 2 with d(U1 ,U2 ) > δ , there exists a w ∈ B such that w |U1 = w1 |U1 and w |U2 = w2 |U2 . 2
On the other hand, we say that a behavior is autonomous if it has no free variables (or inputs). B = ker R(σ , σ −1 ) is autonomous if and only if R(s, s−1 ) has full column rank (over [s, s−1 ]), [13]. In the 1D case, all autonomous behaviors are finite-dimensional vector spaces. For general multidimensional variable behaviors this is no longer true. In fact, an autonomous multidimensional behavior that is finite-dimensional is called strongly autonomous in [6]. As also shown in [13], every 2D behavior B can be decomposed into a sum B = Bc + Ba , where Bc is the controllable part of B (defined as the largest controllable sub-behavior of B) and Ba is a (non-unique) autonomous sub-behavior said to be an autonomous part of B. If the controllable-autonomous decomposition happens to be a direct sum decomposition, i.e., if B = Bc ⊕ Ba , we say that the autonomous part of Ba is an autonomous direct summand of B. An interesting case is when the controllable part Bc is rectifiable. A 2D be2 havior B = ker R(σ , σ −1 ) ⊂ ( q ) is said to be rectifiable if there exists an invertible operator U(σ , σ −1 ), where U(s, s−1 ) is an 2D Laurent-polynomial matrix, such that U(σ , σ −1 )(B) = ker[Il 0], where Il is the l × l identity matrix, for some l ∈ {1, . . . , q}. It has been shown that a behavior B is rectifiable if and only if B is 2 direct summand of ( q ) , (see [10, Lemma 2.12] and [14, Th. 9 and Th. 10, page 819]). When a rectifying operator exists, it is possible to take advantage of the simplified form of the rectified behaviors in order to derive various results. In particular, one can obtain the next proposition, see [9, Prop.1]. Proposition 1. Let B = ker R(σ , σ −1 ) ⊂ ( q ) be a behavior with rectifiable controllable part Bc and U(σ , σ −1 ) be a corresponding rectifying operator such that U(σ , σ −1 )(Bc ) = ker[Il 0]. Then the following are equivalent: 2
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1. B = Bc ⊕ Ba ,
P 0 U , with P(s, s−1 ) such that RU −1 = [P 0] and X(s, s−1 ) 2. Ba = ker X Iq−l an arbitrary Laurent-polynomial matrix of suitable size.
Since the behaviors Ba of Proposition 1 always exist and are autonomous, we have that the previous result shows that every behavior with rectifiable controllable part has autonomous direct summands and, moreover, provides a parametrization for all such summands. Definition 2. A behavior B is said to be regular if Mod(B) is free (as a [s, s−1 ]module), or equivalently if there exists a polynomial matrix R of full row rank such that B = ker R.
3 Control by Regular Interconnections Given two behaviors B1 and B2 their interconnection is defined as the intersection B1 ∩ B2 . This interconnection is said to be regular if Mod(B1 ) ∩ Mod(B2 ) = {0} 2 or equivalently if B1 + B2 = ( q ) , see [10, Lemma 3, pag 115]. If the interconnection of B1 and B2 is regular then we denote it as B1 ∩reg B2 . In a regular interconnection, the controller imposes restrictions which are not already present in the plant. In this sense a feedback controller is a simple example of a regular interconnection where the controller imposes restrictions only on the plant input, which in the plant is unrestricted [7]. This notion of regularity of an interconnection is independent from the concept of a regular behavior. Based on the notion of behavior interconnection it is possible to formulate a control problem in set theoretic terms. Indeed, if B is the behavior of the system to be controlled (the plant) and C is the set of all signals compatible with the additional restrictions to be imposed on w, i.e., the controller, then the resulting controlled behavior is given by the interconnection B ∩ C of the behaviors B and C . Thus, in the behavioral setting, a control problem consists in, given a desired controlled behavior Bd , finding a controller C such that its interconnection with the plant behavior B results in Bd . In case this interconnection is regular, the desired behavior Bd is said to be achievable or implementable by regular interconnection. The following necessary condition for implementation by regular interconnection has been derived in [10, Th. 4.5, pag 124]. Theorem 2. Let B and Bd be two behaviors. Then if Bd is implementable by regular interconnection from B then B = Bc + Bd . Using this result it is possible to show the next useful Lemma.
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Lemma 1. Let B and K be two 2D behaviors. If the interconnection of B and K is regular then also is the interconnection between Bc and K . Proof. Let B ∩ K = Bd with regular interconnection, i.e. Mod(B)⊕ Mod (K ) = Mod(Bd ). Using Theorem 2 we have that B = Bc + Bd or equivalently Mod(B) = Mod(Bc ) ∩ Mod(Bd ) = Mod(Bc ) ∩ (Mod(B) ⊕ Mod(K )). Using that Mod(B) ⊂ Mod(Bc ) one easily show that Mod(Bc ) ∩ (Mod(B) ⊕ Mod(K )) = (Mod(Bc ) ∩ Mod(K )) ⊕ Mod(B). Since Mod(B) ∩ Mod(K ) = {0} we have that Mod(Bc ) ∩ Mod(K ) = {0}. Lemma 1 shows that the controllable part of a behavior plays an important role in the context of regular interconnections. Indeed, a controller which does not interconnect with Bc in a regular way, can not interconnect with B regularly. The next results are crucial for the study of the regular implementation of finite dimensional behaviors. Lemma 2. (see [4, Th.12]) Let B be a behavior. Then there exists a unique regular behavior B+ such that B/B+ is finite dimensional (as a vector space over ), i.e., there exists a unique (up to isomorphism) free [s, s−1 ]-module, denoted by Mod(B)+ , which is contained in 1×q [s, s−1 ] and such that Mod(B)+ /Mod(B) has finite dimension (as a vector space over ). Note that Mod(B)+ is the smallest free module containing Mod(B) and its computation can be effectively implemented, see [4]. Lemma 3. (see [4, Cor.23]) Let B be a behavior and Bd ⊂ B be a sub-behavior. Then there exists a controller C such that (B ∩reg C )/Bd has finite dimension (as a vector space) if and only if Mod(B)+ is direct summand of Mod(Bd )+ . The previous lemma has a clear system theoretical interpretation. It can be seen as the almost implementation of Bd since the controlled behavior and the desired one ’differ’ just in a finite dimensional behavior which in the nD context is considered “small”. The idea that two behaviors are “almost” equal if they differ by finite dimensional behaviors has been considered in several recent papers [2, 4, 5]. Definition 3. Let B be a behavior and Bd ⊂ B be a sub-behavior. If there exists a controller C such that (B ∩reg C )/Bd has finite dimension (as a vector space) then we say that Bd is regularly almost implementable.
Theorem 3. Let B ⊂ ( q ) be a behavior. If there exists a controller behavior C such that B f d = B ∩reg C is finite dimensional (strongly autonomous) then Bc is rectifiable. 2
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Proof. Applying Lemma 3 for Bd = 0 one obtains that B ∩ C = B f d is finite dimensional if and only if Mod(B)+ is direct summand of Mod(0)+ = Mod(0) = 1×q [s, s−1 ]. Define B+ as in Lemma 2. Using Theorem 1 one can obtain that B+ 2 is direct summand of ( q ) . Thus, B+ is a rectifiable behavior and therefore also controllable. We have that for the 2D case Mod(Bc ) is free (see [6, Corollary 4, page 399]) and since Mod(B)+ is the smallest free module containing Mod(B) we have that Bc ⊂ B+ ⊂ B. Further, Bc is, by definition, the largest controllable sub-behavior which implies B+ ⊂ Bc and therefore Bc = B+ is rectifiable. Remark 1. According to [10, 15], rectifiability is equivalent to the possibility of obtaining the zero behavior by regular interconnection. Hence, the possibility of obtaining a finite dimensional behavior by regular interconnection from B can be regarded as almost regular interconnection to zero, since it represents the implementation of the zero behavior up to a finite dimensional one. The following theorem extends the results obtained in [10, Lemma 3.5., pag.117] and [15, Cor.5.2,pag.1083] on the characterization of regular implementation of the zero behavior, and also the related results on the dual problem of decomposition of behaviors obtained in [2]. Theorem 4. Let B be a behavior. Then a finite dimensional behavior is regularly implementable from B if and only if all sub-behaviors B ⊂ B are almost implementable. Proof. (⇒): Let K be a behavior such that B ∩reg K has finite dimension. We apply Lemma 3 to obtain that Mod(B)+ ⊕ Mod(K )+ = 1×q [s, s−1 ]. Further, B ⊂ B is almost regular implementable if and only if Mod(B)+ is direct summand of Mod(B)+ , again by Lemma 3. Now it is easy to check that Mod(K )+ ∩ Mod(B)+ is the direct complement of Mod(B)+ in Mod(B)+ , i.e., Mod(B)+ ⊕(Mod(K )+ ∩ Mod(B)+ ) = Mod(B)+ . The implication Mod(B)+ ⊕ (Mod(K )+ ∩Mod(B)+ ) ⊂ Mod(B)+ is trivial. For the other inclusion, take m ∈ Mod(B)+ . Obviously m = a + b for some a ∈ Mod(B)+ and b ∈ Mod(K )+ . Since a ∈ Mod(B)+ ⊂ Mod(B)+ one has that b = m − a ∈ Mod(B)+ and therefore m ∈ Mod(B)+ ⊕ (Mod(K )+ ∩ Mod(B)+ ). (⇐): Obvious since zero is almost regular implementable.
4 Implementation of Finite Dimensional Behaviors by Partial Regular Interconnections In contrast to previous sections where the interconnections were considered for the case that all system variables are available for interconnection (called full interconnections), we consider, in this section, interconnections in the generality that we are
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allowed to use only some of the system variables for the purpose of interconnection (called partial interconnections), see [1, 3, 8, 11, 12]. The reference to partial and full is sometimes dropped when it is clear from the context which is the case under consideration. The variables whose trajectories we intend to shape, denoted by w, can be controlled through a set of control variables c, over which we can ’attach’ a controller. These are the variables, that can be measured and/or actuated upon. Hence, suppose we have a system behavior, denoted by B(w,c) , with two types of variables, the variable to be controlled w and the variable c through which the system can be interconnected to a controller behavior. To interconnect the behavior to a controller means requiring that the c trajectories in the behavior are also elements of the controller behavior. The space of w trajectories in B(w,c) is called the manifest behavior and the space of w trajectories in the interconnection of the behavior and controller is called the manifest controlled behavior. A given (’desired’) behavior is called regularly implementable by partial interconnection (through c) if it can be obtained as manifest controlled behavior through a regular interconnection. It is immediately apparent that restricting oneself to using only the control variables for interconnection is more involved. In this section we prove, that, under a certain condition, if a finite dimensional behavior is regularly implementable from a given behavior B(w,c) , then the controllable part of the corresponding manifest behavior Bw must be rectifiable. In order to make the notion of partial control more precise we introduce the following notation. If the variables are partitioned as (w, c), we assume that the kernel representation of B(w,c) is partitioned accordingly as B(w,c) = ker(R M). On the other hand, a controller behavior Cc = ker K, with variable c, can be thought as a behavior C(w,c) with extended variable (w, c) and kernel representation ker(0 K). For notation convenience we denote C := C(w,c) . We say that the partial interconnection of B(w,c) and Cc is regular if the full interconnection of B(w,c) and C is regular. We define the w-behavior of B(w,c) as Bw := Πw B(w,c) := {w | ∃ c such that (w, c) ∈ B(w,c) }. This is the the manifest behavior and can be interpreted as the ’to be controlled’ behavior, the behavior we are interested in, see [12]. Using the fundamental principle of Ehrenpreis and Palamodov it can be shown that Bw is again a linear nD behavior. Indeed, if B(w,c) = ker(R M), then a kernel representation of Bw is constructed as follows: take a minimal left annihilator (MLA) F of M, i.e., a polynomial matrix F such that ker F = im M. Then Bw = ker(FR). In an analogous way, we proceed with Bc . Definition 4. Let B(w,c) = ker(R M) be a behavior then we define the hidden behavior H := {(w, c) | (0, c) ∈ B(w,c) }. Clearly H = ker(0 M). The following remark collects some known results in this area, for more information see [8, 11]. Remark 2. Let B(w,c) = ker(R M) be a behavior and C = ker(0 K) a controller behavior. Thus, note that Πw (B(w,c) ∩ C ) = Πw (B(w,c) ∩ (C + H )) and C + H = ker(0 LM) for some polynomial matrix L. Also it is not difficult to see
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that B(w,c) ∩ C is a regular interconnection if and only if B(w,c) ∩ (C + H ) is a regular interconnection. Moreover, the (partial) interconnection ker(R M) ∩ ker(0 K) is regular if and only if the (full) interconnection ker XM ∩ ker K is regular, where X is the MLA of R. Analogously the interconnection ker(R M) ∩ ker(T 0) is regular if and only if the interconnection ker FR ∩ ker T is regular, where F is the MLA of M. Lemma 4. Let B(w,c) be a behavior and C a controller behavior. If the manifest controlled behavior, i.e., Πw (B(w,c) ∩reg C ), has finite dimension then Πw (B(w,c) ∩reg C c ) has finite dimension, where C c is the controllable part of C . Proof. Obviously B(w,c) ∩ C c ⊂ B(w,c) ∩ C implies Πw (B(w,c) ∩ C c ) ⊂ Πw (B(w,c) ∩ C ) and therefore if Πw (B(w,c) ∩ C ) is finite dimensional so it is Πw (B(w,c) ∩ C c ). Finally, the intersection of B(w,c) and C c is regular by Lemma 1 and Remark 2, which concludes the proof. Lemma 5. Let B(w,c) = ker(R M) be a behavior. If a desired behavior Bd is implementable by regular partial interconnection with a regular controller C = ker(0 LM) then Bd = Bw ∩reg ker(LR), i.e., Bd can also be implementable by regular (full) interconnection from Bw . Proof. Without loss of generality we supposed that the matrix LM is fullrow rank I 0 R M R M since C is a regular behavior. Further, · = . Let X L −I 0 LM LR 0 R M XR ) = ker = be the MLA of M. Hence Πw (B(w,c) ∩ C ) = Πw (ker LR 0 LR B(w,c) ∩ ker LR. To see that the interconnection between B(w,c) and ker LR is regular we prove that the interconnection between ker(R M) ∩ ker(LR 0) is regular, i.e., v(R M) = z(LR 0) for some row vectors v and z, implies v(R M) = 0 = z(LR 0). Suppose that v(R M) = z(LR 0). Note that z(LR 0) = z[(0 − LM) + (LR LM)] and then v(R M) − z(LR LM) = (v − zL)(R M) = z(0 − LM). By assumption that the interconnection of B(w,c) and C is regular one has that (v − zL)(R M) = z(0 − LM) = 0 and since LM is full row rank one obtains that z = 0 and therefore v(R M) = z(LR 0) = 0 which proves that the interconnection is regular. Theorem 5. Let B(w,c) be a behavior and assume that H is controllable. If a finite dimensional behavior is regular implementable from B(w,c) by partial interconnection, then the controllable part of the manifest behavior Bw is rectifiable. Proof. Let Bd be a finite dimensional behavior that is regular implementable from B(w,c) = ker(R M), i.e., Πw (B(w,c) ∩reg C ) = Bd with C a controller behavior. By Remark 2 one can assume that C = ker(0 LM), i.e., H ⊂ C . Applying Lemma 4
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one obtains that Πw (B(w,c) ∩reg C c ) = Bd has finite dimension, where C c is the controllable part of C . By [6, Corollary 4, page 399] we have that Mod(C c ) is free since C c is controllable, i.e., C c is a regular behavior. Further, H ⊂ C implies that H c ⊂ C c , where H c is the controllable part of H . Since H c = H ker M controllable we have that and C c = ker(0 LM) for some matrix L. It follows from Lemma 5 that Bd is regular implementable from Bw by full interconnection. Since we are now in the context of full interconnections we can apply Theorem 3 in order to conclude the proof.
5 Conclusions In this paper we have investigated two fundamental notions in the behavioral approach to control theory, namely, regular implementation and controllability. We have shown that if a finite dimensional behavior can be implemented by a regular interconnection from a given behavior B, then the controllable part of B is rectifiable, and therefore it is possible to derive important structural properties of B. We have also treated the situation where not all system variables are available for interconnection. The results rely strongly on the properties of the two dimensional behaviors and n the proofs are not adaptable to the higher dimensional case, i.e., B ⊂ ( q ) with n > 2. We expect some interesting problems arising in the extension of the results presented in this paper to the higher dimensional case, where some new tools will need to be introduced.
References 1. Belur, M.N., Trentelman, H.L.: Stabilization, pole placement, and regular implementability. IEEE Trans. Automat. Control 47(5), 735–744 (2002) 2. Bisiacco, M., Valcher, M.E.: Two-dimensional behavior decompositions with finitedimensional intersection: a complete characterization. Multidimens. Syst. Signal Process 16(3), 335–354 (2005) 3. Julius, A.A., Willems, J.C., Belur, M.N., Trentelman, H.L.: The canonical controllers and regular interconnection. Systems & Control Letters 54(8), 787–797 (2005) 4. Napp Avelli, D.: Almost direct sum decomposition and implementation of 2D behaviors. Math. Control Signals Systems 21(1), 1–19 (2009) 5. Oberst, U.: Almost regular interconnection of multidimensional behaviors. Accepted for publication in SIAM Journal on Control and Optimization (2008) 6. Pillai, H., Shankar, S.: A behavioral approach to control of distributed systems. SIAM J. Control Optim. 37(2), 388–408 (1998) 7. Rocha, P.: Feedback control of multidimensional behaviors. Systems & Control Letters 45, 207–215 (2002) 8. Rocha, P.: Canonical controllers and regular implementation of nD behaviors. In: Proceedings of the 16th IFAC World Congress, Prague, Chech Republic (2005)
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9. Rocha, P.: Stabilization of multidimensional behaviors. Multidimens. Systems Signal Process 19, 273–286 (2008) 10. Rocha, P., Wood, J.: Trajectory control and interconnection of 1D and nD systems. SIAM J. Control Optim. 40(1), 107–134 (2001) 11. Trentelman, H.L., Napp Avelli, D.: On the regular implementability of nD systems. Systems and Control Letters 56(4), 265–271 (2007) 12. Willems, J.C.: On interconnections, control, and feedback. IEEE Trans. Automat. Control 42(3), 326–339 (1997) 13. Zerz, E.: Primeness of multivariate polynomial matrices. Systems Control Lett. 29(3), 139–145 (1996) 14. Zerz, E.: Multidimensional behaviours: an algebraic approach to control theory for (PDE). International Journal of Control 77(9), 812–820 (2004) 15. Zerz, E., Lomadze, V.: A constructive solution to interconnection and decomposition problems with multidimensional behaviors. SIAM J. Control Optim. 40(4), 1072–1086 (2002)
Ordering of Matrices for Iterative Aggregation Disaggregation Methods Ivana Pultarov´a
Abstract. In this short paper we show how the convergence of the iterative aggregation-disaggregation methods for computing the Perron eigenvector of a large sparse irreducible stochastic matrix can be improved by an appropriate ordering of the data and by the choice of a basic iteration matrix. Some theoretical estimates are introduced and a fast algorithm is proposed for obtaining the desired ordering. Numerical examples are presented.
1 Introduction The problem of solving Perron eigenvectors of stochastic matrices appears in many applications in web information retrieval, in computing the reliability of composed security electrical appliances or in the queuing problems. Due to complexities of these tasks, efficient methods are needed. Similarly to another computing disciplines, two- or multi-level approaches are well applicable in this field. These algorithms are called the iterative aggregation-disaggregation (IAD) methods. Properties of basic IAD algorithms were introduced in [13]. Further theoretical results were obtained in [1, 4, 6] and some modifications were devised in order to solve effectively some particular problems [2, 7]. A progress in the convergence estimates was achieved in paper [2] for a special two-level IAD process, and a generalization of this idea was proved in [12]. We continue in studying the convergence properties of the IAD methods. From the results in [8, 12] it follows that a nonzero pattern of a stochastic matrix close to the pattern of a cyclic matrix may cause slowing down the convergence rate of some IAD methods. In spite of it, choosing an appropriate basic iteration method may increase the convergence speed significantly. We address this issue in this paper. Ivana Pultarov´a Department of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague, Thakurova 7, 166 29 Prague, Czech Republic, e-mail: [email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 379–385. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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In the next section, the IAD algorithm is described and its basic properties are recalled. In Section 3 we focus on the cases when the stochastic matrix is cyclic and several types of the IAD method with different basic iterations are theoretically examined. Two theorems are introduced regarding this problem. In Section 4 we present a simple algorithm which yields an ordering of a stochastic matrix which is close to that of a cyclic matrix. Some large scale numerical examples are introduced.
2 IAD Method We assume an N × N irreducible column stochastic matrix B and we want to obtain a Perron eigenvector xˆ of B, i.e. Bxˆ = x, ˆ eT xˆ = 1, where e is a vector with all ones entries. The irreducibility of B implies that xˆ is unique. For a two-level iterative method, the set of indices {1, 2, . . . , N} is divided into n ≤ N subgroups G1 , . . . , Gn and they are considered as a new set of macro-states of a higher level. We will use the following notation. Let R be an n × N matrix for which Ri j = 1 if j ∈ Gi and Ri j = 0 otherwise. For any positive vector x we define an N × n matrix S(x) with the elements xi S(x)i j = ∑k∈G j xk if i ∈ G j and S(x)i j = 0 otherwise. We denote P(x) = S(x)R. Let Ba (x) be the aggregated matrix of size n Ba (x) = RBS(x). Then the algorithm of the IAD method is the following. 1. Starting with some positive vector x0 we solve the equation Ba (x0 )z = z for z. 2. Then z is prolonged to the size N by y = S(x0 )z. 3. Several steps of some basic iterative method are performed. We may use e.g. the power method, Jacobi or Gauss-Seidel methods or their block forms. Let M −W be some weak nonnegative splitting of I − B, where I is an identity matrix. Then the basic iteration matrix will be T = M −1W . Then x1 = T m y for some chosen integer m, and one loop of the IAD method is finished.
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Matrix Ba (x) is an irreducible stochastic matrix for any positive x [6]. Obviously P(x) is a projection.
3 Spectral Radius of the Error Matrix It was shown [6, 9, 11] that for the sequence of computed approximations xk it holds xk+1 − xˆ = J(xk )(xk − x), ˆ where the error matrix J(x) is J(x) = T (I − P(x)Z)−1 (I − P(x)), where Z = B − xe ˆ T. In [2] the authors show that for a certain IAD scheme the global convergence of the algorithm can be proved for a matrix B which has a special nonzero structure with respect to the choice of the aggregation groups Gk . As a generalization of it we can show [12] that we obtain a locally convergent process when it is chosen T = B and the sparsity pattern of B has some special property. Up to now a general satisfactory proof of the convergence of IAD algorithm in dependency on the data (even in local sense) has not been established yet. From some examples [8, 12] we may deduce that cyclicity of the data causes divergence of the algorithm for some choice of the aggregation groups and for some basic iteration matrices T . That is why we study such situations in this paper. Matrix B is assumed to be cyclic, B1,N = 1,
Bi+1,i = 1 for i = 1, 2, . . . , N − 1
(1)
ˆ to whether determine and Bi j = 0 otherwise. We will study the spectral radius of J(x) the asymptotic rate of convergence of the IAD method or to prove its divergence. Since we study the IAD methods in which various basic iterative matrices are used we will denote the corresponding error matrices J(x, ˆ T m ) in the remaining part of the paper. We assume that the numbering of the events and that of the groups are consecutive, i.e. if i ∈ Gk , j ∈ Gm , k < m then i < j. Let us denote B1 a block-diagonal matrix composed from the diagonal blocks of B where the indices of the particular blocks correspond to the aggregation groups G1 , . . . , Gn . Let B2 = B − B1. The asymptotic convergence behavior is determined by the spectral radius of J(x, ˆ T ). The view of the asymptotic convergence for some special cases is provided by the following two theorems. Theorem 1. Let n < N and let B be defined by (1). Asymptotic spectral radii of the error matrices corresponding to the IAD methods for the basic iteration matrices B, BN and (I − B1 )−1 B2 , respectively, are
ρ (J(x, ˆ B)) = 1,
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ρ (J(x, ˆ BN )) = 1, and
ρ (J(x, ˆ (I − B1 )−1 B2 )) = 0,
respectively. Proof. The first proposition ρ (J(x, ˆ B)) = 1 follows from [9, Theorem 1] and from [12, Theorem 3.10]. ˆ BN )) = 1 is equal to Since BN = I, the second proposition ρ (J(x,
ρ ((I − P(x)Z) ˆ −1 (I − P(x))) ˆ = 1. Matrix (I − P(x)Z) ˆ −1 (I − P(x)) ˆ is a projection which is not null for n < N, then we get the assertion. The off-diagonal blocks of (I − B1 )−1 B2 are all rank-one matrices, then the third proposition follows directly from [7]. Theorem 2. [10, Theorem 7.6] Let the block rows of the lower block triangle of B be all rank-one matrices. Then the IAD method with T = (I − B1 )−1 B2 finishes after at most n steps. It might be assumed that the IAD methods converge at least in local sense for great part of the set of irreducible stochastic matrices. But for an illustration we introduce examples, that cyclicity of B increases the spectral radius of J(x, ˆ BN−1 ) close to 2. From the continuous dependency of the spectrum on the elements of the matrix, the convergence can be arbitrarily slow for data which have almost a cyclic structure. In Figures 1 and 2 one can see the spectra (thick dots) of the error matrices ˆ BN/2−1 ) for 20 aggregation groups each including 30 elements. J(x, ˆ BN−1 ) and J(x, Thin lines of two or three circles are also displayed in each figure, which help to recognize the location of the eigenvalues.
1
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Fig. 1 The spectrum of J(x, ˆ BN−1 ), where B is defined by (1) and N = 600, n = 20, each aggregation group with 30 elements.
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From Theorems 1 and 2 and from these two examples of the spectra of the error matrices we may conclude that the choice of the aggregation groups and of the
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1 0.5 0 −0.5 −1 −2
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Fig. 2 The spectrum of J(x, ˆ BN/2−1 ), where B is defined by (1) and N = 600, n = 20, each aggregation group with 30 elements.
basic iteration matrix together with ordering the events have a crucial impact on the convergence of the IAD method. Apparently (Theorem 1) a very effective algorithm can be obtained when big elements of B are concentrated in diagonal blocks and T = (I − B1 )−1 B2 is used as the basic iteration matrix. The inversion of (I − B1 ) may be substituted by I + B1 + B21 + · · · + Bk1 for some integer k. In practical sparse large scale examples we may try to organize the columns and rows of matrix B in order to obtain a nonzero structure similar to that defined by (1). Then we may expect an acceleration of convergence. A simple algorithm for symmetric reordering of B is discussed in the following section.
4 Numerical Examples The set of data available in [3] is examined. It is Stanford web matrix of size 281903. The matrix is very sparse and represents links within a set of web pages. The aim is to find a Perron eigenvector of a stochastic matrix α B + (1 − α )peT , where B is Stanford web matrix, α ∈ (0, 1), p is a positive vector and eT p = 1. Before applying the IAD method either no reordering of B is performed or the rows and columns of B are symmetrically reordered in the following manner. It is started from an arbitrary column and then the largest value of it is found. The corresponding row index is then taken as for the next column. Then in this column the maximal element is found and its row index is taken as for the next column and so on. No column can be gone through twice or more. When it happens that there is no remaining nonzero element in some column, the path then continues in an arbitrary column. We can call this process ”following the maximal column value” (FMCV). In a formal description of the FMCV algorithm we use a vector Perm in which we store the permutation vector, a set C which contains the indices of columns which have not been checked yet and a variable F which points to the last number
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(position) stored in Perm. The FMVC algorithm then can be performed in the following two steps. 1. Set C := {2, . . . , N}, F := 1, Perm(F) := 1. 2. Repeat until C is empty (or equivalently F is equal to N): find m that Bm,Perm(F) = max j∈C B j,Perm(F) , if Bm,Perm(F) = 0 then m := minC, F := F + 1, Perm(F) := m, C := C \ {m}. The desired nonzero pattern of B is then obtained by B := B(Perm, Perm). Of course the resulting permuting of B is not unique. It depends on the choice of the initial column and of the columns when the quantity max j∈C B j,Perm(F) is zero. Complexity of the algorithm is equal to N. We compare sequences of approximations computed by IAD method where T = (I − B1 )−1 B2 for the original matrix B with four applications of the IAD method to the reordered matrix B with different basic iteration matrices: T1 = (I − B1 )−1 B2 , −1 T2 = (I + B1 + B21 + · · · + B10 1 )B2 , T3 = (I − BD,1 ) (BD,2 ), where BD,1 is a block diagonal matrix, the diagonal blocks of which contain only the main diagonals and the second lower diagonals of the corresponding blocks of B and BD,2 = B − BD,1 . The fourth set of approximations is computed for T4 = (I + BD,1 + · · · + B10 D,1 )(BD,2 ). Thus we have five series of approximations. Numbers of steps in each set in which the error is decreased to 10−5 are shown in five columns of Table 1 which are denoted by T , T1 , T2 , T3 , and T4 , respectively. Matrices B are parts of the Stanford web matrix. The number of blocks times the sizes of blocks are 100 × 100, 200 × 200, 300 × 300 and 400 × 400, respectively. Into all of the empty columns one value of 1 is added. We set α = 0.85 and p = (1, 1, . . . , 1)T /N. Table 1 Number of steps of the five variants of the IAD method until the error 10−5 is reached.
number of blocks × block size
T
T1 T2
T3
T4
100 × 100 200 × 200 300 × 300 400 × 400
59 59 55 53
30 34 37 39
33 35 36 37
34 35 36 37
48 42 40 42
As we can see, while for smaller size data the matrix T1 yields the best results, for larger data manipulating with only the diagonal and with the first subdiagonal is more efficient.
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5 Discussion We have shown some difficulties which can be met when applying blindly the IAD method and we propose an approach for their resolution. Achieving a structure of elements of a stochastic matrix similar to that defined by (1) and using the block Jacobi basic iteration usually increases the convergence speed. Let us stress that this way is well applicable in practical large scale computing. The appropriate ordering and the use of the basic iteration matrix T2 = (I + B1 + B21 + · · · + Bk1 )B2 which is the approximation of T1 = (I − B1 )−1 B2 is e.g. exactly performed by the algorithm proposed in [5]. ˇ No. 201/09/P500 and by the reAcknowledgements. Supported by the project of GACR search project CEZ MSM 6840770001.
References 1. Courtois, P.J., Semal, P.: Block iterative algorithms for stochastic matrices. Linear Algebra and its Applications 76, 59–80 (1986) 2. Ipsen, I., Kirkland, S.: Convergence analysis of a PageRank updating algorithm by Langville and Meyer. SIAM Journal on Matrix Analysis and Applications 27, 952–967 (2006) 3. Kamvar, S.: Data sets of Stanford Web Matrix and Stanford-Berkeley Web Matrix, http://www.cise.ufl.edu/research/sparse 4. Krieger, U.R.: Numerical solution of large finite Markov chains by algebraic multigrid techniques. In: Stewart, W.J. (ed.) Computations with Markov Chains, pp. 403–424. Kluwer Academic Publisher, Boston (1995) 5. Litvak, N., Robert, P.: Analysis of an on-line algorithm for solving large Markov chains. In: Proceedings SMCTools 2008, Athens, Greece (2008) 6. Marek, I., Mayer, P.: Convergence analysis of an aggregation/disaggregation iterative method for computation stationary probability vectors of stochastic matrices. Numerical Linear Algebra With Applications 5, 253–274 (1998) 7. Marek, I., Mayer, P.: Convergence theory of some classes of iterative aggregation/ disaggregation methods for computing stationary probability vectors of stochastic matrices. Linear Algebra and its Applications 363, 177–200 (2003) 8. Marek, I., Mayer, P., Pultarov´a, I.: IAD methods based on splittings with cyclic iteration matrices. In: Dagstuhl Seminar Proceedings 07071 (2007) 9. Marek, I., Pultarov´a, I.: A note on local and global convergence analysis of iterative aggregation/disaggregation methods. Linear Algebra and its Applications 413, 327–341 (2006) 10. Marek, I., Pultarov´a, I.: An aggregation variation on the Google matrix (submitted) 11. Pultarov´a, I.: Local convergence analysis of iterative aggregation-disaggregation methods with polynomial correction. Linear Algebra and its Applications 421, 122–137 (2007) 12. Pultarov´a, I.: Necessary and sufficient local convergence condition of one class of iterative aggregation - disaggregation methods. Numerical Linear Algebra with Applications 15, 339–354 (2008) 13. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)
The Positive Servomechanism Problem under LQcR Control Bartek Roszak and Edward J. Davison
Abstract. This paper considers the servomechanism problem for MIMO positive LTI systems. In particular, the servomechanism problem of nonnegative constant reference signals for stable MIMO positive LTI systems with unmeasurable unknown constant nonnegative disturbances under strictly nonnegative control inputs is solved using a clamping LQ regulator.
1 Introduction In this paper we consider the positive servomechanism problem for stable MIMO positive LTI systems using linear quadratic clamping regulators. The servomechanism problem for LTI systems has been nicely captured and solved by [1]; however, the interest of the servomechanism problem for positive systems has had limited consideration [6]-[8]. In the case of positive systems, [7] considers a subclass of the servomechanism problem under measurable disturbances with feedforward compensators and tuning regulators; [6, 8] take into account the tracking/regulation problem for SISO positive LTI systems with almost-positivity and clamping tuning regulators. The above references only consider unknown systems (i.e. where the plant matrices (A, B,C, D) are unknown) and thus the control methods have been limited to tuning regulators with on-line tuning. In this paper, we assume that the plant matrices are known; in this case, we show that our results for tracking/disturbance rejection can incorporate “LQR clamping” control that in general results in improved performance compared to [6]-[8]. Bartek Roszak and Edward J. Davison Systems Control Group, Electrical and Computer Engineering University of Toronto, Toronto, Ontario M5S 1A4, Canada, e-mail: [email protected],[email protected] R. Bru and S. Romero-Viv´o (Eds.): Positive Systems, LNCIS 389, pp. 387–395. c Springer-Verlag Berlin Heidelberg 2009 springerlink.com
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For numerous related topics to this paper and positive systems see [2], [3], [5], [6–8], and references therein. The paper is organized as follows. Preliminaries and background are presented first, where the terminology, the problem of interest, and the control strategy are outlined. Section 3 provides the main theoretical results of the paper. An illustrative example is presented in Section 4 and all concluding remarks finalize the paper.
2 Terminology and Problem Statement Throughout the paper we use standard positive system terminology with several definitions given next. Let the set + := {x ∈ | x ≥ 0}, the set n+ := {x = (x1 , x2 , ..., xn ) ∈ n | xi ∈ + , ∀i = 1, ..., n}. If exclusion of 0 from the sets will be necessary, then we’ll denote the sets in the standard way n+ \ {0}. The i jth entry of a matrix A will be denoted as ai j . A nonnegative matrix A has all of its entries greater or equal to 0, ai j ∈ + ∀i, j. A Metzler matrix A is a matrix for which all off-diagonal elements of A are nonnegative, i.e. ai j ∈ + for all i = j. In this paper a plant is considered stable if all eigenvalues are located in the open left-half plane of the imaginary axis. The plant of interest is defined next. Consider, x˙ = Ax + Bu + E ω y = Cx + Du + F ω
(1)
e := y − yre f r×n where A is an n × n Metzler Hurwitz matrix, B ∈ n×m + , C ∈ + , D = 0, E ω ∈ n r r ω Ω1 ⊂ + , F ω ∈ Ω2 ⊂ + , yre f ∈ Yre f ⊂ + , ω ∈ Ω ⊂ + . Also, assume that m = r, i.e. the number of inputs equals the number of outputs. The problem of interest is outlined below.
Problem 1. Consider the plant (1). Assume that rank(D − CA−1B) = r and that the sets Ω and Yre f are chosen such that E ω ∈ n+ and F ω ∈ r+ , with the steady state values of the plant’s states, outputs (xss and yss )being nonnegative and the steady state of the input (uss )being positive, i.e. for all constant tracking and constant disturbance signals in question, it is assumed that the steady-state of the system (1) is given by
ω E 0 AB xss = (2) − yre f F −I C D uss has the property that xss ∈ n+ , yss = Cxss + Duss + F ω = yre f ∈ r+ , uiss ∈ + \ {0}, ∀i ∈ {1, ..., m}. It is to be noted that a solution to (2) exists if and only if rank(D − CA−1B) = r. Then, find a controller that
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(a) guarantees closed loop stability; (b) ensures the plant (1) is nonnegative, i.e. the inputs u are nonnegative for all time; and (c) ensures tracking of the reference signals, i.e. e = y − yre f → 0, as t → ∞, ∀yre f ∈ Yre f and ∀ω ∈ Ω . In addition, (d) assume that a controller has been found so that conditions (a), (b), (c) are satisfied; then for all perturbations of the nominal plant model which maintain properties (a) and (b), it is desired that the controller can still achieve asymptotic tracking and regulation, i.e. property (c) still holds. Problem 1 will be referred to as the positive robust servomechanism problem. Note that condition (b) above also guarantees, by nonnegativity of u, that the states x and the outputs y will be nonnegative for all time. Notice that the two assumptions in the latter problem are in fact necessary for a positive system to adhere to any type of tracking constraint, i.e. the rank condition (rank(D − CA−1B) = r) is a necessary condition for the servomechanism problem of LTI systems and hence must hold true for positive LTI systems; also, the steadystate condition for xss and yss is clearly a necessary condition due to the restriction of positive systems. Also, it must be pointed out that in Problem 1 we do not make any assumption that the disturbance is known; thus, we are considering unmeasurable disturbances that abide to the steady-state conditions. See [7] for a closer discussion of this steady-state assumption. Next, the linear quadratic clamping regulator (LQcR) is defined. This control law will be used in the sequel to solve Problem 1. Controller 2.1 Assume rank(D − CA−1 B) = r. Given ρ > 0, the controller is described by:
η˙ = y − yre f , η (0) = 0;
α= where Kx ∈
m×n
u = α [Kx Kη ][x η ]T ,
(3)
0 if ∃ i ∈ {1, ..., r} s.t ([Kx Kη ][x η ])i ≤ 0, 1 otherwise
and Kη ∈
m×r
are found by solving the LQ control problem:
∞ 0
eT Qe + ρ 2u˙T ud ˙ τ
(4)
with ρ > 0 and Q = ((D −CA−1 B)−1 )T (D −CA−1 B)−1 ). Notice that with the given Q we still abide to the conditions of stabilizability and detectability, which are needed to proceed with the LQ control problem.
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For convenience, since we will be interested in letting ρ → ∞ we re-write (4) as ∞ 0
ε 2 eT Qe + u˙T ud ˙ τ
(5)
where ε > 0. The latter transformation allows us to treat the problem with ε → 0 and still obtain the same value for K = [Kx Kη ].
3 Main Results In this section, the necessary and sufficient conditions for Problem 1 are presented via the use of the LQcR controller. Theorem 1. Consider system (1), with x0 ∈ n+ , under controller (3). Then Problem 1 is solvable if and only if there exists an ε ∗ for (5) such that for all ε ∈ (0, ε ∗ ] the controller (3) solves Problem 1. Proof. A sketch only of the proof will be given. Let us show that indeed with the given assumptions the LQcR controller will solve Problem 1. We first concentrate on showing that tracking of yre f occurs. In order to show the latter, we will break down the proof into two steps. 1. Our first step will be to show that if u = 0 (clamping occurs), then there exists a time t1 > 0 such that the input will switch to the control law u = [Kx Kη ][x η ]T , i.e. in (3) the input will eventually stop clamping. 2. Then, we’ll show that if there exists a time t2 such that u > 0, then there exists an ε ∗ such that for all time t ≥ t2 and all ε ∈ (0, ε ∗ ] the controller (3) maintains nonnegativity of the input and solves Problem 1. Each of the above steps is given next: By contradiction, assume there does not exist a time t1 , i.e. u = 0 for all time. Therefore, the closed loop system becomes x˙ = Ax + E ω
η˙ = Cx + F ω − yre f
and since A is stable x → −A−1E ω = xss , t → ∞, i.e. if u = 0 for all time t > 0, then the system state tends toward xss as t → ∞, which can also be expressed as: x˙ = 0 = Axss + Buss + E ω −A E ω = xss + A−1Buss −1
xss = xss + A−1Buss . However, this implies that
The Positive Servomechanism Problem under LQcR Control
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u˙ = Kx x˙ + Kη η˙ → Kx (Axss + E ω ) + Kη (Cxss + D(0) + F ω − yre f ) → Kx (0) + Kη (C(xss + A−1Buss ) + D(0) + F ω − yre f ) → Kη (C(xss + A−1 Buss ) + D(uss − uss) + F ω − yre f ) → Kη (Cxss + Duss + F ω − yre f ) − Kη (D − CA−1B)uss ) → 0 − Kη (D − CA−1B)uss → ε uss > 0 component-wise; the second last line comes from the fact as ε → 0, Kη → −ε (D − CA−1 B)−1 (this can be shown via manipulation of the ARE and uniqueness of the gain matrix; the details are omitted.). Next, we proceed to show that if for some time t2 ≥ 0 (which exists from above and satisfies t2 ≥ t1 ), u(t2 ) > 0, then there exists an ε ∗ such that for all time t ≥ t2 and all ε ∈ (0, ε ∗ ] the controller (3) maintains nonnegativity of the input. In order to prove the above, we use the results of singular perturbation [4]. The closed loop system with the controller in place for u > 0 and shifted by its equilibrium
z x x = − ss uss q u becomes
z z˙ A B . = ε (K x A + K η C) ε (K x B + K η D) q q˙
(6)
Note that K = ε K (this can be shown via manipulation of the sequence that solves the ARE of (5) [9][Ch.12], details are omitted due to space limitations). For convenience, rewrite (6) as
q˙ ε (K x B + K η D) ε (K x A + K η C) q = . (7) z˙ B A z Next, let’s scale the derivatives (i.e. scaling of time) by ε dt = d τ , resulting in the transformed system * + +
q (K x B + K η D) (K x A + K η C) q , (8) + = B A z εz +
+
with ε ddqτ = ε q = q˙ and ε ddzτ = ε z = z˙. We have now transformed our model into the standard singular perturbation model. Now, since (8) is linear and time invariant and we are only interested in u it suffices to show that the reduced model by singular
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perturbation yields exponential stability [4]; all other assumptions clearly hold. The reduced model obtained (we omit details) results in: u = q + uss = uss + eKη = uss + e = uss + e
(9) (D−CA−1 B)τ
(u(t2 ) − uss)
ε Kη (D−CA−1 B)t −ε t
(u(t2 ) − uss)
(u(t2 ) − uss)
and since u(t2 ) > 0, then for all time t ≥ t2 , there exists an ε ∗ such that u > 0 for all ε ∈ (0, ε ∗ ] and t ≥ t2 since u is monotonically approaching uss . Thus, y → yre f as t → ∞ if uss > 0. Finally, nonnegativity trivially holds since u ≥ 0 for all time by the definition of the control law, and the fact that all other conditions of Problem 1 hold are also trivially satisfied by similar arguments as used in [7] (details are omitted). Necessity can be deduced from the latter result and is omitted. The latter Theorem can be easily interpreted; mainly, it states that one can accomplish nonnegativity, tracking and disturbance rejection, and robustness with an LQcR control law. Theorem 1 does not tell us how small or how large the ε ∗ can be, thus we cannot guarantee a settling time.
4 LQcR MIMO Example In this section we illustrate via an example the use of the LQcR controller. Example 1. Consider the system of reservoirs of Figure 1; note that each reservoir is identified by a number (1, 2, ..., 6) where the water storage level (x1 , x2 , ..., x6 ) is a state of the system. Also γ and φ are the splitting coefficients of the flows at the branching points. The system is of order 6, as we assume the pump dynamics can be neglected. As pointed out in [2], the dynamics of each reservoir can be captured by a single differential equation: x˙i = −αi xi + v + ei ω , z = αi xi for all i = 1, ..., 6, where xi is the water storage (in L) and α > 0 is the ratio between outflow rate z and storage, with ei ω being the disturbance rate into the storage. The input into the reservoir is designated by v and is in (L/s). Consider the case where γ = 0.5, φ = 0.7, α1 , ..., α6 = 0.8, 0.7, 0.5, 1, 2, 0.8. Note that all the rates are measured in L/s. This results in the following system:
The Positive Servomechanism Problem under LQcR Control
393
u1 + ω γ
1−γ
1
2 pump
u2
3
φ
1−φ
6
4
5 Fig. 1 System set up for Example 1.
⎡ ⎤ 0.5 −0.8 0 0 0 2 0 ⎢ 0.5 ⎢ 0 −0.7 0 0 0 0⎥ ⎢ ⎢ ⎥ ⎢ ⎢ 0.8 0.7 −0.5 0 0 0⎥ ⎥x+⎢ 0 x˙ = ⎢ ⎢ 0 ⎢ ⎥ 0 0 0.15 −1 0 0⎥ ⎢ ⎢ ⎣ 0 ⎣ ⎦ 0 0 0 1 −2 0 0 0 0 0.35 0 0 −0.8 ⎡
y=
001000 x 100001
⎡ ⎤ ⎤ 0.5 0 ⎢ 0.5 ⎥ 0⎥ ⎢ ⎥ ⎥ ⎢ 0 ⎥ ⎥ 0⎥ ⎢ ⎥ω u+⎢ ⎥ 0⎥ ⎢ 0 ⎥ ⎥ ⎣ ⎦ 0 ⎦ 0 0 1
(10)
(11)
It is now desired to solve Problem 1 for this stable system. Assume the initial condition and the disturbance are x0 = [2 2 2 2 2 2] and ω = 0.5, respectively. Additionally, assume that we would like to track the reference input yre f = [5 5]T . With the choice of ε = 10, the desired result is obtained. Figure 2 illustrates the simulated input and output response, and since the input is nonnegative it is clear that the states must also be nonnegative for all time. Notice that initially (0-3 seconds) the controller clamps to avoid negativity of the control input.
394
B. Roszak and E.J. Davison outputs and inputs y 6
5 y2
y and u
4
3 y1 2
u1
1 u2 0 0
5
10
15 time (s)
20
25
30
Fig. 2 Input and output response for Example 1.
5 Conclusion In this paper, we have provided the necessary and sufficient conditions for the positive MIMO servomechanism problem (Problem 1). The results extended the SISO case of [6, 8] to the MIMO case under a new clamping LQcR control strategy. Acknowledgements. Supported by NSERC under Grant No A4396.
References 1. Davison, E.J., Goldenberg, A.: Robust control of a general servomechanism problem: the servo-compensator. Automatica 11, 461–471 (1975) 2. Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Pure and Applied mathematics. John Wiley & Sons, Inc., New York (2000) 3. Kaczorek, T.: 1D and 2D systems. Springer, New York (2002) 4. Khalil, H.K.: Nonlinear Systems. Prentice Hall, New Jersey (2002) 5. Luenburger, D.: Introduction to Dynamic Systems: Theory, Models and Applications. Wiley, New York (1979) 6. Roszak, B., Davison, E.J.: Tuning regulators for tracking SISO positive linear systems. In: Proceedings of the European Control Conference 2007, pp. 540–547 (2007)
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7. Roszak, B., Davison, E.J.: The Servomechanism Problem for Unknown MIMO LTI Positive Systems: Feedforward and Robust Tuning Regulators. In: Proceedings of the American Control Conference 2008, pp. 4821–4826 (2008) 8. Roszak, B., Davison, E.J.: The servomechanism problem for unknown SISO positive systems using clamping. In: Proceedings of the 17th IFAC World Congress, pp. 353–358 (2008) 9. Wonham, W.M.: Linear Multivariable Control A Geometric Approach, 3rd edn. Springer, New York (1985)
Author Index
Abad, Manuel F. 185 Ahmane, Mourad 195 Ait Rami, Mustapha 205 Alfidi, Mohammed 217 Audoly, Stefania 269 Beauthier, Charlotte 45 Bellu, Giuseppina 269 Benoˆıt, Eric 233 Benvenuti, Luca 55 Bokharaie, Vahid S. 101 Cant´ o, Bego˜ na 243 Cant´ o, Rafael 251 Chotijah, Siti 141 Coll, Carmen 243 Conradi, Carsten 163 Cordero, Alicia 259 Corless, Martin 339 Cottenceau, Bertrand 349 D’Angi , Leontina 269 Damm, Tobias 63 Davison, Edward J. 387 Declerck, Philippe 279 de Kerchove, Cristobald 3 Ethington, Cristina Farina, Lorenzo
63
55
Gass´ o, Mar´ıa T. 185 Gaubert, St´ephane 291 Gouz´e, Jean-Luc 233 Guezzi, Abdelhak 279
Hardouin, Laurent 349 Herrera, Manuel 305, 321 Herrero, Alicia 315 Hinrichsen, Diederich 71 Hmamed, Abdelaziz 217 Hueso, Jos´e L. 259 Izquierdo, Joaqu´ın
305, 321
Kaltenbach, Hans-Michael Katkova, Olga M. 83 Knorn, Florian 331, 339
163
Le Corronc, Euriell 349 Lorenz, Dirk A. 91 Lorenz, Jan 91 Luh, Peter B. 151 Marek, Ivo 359 Mart´ınez, Eulalia 259 Mason, Oliver 101, 331 Montalvo, Idel 305, 321 Napp Avelli, Diego 369 Ngoc, Pham Huu Anh 111 Pe˜ na, Juan Manuel 123 P´erez-Garc´ıa, Rafael 305, 321 Pultarov´ a, Ivana 379 Ram´ırez, Francisco J. Reis, Timo 131 Ricarte, Beatriz 251 Rocha, Paula 369
315
398
Author Index
Roszak, Bartek 387 Rumchev, Ventsi 141
Torregrosa, Juan R. 185, 259 Truffet, Laurent 195
Saccomani, Maria Pia 269 Salceanu, Paul Leonard 17 S´ anchez, Elena 243 Sharify, Meisam 291 Shorten, Robert 101, 331, 339 Smith, Hal L. 17 Stanojevic, Rade 339 Stelling, J¨ org 163 Sun, Tao 151
Uhr, Markus 163 Urbano, Ana M. 251
Tadeo, Fernando 217 Thome, N´estor 315
Valcher, Maria Elena 29, 173 Van Dooren, Paul 3 Virnik, Elena 131 Vishnyakova, Anna M. 83 Winkin, Joseph J.
45
Zhao, Qianchuan
151
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Vol. 354: Witczak M.: Modelling and Estimation Strategies for Fault Diagnosis of Non-Linear Systems 215 p. 2007 [978-3-540-71114-8]
Vol. 367: Turner M.C.; Bates D.G. (Eds.) Mathematical Methods for Robust and Nonlinear Control 444 p. 2007 [978-1-84800-024-7]
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Vol. 352: Chiasson, J.; Loiseau, J.J. (Eds.) Applications of Time Delay Systems 358 p. 2007 [978-3-540-49555-0]
Vol. 365: Bates D.; Hagström M. (Eds.) Nonlinear Analysis and Synthesis Techniques for Aircraft Control 360 p. 2007 [978-3-540-73718-6] Vol. 364: Chiuso A.; Ferrante A.; Pinzoni S. (Eds.) Modeling, Estimation and Control 356 p. 2007 [978-3-540-73569-4] Vol. 363: Besançon G. (Ed.) Nonlinear Observers and Applications 224 p. 2007 [978-3-540-73502-1] Vol. 362: Tarn T.-J.; Chen S.-B.; Zhou C. (Eds.) Robotic Welding, Intelligence and Automation 562 p. 2007 [978-3-540-73373-7] Vol. 361: Méndez-Acosta H.O.; Femat R.; González-Álvarez V. (Eds.): Selected Topics in Dynamics and Control of Chemical and Biological Processes 320 p. 2007 [978-3-540-73187-0] Vol. 360: Kozlowski K. (Ed.) Robot Motion and Control 2007 452 p. 2007 [978-1-84628-973-6] Vol. 359: Christophersen F.J. Optimal Control of Constrained Piecewise Affine Systems 190 p. 2007 [978-3-540-72700-2] Vol. 358: Findeisen R.; Allgöwer F.; Biegler L.T. (Eds.): Assessment and Future Directions of Nonlinear Model Predictive Control 642 p. 2007 [978-3-540-72698-2] Vol. 357: Queinnec I.; Tarbouriech S.; Garcia G.; Niculescu S.-I. (Eds.): Biology and Control Theory: Current Challenges 589 p. 2007 [978-3-540-71987-8] Vol. 356: Karatkevich A.: Dynamic Analysis of Petri Net-Based Discrete Systems 166 p. 2007 [978-3-540-71464-4]
Vol. 351: Lin, C.; Wang, Q.-G.; Lee, T.H., He, Y. LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay 204 p. 2007 [978-3-540-49552-9] Vol. 350: Bandyopadhyay, B.; Manjunath, T.C.; Umapathy, M. Modeling, Control and Implementation of Smart Structures 250 p. 2007 [978-3-540-48393-9] Vol. 349: Rogers, E.T.A.; Galkowski, K.; Owens, D.H. Control Systems Theory and Applications for Linear Repetitive Processes 482 p. 2007 [978-3-540-42663-9] Vol. 347: Assawinchaichote, W.; Nguang, K.S.; Shi P. Fuzzy Control and Filter Design for Uncertain Fuzzy Systems 188 p. 2006 [978-3-540-37011-6] Vol. 346: Tarbouriech, S.; Garcia, G.; Glattfelder, A.H. (Eds.) Advanced Strategies in Control Systems with Input and Output Constraints 480 p. 2006 [978-3-540-37009-3] Vol. 345: Huang, D.-S.; Li, K.; Irwin, G.W. (Eds.) Intelligent Computing in Signal Processing and Pattern Recognition 1179 p. 2006 [978-3-540-37257-8] Vol. 344: Huang, D.-S.; Li, K.; Irwin, G.W. (Eds.) Intelligent Control and Automation 1121 p. 2006 [978-3-540-37255-4] Vol. 341: Commault, C.; Marchand, N. (Eds.) Positive Systems 448 p. 2006 [978-3-540-34771-2] Vol. 340: Diehl, M.; Mombaur, K. (Eds.) Fast Motions in Biomechanics and Robotics 500 p. 2006 [978-3-540-36118-3] Vol. 339: Alamir, M. Stabilization of Nonlinear Systems Using Receding-horizon Control Schemes 325 p. 2006 [978-1-84628-470-0]
