Control and Estimation of Systems with Input Output Delays (Lecture Notes in Control and Information Sciences)

Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari 355 Huanshui Zhang, Lihua Xie Contro...

Author: Huanshui Zhang; Lihua Xie

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Lecture Notes in Control and Information Sciences Editors: M. Thoma, M. Morari

355

Huanshui Zhang, Lihua Xie

Control and Estimation of Systems with Input/Output Delays

ABC

Series Advisory Board F. Allgöwer, P. Fleming, P. Kokotovic, A.B. Kurzhanski, H. Kwakernaak, A. Rantzer, J.N. Tsitsiklis

Authors Professor Lihua Xie School of EEE, BLK S2 Nanyang Technological University Nanyang Ave. Singapore 639798 Sinagapore Email: [email protected]

Professor Huanshui Zhang School of Control Science and Engineering Shandong University Jinan P.R. China Email: [email protected]

Library of Congress Control Number: 2007922362 ISSN print edition: 0170-8643 ISSN electronic edition: 1610-7411 ISBN-10 3-540-71118-X Springer Berlin Heidelberg New York ISBN-13 978-3-540-71118-6 Springer Berlin Heidelberg New York 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. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 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. Typesetting: by the authors and SPS using a Springer LATEX macro package Printed on acid-free paper

SPIN: 11903901

89/SPS

543210

Preface

Time delay systems exist in many engineering fields such as transportation, communication, process engineering and more recently networked control systems. In recent years, time delay systems have attracted recurring interests from research community. Much of the research work has been focused on stability analysis and stabilization of time delay systems using the so-called LyapunovKrasovskii functionals and linear matrix inequality (LMI) approach. While the LMI approach does provide an efficient tool for handling systems with delays in state and/or inputs, the LMI based results are mostly only sufficient and only numerical solutions are available. For systems with known single input delay, there have been rather elegant analytical solutions to various problems such as optimal tracking, linear quadratic regulation and H∞ control. We note that discrete-time systems with delays can usually be converted into delay free systems via system augmentation, however, the augmentation approach leads to much higher computational costs, especially for systems of higher state dimension and large delays. For continuous-time systems, time delay problems can in principle be treated by the infinite-dimensional system theory which, however, leads to solutions in terms of Riccati type partial differential equations or operator Riccati equations which are difficult to understand and compute. Some attempts have been made in recent years to derive explicit and efficient solutions for systems with input/output (i/o) delays. These include the study on the H∞ control of systems with multiple input delays based on the stable eigenspace of a Hamlitonian matrix [46]. It is worth noting that checking the existence of the stable eigenspace and finding the minimal root of the transcendent equation required for the controller design may be computationally expensive. Another approach is to split a multiple delay problem into a nested sequence of elementary problems which are then solved based on J-spectral factorizations [62]. In this monograph, our aim is to present simple analytical solutions to control and estimation problems for systems with multiple i/o delays via elementary tools such as projections. We propose a re-organized innovation analysis approach which allows us to convert many complicated delay problems into delay

VI

Preface

free ones. In particular, for linear quadratic regulation of systems with multiple input delays, the approach enables us to establish a duality between the LQR problem and a smoothing problem for a delay free system. The duality contains the well known duality between the LQR of a delay free system and Kalman filtering as a special case and allows us to derive an analytical solution via simple projections. We also consider the dual problem, i.e. the Kalman filtering for systems with multiple delayed measurements. Again, the re-organized innovation analysis turns out to be a powerful tool in deriving an estimator. A separation principle will be established for the linear quadratic Gaussian control of systems with multiple input and output delays. The re-organized innovation approach is further applied to solve the least mean square error estimation for systems with multiple state and measurement delays and the H∞ control and estimation problems for systems with i/o delays in this monograph. We would like to acknowledge the collaborations with Professors Guangren Duan, Yeng Chai Soh and David Zhang on some of the research works reported in the monograph and Mr Jun Xu and Mr Jun Lin for their help in some simulation examples.

Huanshui Zhang Lihua Xie

Symbols and Acronyms

i/o:

input/output.

LQG:

linear quadratic Gaussian.

LQR:

linear quadratic regulation.

PDE:

partial differential equation.

RDE:

Riccati difference (differential) equation.

col{X1 , · · · , Xn }:

the column vector formed by vectors X1 , · · · , Xn .

Rn :

n-dimensional real Euclidean space.

Rn×m :

set of n × m real matrices.

In :

n × n identity matrix.

diag{A1 , A2 , · · · , An }: block diagonal matrix with Aj ( not necessarily square) on the diagonal.

X:

transpose of matrix X.

P ≥ 0:

symmetric positive semidefinite matrix P ∈ Rn×n .

P > 0:

symmetric positive definite matrix P ∈ Rn×n .

P −1 :

the inverse of the matrix P .

X, Y :

inner product of vectors X and Y .

E:

mathematical expectation.

L{y1 , · · · , yn }:

linear space spanned by y1 , · · · , yn .

=:

definition.

dim(x):

the dimension of the vector x.

P roj:

projection

VIII

Symbols and Acronyms

M D: δij : e2 :

the total number of Multiplications and Divisions. ⎧ ⎨ 1, i = j, δij = ⎩ 0, i = j. 2 -norm of a discrete-time signal {e(i)}, ∞ 2 i.e., e(i) . i=0

2 [0, N ]:

space of square summable vector sequences on [0, N ] with values on Rn .

L2 [0, tf ]:

space of square integrable vector functions on [0, tf ] with values on Rn .

Contents

1.

Krein Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Definition of Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Projections in Krein Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Kalman Filtering Formulation in Krein Spaces . . . . . . . . . . . . . . . 1.4 Two Basic Problems of Quadratic Forms in Krein Spaces . . . . . . 1.4.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 5 6 6

2.

Optimal Estimation for Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Single Measurement Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Re-organized Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 2.2.3 Riccati Difference Equation . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Optimal Estimate x 2.2.5 Computational Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Multiple Measurement Delays Case . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Re-organized Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 2.3.3 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Optimal Estimate x 2.3.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 9 11 12 13 15 17 18 19 20 22 24 26

3.

Optimal Control for Systems with Input/Output Delays . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Linear Quadratic Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Duality Between Linear Quadratic Regulation and Smoothing Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Solution to Linear Quadratic Regulation . . . . . . . . . . . . . .

27 27 28 29 34

X

4.

5.

6.

Contents

3.3 Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 44 50

H∞ Estimation for Discrete-Time Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 H∞ Fixed-Lag Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 An Equivalent H2 Estimation Problem in Krein Space . . 4.2.2 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 4.2.3 Calculation of the Innovation Covariance . . . . . . . . . . . . . . 4.2.4 H∞ Fixed-Lag Smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Computational Cost Comparison and Example . . . . . . . . . 4.2.6 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 H∞ d-Step Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 An Equivalent H2 Problem in Krein Space . . . . . . . . . . . . 4.3.2 Re-organized Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Calculation of the Innovation Covariance . . . . . . . . . . . . . . 4.3.4 H∞ d-Step Predictor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 H∞ Filtering for Systems with Measurement Delay . . . . . . . . . . . 4.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 An Equivalent Problem in Krein Space . . . . . . . . . . . . . . . . 4.4.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 4.4.4 Calculation of the Innovation Covariance Qw (t) . . . . . . . . 4.4.5 H∞ Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 55 58 59 63 67 69 69 70 73 74 76 77 77 78 80 82 84 85

H∞ Control for Discrete-Time Systems with Multiple Input Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 H∞ Full-Information Control Problem . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Calculation of v ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Maximizing Solution of JN with Respect to Exogenous Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 H∞ Control for Systems with Preview and Single Input Delay . 5.3.1 H∞ Control with Single Input Delay . . . . . . . . . . . . . . . . . . 5.3.2 H∞ Control with Preview . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 87 88 89 91 96 104 106 106 108 111 113

Linear Estimation for Continuous-Time Systems with Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Contents

6.2 Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Re-organized Measurement Sequence . . . . . . . . . . . . . . . . . . 6.2.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.2.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ (t | t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Optimal Estimate x 6.2.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 H∞ Filtering for Systems with Multiple Delayed Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 An Equivalent Problem in Krein Space . . . . . . . . . . . . . . . . 6.3.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.3.4 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 H∞ Fixed-Lag Smoothing for Continuous-Time Systems . . . . . . 6.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 An Equivalent H2 Problem in Krein Space . . . . . . . . . . . . 6.4.3 Re-organized Innovation Sequence . . . . . . . . . . . . . . . . . . . . 6.4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.

8.

XI

116 116 117 118 119 120 121 122 123 124 126 127 129 130 132 132 133 136 137 140 141

H∞ Estimation for Systems with Multiple State and Measurement Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 H∞ Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Stochastic System in Krein Space . . . . . . . . . . . . . . . . . . . . 7.3.2 Sufficient and Necessary Condition for the Existence of an H∞ Smoother . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The Calculation of an H∞ Estimator zˇ(t, d) . . . . . . . . . . . 7.4 H∞ Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 149 157 162

Optimal and H∞ Control of Continuous-Time Systems with Input/Output Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Linear Quadratic Regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Problem Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Solution to the LQR Problem . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Measurement Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 163 164 164 165 169 175 178 179

143 143 144 145 146

XII

Contents

8.3.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 H∞ Full-Information Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Calculation of v ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 H∞ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 185 185 187 190 195 199 203

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

1. Krein Space

In this monograph, linear estimation and control problems for systems with i/o delays are investigated under both the H2 and H∞ performance criteria. The key to our development is re-organized innovation analysis in Krein spaces. The reorganized innovation analysis approach, which forms the basis of the monograph, will be presented in Chapter 2. Krein space theory including innovation analysis and projections which can be found in [30], plays an important role in dealing with the H∞ control and estimation problems as it provides a powerful tool to convert an H∞ problem into an H2 one under an appropriate Krein space. Thus, simple and intuitive techniques such as projections can be applied to derive a desired estimator or controller. For the convenience of discussions, in the rest of the monograph we shall first give a brief introduction to Krein space in this chapter.

1.1 Definition of Krein Spaces We briefly introduce the definition and some basic properties of Krein spaces, focusing only on a number of key results that are needed in our later development. Much of the material of this section and more extensive expositions can be found in the book [30]. Finite-dimensional (often called Minkowski) and infinite-dimensional Krein spaces share many of the properties of Hilbert spaces, but differ in some important ways that we highlight below. Definition 1.1.1. (Krein Spaces) An abstract vector space K, that satisfies the following requirements is called a Krein space. 1. K is a linear space over the field of complex numbers C. 2. There exists a bilinear form < ·, · >∈ C on K such that 1) < y, x >=< x, y > , 2) < ax + by, z >= a < x, z > +b < y, z > for any x, y, z ∈ K, a, b ∈ C, where “ ” denotes complex conjugation. 3. The vector space K admits a direct orthogonal sum decomposition K = K+ ⊕ K− , H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 1–6, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

(1.1)

2

1. Krein Space

such that K+ , < ·, · > and K− , − < ·, · > are Hilbert spaces, and x, y = 0,

(1.2)

for any x ∈ K+ and y ∈ K− . Remark 1.1.1. Some key differences between Krein spaces and Hilbert spaces include: • Hilbert spaces satisfy not only the above 1)-3), but also the requirement that x, x > 0 when x = 0. • The fundamental decomposition of K defines two projection operators P+ and P− such that P+ K = K+ and P− K = K− . Therefore, for every x ∈ K, we can write x = x+ + x− , where x± = P± x ∈ K± . Note that for every x ∈ K+ , we have x, x ≥ 0, but the converse is generally not true since x, x ≥ 0 does not necessarily imply that x ∈ K+ . • A vector x ∈ K is said to be positive if x, x > 0, neutral if x, x = 0, or negative if x, x < 0. Correspondingly, a subspace M ⊂ K can be positive, neutral, or negative, if all its elements are so, respectively. We now focus on linear subspaces of K. We shall define L{y0 , · · · , yN } as the linear subspace of K spanned by the elements {y0 , y1 , · · · , yN } in K. The Gramian of the collection of the elements {y0 , · · · , yN } is defined as the (N + 1) × (N + 1) (block) matrix

Ry = [yi , yj ]i,j=0,···,N .

(1.3)

The reflexivity property, yi , yj = yj , yi , shows that the Gramian is a Hermitian matrix. Denote the column vector formed by the vectors y0 , y1 , · · · , yN as y = col{y0 , y1 , · · · , yN }. Then the Gramian matrix of the above vector is

Ry = y, y . Similar to the case in Hilbert space, y is called the “random variables” and their Gramian as the “covariance matrix”, i.e., Ry = E yi yj = E yy , where E(·) denotes the mathematical expectation.

1.2 Projections in Krein Spaces

3

In a similar way, if we have two sets of elements {z0 , · · · , zM } and {y0 , · · · , yN }, we shall write z={z0 , z1 , · · · , zM } and y={y0 , y1 , · · · , yN }, and introduce the (M + 1) × (N + 1) cross-Gramian matrix

Rzy = [zi , yj ]i=0,···,M,j=0,···,N = z, y . Note that

Rzy = Ryz .

1.2 Projections in Krein Spaces We now discuss projections in Krein spaces. Definition 1.2.1. (Projections in Krein Spaces) Given the element z in K, and the elements {y0 , · · · , yN } also in K, we define

ˆ z to be the projection of z onto L{y} = L{y0 , · · · , yN } if z=ˆ z + z˜,

(1.4)

where zˆ ∈ L{y} and z˜ satisfies the orthogonality condition ˜ z ⊥ L{y}, i.e., ˜ z, yi = 0 for i = 0, 1, · · · , N. In Hilbert spaces, projections always exist and are unique. However, in Krein spaces, this is not always the case. Indeed, we have the following result Lemma 1.2.1. (Existence and Uniqueness of Projections) In the Hilbert space setting: 1. If the Gramian matrix Ry = y, y is nonsingular, then the projection of z onto L{y} exists, is unique, and is given by −1 ˆ z = z, y y, y y = Rzy Ry−1 y.

(1.5)

2. If the Gramian matrix Ry = y, y is singular, then either 1) R(Ryz ) ⊆ R(Ry ) (where R(A) denotes the column range space of the matrix A). Here the projection zˆ exists but is not unique. In fact, zˆ = k0 y, where k0 is any solution to the linear matrix equation Ry k0 = Ryz ,

(1.6)

or z does not exist. 2) R(Ryz ) ∈ R(Ry ). Here the projection ˆ Since existence and uniqueness will be important for all future results, we shall make the standing assumption that the Gramian Ry is nonsingular. As is well known, the singularity of Ry implies that y0 , y1 , · · · , yN are linearly dependent, i.e.,

det(Ry )= 0 ⇔ k y = 0 for some nonzero vector k ∈ C N +1 .

4

1. Krein Space

In the Krein space setting, all we can deduce from the singularity of Ry is that there exists a linear combination of {y0 , y1 , · · · , yN } that is orthogonal to every vector in L{y}, i.e., that L{y} contains an isotropic vector. This follows by noting that for any complex matrix k1 and for any k in the null space of Ry , we have

k1 Ry k = k1 y, k y = 0,

which shows that the linear combination k y is orthogonal to k1 y for every k1 , i.e., k y is an isotropic vector in L{y}.

1.3 Kalman Filtering Formulation in Krein Spaces Given the state-space model in Krein space x(t + 1) = Φt x(t) + Γt u(t), 0 ≤ t ≤ N, y(t) = Ht x(t) + v(t),

(1.7) (1.8)

and ⎤ ⎡ ⎤ ⎡ ⎡ u(t) u(s) Q(t)δts ⎣ v(t) ⎦ , ⎣ v(s) ⎦ = ⎣ 0 x0 x0 0

⎤ 0 0 ⎦ Π0

0 R(t)δts 0

(1.9)

where x0 is the initial state of the system. Now we shall show that the state-space representation allows us to efficiently compute the innovations by an immediate extension of the standard Kalman filter. Theorem 1.3.1. (Kalman Filter in Krein Space) Consider the Krein statespace representation (1.7)-(1.9). Assume that Ry = [< y(t), y(s) >] is strongly regular1 . Then the innovation can be computed via the formulae ˆ (t), 0 ≤ t ≤ N, w(t) = y(t) − Ht x ˆ (t + 1) = Φt x ˆ (t) + Kp,t (y(t) − Ht x ˆ (t)), x ˆ 0 = 0, x

Kp,t = Φt P (t)Ht [Rw (t)]

−1

,

(1.10) (1.11) (1.12)

where x ˆ(t) is the one-step prediction estimate of x(t), x ˆ0 is the initial estimate,

Rw (t) =< w(t), w(t) >= R(t) + Ht P (t)Ht ,

(1.13)

and P (t) can be recursively computed via the Riccati recursion

P (t + 1) = Φt P (t)Φt − Kp,t Rw (t)Kp,t + Γt Q(t)Γt , P (0) = Π0 . 1

(1.14)

Ry is said to be strongly regular if Ry is nonsingular and all its leading submatrices are nonsingular.

1.4 Two Basic Problems of Quadratic Forms in Krein Spaces

5

1.4 Two Basic Problems of Quadratic Forms in Krein Spaces 1.4.1

Problem 1

Consider the minimization of the following quadratic function over z −1 z Rz Rzy z J(z, y) = , y y Ryz Ry where the central matrix is the inverse of the Gramian matrix z z Rz Rzy , = . y y Ryz Ry

(1.15)

(1.16)

It is known that the above minimization problem is related to the H∞ estimation [30]. Now we give the solution in the following theorem. Theorem 1.4.1. [30] Suppose that both Ry and (1.16) are nonsingular. Then 1. The stationary point zˆ of J(z, y) over z is given by zˆ = Rzy Ry−1 y. 2. The value of J(z, y) at the stationary point is

J(ˆ z , y) = y Ry−1 y. 3. zˆ yields a unique minimum of J(z, y) if and only if Rz − Rzy Ry−1 Ryz > 0. Proof: We note that −1 I 0 Rz Rzy Rz − Rzy Ry−1 Ryz = −1 −Ry Ryz I 0 Ryz Ry −1 I −Rzy Ry . × 0 I

0 Ry

−1

(1.17)

Hence, we can write J(z, y) as

Rz − Rzy Ry−1 Ryz J(z, y) = [ z − y y ] 0 −1 z − Rzy Ry y × . y

Ry−1 Ryz

0 Ry

−1

(1.18)

It now follows by differentiation that the stationary point of J(z, y) is equal to z , y) = y Ry−1 y. From (1.18), it follows that zˆ yields zˆ = Rzy Ry−1 y, and that J(ˆ a unique minimum of J(z, y) if and only if Rz − Rzy Ry−1 Ryz > 0. It should be pointed out that the above minimization problem is related to estimator design, and the presented results in the theorem will be used throughout the monograph.

6

1. Krein Space

1.4.2

Problem 2

Consider the minimization of the following quadratic function over u x0 Rxc0 x0 Rxc0 yc J(x0 , u) = . u Ryc xc0 Ryc u

(1.19)

We have the following results. Theorem 1.4.2. [30] Suppose Ryc and (1.19) are nonsingular. 1. The stationary point u ˆ of J(x0 , u) over u is given by u ˆ = −Ry−1 c Ry c xc x0 . 0 2. The value of J(x0 , u) at the stationary point is ˆ) = x0 Rxc0 − Rxc0 yc Ry−1 J(x0 , u c Ry c xc x0 . 0 3. uˆ yields an unique minimum of J(x0 , u) if and only if Ryc > 0. Proof: Observe that Rxc0 I Rxc0 yc Rxc0 yc Rxc0 − Rxc0 yc Ry−1 c Ry c xc 0 = 0 Ryc xc0 Ryc 0 I I 0 × , I Ry−1 c Rxc y c 0 so we can write J(x0 , u) as

J(x0 , u) = [ x0 x0 Rxc0 yc Ry−1 c + u ] x0 × . Ry−1 c Ry c xc x0 + u 0

Rxc0 − Rxc0 yc Ry−1 c Ry c xc 0 0

0 Ryc

(1.20)

0 Ryc

(1.21)

The desired result then follows. It should be pointed out that the above minimization problem is related to controller design, and the presented results in the theorem will be used later.

1.5 Conclusion In this chapter we have introduced Krein spaces and some basic properties of Krein spaces. Although Krein spaces and Hilbert spaces share many characteristics, they differ in a number of ways, especially the special features of the inner product in a Krein space make it applicable to H∞ problems. The related concepts such as projections and Kalman filtering formulation in Krein spaces have been given which will be applied as important tools to the estimation and control problems to be studied in this monograph. We have also presented two minimization problems associated with two quadratic forms.

2. Optimal Estimation for Systems with Measurement Delays

This chapter studies the optimal estimation for systems with instantaneous and delayed measurements. A new approach termed as re-organized innovation analysis is proposed to handle delayed measurements. The re-organized innovation will play an important role not only in this chapter but also in the rest of the monograph.

2.1 Introduction In this chapter we investigate the minimum mean square error (MMSE) estimation problem for systems with instantaneous and delayed measurements. Such problem has important applications in many engineering fields such as communications and sensor fusion [42]. As is well known, the optimal estimation problem for discrete-time systems with known delays can be approached by using system augmentation in conjunction with standard Kalman filtering [3]. However, the augmentation method generally leads to a much expensive computational cost, especially when the delays are large. Our aim here is to present a simple Kalman filtering solution to such problem by adopting the so-called re-organized innovation approach developed in our work [102][107]. It will be shown that the optimal estimator can be computed in terms of the same number of Riccati difference equations (RDEs) (with order of the system ignoring the delays) as that of the measurement channels. The proposed approach in this chapter forms the basis for solving other related problems that are to be studied in the rest of the monograph.

2.2 Single Measurement Delay Case In this section we shall study the Kalman filtering problem for time-varying systems with measurement delays. We consider the system with instantaneous and single delayed measurements described by x(t + 1) = Φx(t) + Γ e(t), H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 7–26, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

(2.1)

8


y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t),

(2.2) (2.3)

where x(t) ∈ Rn is the state, e(t) ∈ Rr is the input noise, y(0) (t) ∈ Rp0 and y(1) (t) ∈ Rp1 are respectively the instantaneous and delayed measurements, v(0) (t) ∈ Rp0 and v(1) (t) ∈ Rp1 are the measurement noises. d is the measurement delay which is a known integer. The initial state x(0) and e(t), v(0) (t) and v(1) (t) are mutually uncorrelated white noises with zero means and known co-

(s) = variance matrices E [x(0)x (0)] = P0 , E [e(t)e (s)] = Qe δts , E v(0) (t)v(0) Qv(0) δts and E v(1) (t)v(1) (s) = Qv(1) δts , respectively. In the above, E(·) denotes the mathematical expectation and ⎧ ⎨ 1, i = j, δij = ⎩ 0, i = j.

With the delayed measurement in (2.3), the system (2.1)-(2.3) is not in a standard form to which the standard Kalman filtering is applicable. Let y(t) denote the observation of the state x(t) of the system (2.1)-(2.3): ⎧ ⎪ ⎨ y(0) (t), 0≤t t1 , w(s, 1) = y1 (s) − y

ˆ 2 (s, 2), w(s, 2) = y2 (s) − y

ˆ 2 (0, 2) = 0, 0 ≤ s ≤ t1 , y

(2.22) (2.23)

ˆ 1 (s, 1) with s > t1 +1 is the projection of y1 (s) where, similar to Definition 2.2.1, y onto the linear space formed by {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s − 1)} ˆ 2 (s, 2) with s ≤ t1 + 1 is the projection of y2 (s) onto the linear space and y formed by {y2 (0), · · · , y2 (s − 1)}. We then have the following relationships ˜ (s, 1) + v1 (s), w(s, 1) = H1 x ˜ (s, 2) + v2 (s), w(s, 2) = H2 x

(2.24)

˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ (s, 2) = x(s) − x ˆ (s, 2), x

(2.26) (2.27)

(2.25)

where

ˆ (s, 1) and x ˆ (s, 2) are as in Definition 2.2.1. It is clear that x ˜ (s + 1, 1) = and x ˜ (s + 1, 2) when s = t1 . The following lemma shows that {w(·, ·)} is in fact an x innovation sequence. Lemma 2.2.1. {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is the innovation sequence which spans the same linear space as that of (2.13) or equivalently L {y(0), · · · , y(t)}. Proof: First, it is readily seen from (2.23) that w(s, 2), s ≤ t1 ( or w(s, 1), s > t1 ) is a linear combination of the observation sequence {y2 (0), · · · , y2 (s)} (or {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s)}). Conversely, y2 (s), s ≤ t1 , (or y1 (s), s > t1 ) can be given in terms of a linear combination of {w(0, 2), · · · , w(s, 2)} ( or {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}). Thus, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} spans the same linear space as L {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (t)} or equivalently L {y(0), · · · , y(t)}. Next, we show that w(·, ·) is an uncorrelated sequence. In fact, for any s > t1 and τ ≤ t1 , from (2.24) we have ˜ (s, 1)w (τ, 2)] + E [v2 (s)w (τ, 2)] . E [w(s, 1)w (τ, 2)] = E [H1 x

(2.28)

˜ (s, 1) is the state prediction error, it folNote that E [v2 (s)w (τ, 2)] = 0. Since x lows that E [˜ x(s, 1)w (τ, 2)] = 0, and thus E [w(s, 1)w (τ, 2)] = 0, which implies that w(τ, 2) (τ ≤ t1 ) is uncorrelated with w(s, 1) (s > t1 ). Similarly, it can be verified that w(s, 2) is uncorrelated with w(τ, 2) and w(s, 1) is uncorrelated with w(τ, 1) for s = τ . Hence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is an innovation sequence. ∇

12


The white noise sequence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)} is termed as re-organized innovation sequence associated with the measurement sequence {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (t)}. Similarly, for any s > t1 , the sequence {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} is also termed as re-organized innovation sequence associated with the measurement sequence {y2 (0), · · · , y2 (t1 ); y1 (t1 + 1), · · · , y1 (s)}. The re-organized innovation sequence will play a key role in deriving the optimal estimator in the later subsections. 2.2.3

Riccati Difference Equation

For a given t > d, denote

P2 (s) = E [˜ x(s, 2)˜ x (s, 2)] , 0 ≤ s ≤ t1 ,

P1 (s) = E [˜ x(s, 1)˜ x (s, 1)] , s > t1 .

(2.29) (2.30)

˜ (t1 + 1, 2), it is obvious ˜ (t1 + 1, 1) = x We note that in view of the fact that x that P1 (t1 + 1) = P2 (t1 + 1). Now, it follows from (2.24) and (2.25) that the innovation covariance matrix of w(·, ·) is given by

Qw (s, 2) = E [w(s, 2)w (s, 2)] = H2 P2 (s)H2 + Qv2 , 0 ≤ s ≤ t1 (2.31) and

Qw (s, 1) = E [w(s, 1)w (s, 1)] = H1 P1 (s)H1 + Qv1 , s > t1 . (2.32) We have the following results Theorem 2.2.1. For a given t > d, the covariance matrices P2 (·) and P1 (·) can be calculated as follows. – P2 (s), 0 < s ≤ t1 , is calculated by the following standard RDE: P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qw (s, 2)K2 (s) + Γ Qe Γ , P2 (0) = P0 ,

(2.33)

where Qw (s, 2) is as in (2.31) and K2 (s) = ΦP2 (s)H2 Q−1 w (s, 2).

(2.34)

– P1 (s), t1 + 1 < s ≤ t, is given by P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s, 1)K1 (s) + Γ Qe Γ , P1 (t1 + 1) = P2 (t1 + 1),

(2.35)

where Qw (s, 1) is as in (2.32) and K1 (s) = ΦP1 (s)H1 Q−1 w (s, 1).

(2.36)

2.2 Single Measurement Delay Case

13

Proof: It is obvious that P2 (s + 1) is the covariance matrix of the one step ahead prediction error of the state x(s + 1) associated with the system (2.1) and (2.18). Thus following the standard Kalman filtering theory, P2 (s + 1) satisfies the RDE(2.33). ˆ (s + 1, 1) (s > t1 ) is the projection of the state On the other hand, note that x x(s + 1) onto the linear space L {w(0, 2), · · · , w(t1 , 2), w(t1 + 1, 1), · · · , w(s, 1)} . ˆ (s + 1, 1) can be calculated by using Since w(·, ·) is a white noise, the estimator x the projection formula as ˆ (s + 1, 1) x = P roj {x(s + 1 | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s − 1, 1)} +P roj {x(s + 1 | w(s, 1)} = Φˆ x(s, 1) + ΦE [x(s)˜ x (s, 1)] H1 Q−1 w (s, 1)w(s, 1) = Φˆ x(s, 1) + ΦP1 (s)H1 Q−1 w (s, 1)w(s, 1).

(2.37)

It is readily obtained from (2.1) and (2.37) that ˜ (s + 1, 1) = x(s + 1) − x ˆ (s + 1, 1) x = Φ˜ x(s, 1) + Γ e(s) − ΦP1 (s)H1 Q−1 w (s, 1)w(s, 1). (2.38) ˜ (s+ 1, 1) is uncorrelated with w(s, 1) and so is x ˜ (s, 1) with e(s), it follows Since x from the above equation that P1 (s + 1) + ΦP1 (s)H1 Q−1 w (s, 1)H1 P1 (s)Φ = ΦP1 (s)Φ + Γ Qe Γ ,

∇

which is (2.35). 2.2.4

Optimal Estimate x ˆ(t | t)

In this section we shall give a solution to the optimal filtering problem. Based on the discussion in the previous subsection, the following results are obtained by applying the re-organized innovation sequence. Theorem 2.2.2. Consider the system (2.1)-(2.3). Given d > 0, the optimal ˆ (t | t) is given by filter x ˆ (t, 1)] , ˆ (t | t) = x ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1) [y1 (t) − H1 x

(2.39)

ˆ (t, 1) is calculated recursively as where x ˆ (s + 1, 1) = Φ1 (s)ˆ x x(s, 1) + K1 (s)y1 (s), t1 + 1 ≤ s < t,

(2.40)

K1 (s) is as in (2.36) and Φ1 (s) = Φ − K1 (s)H1 ,

(2.41)

14


while Qw (s, 1) is as in (2.32) and P1 (s) is computed by (2.35). The initial value ˆ (t1 + 1, 1) = x ˆ (t1 + 1, 2), and x ˆ (t1 + 1, 2) is calculated by the following Kalman x filtering ˆ (s + 1, 2) = Φ2 (s)ˆ ˆ (0, 2) = 0, x x(s, 2) + K2 (s)y2 (s), 0 ≤ s ≤ t1 , x (2.42) where K2 (s) is as in (2.34) and Φ2 (s) = Φ − K2 (s)H2 .

(2.43)

ˆ (t | t) is the projection of the state x(t) onto Proof: By applying Lemma 2.2.1, x the linear space L{w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(t, 1)}. Since w(·, ·) is ˆ (t | t) is calculated by using the projection formula as a white noise, the filter x ˆ (t | t) x = P roj {x(t) | w(0, 2), · · · , w(t1 , 2), w(t1 + 1, 1), · · · , w(t − 1, 1)} +P roj {x(t) | w(t, 1)} ˆ (t, 1) + E [x(t)w (t, 1)] Q−1 =x w (t, 1)w(t, 1) ˆ (t, 1)] , ˆ (t, 1) + P1 (t)H1 Q−1 (2.44) (t, 1) [y1 (t) − H1 x =x w ˆ (s+1, 1) (s > t1 ) is the projection which is (2.39). Similarly, from Lemma 2.2.1, x of the state x(s + 1) onto the linear space L{w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}. Thus, it follows from the projection formula that ˆ (s + 1, 1) x = P roj {x(s + 1) | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} = Φˆ x(s, 1) + Γ P roj {e(s) | w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)} . (2.45) Noting that e(s) is uncorrelated with the innovation sequence, {w(0, 2), · · · , w(t1 , 2); w(t1 + 1, 1), · · · , w(s, 1)}, we have ˆ (s + 1, 1) = Φˆ x x(s, 1) + ΦE [x(s)w (s, 1)] Q−1 w (s, 1)w(s, 1)

ˆ (s, 1)] = Φˆ x(s, 1) + ΦP1 (s)H1 Q−1 w (s, 1) [y1 (s) − H1 x = Φ1 (s)ˆ x(s, 1) + K1 (s)y1 (s),

(2.46)

ˆ (t1 +1, 1) = x ˆ (t1 +1, 2), which is the which is (2.40). The initial value of (2.40) is x standard Kalman filter of (2.1) and (2.18), and is obviously given by (2.42). ∇

2.2 Single Measurement Delay Case

15

Remark 2.2.3. For k ≥ m, denote Φ1 (k, m) = Φ1 (k − 1) · · · Φ1 (m), Φ1 (m, m) = In ,

(2.47)

where Φ1 (s) is as in (2.41). Then the recursion of (2.40) can be easily rewritten as ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1

Φ1 (t, s + 1)K1 (s)y1 (s). (2.48)

s=t1 +1

ˆ (t, 1) can be obtained with the From (2.48), it is clear that the estimator x ˆ (t1 + 1, 2), which is given by (2.42). initial value of x Remark 2.2.4. The Kalman filtering solution for system (2.1)-(2.3) with delayed measurement has been given by applying the re-organized innovation analysis. Different from the standard Kalman filtering approach, our approach consists of two parts. The first part is given by (2.40) and (2.35), which is the Kalman filtering for the system (2.1) and (2.19). The second part is given by (2.42) and (2.33), which is the Kalman filtering for the system (2.1) and (2.18). Observe that the solution only relies on two RDEs of dimension n × n. This is in comparison with the traditional augmentation method where one Riccati equation of dimension (n + d × p1 ) × (n + d × p1 ) is involved. In the following subsection, we shall demonstrate that the proposed method indeed possesses computational advantages over the latter. Remark 2.2.5. The above re-organized innovation analysis in Hilbert space can be extended to Krein space to address the more complicated H∞ fixed-lag smoothing and other H∞ estimation problems for time-delay systems which will be demonstrated in the later chapters. 2.2.5

Computational Cost

W shall now compare the computational cost of the presented approach and the system augmentation method. As additions are much faster than multiplications and divisions, it is the number of multiplications and divisions that is used as the operation count. Let M D denote the number of multiplications and divisions. First, note that the algorithm by Theorem 2.2.2 can be summarized as 1. Compute matrix P2 (t1 ) using the RDE (2.33). 2. Compute P1 (s) for t1 + 1 ≤ s < t using (2.35). ˆ (t | t) using (2.39)-(2.42). 3. Compute x ˆ (t | t) in one It is easy to know that the total M D count for obtaining x iteration, denoted as M Dnew , is given by ! M Dnew = 3n3 + (3p0 + r)n2 + 2p20 n + p30 d + (6p1 + 1)n2 ! + 2(p0 + p1 )2 − 2p20 + 4p0 p1 + 2p21 + p0 + p1 n + 2(p0 + p1 )3 + (p0 + p1 )2 − 2p30 .

(2.49)

16


On the other hand, recall the Kalman filtering for the augmented state-space ˆ a (t | t) is computed by model (2.5)-(2.10). The optimal filter x ˆ a (t + 1 | t + 1) = Φa x ˆ a (t | t) + Pa (t)HL a (t)Q−1 x w (t) y(0) (t) ˆ a (t | t) , × − Ha (t + 1)Φa x y(1) (t)

(2.50)

where the matrix Pa (t) satisfies the following RDE Pa (t + 1) = Φa Pa (t)Φa − Φa Pa (t)HL a (t)Q−1 w (t)HLa (t)Pa (t)Φa + Γa Qe Γa , (2.51) Qv(0) 0 with Qw (t) = HLa (t)Pa (t)HLa (t) + Qvs and Qvs = Qv2 = . In 0 Qv(1) view of the special structure of the matrices Φa , Γa , HLa (t), the calculation burden for the RDE (2.51) can be reduced. We partition Pa (t) as

Pa (t) = {Pa,ij (t), 1 ≤ i ≤ d + 1, 1 ≤ j ≤ d + 1}, where Pa,11 (t) and Pa,ii (t), i > 1 are of the dimensions n × n and p1 × p1 , respectively. The RDE (2.51) can be simplified as [91]: −1 Πa (t) = Pa (t) − Pa,1 (t)H(0) [Qv(0) + H(0) Pa,11 (t)H(0) ] H(0) Pa,1 (t), (t) Σa (t) = Πa (t) − Πa,d+1 (t)[Qv(1) + Πa,(d+1)(d+1) (t)]−1 Πa,d+1

Pa (t + 1) = Γa Qe Γa + Φa Σa (t)Φa ,

(2.52)

where Pa,i (t) and Πa,i (t) represent the ith column blocks of Pa (t) and Πa (t), respectively. Suppose that Σa (t) is partitioned similarly to Pa (t) and Πa (t). By taking into account the structure of the matrices Φa and Γa , (2.52) can be rewritten as Pa (t + 1) = ΦΣa,11 (t)Φ + Γ Qe Γ ⎢ H(1) Σa,11 (t)Φ ⎢ ⎢ Σa,21 (t)Φ ⎢ ⎢ .. ⎣ . Σa,d1 (t)Φ ⎡

ΦΣa,11 (t)H(1) H(1) Σa,11 (t)H(1) Σa,21 (t)H(1) .. . Σa,d1 (t)H(1)

ΦΣa,12 (t) H(1) Σa,12 (t) Σa,22 (t) .. . Σa,d2 (t)

··· ··· ··· ··· ···

⎤ ΦΣa,1d (t) H(1) Σa,1d (t) ⎥ ⎥ Σa,2d (t) ⎥ , ⎥ ⎥ .. ⎦ . Σa,dd (t) (2.53)

and the optimal filter (2.50) is then ˆ a (t + 1 | t + 1) x ⎡ Φˆ x (t | t) ⎤

⎡ a1 Pa,11 (t)H(0) ˆ a1 (t | t) ⎥ ⎢ P ⎢ H(1) x a,21 (t)H(0) ⎢ ⎥ ⎢ ˆ x (t | t) ⎢ ⎥ a2 =⎢ .. ⎥+⎢ .. ⎣ ⎦ ⎣ . . (t)H(0) P a,(d+1)1 ˆ ad (t | t) x y(0) (t) xa1 (t | t) H(0) Φˆ × − , ˆ ad (t | t) x y(1) (t)

⎤ Pa,1(d+1) (t) Pa,2(d+1) (t) ⎥ ⎥ −1 ⎥ Qw (t) .. ⎦ . Pa,(d+1)(d+1) (t) (2.54)

2.3 Multiple Measurement Delays Case

17

ˆ a (t | t) = [ x ˆ a1 (t | t) x ˆ a2 (t | t) · · · x ˆ a d+1 (t | t) ] . Let M Daug denote where x ˆ a (t + 1 | t + 1) by (2.52), (2.53) and (2.54). the operation number of calculating x We have M Daug = p21 (n + p1 )d2 + 4p1 n2 + (4p21 + 3p0 p1 )n ! +(p20 + p0 + p21 + p1 )p1 d + 4n3 + (4p0 + 3p1 + 1)n2 +(2p20 + 2p0 − p21 + 2p1 )n + p30 + p20 + p31 .

(2.55)

From (2.49) and (2.55), it is clear that M Daug is of magnitude O(d2 ) whereas M Dnew is linear in d. Thus, when the delay d is sufficiently large, it is easy to know that M Daug > M Dnew . Moreover, the larger the d, the larger the ratio MDaug MDnew . To see this, we consider one example. Example 2.2.1. Consider the system (2.1)-(2.3 ) with n = 3, p0 = 1, r = 1 and p1 = 3. The M D numbers of the proposed approach and the system augmentation approach are compared in Table 2.1 for various values of d.

Table 2.1. Comparison of the Computational Costs d M Dnew M Daug M Daug M Dnew

1 629 605 0.9618

2 753 1052 1.3971

3 877 1607 1.8324

6 1249 3920 3.1385

12 1993 11462 5.7511

2.3 Multiple Measurement Delays Case In this section we shall extend the study of the last section to consider systems with multiple delayed measurements. We consider the linear discrete-time system x(t + 1) = Φx(t) + Γ e(t),

(2.56)

where x(t) ∈ Rn is the state and e(t) ∈ Rr is the system noise. The state x(t) is observed by l + 1 different channels with delays as described by y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, 1, · · · , l,

(2.57)

where, without loss of generality, the time delays di , i = 0, 1, · · · , l are assumed to be in a strictly increasing order: 0 = d0 < d1 < · · · < dl , y(i) (t) ∈ Rpi is the ith delayed measurement, and v(i) (t) ∈ Rpi is the measurement noise. The initial state x(0) and the noises e(t) and v(i) (t), i = 0, 1, · · · , l are mutually uncorrelated white noises with zero means and covariance matrices as E [x(0)x (0)] = P0 , (j) = Qv(i) δkj , respectively. E [e(k)e (j)] = Qe δkj , and E v(i) (k)v(i)

18


Observe from (2.57) that y(i) (t) is in fact an observation of the state x(t − di ) at time t, with delay di . Let y(t) denote all the observations of the system (2.56)-(2.57) at time t, then we have ⎡ ⎤ y(0) (t) ⎢ ⎥ .. y(t) = ⎣ (2.58) ⎦ , di−1 ≤ t < di , . y(i−1) (t) and for t ≥ dl , ⎤ y(0) (t) ⎥ ⎢ y(t) = ⎣ ... ⎦ . y(l) (t) ⎡

(2.59)

The linear optimal estimation problem can be stated as: Given the observation ˆ (t | t) sequence {{y(s)}ts=0 }, find a linear least mean square error estimator x of x(t). Since the measurement y(t) is associated with states at different time instants due to the delays, the standard Kalman filtering is not applicable to the estimation problem. Similar to the single delayed measurement case, one may convert the problem into a standard Kalman filtering estimation by augmenting the state. However, the computation cost of the approach may be very high due to a much increased state dimension of the augmented system [3]. In this section, we shall extend the re-organized innovation approach of the last section to give a simpler derivation and solution for the optimal estimation problem associated with systems of multiple delayed measurements. Throughout the section we denote that

ti = t − di , i = 0, 1, · · · , l and assume that t ≥ dl for the convenience of discussions. 2.3.1

Re-organized Measurements

In this subsection, the instantaneous and l-delayed measurements will be reorganized as delay free measurements so that the Kalman filtering is applicable. As is well known, given the measurement sequence {y(s)}ts=0 , the optiˆ (t | t) is the projection of x(t) onto the linear space mal state estimator x L {{y(s)}ts=0 } [3, 38]. Note that the linear space L {{y(s)}ts=0 } is equivalent to the following linear space L {yl+1 (0), · · · , yl+1 (tl ); · · · ; yi (ti + 1), · · · , yi (ti−1 ); · · · ; y1 (t1 + 1), · · · , y1 (t)} ,

(2.60)


19

where ⎡

⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ ⎦. . y(i−1) (s + di−1 )

(2.61)

It is clear that yi (s) satisfies yi (t) = Hi x(t) + vi (t),

i = 1, · · · , l + 1,

(2.62)

with ⎡

⎡ ⎤ ⎤ H(0) v(0) (t) ⎢ ⎢ ⎥ ⎥ .. .. Hi = ⎣ ⎦ , vi (t) = ⎣ ⎦. . . v(i−1) (t + di−1 ) H(i−1) (t + di−1 )

(2.63)

It is easy to know that vi (t) is a white noise of zero mean and covariance matrix Qvi = diag{Qv(0) , · · · , Qv(i−1) }, i = 1, 2, · · · , l + 1.

(2.64)

Note that the measurements in (2.62), termed as re-organized measurements of {{y(s)}ts=0 }, are no longer with any delay. 2.3.2

Re-organized Innovation Sequence

In this subsection we shall define the innovation associated with the re-organized measurements (2.60). First, we introduce a similar definition of projection as in Definition 2.2.1. Definition 2.3.1. The estimator x ˆ(s, i) for ti + 1 < s ≤ ti−1 is the optimal estimation of x(s) given the observation sequence: {yl+1 (0), · · · , yl+1 (tl );

· · · ; yi (ti + 1), · · · , yi (s − 1)}.

(2.65)

For s = ti + 1, x ˆ(s, i) is the optimal estimation of x(s) given the observation sequence: {yl+1 (0), · · · , yl+1 (tl );

· · · ; yi+1 (ti+1 + 1), · · · , yi+1 (ti )}.

(2.66)

From the above definition we can introduce the following stochastic sequence.

ˆ i (s, i), wi (s, i) = yi (s) − y

(2.67)

ˆ i (s, i) is the optimal estimation of yi (s) given the observawhere for s > ti + 1, y ˆ i (s, i) is the optimal estimation of yi (s) tion sequence of (2.65) and for s = ti +1, y given the observation sequence (2.66). For i = l + 1, it is clear that wl+1 (s, l + 1)

20


is the standard Kalman filtering innovation sequence for the system (2.56) and (2.62) for i = l + 1. In view of (2.62), it follows that ˜ (s, i) + vi (s), i = 1, · · · , l + 1, wi (s, i) = Hi x

(2.68)

where ˜ (s, i) = x(s) − x ˆ (s, i), x

i = 1, · · · , l + 1

(2.69)

is the one step ahead prediction error of the state x(s) based on the observations (2.65) or (2.66). The following lemma shows that wi (s, i), i = 1, · · · , l + 1 form an innovation sequence associated with the re-organized observations (2.60). Lemma 2.3.1 {wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (ti−1 , i); · · · ; w1 (t1 + 1, 1), · · · , w1 (t, 1)}

(2.70)

is an innovation sequence which spans the same linear space as: L {yl+1 (0), · · · , yl+1 (tl ); · · · ; yi (ti + 1), · · · , yi (ti−1 ); · · · ; y1 (t1 + 1) · · · , y1 (t)} , or equivalently L{y(0), · · · y(t)}. Proof: The proof is very similar to the single delay case of the last section.

∇

The white noise sequence, {wl+1 (0), · · · , wl+1 (tl ); · · · ; wi (ti + 1), · · · , wi (ti−1 ); · · · ; w1 (t1 + 1), · · · , w1 (t)} is termed as re-organized innovation sequence associated with the measurement sequence {y(0), · · · y(t)}. 2.3.3

Riccati Equation

Let

˜ (s, i)] , i = l + 1, · · · , 1 Pi (s) = E [˜ x(s, i) x

(2.71)

be the covariance matrix of the state estimation error. For delay free systems, it is well known that the covariance matrix of the state filtering error satisfies a Riccati equation. Similarly, we shall show that the covariance matrix Pi (s) defined in (2.71) obeys certain Riccati equation.


21

Theorem 2.3.1. For a given t > dl , the matrix of Pl+1 (tl + 1) is calculated as Pl+1 (tl + 1) = ΦPl+1 (tl )Φ − Kl+1 (tl )Qw (tl , l + 1)Kl+1 (tl ) + Γ Qe Γ , (2.72) Pl+1 (0) = E[x(0)x (0)] = P0 ,

where Kl+1 (tl ) = ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1), + Qvl+1 . Qw (tl , l + 1) = Hl+1 Pl+1 (tl )Hl+1

(2.73) (2.74)

With the calculated Pl+1 (tl + 1), the matrices of Pi (s) for i = l, · · · , 1 and ti < s ≤ ti + di − di−1 = ti−1 are calculated recursively as Pi (s + 1) = ΦPi (s)Φ − Ki (s)Qw (s, i)Ki (s) + Γ Qe Γ , Pi (ti + 1) = Pi+1 (ti + 1), i = l, · · · , 1,

(2.75)

where Ki (s) = ΦPi (s)Hi Q−1 w (s, i),

Qw (s, i) = E [wi (s, i)wi (s, i)] = Hi Pi (s)Hi + Qvi .

(2.76) (2.77)

ˆ (s + 1, i) is the projection of the state x(s + 1) onto the linear Proof: Note that x space L{wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)}. Since ˆ (s + 1, i) is calculated by using the projection w is a white noise, the estimator x formula as ˆ (s + 1, i) x = P roj {x(s + 1) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s − 1, i)} +P roj{x(s + 1) | wi (s, i)} = Φˆ x(s, i) + ΦE [x(s)x (s, i)] Hi Q−1 w (s, i)wi (s, i) −1 = Φˆ x(s, i) + ΦPi (s)Hi Qw (s, i)wi (s, i).

(2.78)

It is easily obtained from (2.56) and (2.78) that ˜ (s + 1, i) = x(s + 1) − x ˆ (s + 1, i) x = Φ˜ x(s, i) + Γ e(s) − ΦPi (s)Hi Q−1 w (s, i)wi (s, i).

(2.79)

˜ (s, i) with e(s), it ˜ (s + 1, i) is uncorrelated with wi (s, i) and so is x Since x follows from (2.79) that Pi (s + 1) + ΦPi (s)Hi Q−1 w (s, i)Hi Pi (s)Φ = ΦPi (s)Φ + Γ Qe Γ ,

which is (2.75). Similarly, we can prove (2.72).

(2.80) ∇

22


2.3.4

Optimal Estimate x ˆ(t | t)

In this subsection we shall give a solution to the optimal filtering problem by applying the re-organized innovation sequence and the Riccati equations obtained in the last subsection. ˆ (t | t) Theorem 2.3.2. Consider the system (2.56)-(2.57). The optimal filter x is given by ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t),(2.81) ˆ (t, 1) is computed through where Qw (t, 1) = H1 P1 (t)H1 +Qv1 and the estimator x the following steps ˆ (tl + 1, l + 1) with the following standard Kalman filtering – Step 1: Calculate x ˆ (tl + 1, l + 1) = Φl+1 (tl )ˆ x x(tl , l + 1) + Kl+1 (tl )yl+1 (tl ), ˆ (0, l + 1) = 0, x

(2.82)

where Φl+1 (tl ) = Φ − Kl+1 (tl )Hl+1 , Kl+1 (tl ) = ΦPl+1 (tl )Hl+1 Q−1 w (tl , l + 1), and Pl+1 (tl ) is computed by (2.72), with Pl+1 (0) = P0 . ˆ (t, 1) = x ˆ (t0 , 1) is calculated by the following backward itera– Step 2: Next, x ˆ (tl + 1, l + 1): tion with the initial condition x

ˆ (ti−1 , i) = Φi (ti−1 , ti + 1)ˆ x x(ti + 1, i + 1) ti−1 −1

+

Φi (ti−1 , s + 1)Ki (s)yi (s), i = l, l − 1, · · · , 1

s=ti +1

(2.83) where for k ≥ m, Φi (k, m) = Φi (k − 1) · · · Φi (m), Φi (m, m) = In , Ki (s) = ΦPi (s)Hi Q−1 w (s, i),

(2.84) (2.85)

while Φi (k) = Φ − Ki (k)Hi , Qw (s, i) = Hi Pi (s)Hi + Qvi ,

(2.86) (2.87)

and Pi (s) is calculated by (2.75). ˆ (t | t) is the projection of the state x(t) onto Proof: By applying Lemma 2.3.1, x ˆ (t | t) is calculated the linear space of (2.70). Since w is a white noise, the filter x by using the projection formula as ˆ (t | t) = P roj {x(t) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); x · · · ; w1 (t1 + 1, 1), · · · , w1 (t − 1, 1)} + P roj{x(t) | w1 (t, 1)}. ˆ (t, 1) + E [x(t)w1 (t, 1)] Q−1 =x w (t, 1)w1 (t, 1) −1 ˆ (t, 1)] ˆ (t, 1) + P1 (t)H1 Qw (t, 1) [y1 (t) − H1 x =x ! −1 ˆ = In −P1 (t)H1 Q−1 x (t, 1) + P (t, 1)H 1 1 (t)H1 Qw (t, 1)y1 (t), (2.88) w


23

ˆ (s + 1, i) is the projection of the state which is (2.81). From Lemma 2.3.1, x x(s + 1) onto the linear space L{wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)}, it follows from the projection formula that ˆ (s + 1, i) x = P roj {x(s + 1) | wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)} = Φˆ x(s, i) + P roj{x(s + 1) | wi (s, i)} + Γ P roj {e(s) | wl+1 (0, l + 1), · · · , (2.89) wl+1 (tl , l + 1); · · · ; wi (ti + 1, i), · · · , wi (s, i)} . Noting that e(s) is uncorrelated with the innovation {wl+1 (0, l + 1), · · · , wl+1 (tl , l + 1), · · · ; wi (ti + 1, i), · · · , wi (s, i)}, we have ˆ (s + 1, i) = Φˆ x x(s, i) + ΦE [x(s)wi (s, i)] Q−1 w (s, i)wi (s) −1 ˆ (s, i)] , = Φˆ x(s, i) + ΦPi (s)Hi Qw (s, i) [yi (s) − Hi x which can be rewritten as ˆ (s + 1, i) = Φi (s)ˆ x x(s, i) + Ki (s)yi (s),

(2.90)

ˆ (ti + 1, i) = x ˆ (ti + 1, i + 1). For each i, it follows from with the initial condition x (2.90) that ˆ (ti−1 , i) x = Φi (ti−1 − 1)ˆ x(ti−1 − 1, i) + Ki (ti−1 − 1)yi (ti−1 − 1) x(ti−1 − 2, i) = Φi (ti−1 − 1)Φi (ti−1 − 2)ˆ +Φi (ti−1 − 1)Ki (ti−1 − 2)yi (ti−1 − 2) + Ki (ti−1 − 1)yi (ti−1 − 1) = ··· = Φi (ti−1 − 1)Φi (ti−1 − 2) · · · Φi (ti + 1)ˆ x(ti + 1, i) + ti−1 −1

Φi (ti−1 − 1)Φi (ti−1 − 2) · · · Φi (s + 1)Ki (s)yi (s)

s=ti +1 ti−1 −1

= Φi (ti−1 , ti + 1)ˆ x(ti + 1, i) +

Φi (ti−1 , s + 1)Ki (s)yi (s).

(2.91)

s=ti +1

The proof is completed.

∇

Remark 2.3.1. The Kalman filtering solution for system (2.56)-(2.57) has been derived by applying the re-organized innovation sequence. Different from the standard Kalman filtering approach, the computation procedure at a given time instant t is summarized as: ˆ (tl + 1, l + 1) by (2.72) and (2.82) with initial – Calculate Pl+1 (tl + 1) and x ˆ (tl , l + 1), respectively. values Pl+1 (tl ) and x

24


ˆ (ti−1 , i) by the backward recursive iteration (2.83) for i = l, · · · , 1 – Calculate x ˆ (t, 1) = x ˆ (t0 , 1). and set x ˆ (t | t) using (2.81). – Finally, compute x Observe that the above solution relies on l + 1 Riccati recursions of dimension n× n. 2.3.5

Numerical Example

In this subsection, we present one numerical example to illustrate the computation procedure of the proposed Kalman filtering. Consider the system (2.56)(2.57) with l = 2, d1 = 20, d2 = 40 and 0.8 0 0.6 Φ= , Γ = , 0.9 0.5 0.5 H(0) = [ 1

2 ] , H(1) = [ 2

0.5 ] , H(2) = [ 3 1 ] .

The initial state x(0), the noises e(t), v(0) (t), v(1) (t) and v(2) (t) are with zero means and unity covariance matrices, i.e., P0 = I2 , Qe = 1, Qv(0) = Qv(1) = Qv(2) = 1. Then, it is easy to know that ⎡ ⎤ 1 0 0 1 0 Qv1 = 1, Qv2 = , Qv3 = ⎣ 0 1 0 ⎦ , 0 1 0 0 1 ⎡ ⎤ 1 2 1 2 , H3 = ⎣ 2 0.5 ⎦ , H1 = [ 1 2 ] , H2 = 2 0.5 3 1 ⎡ ⎤ y(0) (s) y(0) (s) y1 (s) = y(0) (s), y2 (s) = , y3 (s) = ⎣ y(1) (s + 20) ⎦ . y(1) (s + 20) y(2) (s + 40) – For 0 ≤ t < d1 = 20, only one channel measurement is available and is ˆ (t | t) is computed by delay-free, the optimal estimator x ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.92) ˆ (t, 1) is the standard Kalman filter which is given by where x ˆ (t + 1, 1) = Φ1 (t)ˆ ˆ (0, 1) = 0, x x(t, 1) + K1 (t)y1 (t), x

(2.93)

while Φ1 (t) = Φ − K1 (t)H1 , K1 (t) = ΦP1 (t)H1 Q−1 w (t, 1), Qw (t, 1) = H1 P1 (t)H1 + 1,

(2.94) (2.95) (2.96)

and P1 (t) is the solution to the following Riccati equation P1 (t + 1) = ΦP1 (t)Φ −K1 (t)Qw (t, 1)K1 (t)+Γ Γ , P1 (0) = P0 . (2.97)


25

– For 20 = d1 ≤ t < 40, there are two channel measurements available, and ˆ (t | t) is given by (2.81) as the optimal estimator x ! ˆ (t, 1) + P1 (t)H1 Q−1 ˆ (t | t) = In − P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.98) ˆ (t, 1) is computed in the following steps: where x ˆ (t1 + 1, 2) (t1 = t − d1 = t − 20) by the Kalman filtering (2.82) i) Calculate x with l = 1 as ˆ (0, 2) = 0, ˆ (t1 + 1, 2) = Φ2 (t1 )ˆ x(t1 , 2) + K2 (t1 )y2 (t1 ), x x

(2.99)

where Φ2 (t1 ) = Φ − K2 (t1 )H2 , K2 (t1 ) = ΦP2 (t1 )H2 Q−1 w (t1 , 2) and P2 (t1 ) is computed by P2 (t1 + 1) = ΦP2 (t1 )Φ − K2 (t1 )Qw (t1 , 2)K2 (t1 ) + Γ Γ , P2 (0) = P0 . (2.100) ˆ (t, 1) in (2.98) is then calculated by (2.83) with i = 1 as ii) x ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1

Φ1 (t, s + 1)K1 (s)y1 (s),

s=t1 +1

(2.101) ˆ (t1 + 1, 2) is given by (2.99), where x Φ1 (t, s + 1) = Φ1 (t − 1) · · · Φ1 (s + 1),

(2.102)

Φ1 (s), K1 (s) and Qw (s, 1) are respectively as in (2.94)-(2.96), and P1 (s) is the solution to P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s, 1)K1 (s) + Γ Γ , P1 (t1 + 1) = P2 (t1 + 1).

(2.103)

– For t ≥ 40, the optimal estimator x(t | t) is given from (2.81) as ! ˆ (t | t) = In − P1 (t)H1 Q−1 ˆ (t, 1) + P1 (t)H1 Q−1 x w (t, 1)H1 x w (t, 1)y1 (t), (2.104) ˆ (t, 1) is computed in the following where x ˆ (t2 + 1, 3) (t2 = t − d2 = t − 40) with the following standard i) Calculate x Kalman filtering (2.82) with l = 2 as ˆ (t2 + 1, 3) = Φ3 (t2 )ˆ ˆ (0, 3) = 0, (2.105) x x(t2 , 3) + K3 (t2 )y3 (t2 ), x

26


where Φ3 (t2 ) = Φ − K3 (t2 )H3 , K3 (t2 ) = ΦP3 (t2 )H3 Q−1 w (t2 , 3) and P3 (t2 ) is computed by P3 (t2 + 1) = ΦP3 (t2 )Φ − K3 (t2 )Qw (t2 , 3)K3 (t2 ) + Γ Γ , (2.106) P3 (0) = E[x(0)x (0)] = P0 . ˆ (t1 + 1, 2) is calculated by (2.83) with i = 2 ii) The estimator x ˆ (t1 + 1, 2) = Φ2 (t1 + 1, t2 + 1)ˆ x(t2 + 1, 3) x t1 + Φ2 (t1 + 1, s + 1)K2 (s)y2 (s),

(2.107)

s=t2 +1

ˆ (t2 + 1, 3) is obtained from (2.105), where x Φ2 (t1 + 1, s + 1) = Φ2 (t1 ) · · · Φ2 (s + 1), Φ2 (s) = Φ − K2 (s)H2 , K2 (s) = ΦP2 (s)H2 Q−1 w (s, 2), Qw (s, 2) = H2 P2 (s)H2 + I2 , and P2 (s) is calculated by P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qw (s, 2)K2 (s) + Γ Γ , P2 (t2 + 1) = P3 (t2 + 1). ˆ (t, 1) of (2.81) is then calculated by (2.83) with i = 1 as iii) x ˆ (t, 1) = Φ1 (t, t1 + 1)ˆ x x(t1 + 1, 2) +

t−1

Φ1 (t, s + 1)K1 (s)y1 (s),

s=t1 +1

(2.108) ˆ (ti + 1, 2) where Φ1 (t, s+ 1) is as in (2.102) and K1 (s) is as in (2.95), and x is calculated by (2.107).

2.4 Conclusion In this chapter we have studied the optimal filtering for systems with instantaneous and single or multiple delayed measurements. By applying the so-called re-organized innovation approach [102], a simple solution has been derived. It includes solving a number of Kalman filters with the same dimension as the original system. As compared with the system augmentation approach, the presented approach is much more computationally attractive, especially when the delays are large. The proposed results in this section will be useful in applications such as sensor fusion [42]. It will also be useful in solving the H∞ fixed-lag smoothing and multiple-step ahead prediction as demonstrated in later chapters.

3. Optimal Control for Systems with Input/Output Delays

In this chapter we study both the LQR control for systems with multiple input delays and the LQG control for systems with multiple i/o delays. For the LQR problem, an analytical solution in terms of a standard RDE is presented whereas the LQG control involves two RDEs, one for the optimal filtering and the other for the state feedback control. The key to our approach is a duality between the LQR control and a smoothing estimation which extends the well known duality between the LQR for delay free systems and the Kalman filtering.

3.1 Introduction We present in this chapter a complete solution to the LQR problem for linear discrete-time systems with multiple input delays. The study of discrete-time delay systems has gained momentum in recent years due to applications in emerging fields such as networked control and network congestion control [96, 82, 2]. For discrete-time systems with delays, one might tend to consider augmenting the system and convert a delay problem into a delay free problem. While it is possible to do so, the augmentation approach, however, generally results in higher state dimension and thus high computational cost, especially when the system under investigation involves multiple delays and the delays are large. Further, in the state feedback case, the augmentation approach generally results in a static output feedback control problem which is non-convex; see the work of [96]. This has motivated researchers in seeking more efficient methods for control of systems with i/o delays. We note that the optimal tracking problem for discrete-time systems with single input delay has been studied in [75]. Our aim in this chapter is to give an intuitive and simple derivation and solution to the LQR problem for systems with multiple input delays. We present an approach based on a duality principle and standard smoothing estimation. We shall first establish a duality between the LQR problem for systems with multiple input delays and a smoothing problem for a backward stochastic delay free system, which extends the well known duality between the LQR of delay free H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 27–51, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

28


systems and the Kalman filtering. With the established duality, the complicated LQR problem for systems with multiple input delays is converted to a smoothing estimation problem for the backward delay free system and a simple solution based on a single RDE of the same order as the original plant (ignoring the delays) is obtained via standard projection in linear space. In this chapter, we shall also consider the LQG problem for systems with multiple i/o delays. By invoking the separation principle and the Kalman filtering for systems with multiple measurement delays studied in the last chapter, a solution to the LQG problem is easily obtained.

3.2 Linear Quadratic Regulation We consider the following linear discrete-time system with multiple input delays x(t + 1) = Φx(t) +

l

Γ(i) ui (t − hi ), l ≥ 1,

(3.1)

i=0

where x(t) ∈ Rn and ui (t) ∈ Rmi , i = 0, 1, · · · , l represent the state and the control inputs, respectively. Although Φ and Γ(i) , i = 0, 1, · · · , l are allowed to be time-varying, for the sake of simplicity of notations, we confine them to be constant matrices. We assume, without loss of generality, that the delays are in an increasing order: 0 = h0 < h1 < · · · < hl and the control inputs ui , i = 0, 1, · · · , l have the same dimension, i.e., m0 = m1 = · · · = ml = m. Consider the following quadratic performance index for the system (3.1): JN = xN +1 P xN +1 +

−hi l N i=0 t=0

ui (t)R(i) ui (t) +

N

x (t)Qx(t),

(3.2)

t=0

where N > hl is an integer, xN +1 is the terminal state, i.e., xN +1 = x(N + 1), P = P ≥ 0 is the penalty matrix for the terminal state, the matrices R(i) , i = 0, 1, · · · , l, are positive definite and the matrix Q is non-negative definite. The LQR problem is stated as: find the control inputs: ui (t) = Fi (x(t), uj (s), j = 0, 1, · · · , l; −hj ≤ s < t), 0 ≤ t ≤ N − hi , i = 0, 1, · · · , l such that the cost function JN of (3.2) is minimized. We note that in the absence of input delays, the solution to the LQR problem is well known and is related to one backward RDE. In the case of single input delay, the optimal tracking problem has been studied in [75] and a solution is also given in terms of one backward RDE. Here, we aim to give a similar solution for systems with multiple input delays. It is worth pointing out that the presence of delays in multiple input channels makes the control problem much more challenging due to interactions among various input channels.

3.2 Linear Quadratic Regulation

3.2.1

29

Duality Between Linear Quadratic Regulation and Smoothing Estimation

In this section, we shall convert the LQR problem into an optimization problem in a linear space for an associated stochastic system and establish a duality between the LQR problem and a smoothing problem. The duality will allow us to derive a solution for the LQR problem via standard smoothing estimation in the next section. First, the system (3.1) can be rewritten as ⎧ ⎨ Φx(t) + Γ u(t) + u˜(t), h ≤ t < h , t i i+1 (3.3) x(t + 1) = ⎩ Φx(t) + Γ u(t), t≥h, t

where

l

⎧⎡ ⎤ u0 (t − h0 ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ui (t − hi ) u(t) =

u ˜(t) =

⎪ ⎤ ⎡ ⎪ ⎪ u0 (t − h0 ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl , ⎣ . ⎪ ⎪ ⎩ ul (t − hl ) ⎧ l ⎪ ⎨ Γ(j) uj (t − hj ), hi ≤ t < hi+1 , j=i+1

⎪ ⎩ 0, ⎧ ⎨[Γ (0) Γt = ⎩[Γ

(3.4)

(3.5)

t ≥ hl , · · · Γ(i) ] , hi ≤ t < hi+1 ,

(3.6)

· · · Γ(l) ] , t ≥ hl .

(0)

Using the above notations, the cost function (3.2) can be rewritten as JN = x (N + 1)P x(N + 1) +

N

u (t)Rt u(t) +

t=0

where

Rt =

N

x (t)Qx(t),

⎧ ⎨ diag{R

· · · , R(i) }, hi ≤ t < hi+1 ,

⎩ diag{R

· · · , R(l) }, t ≥ hl .

(0) , (0) ,

(3.7)

t=0

(3.8)

It should be noted that while the optimal control of the system (3.3) associated with the cost (3.7) seems to be a standard LQR problem, its direct LQR solution does not lead to a causal control law for the system (3.1) in view of u(t) of the form (3.4).

30


Now we define the following backward stochastic state-space model associated with (3.3) and performance index (3.7): x(t) = Φ x(t + 1) + q(t), y(t) =

Γt x(t

(3.9)

+ 1) + v(t), t = 0, · · · , N,

(3.10)

where x(t) is the state and y(t) is the measurement output. The initial state x(N + 1), q(t) and v(t) are white noises with zero means and covariance matrices x(N + 1), x(N + 1) = P , q(t), q(s) = Qδt,s and v(t), v(s) = Rt δt,s , respectively. It can be seen that the dimensions of q(t) and y(t) are respectively dim{q(t)} = n × 1 and ⎧ ⎨ (i + 1)m × 1, h ≤ t < h , i i+1 dim {y(t)} = (3.11) ⎩ (l + 1)m × 1, t ≥ h , l

and v(t) has the same dimension as y(t). Introduce the column vectors:

⎫ ⎪ x = col{x(0), · · · , x(N )}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u = col{u(0), · · · , u(N )}, ⎪ ⎪ ⎬ y = col{y(0), · · · , y(N )}, ⎪ ⎪ ⎪ ⎪ q = col{q(0), · · · , q(N )}, ⎪ ⎪ ⎪ ⎪ ⎪ v = col{v(0), · · · , v(N )}. ⎭

(3.12)

In the above, x(i), u(i), i = 0, 1, · · · , N are associated with the system (3.3) and y(i), q(i) and v(i) with the backward stochastic system (3.9)-(3.10). Then we have the following result. Lemma 3.2.1. By making use of the stochastic state-space model (3.9)-(3.10), the cost function JN of (3.2) can be put in the following quadratic form ξ ξ JN = Π , (3.13) u u where ξ = [ x (0) u ˜ (0) · · · u ˜ (hl − 1) ] , Rx0 Rx0 y x0 x0 = Π= , , y y Ryx0 Ry x0 = [ x (0)

x (1)

···

x (hl ) ] ,

(3.14) (3.15) (3.16)

with u ˜(i), i = 0, 1, · · · , hl − 1 as defined in (3.5), x0 , x0 = Rx0 , x0 , y = Rx0 y , and y, y = Ry . Proof: By applying (3.3) repeatedly, it is easy to know that the terminal state x(N + 1) and x of (3.12) can be given in terms of the initial state x(0), u˜, and u of (3.12) as

3.2 Linear Quadratic Regulation h l −1

x(N + 1) = ΨN x(0) +

ΨN −i−1 u˜(i) + Cu,

31

(3.17)

i=0

x = ON x(0) +

h l −1

ON −i−1 u ˜(i) + Bu,

(3.18)

i=0

where u ˜(i) is as in (3.5) and ΨN −i−1 = ΦN −i , C = [ ΦN Γ0 , ΦN −1 Γ1 , · · · , ΓN ] , ⎡ ⎤ In ⎢ Φ ⎥ ⎥ ON = ⎢ ⎣ ... ⎦ , ⎡

ON −i−1

ΦN

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

0 In Φ .. .

ΦN −i−1 0 0 ⎢ Γ0 ⎢ ΦΓ Γ ⎢ 0 1 B=⎢ .. ⎣ ... .

(3.22)

⎤

⎡

∆1 while

(3.21)

⎤

0 .. .

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

(3.19) (3.20)

∆2

⎥ ⎥ ⎥, ⎥ ⎦

..

. · · · ΓN −1

(3.23)

∆1 = ΦN −1 Γ0 , ∆2 = ΦN −2 Γ1 .

Next, by using (3.17)-(3.18), the cost function, JN , can be rewritten as the following quadratic form of x(0), u and u ˜: % & h l −1 ΨN −i−1 u˜(i) + Cζ P JN = ΨN x(0) + %

i=0

× ΨN x(0) + % + ON x(0) + % × ON x(0) +

h l −1

& ΨN −i−1 u ˜(i) + Cu

i=0 h l −1 i=0 h l −1 i=0

& ON −i−1 u ˜(i) + Bu

Q &

ON −i−1 u ˜(i) + Bu + u Ru,

(3.24)

32


where R = diag{R0 , · · · , RN }, N + 1 blocks ' () * Q = diag{ Q, · · · , Q }.

(3.25) (3.26)

We now show that JN can be rewritten as in (3.13). Note that it follows from (3.24) that

JN = [Ψ ξ + Cu] P [Ψ ξ + Cu] + [Oξ + Bu] Q [Oξ + Bu] + u Ru, ξ Ψ P Ψ + O QO Ψ P C + O QB ξ = , u C P Ψ + B QO C P C + B QB + R u (3.27) where ξ is as in (3.14) and Ψ = [ ΨN

· · · ΨN −i−1

O = [ ON

· · · ON −i−1

· · · ΨN −hl ] , · · · ON −hl ] .

On the other hand, from (3.9), it is easy to know that x(i) = ΨN −i x(N + 1) + ON −i q, i = 0, · · · , hl ,

(3.28)

where ΨN −i and ON −i are given in (3.19) and (3.22), respectively, and q is defined in (3.12). Putting together (3.28) for i = 0, 1, · · · , hl yields x0 = Ψ x(N + 1) + O q,

(3.29)

where x0 is defined in (3.16). Similarly, from (3.10), it follows that y = C x(N + 1) + B q + v,

(3.30)

where y is defined in (3.12). Combining (3.29) with (3.30) yields O 0 Ψ x0 x(N + 1) + q + . = v y C B Thus we obtain that x0 x0 Π= , y y Ψ P Ψ + O QO Ψ P C + O QB = . C P Ψ + B QO C P C + B QB + R Hence, the required result follows from (3.27) and (3.31).

(3.31) ∇

Since the weighting matrices R(i) , i = 0, 1, · · · , l in (3.2) are positive definite, R > 0 and Ry > 0. Thus, by completing the squares for (3.13), we have the following result.


33

Lemma 3.2.2. JN can further be rewritten as JN = ξ Pξ + (u − u∗ ) Ry (u − u∗ ),

(3.32)

where ∗

u =

−Ry−1 Ryx(0) x(0)

−

hl

Ry−1 Ryx(i) u ˜(i − 1),

(3.33)

i=1

ˆ 0 , x0 − x ˆ 0 , P = x0 − x

(3.34)

ˆ 0 is the projection of x0 onto the linear space L{y(0), · · · , y(N )}. The and x minimizing solution of JN with respect to control input ui (t) is given by

u∗i (t)

i + 1 blocks () * ' = [0 · · · 0 Im ] u∗ (t + hi ),

where, in view of (3.12), u∗ (t + hi ) is the (t + hi + 1)-th block of u∗ in (3.33). Proof: In view of Lemma 3.2.1, the minimizing solution of JN with respect to control input ui (t), i = 0, 1, · · · , l, is readily given by [30] u∗ = −Ry−1 Ryx0 ξ and JN can be written as in (3.32). Further, since Ryx0 =

⎡ x(0) ⎢ x(1) y, ⎢ ⎣ .. .

⎤ ⎥ ⎥ = [ Ryx(0) ⎦

Ryx(1)

· · · Ryx(hl ) ] ,

x(hl ) where Ryx(i) = y, x(i) , i = 0, 1, · · · , hl , the minimizing solution of (3.33) thus follows.

∇

Observe from linear estimation theory [38] that Ry−1 Ryx(i) is the transpose of the gain matrix of the optimal smoothing (or filtering when i = 0) estimate ˆ (i | 0) of the backward system (3.9)-(3.10) which is the projection of the of x state x(i) onto the linear space L{y(0), · · · , y(N )}. Thus, we have converted the LQR problem for the delay system (3.1) into the optimal smoothing problem for the associated system (3.9)-(3.10). In another words, in order to calculate the ˆ (i | 0) optimal controller u∗ , we just need to calculate the smoothing gain of x associated with the backward system (3.9)-(3.10). Therefore, we have established a duality between the LQR of the delay system (3.1) and a smoothing problem for an associated delay free system.

34


Remark 3.2.1. Observe that when the system (3.1) is delay free, i.e. hi = 0, i = 1, 2, · · · , l, (3.33) becomes u∗ = −Ry−1 Ryx(0) x(0), where Ry−1 Ryx(0) is the transpose of the optimal filtering gain matrix, which is the well known duality between the LQR problem and the Kalman filtering and has played an important role in linear systems theory. Thus, the established duality between the LQR for systems with input delays and the optimal smoothing contains the duality between the LQR problem for delay free systems and the Kalman filtering as a special case. It is expected that the duality will play a significant role in control design for systems with i/o delays. By using the duality of (3.33), we shall present a solution to the LQR problem for systems with multiple input delays via the standard projection in linear space in the next section. 3.2.2

Solution to Linear Quadratic Regulation

In view of the result of the previous section, to give a solution to the LQR problem, we need to calculate the gain matrix Rx(i)y Ry−1 of the smoothing problem associated with the stochastic backward system (3.9)-(3.10). First, define the RDE associated with the Kalman filtering of the backward stochastic system (3.9)-(3.10): Pj = Φ Pj+1 Φ + Q − Kj Mj Kj , j = N, N − 1, · · · , 0,

(3.35)

where the initial condition is given by PN +1 = P with P the penalty matrix of (3.2), Kj = Φ Pj+1 Γj Mj−1 , Mj = Rj +

Γj Pj+1 Γj ,

(3.36) (3.37)

and Γj is defined in (3.6). It follows from the standard Kalman filtering that the optimal filtering estiˆ (i | i), of the backward system (3.9)-(3.10) is given by [30]: mate, x ˆ (i | i) = x

N

Φi,k Kk y(k),

(3.38)

k=i

where Φj,m = Φj · · · Φm−1 , m ≥ j, Φm,m = In , j = i, · · · , N − 1. Φj = Φ − Kj Γj ,

(3.39) (3.40)


35

Also, by applying standard projection, the smoothing estimate is given below. Lemma 3.2.3. Consider the stochastic backward state space model (3.9)-(3.10), ˆ (i | 0) is given by the optimal smoothing estimate x ˆ (i | 0) = x

i−1

Si (k)y(k) +

k=0

where

N

Fi (k)y(k)

⎫ Si (k) = Pi Φk+1,i Γk Mk−1 − Φk,i G(k)Kk , 0 ≤ k < i, ⎬ F (k) = [I − P G(i)] Φ K , i ≤ k ≤ N, ⎭ i

n

i

(3.41)

k=i

i,k

(3.42)

k

and G(k) =

k

−1 Φj,k Γj−1 Mj−1 Γj−1 Φj,k ,

(3.43)

j=1

while Φj,m is given in (3.39). In the above, Pi is the solution to the RDE (3.35) and Mj is given by (3.37). ˆ (i | 0) is the projection of x(i) onto the linear space of Proof: Note that x L{y(0), · · · , y(N )}. By applying the projection formula, we have [30] ˆ (i | 0) = x ˆ (i | i) + x

i

−1 x(i), w(j − 1)Mj−1 w(j − 1)

j=1

ˆ (i | i) + =x

i

! −1 ˆ (j | j) , ˜ (j)Γj−1 Mj−1 x y(j − 1) − Γj−1 x(i), x

j=1

(3.44) ˜ (j | j) = x(j) − x ˆ (j | j) is the filtering error, w(j − 1) = y(j − 1) − where x ˆ (j | j) is the innovation and Mj−1 is the covariance matrix of w(j − 1) Γj−1 x ˜ (j | j) for j < i. which can be computed using (3.37). Now we calculate x(i), x In consideration of (3.9)-(3.10) and by applying the projection lemma, it is easy to know that ˜ (j + 1 | j + 1) + q(j) − Φ Pj+1 Γj Mj−1 v(j), ˜ (j | j) = Φj x x where Φj is as in (3.40). In view of the fact that q(j) and v(j) are independent of x(i) for j < i, it follows that ˜ (j | j) = Pi Φi−1 · · · Φj = Pi Φj,i , x(i), x

(3.45)

where Φj,i is given by (3.39). By taking into account (3.38) and (3.45), it follows from (3.44) that

36


ˆ (i | 0) = x ˆ (i | i) − x

i

−1 ˆ (j | j) x Pi Φj,i Γj−1 Mj−1 Γj−1

j=1

+

i

−1 Pi Φj,i Γj−1 Mj−1 y(j − 1)

j=1

=

N

Φi,k Kk y(k) +

i−1

Pi Φj+1,i Γj Mj−1 y(j)

j=0

k=i

−

i

⎡ ⎤ N −1 ⎣ Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)⎦

j=1

=

N

k=j

Φi,k Kk y(k) +

i−1

Pi Φj+1,i Γj Mj−1 y(j)

j=0

k=i

−

k N

−1 Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)

k=i j=1

−

k i−1

−1 Pi Φj,i Γj−1 Mj−1 Γj−1 Φj,k Kk y(k)

k=1 j=1

=

N

[In − Pi G(i)] Φi,k Kk y(k)

k=i

+

i−1

! Pi Φk+1,i Γk Mk−1 − Φk,i G(k)Kk y(k) + Pi Φ1,i Γ0 M0−1 y(0),

k=1

∇

where G(i) is given by (3.43). We complete the proof of the lemma.

Therefore, by the duality established in the previous section, the solution to the LQR problem for the discrete-time system (3.1) is given in the following theorem. Theorem 3.2.1. Consider the system (3.1) and the associated cost (3.2). The optimal controller u∗i (t), i = 0, · · · , l; t = 0, · · · , N − hi that minimizes the cost is given by

u∗i (t)

i + 1 blocks ' () * = [0 · · · 0 Im ] u∗ (t + hi ),

(3.46)

where for t < hl , ∗

u (t) = − [F0 (t)] x(0) −

t s=1

[Fs (t)] u ˜(s − 1) −

hl

[Ss (t)] u ˜(s − 1),

s=t+1

(3.47)


37

and for t ≥ hl ,

u∗ (t) = − [F0 (t)] x(0) −

hl

[Fs (t)] u˜(s − 1),

(3.48)

s=1

while u ˜(·) is as in (3.5) and Ss (t) and Fs (t) in (3.42). Proof: From Lemmas 3.2.2 and 3.2.3, we observe that Ry−1 Ryx(0) = [F0 (0), · · · , F0 (N )] , Ry−1 Ryx(s) = [Ss (0), · · · , Ss (s − 1); Fs (s), · · · , Fs (N )] , s = 1, 2, · · · , hl , where Fs (t) and Ss (t) are as in (3.42). In view of (3.33) and (3.4), it is easy to know that u∗ (t) is given by (3.47)-(3.48) and u∗i (t) by (3.46). Thus the proof is completed. ∇ Note, however, that for t > 0, the optimal controller u∗i (t) of (3.46) is given in terms of the initial state x(0) and the past control inputs and is an open-loop control in nature. Our aim is to find the optimal controller of ui (t) in terms of the current state x(t). This problem can be addressed by shifting the time interval from [0, hl ] to [τ, τ + hl ]. Note that for any given τ ≥ 0, the system (3.1) and the cost (3.2) can be rewritten respectively as ⎧ ⎨ Φx(t + τ ) + Γ uτ (t) + u ˜τ (t), hi ≤ t < hi+1 , t x(t + τ + 1) = ⎩ Φx(t + τ ) + Γ uτ (t), t≥h, t

l

(3.49) τ + JN = JN

l τ −1

ui (t)R(i) ui (t) +

i=0 t=0

τ

x (t)Qx(t),

(3.50)

t=1

where τ = x (N + 1)P x(N + 1) + JN

N −τ

[uτ (t)] Rt uτ (t)

t=0

+

N −τ

x (t + τ )Qx(t + τ ),

(3.51)

t=1

while

uτ (t) =

⎧⎡ ⎤ u0 (t − h0 + τ ) ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎪ ⎪ (t − h + τ ) u ⎪ i i ⎨ ⎪ ⎤ ⎡ ⎪ ⎪ u0 (t − h0 + τ ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl , ⎣ . ⎪ ⎪ ⎩ ul (t − hl + τ )

(3.52)

38


u ˜τ (t) =

⎧ ⎨ l

j=i+1

Γ(j) uj (t − hj + τ ), hi ≤ t < hi+1 , t + τ ≤ N,

⎩ 0,

(3.53)

otherwise.

Remark 3.2.2. It is easy to see that the above uτ (t) and u ˜τ (t) are related to u(t) and u˜(t) of (3.5) as follows: – For t ≥ hl , uτ (t) ≡ u(t + τ ) and u ˜τ (t) ≡ u ˜(t + τ ). – For τ = 0 and any t, uτ (t) ≡ u(t + τ ) and u ˜τ (t) ≡ u ˜(t + τ ). ˜τ (t) = u ˜(t + τ ). – For t < hl and τ = 0, uτ (t) = u(t + τ ) and u Let the matrix Pjτ (j = N − τ + 1, · · · , 0) obey the following backward RDE: τ Φ + Q − Kjτ Mjτ (Kjτ ) , PNτ −τ +1 = PN +1 = P, Pjτ = Φ Pj+1

(3.54)

τ Γj (Mjτ )−1 , Kjτ = Φ Pj+1

(3.55)

where Mjτ

= Rj +

τ Γj Pj+1 Γj .

(3.56)

Remark 3.2.3. When j ≥ hl , it is not difficult to know that Pjτ = Pτ +j , where Pτ +j is the solution to Riccati equation (3.35). Thus, we just need to calculate Pjτ for j = hl − 1, hl − 2, · · · , 0 using (3.54) with the initial condition of Phτl = Pτ +hl . Similar to (3.39)-(3.40), denote that

Φτj = Φ − Kjτ Γj ,

(3.57)

Φτj,m = Φτj · · · Φτm−1 , m ≥ j,

(3.58)

with Φτj,j = In . By using a similar line of arguments as for the case of τ = 0 in the above, we have the following result. Lemma 3.2.4. Consider the system (3.49) and the associated cost (3.51). The optimal control associated with uτi (t), i = 0, · · · , l, 0 ≤ t ≤ N − τ − hi that τ minimizes JN , denoted by uτi ∗ (t), is given by uτi ∗ (t)

i + 1 blocks ' () * = [0 · · · 0 Im ] uτ ∗ (t + hi ),

(3.59)

where for t < hl ,

uτ ∗ (t) = − [F0τ (t)] x(τ ) −

t

hl

[Fsτ (t)] u˜τ (s − 1) −

s=1

[Ssτ (t)] u˜τ (s − 1),

s=t+1

(3.60) and for t ≥ hl ,

uτ ∗ (t) = − [F0τ (t)] x(τ ) −

hl s=1

[Fsτ (t)] u˜τ (s − 1),

(3.61)


39

while Ssτ (·) and Fsτ (·) are given by

⎫ ! Ssτ (t) = Psτ (Φτt+1,s ) Γt (Mtτ )−1 − (Φτt,s ) Gτ (t)Ktτ , 0 < t < s, ⎬ F τ (t) = [I − P τ Gτ (s)] Φτ K τ , s ≤ t ≤ N, ⎭ s

n

s

s,t

t

(3.62) and Gτ (t) =

t

τ (Φτj,t ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,t .

(3.63)

j=1

∇

Proof: The proof is similar to that of Theorem 3.2.1.

Remark 3.2.4. Observe that u∗i (t) is the optimal controller associated with the cost (3.2) while uτi ∗ (t) is the optimal controller associated with the cost (3.51). Furthermore, u∗i (t) is given in terms of initial state x(0) while uτi ∗ (t) in terms of state x(τ ). Corollary 3.2.1. For t = 0, the controller uτi ∗ (0) associated with the cost (3.51) is given by uτi ∗ (0)

i + 1 blocks ' () * = [0 · · · 0 Im ] uτ ∗ (hi ), i = 0, 1, · · · , l,

(3.64)

where

uτ ∗ (hi ) = − [F0τ (hi )] x(τ ) −

hi

[Fsτ (hi )] u ˜τ (s − 1)

s=1

−

hl

[Ssτ (hi )] u ˜τ (s − 1),

(3.65)

s=hi +1

and u˜τ (·) is defined in (3.53). It is clear that uτi ∗ (0) is given in terms of the current state x(τ ) and control inputs ui (t), τ − hi ≤ t ≤ τ − 1, i = 1, · · · , l. ˜0 (t) ≡ u ˜(t) and the RDE (3.54) Remark 3.2.5. Since for τ = 0, u0 (t) ≡ u(t), u reduces to the RDE (3.35), thus uτi ∗ (0) for τ = 0 is the same as u∗i (0) of (3.46). Note that u∗i (τ ) is given in terms of the initial state x(0) while uτi ∗ (0) is given in terms of current state x(τ ). The following result follows similarly from the well known dynamic programming. Lemma 3.2.5. If ui (t) = u∗i (t) for t = 0, · · · , τ − 1; i = 0, · · · , l, then u∗i (τ ) ≡ uτi ∗ (0) |uj (t)=u∗j (t)(

0≤t 0, an integer d > 0 and the observation {y(j)}tj=0 , find an estimate zˇ(t − d | t) of z(t − d), if it exists, such that the following inequality is satisfied: N

sup (x0 ,u,v)=0

[ˇ z (t − d | t) − z(t − d)] [ˇ z (t − d | t) − z(t − d)]

t=d

x0 P0−1 x0

+

N −1 t=0

e (t)e(t)

+

N

< γ2

(4.4)

v (t)v(t)

t=0

where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises. Remark 4.2.1. In the performance criterion (4.4), we have assumed that the initial estimate x ˆ(0|d) is zero and P0 reflects how accurate the estimate is. It is worth pointing out that the above criterion can be easily modified to accommodate the situation where the initial estimate is not zero. Remark 4.2.2. Note that the H∞ fixed-lag smoothing has been addressed in [91] through system augmentation and standard H∞ filtering. The approach is easy to understand but suffers from heavy computational requirement, especially for the case of large smoothing lag and high dimension of the signal to be estimated. Very recently, [17, 16] have presented a J-factorization approach to the steadystate H∞ fixed-lag smoothing without resorting to system augmentation. In the present section, we shall discuss the finite horizon H∞ fixed-lag smoothing for systems using an innovation analysis approach in Krein space.

4.2 H∞ Fixed-Lag Smoothing

4.2.1

55

An Equivalent H2 Estimation Problem in Krein Space

In this section we shall demonstrate that the deterministic H∞ fixed-lag smoothing problem can be converted to an innovation analysis for an associated stochastic system in Krein space. First, in view of (4.4) we define

Jd,N =

x0 P0−1 x0

+

N −1 t=0

e (t)e(t) +

N

v (t)v(t) − γ

−2

N

t=0

vz (t)vz (t),

t=d

(4.5) where vz (t) = zˇ(t − d | t) − Lx(t − d), t ≥ d.

(4.6)

The H∞ fixed-lag smoothing problem is apparently a minimax optimization problem of the above cost function, that is, Jd,N has a minimum over {x(0), e} and the smoother is such that the minimum is positive. To convert the optimization into an innovation analysis problem, in view of (4.1)-(4.2) and (4.6), we introduce the following stochastic system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t), zˇ(t − d | t) = Lx(t − d) + vz (t), t ≥ d,

(4.7) (4.8) (4.9)

where e(t), v(t) and vz (t) are assumed to be uncorrelated white noises, with x(0), x(0) = P0 , e(t), e(s) = Qe δts , v(t), v(s) = Qv δts , vz (t), vz (s) = Qvz δts , while Qe = Ir , Qv = Ip , Qvz = −γ 2 Iq .

(4.10)

ˇ(t − d | t) are respectively the observations for In the above system, y(t) and z x(t) and x(t − d) at the time instant t. Since vz (t) is of negative covariance, the above stochastic system should be considered in Krein space [30] rather than Hilbert space. Remark 4.2.3. Note that the elements in (4.1)-(4.3), which are denoted in normal letters, are from Euclidean space while the elements in (4.7)-(4.9), denoted by bold face letters, are from Krein space. They satisfy the same constraints. Combining (4.8) with (4.9) yields, ⎧ ⎪ ⎨ Hx(t) + vo (t), 0≤t 0 and an integer d > 0, an H∞ smoother zˇ(t − d + 1 | t + 1) that achieves (4.4) exists if and only if, for each t = d − 1, · · · , Ip 0 N − 1, Qwo (t + 1) and Qvo (t + 1) = have the same inertia1 , 0 −γ 2 Iq where Qwo (t + 1) is calculated by (4.58). In this situation, the central smoother is given by zˇ(t − d + 1 | t + 1) = Lˆ x(t − d + 1 | t + 1, t − d),

(4.67)

where x ˆ(t−d+1 | t+1, t−d) is defined as the Krein space projection of x(t−d+1) onto the linear space L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t + 1)}, which can be computed by xˆ(t − d + 1 | t + 1, t − d) = xˆ(t − d + 1, 2) +

d+1

R(t − d + 1, i − 1)H

i=1

×Q−1 ˆ(t − d + i, 1)] , w (t − d + i) [y(t − d + i) − H x 1

(4.68)

The inertia of a matrix is the numbers of positive and negative eigenvalues of the matrix.

64

4. H∞ Estimation for Discrete-Time Systems with Measurement Delays

while x ˆ(s + 1, 1) (s > t − d) in (4.68) is calculated recursively as x ˆ(s + 1, 1) = Φˆ x(s, 1) + K1 (s) [y(s) − H x ˆ(s, 1)] , x ˆ(t − d + 1, 1) = x ˆ(t − d + 1, 2),

(4.69)

ˆ(t − d + 1, 2) in (4.68) and with K1 (s) computed by (4.51) and the initial value x (4.69) given by the Kalman recursion: H x ˆ(t − d + 1, 2) = Φˆ x(t − d, 2) + K2 (t − d) yf (t − d) − x ˆ(t − d, 2) , L x ˆ(0, 2) = 0. (4.70) y(t − d) where K2 (t − d) is as in (4.49) and yf (t − d) = . zˇ(t − d | t) Proof: First, in view of (4.18), the H∞ estimator zˇ(t − d + 1 | t + 1) (t ≥ −1) 0 with respect that achieves (4.4) exists if and only if Jd,N has a minimum Jd,N 0 to (x(0), e) and zˇ(t−d+1 | t+1) can be chosen such that Jd,N > 0, where the 0 can be given in terms of the original innovation wo (t + 1) as minimum Jd,N [31, 104] 0 = Jd,N

N −1

wo (t + 1)Q−1 wo (t + 1)wo (t + 1).

(4.71)

t=−1

Recall that Jd,N has a minimum over {x(0); e} iff Qwo (t + 1) and Qvo (t + 1) for −1 ≤ t ≤ N − 1 have the same inertia [31, 104]. Note that for −1 ≤ t < d − 1, Qvo (t + 1) > 0 and Qwo (t + 1) > 0. Thus, an H∞ estimator zˇ(t − d + 1 | t + 1) exists iff Qwo (t + 1) and Qvo (t + 1) for d − 1 ≤ t ≤ N − 1 have the same inertia. Next, we prove (4.67) based on (4.27). In light of (4.58), Qwo (t + 1) (t ≥ d − 1) has the following factorization Q11 (t) Q12 (t) Qwo (t + 1) = Q21 (t) Q22 (t) Ip Q11 (t) 0 0 Ip Q−1 (t)Q12 (t) 11 = , Q21 (t)Q−1 0 ∆(t) 0 Iq 11 (t) Iq (4.72) where Q11 (t) = HP1 (t + 1)H + Ip , Q21 (t) = LR(t − d + 1, d)H ; Q12 (t) = Q21 (t), Q22 (t) = LP(t − d)L − γ 2 Iq ,

(4.73)

and ∆(t) = Q22 (t) − Q21 (t)Q−1 11 (t)Q12 (t).

(4.74)


65

Note that for −1 ≤ t < d − 2, wo (t + 1) is as wo (t + 1) ≡ woy (t + 1) = y(t + 1) − yˆ(t + 1 | t),

(4.75)

and for t ≥ d − 1, wo (t + 1) is as woy (t + 1) y(t + 1) yˆ(t + 1 | t) wo (t + 1) ≡ = − woz (t + 1) zˇ(t − d + 1 | t + 1) zˆ(t − d + 1 | t) (4.76) where zˆ(t−d+1 | t) and yˆ(t−d+1 | t) are obtained from the projections of ˇz(t−d) and y(t − d + 1) onto the Krein space L{yo (0), · · · , yo (t)} = L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t)}, respectively. 0 is equivalently written By applying the factorization (4.72), the minimum Jd,N as 0 = Jd,N

d−2

wo (t + 1)Q−1 wo (t + 1)wo (t + 1)

t=−1

+

N −1

wo (t + 1)Q−1 wo (t + 1)wo (t + 1)

t=d−1

=

d−1

wo (t)Q−1 wo (t)wo (t) +

t=0

N −1

[y(t + 1) − yˆ(t + 1 | t)] Q−1 11 (t)

t=d−1

×[y(t + 1) − yˆ(t + 1 | t)] +

N −1

zy (t + 1)∆−1 (t)zy (t + 1), (4.77)

t=d−1

with zy (t + 1) = zˇ(t − d + 1 | t + 1) − zˆ(t − d + 1 | t) − Q21 (t)Q−1 11 (t) × [y(t + 1) − yˆ(t + 1 | t)] .

(4.78)

It is not difficult to show that if Qwo (t + 1) and Qvo (t + 1) have the same inertia, then Q11 (t) > 0. Thanks to the fact that Qwo (t + 1) and Qvo (t + 1) have the same inertia, from (4.72) and (4.14), Q11 (t) > 0 implies ∆(t) < 0. We note that 0 any choice of estimator that renders Jd,N > 0 is an acceptable one, and that Qwo (t + 1) > 0 for −1 ≤ t ≤ d − 2. Thus the estimator can be obtained from (4.77) by setting zy (t + 1) = 0,

(4.79)

which implies that zˇ(t − d + 1 | t + 1) = zˆ(t − d + 1 | t) + Q21 (t)Q−1 ˆ(t + 1 | t)] 11 (t) [y(t + 1) − y −1 = zˆ(t − d + 1 | t) + Q21 (t)Q11 (t) × [y(t + 1) − yˆ(t + 1 | t, t − d)] .

(4.80)

66


Furthermore, by considering that 3 4 Q11 (t) = woy (t + 1), woy (t + 1) = Qwy (t + 1), 3 4 Q21 (t) = woz (t + 1), woy (t + 1) 4 3 = ˇ z(t − d + 1 | t + 1), woy (t + 1) , and that zˆ(t − d + 1 | t) is obtained from the projection of zˇ(t − d + 1) onto the Krein space L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t)}, it follows from (4.80) that zˇ(t − d + 1 | t + 1) = zˆ(t − d + 1 | t + 1, t − d),

(4.81)

where zˆ(t − d + 1 | t + 1, t − d) is obtained from the projection of zˇ(t − d + 1) onto L{yf (0), · · · , yf (t − d); y(t − d + 1), · · · , y(t + 1)}. Thus zˇ(t − d + 1 | t + 1) = Lˆ x(t − d + 1 | t + 1, t − d), which is (4.67). Finally we prove that the projection x ˆ(t − d + 1 | t + 1, t − d) is calculated by (4.68). Note that x ˆ(t − d + 1 | t + 1, t − d) is obtained from the projection of the state x(t − d + 1) onto L{wf (0), · · · , wf (t − d); w(t − d + 1), · · · , w(t + 1)}. Since w is a white noise, the estimator x ˆ(t − d + 1 | t + 1, t − d) is calculated by using the projection formula as ˆ (t − d + 1 | t + 1, t − d) x = P roj {x(t − d + 1) | wf (0), · · · , wf (t − d)} + P roj {x(t − d + 1) | w(t − d + 1), · · · , w(t + 1)} ˆ (t − d + 1, 2) + =x

d+1

x(t − d + 1), w(t − d + i)

i=1

×Q−1 w (t − d + i)w(t − d + i) ˆ (t − d + 1, 2) + =x

d+1

R(t − d + 1, i − 1) ×

i=1

ˆ (t − d + i, 1)] , H Q−1 w (t − d + i) [y(t − d + i) − H x

(4.82)

which is (4.68). Similarly, by applying the projection formula and the re-organized innovation sequence, for s > t − d, it follows that ˆ (s + 1, 1) = P roj {x(s + 1) | wf (0), · · · , wf (t − d); w(t − d + 1), x · · · , w(s)} = P roj {x(s + 1) | wf (0), · · · , wf (t − d); w(t − d + 1), · · · , w(s − 1)} + x(s + 1), w(s) Q−1 w (s)w(s) −1 ˆ (s, 1)] = Φˆ x(s, 1) + ΦP1 (s)H Qw (s) [y(s) − H x which is (4.69). Similarly, we derive (4.70) immediately.

(4.83) ∇


67

Remark 4.2.5. In Theorem 4.2.2, we have presented a solution to the H∞ fixed-lag smoothing. Unlike [91] where the problem is converted to the H∞ filtering for an augmented system, our solution is given in terms of two RDEs of the same dimension as that of the original plant and can be considered as the H∞ counterpart of the forward and backward algorithm of the H2 fixed-lag smoothing [65]. 4.2.5

Computational Cost Comparison and Example

We now analyze the computational cost of the H∞ fixed-lag smoothing algorithm presented in Theorem 4.2.2, in comparison with the state augmentation method in [91]. To this end, we recapture the procedures for computing an H∞ smoother using the above two methods. From Theorem 4.2.2 of this section, the procedure for computing an H∞ fixed-lag smoother zˇ(t − d + 1 | t + 1) is as follows – First, compute the matrix P2 (t − d + 1) using the RDE (4.48), with the value of P2 (t − d) obtained from the last iteration as the initial value. – Calculate P1 (s) (t − d + 1 ≤ s ≤ t) recursively using (4.50) with the initial condition P1 (t − d + 1) = P2 (t − d + 1) obtained in the last step. – Calculate innovation covariance Qwo (t + 1) using Theorem 4.2.1 and then check the existence condition of an H∞ smoother. – Compute the H∞ smoother zˇ(t − d | t) using Theorem 4.2.2. Recall the result in [91]. The smoother is computed with the help of the following augmented model xa (t + 1) = Φa xa (t) + Γa e(t), y(t) = Ha xa (t) + v(t), zˇ(t − d | t) = Lx(t − d) = La (t)xa (t),

(4.84) (4.85) (4.86)

where

xa (t) = [ x (t) z (t − 1) · · · z (t − d) ] , Ha = [ H 0 · · · 0 ] , La (t) = [ 0

(4.87) (4.88)

· · · 0, Iq ] f or t > d, and La (t) = 0 f or t ≤ d,

⎡

Φ ⎢L ⎢ Iq Φa = ⎢ ⎣

⎤

··· Iq

(4.89) ⎡

⎤

Γ ⎥ ⎢ 0⎥ ⎥ .. ⎥ . ⎥ , Γa = ⎢ ⎣ ⎦ . ⎦ 0 0

(4.90)

Then the H∞ fixed-lag smoother is equivalent to the standard H∞ filtering problem of the above augmented system (4.84)-(4.86), which has been studied in [91]. Computational Costs. As additions are much faster than multiplications and divisions, it is the number of multiplications and divisions, counted together,

68


that is used as the operation count. Let M D(t) denote the number of multiplications and divisions at time instant t. A. Re-organized innovation approach It follows that the operation count at time instant t with the new approach, denoted as M Dnew (t) (the multiplication is from right to left), is given as M Dnew (t) = (d + 1)(p + n)2 + d(2n3 + n2 + pn + p2 ) +d(3n2 + 2pn + p2 ) + d(3n3 + 3pn2 + rn2 + 2p2 n + p3 ) +3n2 + 2(p + q)n + (p + q)2 + 3n3 +3(p + q)n2 + rn2 + 2(p + q)2 n + (p + q)3 + pn.

(4.91)

B. Augmentation approach Note that the augmented system (4.84)-(4.86) is of dimension of n + dp, the MD count for computing an smoother by using the algorithm of [91] in each iteration, denoted by M Daug (t), is given by M Daug (t) = 2(n + dq)3 + (n + q + r)(n + dq)2 + (2pn + p2 + q 2 )(n + dq) +pn2 + p2 n + p3 + q 3 + ! d (2p + 4)n2 + (2p2 + 4p + q)n + 2p3 + 2p2 + qn. (4.92) It is clear that the order of d in M Dnew (t) is 1 and the order in M Daug (t) is 3. That is, when the smoothing lag d is sufficiently large, it is not difficult to see that M Daug (t) M Dnew (t). This can also be seen from the examples below. Example 4.2.1. Consider the system (4.1)-(4.3) with n = 3, p = 1, q = 3 and r = 1. We investigate the relationship between the lag d and the MD number. Table 4.1. Comparison of the Computational Costs d M Daug (t) M Dnew (t)

1 1022 691

2 2631 932

3 5392 1173

4 9629 1414

5 15666 1655

10 84191 2860

20 542541 5270

Assume that n, m and r are as in the above, but p = 2, the MD number in relation to the lag d is given in Table 4.2. Table 4.2. Comparison of the Computational Costs d M Daug (t) M Dnew (t)

1 639 569

2 1395 810

3 2559 1051

4 4227 1292

5 6495 1533

10 30195 2738

20 177795 5148

4.3 H∞ d-Step Prediction

69

Clearly, the presented new approach is more efficient than the augmented model approach, especially when the lag d is large. 4.2.6

Simulation Example

Example 4.2.2. Consider the discrete-time system (4.1)-(4.3) with ⎡ ⎤ ⎡ ⎤ 0.8 −0.2 0 0 0 Φ=⎣ 0 0.3 0.5 ⎦ , Γ = ⎣ 0.5 0 ⎦ 0 0 0.9 1 0 ⎡ ⎤ 0.1 0 0 0.5 0.8 0 H = [ 1 −0.8 0.6 ] , L = , P0 = ⎣ 0 0.1 0 ⎦ . 0 0 0.6 0 0 0.1 We shall investigate the relationship between the lag d and the achievable optimal γ for all 0 ≤ t ≤ N where N = 20. By applying the method presented in this section, the results are shown in Table 4.3. Table 4.3. The Relationship between the lag d and γ d γopt

0 2.1

1 1.76

2 1.49

3 1.35

4 1.26

5 1.25

By using the state augmentation approach [91], the same relationship between d and γ is obtained as in the above table.

4.3 H∞ d-Step Prediction In the above section, we have studied the H∞ fixed-lag smoothing problem. In this section we shall investigate the H∞ multiple-step-ahead prediction problem. Consider the following linear system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t) z(t) = Lx(t)

x(0) = x0

(4.93) (4.94) (4.95)

where x(t) ∈ Rn , e(t) ∈ Rr , y(t) ∈ Rp , v(t) ∈ Rp and z(t) ∈ Rq represent the state, input noise, measurement output, measurement noise and the signal to be estimated, respectively. It is assumed that the input and measurement noises are deterministic signals and are from 2 [0, N ] where N is the time-horizon of the prediction problem under investigation.

70


The H∞ d-step prediction problem under investigation is stated as follows: Given a scalar γ > 0, an integer d > 0 and the observation {y(j)}tj=0 , find an estimate zˇ(t + d | t) of z(t + d), if it exists, such that the following inequality is satisfied: N

sup (x0 ,u,v)=0

[ˇ z (t + d | t) − z(t + d)] [ˇ z (t + d | t) − z(t + d)]

t=−d

x0 P0−1 x0

+

N +d−1

e (t)e(t) +

t=0

N

< γ 2 (4.96) v (t)v(t)

t=0

where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises. 4.3.1

An Equivalent H2 Problem in Krein Space

Similar to the discussion for the H∞ fixed-lag smoothing, we define

Jd,N = x0 P0−1 x0 +

N +d−1

e (t)e(t) +

t=0

−γ −2

N

N

v (t)v(t)

t=0

vz (t + d)vz (t + d),

(4.97)

t=−d

where vz (t + d) = zˇ(t + d | t) − Lx(t + d), t ≥ −d.

(4.98)

Introduce the following stochastic system x(t + 1) = Φx(t) + Γ e(t), y(t) = Hx(t) + v(t), ˇ z(t + d | t) = Lx(t + d) + vz (t + d), t ≥ −d,

(4.99) (4.100) (4.101)

where e(t), v(t) and vz (t) are assumed to be uncorrelated white noises, with x(0), x(0) = P0 , e(t), e(s) = Qe δts , v(t), v(s) = Qv δts , vz (t), vz (s) = Qvz δts , while Qe = Ir , Qv = Ip , Qvz = −γ 2 Iq .

(4.102)

ˇ(t + d | t) are respectively the observations for In the above system, y(t) and z x(t) and x(t + d) at the time instant t. Since vz (t) is of negative covariance, the above stochastic system should be considered in Krein space [30] rather than Hilbert space.


71

Let yo (t) be the measurement at time t and vo (t) be associated measurement noise at time t, then we have ⎧ ⎪ ⎨ zˇ(t + d | t), −d ≤ t < 0 yo (t) = (4.103) y(t) ⎪ ,t≥0 ⎩ ˇ z(t + d | t) ⎧ ⎪ ⎨ vz (t + d), −d ≤ t < 0 vo (t) = (4.104) v(t) ⎪ , t ≥ 0. ⎩ vz (t + d) Combining (4.100) with (4.101) yields, ⎧ ⎪ ⎨ Lx(t + d) + vo (t), −d ≤ t < 0 yo (t) = H 0 x(t) ⎪ + vo (t), t ≥ 0. ⎩ 0 L x(t + d)

(4.105)

It is easy to know that vo (t) is a white noise with vo (t), vo (s) = Qvo (t)δts , where ⎧ ⎪ ⎨ −γ 2 Iq , −d ≤ t < 0 Qvo (t) = (4.106) 0 I ⎪ , t ≥ 0. ⎩ p 2 0 −γ Iq Let yo be the collection of the measurements of system (4.93)-(4.94) up to time N , i.e., yo = col {yo (−d), · · · , yo (N )} ,

(4.107)

and yo be the collection of the Krein space measurements of system (4.99)(4.101) up to time N , i.e., yo = col {yo (−d), · · · , yo (N )} .

(4.108)

With the Krein space state-space model of (4.99)-(4.101), we can show that Jd,N of (4.97) can be rewritten as (see also [104]) ⎤ ⎡ ⎤ ⎡ ⎤−1 ⎡ ⎤ x(0) x(0) x(0) x(0) ⎣ e ⎦ , (4.109) Jd,N (x(0), e; yo ) = ⎣ e ⎦ ⎣ e ⎦ , ⎣ e ⎦ yo yo yo yo ⎡

where e = col{e(0), e(1), · · · , e(N + d − 1)},

(4.110)

e = col{e(0), e(1), · · · , e(N + d − 1)},

(4.111)

with e(·) from system (4.93) and e(·) from the Krein space system (4.99).

72


From (4.109), it is clear that the H∞ prediction problem is equivalent to that Jd,N has a minimum over {x(0), e} and the predictor is such that the minimum is positive. Now the problem is to derive conditions under which Jd,N (x(0), e; yo ) is minimum over {x(0), e}, and find a predictor such that the minimum of Jd,N (x(0), e; yo ) is positive. Similar to the H∞ fixed-lag smoothing discussed in the last section, we introduce the innovation associated with the measurements yo (t).

ˆ o (t | t − 1), wo (t) = yo (t) − y

(4.112)

ˆ o (t | t − 1) is the projection of yo (t) onto where y L{yo (−d), · · · , yo (0), · · · , yo (t − 1)}.

(4.113)

For −d ≤ t < 0, the innovation wo (t) is given from (4.105) as wo (t) = ˇ z(t + d | t) − Lˆ x(t + d | t − 1),

(4.114)

ˆ (t + d | t − 1) is the projection of x(t + d) onto L{y(−d), · · · , y(t − 1)}. where x For t ≥ 0, the innovation wo (t) is given from (4.105) as

H wo (t) = yo (t) − 0

0 L

ˆ (t | t − 1) x + vo (t), ˆ (t + d | t − 1) x

(4.115)

ˆ (t | t − 1) and x ˆ (t + d | t − 1) are respectively the projections of x(t) and where x x(t + d) onto (4.113). Furthermore, let

Qwo (t) = wo (t), wo (t).

(4.116)

Qwo (t) is the covariance matrix of the innovation wo (t). Note that Qvo (t) < 0 for −d ≤ t < 0, it is easy to know that Qwo (t) < 0,

− d ≤ t < 0.

(4.117)

With the innovation and covariance matrix, we have the following results [31, 104], Lemma 4.3.1. 1) The H∞ estimator zˇ(t + d | t) that achieves (4.96) exists if and only if Qwo (t) and Qvo (t) for −d ≤ t ≤ N have the same inertia. 0 , if exists, can be given in terms of the innovation 2)The minimum of Jd,N wo (t) as 0 Jd,N =

N t=−d

wo (t)Q−1 wo (t)wo (t).

(4.118)


4.3.2

73

Re-organized Innovation

Similar to the line of arguments for fixed-lag smoothing, we shall apply the re-organized innovation analysis approach to derive the main results. First, the measurements up to time t is denoted by {yo (−d), · · · yo (t)},

(4.119)

which, for t ≥ 0, can be equivalently re-organized as {yf (s)}ts=0 , zˇ(t + 1 | t − d + 1), · · · , ˇz(t + d | t) , where

y(s) yf (s) = ˇ z(s | s − d) satisfies:

yf (s) =

(4.121)

H x(s) + vf (s), s = 0, 1, · · · , t, L

with

v(s) vf (s) = vz (s)

(4.120)

(4.122)

(4.123)

Ip 0 . 0 −γ 2 Iq It is now clear that the measurements in (4.120) are no longer with time delays. Secondly, we introduce the innovation associated with the reorganized measurements (4.120). Given time instant t and t + 1 ≤ s ≤ t + d, let

a white noise satisfying that vf (s), vf (t) = Qvf δst and Qvf =

w(s) = zˇ(s | s − d) − zˆ(s),

(4.124)

where ˆ z(s) for s > t + 1 is the projection of zˇ(s | s − d) onto the linear space z(t + 1 | t − d + 1), · · · , zˇ(s − 1 | s − d − 1) , (4.125) L {yf (s)}ts=0 , ˇ and for s = t + 1, ˆ z(s) is the projection of ˇ z(s | s − d) onto the linear space L{yf (0), · · · , yf (t)}.

(4.126)

Similarly, we define that

ˆ f (s), 0 ≤ s ≤ t, wf (s) = yf (s) − y

(4.127)

ˆ f (s) is the projection of yf (s) onto the linear space where y L{yf (0), · · · , yf (s − 1)}.

(4.128)

74


Recall the discussion in Section 2.2, it is easy to know that {wf (0), · · · , wf (t), w(t + 1), · · · , w(t + d)}

(4.129)

is a white noise sequence and spans the same linear space as (4.120) or equivalently (4.119). Different from the innovation wo (s) (−d ≤ s ≤ t), w(s) (0 ≤ s ≤ t) and wf (s) (t + 1 ≤ s ≤ t + d) are termed as reorganized innovations. 4.3.3

Calculation of the Innovation Covariance

In this subsection, we shall calculate the covariance matrix Qwo (t) of the innovation wo (t) using the re-organized innovation. To compute the covariance matrix, we first derive the Riccati equation associated with the reorganized innovation. By considering (4.100) and (4.122), it follows from (4.124)-(4.127) that w(s) = L˜ x(s, 1) + vz (s), t + 1 ≤ s ≤ t + d, H ˜ (s, 2) + vf (s), 0 ≤ s ≤ t, x wf (s) = L

(4.130) (4.131)

where ˜ (s, 1) = x(s) − x ˆ (s, 1), x ˜ ˆ x(s, 2) = x(s) − x(s, 2).

(4.132) (4.133)

ˆ (s, 1), s > t + 1 is the projection of x(s) onto the linear space In the above, x L {yf (s)}ts=0 , zˇ(t + 1 | t − d + 1), · · · , ˇz(s − 1 | s − d − 1) (4.134) ˆ (s, 2) (s ≤ t + 1) the projection of x(s) onto the linear space and x L{yf (0), · · · , yf (s − 1)}.

(4.135)

ˆ (s, 1) is the projection of x(t + 1) onto the linear space For s = t + 1, x L{yf (0), · · · , yf (t)}.

(4.136)

˜ (t + 1, 1) = x ˜ (t + 1, 2). Obviously, x Let

˜ (s, 2), 0 ≤ s ≤ t + 1, P2 (s) = ˜ x(s, 2), x

˜ (s, 1), t + 1 ≤ s ≤ t + d. P1 (s) = ˜ x(s, 1), x

(4.137) (4.138)


75

Then, we have the following Lemma Lemma 4.3.2. Given time instant t, the matrix P2 (s) for 0 ≤ s ≤ t + 1 obeys the following Riccati equation associated with the system (4.99) and (4.122): P2 (s + 1) = ΦP2 (s)Φ − K2 (s)Qwf (s)K2 (s) + Γ Γ , P2 (0) = P0 , (4.139) where K2 (s) = ΦP2 (s)

H L

Q−1 wf (s),

(4.140)

while Qwf (s) is given by Qwf (s) =

H H I + p P2 (s) 0 L L

0 . −γ 2 Iq

The matrix P1 (s) for t + 1 ≤ s < t + d obeys the following Riccati equation associated with the system (4.99)-(4.100): P1 (s + 1) = ΦP1 (s)Φ − K1 (s)Qw (s)K1 (s) + Γ Γ , P1 (t + 1) = P2 (t + 1),

(4.141)

where K1 (s) = ΦP1 (s)L Q−1 w (s),

(4.142)

Qw (s) = LP1 (s)L − γ 2 Iq .

(4.143)

and

Proof: The proof is similar to the case of the optimal H2 estimation discussed in Chapter 2. ∇ It should be noted that the terminal solution of the Riccati equation (4.139) is the initial condition of the Riccati equation (4.141). Denote

˜ (s + i, 1), i ≥ 0, s ≥ t + 1. R(s, i) = x(s), x

(4.144)

By a similar line of discussions as for the fixed-lag smoothing, it is easy to know that R(s, i) satisfies the following difference equation: R(s, i + 1) = R(s, i)A (s, i), R(s, 0) = P1 (s),

(4.145)

A(s, i) = Φ In − P1 (s + i)L Q−1 w (s + i)L .

(4.146)

where

Now we are in the position to give the computation of innovation covariance matrix of wo (t).

76


Theorem 4.3.1. For t + 1 ≥ 0, Qwo (t + 1) is given by HR(t + 1, d)L HP(t)H + Ip Qws (t + 1) = , LR (t + 1, d)H P1 (t + d + 1) − γ 2 Iq

(4.147)

where P(t) = P2 (t + 1) −

d

R(t + 1, i − 1)L Q−1 w (t + i)LR (t + 1, i − 1),(4.148)

i=1

while R(t + 1, i) is computed by (4.145) and Qw (t + i) = LP1 (t + i)L − γ 2 Iq . Proof: The proof is similar to the case for the H∞ fixed-lag smoothing in the last section. ∇ 4.3.4

H∞ d-Step Predictor

Having presented the Riccati equations and calculated the covariance matrix of innovation in the last subsection, we are now in the position to give the main result of the H∞ d-step prediction. Theorem 4.3.2. Consider the system (4.93)-(4.95) and the associated performance criterion (4.96). Then for a given scalar γ > 0 and d > 0, an H∞ predictor zˇ(t+d | t) that achieves (4.96) exists if and only if, for each t = −1, · · · , N −1, 0 Ip have the same inertia, where Qwo (t+1) Qwo (t+1) and Qvo (t+1) = 0 −γ 2 Iq is calculated by (4.147). In this situation, the central predictor is given by zˇ(t + d | t) = Lˆ x(t + d, 1),

(4.149)

where x ˆ(t + d, 1) is obtained from the Krein space projection of x(t + d) onto the z(t + 1 | t − d + 1), · · · , ˇz(t + d − 1 | t − 1)}, linear space of L{yf (0), · · · , yf (t); ˇ which can be computed recursively as z (s | s − d) − Lˆ x(s, 1)] , x ˆ(s + 1, 1) = Φˆ x(s, 1) + K1 (s) [ˇ s = t + 1, · · · , t + d − 1, xˆ(t + 1, 1) = xˆ(t + 1, 2). (4.150) ˆ(t + 1, 2) in (4.150) In the above, K1 (s) is as in (4.142) and the initial value x is given by the Kalman recursion: H x ˆ(t + 1, 2) = Φˆ x(t, 2)+K2(t) yf (t)− x ˆ(t, 2) , x ˆ(0, 2) = 0, (4.151) L y(t) where K2 (t) is as in (4.140) and yf (t) = . zˇ(t | t − d) Proof: The proof is similar to the case for the H∞ fixed-lag smoothing in the last section. ∇

4.4 H∞ Filtering for Systems with Measurement Delay

77

4.4 H∞ Filtering for Systems with Measurement Delay In this section we study the H∞ filtering for measurement delayed systems. By identifying an associated stochastic system in Krein space, the H∞ filtering is shown to be equivalent to the Krein space Kalman filtering with measurement delay. Necessary and sufficient existence conditions of the H∞ filtering are given in terms of the innovation covariance matrix of the identified stochastic system. 4.4.1

Problem Statement

We consider the following linear system for the H∞ filtering problem. x(t + 1) = Φx(t) + Γ e(t), x(0) = x0 , z(t) = Lx(t),

(4.152) (4.153)

where x(t) ∈ Rn , e(t) ∈ Rr , and z(t) ∈ Rq represent the state, input noise, and the signal to be estimated, respectively. Φ, Γ , L are bounded matrices with dimensions of n × n, n × r and q × n, respecively. Assume that the state x(t) is observed by two channels described by y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t),

(4.154) (4.155)

where H(i) (i = 0, 1) is of dimension pi × n. In (4.154)-(4.155), y(0) (t) is a measurement without delay whereas y(1) (t) ∈ Rp1 is a delayed measurement and v(i) (t) ∈ Rpi , i = 0, 1 are measurement noises. It is assumed that the input noise e is from L2 [0, N ] and measurement noises v(0) and v(1) respectively from L2 [0, N ] and L2 [d, N ], where N > 0 is the time-horizon of the filtering problem under consideration. Let y(t) denote the observation of the system (4.154)-(4.155) at time t and v(t) the related observation noise at time t, then we have ⎧ ⎨ y (t), 0 ≤ t < d, (0) y(t) = (4.156) ⎩ col y (t), y (t) , t ≥ d (0)

v(t) =

⎧ ⎨v

(1)

0 ≤ t < d, ⎩ col v (t), v (t) , t ≥ d. (0) (1) (0) (t),

(4.157)

The H∞ filtering problem is stated as: Given a scalar γ > 0 and the observation {y(s), 0 ≤ s ≤ t}, find a filtering estimate zˇ(t | t) of z(t), if it exists, such that the following inequality is satisfied: N z (t | t) − z(t)] [ˇ z (t | t) − z(t)] t=0 [ˇ < γ 2 , (4.158) sup N −1 N P −1 x + x e (t)e(t) + v (t)v(t) (x0 ,u,vd )=0 0 0 0 t=0 t=0 where P0 is a given positive definite matrix which reflects the relative uncertainty of the initial state to the input and measurement noises.

78


4.4.2

An Equivalent Problem in Krein Space

First, in view of (4.158) we define

Jd,N = x0 P0−1 x0 +

N −1

N

t=0

t=0

e (t)e(t)+

v (t)v(t)−γ −2

N

vz (t)vz (t), (4.159)

t=d

where vz (t) = zˇ(t | t) − Lx(t).

(4.160)

The H∞ filtering problem is equivalently stated as: 1) Jd,N has a minimum over {x0 , e}, and 2) the filter zˇ(t | t) is such that the minimum is positive. We investigate the above minimization problem by introducing a Krein space model as discussed in the last section. Define the following stochastic system associated with (4.152)-(4.155): x(t + 1) = Φx(t) + Γ e(t), y(0) (t) = H(0) x(t) + v(0) (t), y(1) (t) = H(1) x(t − d) + v(1) (t).

(4.161) (4.162) (4.163)

We also introduce a ‘fictitious’ observation system: ˇ z(t | t) = Lx(t) + vz (t), t ≥ 0,

(4.164)

where zˇ(t | t) ∈ Rq implies the observation of the state x(t) at time t. In the above, the matrices Φ, Γ , H(i) and L are the same as in (4.152)-(4.155). Note that x(·), e(·), yi (·), vi (·), ˇ z(·) and vz (·) , in bold faces, are Krein space elements. Assumption 4.4.1. The initial state x(0) and the noises e(t), v(i) (t) (i = 0, 1) and vz (t) are mutually uncorrelated white noises with zero means and known covariance matrices as P0 , Qe = Ir , Qv(i) = Ipi and Qvz = −γ 2 Iq , respectively. Combining (4.162) with (4.164) yields, ¯ ¯ (t) = Hx(t) ¯ (t), y +v where

(4.165)

y(0) (t) , ˇ z(t | t) v(0) (t) ¯ (t) = v , vz (t) ¯ = H(0) . H L

¯ (t) = y

(4.166) (4.167) (4.168)

¯ is as The covariance of v Qv¯ = diag{Qv(0) , Qvz } = diag{Ip0 , −γ 2 Iq }.

(4.169)

4.4 H∞ Filtering for Systems with Measurement Delay

79

Let y(t) be the measurement at time t for system (4.163) and (4.165), i.e., ⎧ ⎪ ⎨y ¯ (t), 0≤t 0, the RDE (5.46) admits a bounded solution Pjτ , j = min{hl − 1, N − τ }, · · · , 1, 0. If vi (t) = vi∗ (t) for t = 0, · · · , τ − 1; i = 0, · · · , l, then vi∗ (τ ) ≡ viτ ∗ (0) |vj (t)=vj∗ (t)(

0≤t 0 exists if and only if achieves JN Π −1 − γ −2 P0 > 0, ˜ 2,2 (t) < 0. M

(5.118) (5.119)

In view of (5.116), the suitable controller can be chosen such that u ¯(τ ) = u ¯∗ (τ ).

(5.120)

Therefore, the controller is u∗i (τ ), which is given by (5.113) and (5.114).

∇

Remark 5.2.4. From Theorem 5.2.4, the solution to the H∞ control of systems with multiple input delays requires solving a standard RDE (5.26). In addition, for every τ > 0, the RDE (5.46) is to be computed for Pjτ , j = min{hl − 1, N − τ }, · · · , 1, 0. Both the RDEs (5.26) and (5.46) have the same order as the original system (ignoring the delays). The result shares some similarity with the H∞ fixed-lag smoothing in [37, 111, 106] where an H∞ -type of RDE together with some Riccati recursions related to the length of the lag are to be solved. The solution has clear computational advantage over the traditional approach of state augmentation even if the latter can be applied to address our problem as the latter usually leads to higher system dimension. It is worthy pointing out that it is not clear at this stage if the existence of bounded solution of (5.46) is also necessary for the solvability of the H∞ control problem. Remark 5.2.5. In the delay free case, i.e., h1 = · · · = hl = 0, it is obvious that the H∞ controller (5.113)-(5.114) reduces to: u∗0 (τ ) = [Im , 0]v0∗ (τ ) = −[Im , 0] × [F0τ (0)] x(τ ) = −[Im , 0] × Kτ x(τ ),

(5.121)

106

5. H∞ Control for Discrete-Time Systems with Multiple Input Delays

where Kτ is as in (5.27). And the existence condition becomes Π −1 − γ −2 P0 > 0, M2,2 (t) < 0,

(5.122) (5.123)

where M2,2 (t) is the (2, 2)-block of Mt which is given by (5.28). Thus the obtained result is the same as the well-known full-information control solution [30].

5.3 H∞ Control for Systems with Preview and Single Input Delay 5.3.1

H∞ Control with Single Input Delay

The linear single input delay system is described by x(t + 1) = Φx(t) + Ge(t) + Bu(t − h) ⎡ ⎤ Cx(t) ⎦. z(t) = ⎣ Du(t − h)

(5.124) (5.125)

The H∞ control with single input delay is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy u(t) = F (x(t), u(τ ) |t−h≤τ 0, ¯ 1,1 (t) < 0, M

(5.138) (5.139)

¯ 1,1 (t) is where P0 is the terminal value of Pτ of (5.131), and for t ≤ N − h, M ¯ the (1,1)-block of matrix Mt which is given by ! ¯ t = diag{G , B } P¯t+1 (i, j) M diag{G, B} 2×2 +diag{−γ 2Ir , R(1) }

(5.140)

108


and for N − h < t ≤ N , ¯ 1,1 (t) = M ¯ t = G P¯t+1 (0, 0)G − γ 2 Ir , M (5.141) where P¯t+1 (i, j) is given by P¯τ (0, 0) = P0τ , P¯τ (1, 0) = P¯τ (0, 1) = Phτ (Φτ0,h ) , P¯τ (1, 1) = Phτ [I − Gτ (h)Phτ ] .

(5.142) (5.143) (5.144)

In this situation, a suitable H∞ controller is given by + , h ∗ τ τ ∗ u (τ ) = −[0, Im ] × [F0 (h)] x(τ ) + [Fk (h)] Gu (τ + k − h − 1) , k=1

(5.145) where Fkτ (h) is as Fkτ (h) = [In − Pkτ Gτ (k)] Φτk,h Khτ ,

(5.146)

and Gτ (h) =

h

τ (Φτj,h ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,h ,

(5.147)

Φτj,t = Φτj · · · Φτt−1 , t ≥ j,

(5.148)

j=1

Φτj,j = I,

while Φτs · · · Φτt = I for t < s, and Φτj = Φ − Kjτ Γj . 5.3.2

(5.149)

H∞ Control with Preview

We consider the following system for the H∞ control with preview: x(t + 1) = Φx(t) + Ge(t − h) + Bu(t), ⎤ ⎡ Cx(t) ⎦. z(t) = ⎣ Du(t)

(5.150) (5.151)

The H∞ control with preview is stated as follows: for a given positive scalar γ, find a finite-horizon full-information control strategy u(t) = F (x(t), e(τ ) |t−h≤τ 0, ¯ 2,2 (t) < 0, M

(5.164)

0≤t≤N −h (5.165) ¯ where P0 is the terminal value of Pτ of (5.157), and M2,2 (t) is the (2,2)-block ¯ t which, for t ≤ N − h, is given by of matrix M ! ¯ t = diag{B , G } P¯t+1 (i, j) M diag{B, G} 2×2

+diag{D D, −γ I}, ¯ where Pt+1 (i, j) is computed by P¯τ (0, 0) = P τ , 2

0

P¯τ (1, 0) = P¯τ (0, 1) = Phτ (Φτ0,h ) , P¯τ (1, 1) = Phτ [I − Gτ (h)Phτ ] .

(5.166)

(5.167) (5.168) (5.169)

In this situation, a suitable H∞ controller is given by + , h u∗0 (τ ) = − [F0τ (0)] x(τ ) + [Skτ (0)] Ge∗ (τ + k − h − 1) , k=1

(5.170) where

F0τ (0)

=

K0τ

Skτ (h)

and

! = Pkτ (Φτh+1,k ) Γh (Mhτ )−1 − (Φτh,k ) Gτ (h)Khτ ,

(5.171)

with Gτ (h) =

h

τ (Φτj,h ) Γj−1 (Mj−1 )−1 Γj−1 Φτj,h ,

(5.172)

j=1

Φτj,j = I,

Φτj,t = Φτj · · · Φτt−1 , t ≥ j

(5.173)

and Φτj = Φ − Kjτ Γj . Note that

Φτs

· · · Φτt

= I for t < s.

(5.174)

5.4 An Example

111

5.4 An Example Consider the mathematical model of congestion control taken from [1]: qt+1 = qt +

d

v¯i,t−hi − µt

(5.175)

i=1

µt = µ + ξt τ ξt = li ξt−i + ηt−1

(5.176) (5.177)

i=1

where qt is the queue length at the bottleneck, ξt is the higher priority source (interference) which is modelled as a stable auto-regressive (AR) process as in (5.177) with ηt the noise input. µt denotes the effective service rate available for the traffic of the given source, µ is the constant nominal service rate, and v¯i,t−hi is the input rate of the i-th source. hi is the round trip delay of the i-th source consisting of two path delays, one is the return path (from the switch to the source) delay and the other is forward path (from the source through the congested switch) delay. We note that round trip delay in transmission is one reason for the disagreement between the switch input and output. For a prescribed γ > 0, the H∞ congestion control problem is to find source rate v¯ such that sup {q0 ,¯ vi,t−hi |0≤i≤d,0≤k≤N −hi }

J(q0 , v¯i,t−hi , µt ) < γ 2 ,

(5.178)

for any non-zero η ∈ 2 [0, N ], where J(q0 , v¯i,t−hi , µt ) N d λ2 (¯ vi,t−hi − ai µ)2 (qt − q¯)2 + =

t=1

i=1 N t=1

(5.179) ηt2

with q¯ the target queue length and λ a weighting factor. ai satisfying

d

ai = 1 is

i=1

the weight for different source rates which determines the allocation of bandwidth for each channel. The objective is to make the queue buffer close to the desired level while the difference between the source rate and the nominal service rate should not be too large. We adopt a second order AR model for the higher priority source, i.e. τ = 2 [82]. Denote ⎛ ⎞ qt − q¯ x(t) = ⎝ ξt−1 ⎠ , ξt

112


ui (t − hi ) = v¯i,t−hi − ai µ, ⎛ q − q¯ ⎞ t

⎜ λu1 (t − h1 ) ⎟ ⎟, z(t) = ⎜ .. ⎝ ⎠ . λud (t − hd ) e(t) = ηt .

The system (5.175)-(5.177) can be put into the form of (5.1)-(5.2) with ⎞ ⎛ ⎛ ⎞ 1 0 −1 0 Φt = ⎝ 0 0 1 ⎠ , G0,t = ⎝ 0 ⎠ , 0 l2 l1 1 Gi,t = 0 (i > 0), B0,t = 0, ⎛ ⎞ 1 Bi,t = ⎝ 0 ⎠ (i = 1, · · · , d), Ct = 1, 0 and the cost function (5.6) with ⎛ ⎞ 1 0 0 Di,t = λ2 . Qt = ⎝ 0 0 0 ⎠ , Di,t 0 0 0 The initial state is assumed to be known. We adopt the similar parameters as given by [82]: The buffer length ymax = 10000 cells/s The buffer set point q¯ = (1/2)ymax = 5000 cells The controller cycle time T = 1 ms. Assume that there are 10 sources with round trip delays from 1 to 10, respectively, i.e., d = 10, hi = i, i = 1, 2, · · · , 10. We also assume that l1 = l2 = 0.4 and the Gaussian white noise process ηt ’s variance is equal to 1 [82]. The time horizon N is 100 and we set γ = 15. The weighting between the queue length and the transmission rate is λ = 1 and ai = 1/d(i = 1, 2, · · · , d) are the source sharing. By applying Theorem 5.2.4, we design an H∞ controller. Simulation result for the designed controller is shown in Figure 5.1 where the vertical axis is the queue length qt . The initial queue length of the congested switch is set to be 5100. From the graph we can see that the queue length quickly converges to the target queue length. As mentioned in [1], the congestion control problem considered in this example can be studied with the state augmentation and standard H∞ control theory. In what follows, we shall show that the augmentation approach will lead to more expensive computation than the one of the presented approach, especially when the delays are large. In fact, if we apply the state augmentation approach to the problem, a RDE with dimension of hd + τ + 1 [1] will be solved where hd is the maximum delay in (5.175) and τ is the order of the AR process of (5.176), and thus the operation number (the total number of multiplication and division) can

5.5 Conclusion

113

be roughly estimated as of O((hd + τ + 1)3 ) (note that the computation cost of controller design is mainly boiled down to the solving of the RDE) for each iteration step. While if we apply the presented approach in this paper, we require to solve the two RDEs with dimension of τ + 1, in which the computation cost can be roughly computed as O((τ + 1)3 + hd (τ + 1)3 ). It is easy to know that (hd +τ +1)3 >> (1+hd )(τ +1)3 when the delay hd is large enough. In the example, we have hd = 10, τ = 2 and thus (hd +τ +1)3 = 2197 where (1+hd)(τ +1)3 = 297. In other words, the augmentation approach requires a much larger operation number (the sum of multiplications and divisions) than the presented approach. On the other hand, it should be noted that the augmentation approach generally leads to a static output feedback control problem of non-convex [96]. 5120

5100

5080

5060

5040

qt 5020

5000

4980

4960

4940

0

10

20

30

40

50

60

70

80

90

100

t (sample time)

Fig. 5.1. Queue length response

5.5 Conclusion We have presented a simple solution to the H∞ control of systems with multiple input delays. We solved the problem by converting it to an optimization associated with a stochastic system in Krein space and thus establishing a duality between the H∞ full-information control and an H∞ smoothing problem. The duality enables us to address the full-information control problem via a simple approach. Our solvability condition and the construction of the controller are given in terms of the solutions of two Riccati difference equations of the same order as the original plant, which is advantageous over methods such as system augmentation. Our result assumes the existence of bounded solutions of two Riccati difference equations whose necessity deserves further studies. While this chapter only deals with systems with input delays via state feedback, the presented approach together with the existing studies on H∞ estimation for systems with delayed measurements [109] would make it possible to solve the H∞ control of systems with multiple i/o delays via dynamic output feedback.

6. Linear Estimation for Continuous-Time Systems with Measurement Delays

In this chapter, we shall study the optimal estimation problems for continuoustime systems with measurement delays under both the minimum variance and H∞ performance criteria. We also investigate a related H∞ fixed-lag smoothing problem. The key technique that will be applied in this chapter is the reorganized innovation analysis approach in both Hilbert spaces and Krein spaces.

6.1 Introduction This chapter is concerned with the H2 and H∞ estimation problems for continuous-time systems with measurement delays. This problem may be approached by a partial differential equation approach or an infinite dimensional system theory approach. However, these approaches lead to solutions either in terms of partial Riccati differential equations or operator Riccati equations which are difficult to understand and implement in practice [4, 13, 12, 52, 54]. In order to obtain an effective approach to the problems, some attempts have been made. In [72], the H∞ estimation for systems with measurement delay is studied and the derived filter is given in terms of one H∞ Riccati equation and one H2 Riccati equation. Notice that only single measurement delay was considered in [72]. Our aim here is to present a simple approach for the problems in more general case without involving Riccati Partial Differential Equation (PDE) or operator Riccati equation. First, we study the linear minimum mean square error (H2 ) estimation for systems with multiple delayed measurements. By applying the re-organized innovation analysis, an estimator is derived in a simple manner. The estimator is designed by performing two standard differential Riccati equations. Secondly, we shall investigate the H∞ filtering problem for systems with measurement delays. By introducing an appropriate dual stochastic system in Krein space, it will be shown that the calculation of the central H∞ estimator is the same as that of the H2 estimator associated with the dual system. Thirdly, we shall study the complicated H∞ fixed-lag smoothing problem which has been an open problem for many years. By defining a dual stochastic system H. Zhang and L. Xie: Cntrl. and Estim. of Sys. with I/O Delays, LNCIS 355, pp. 115–141, 2007. c Springer-Verlag Berlin Heidelberg 2007 springerlink.com

116


in Krein space, the H∞ fixed-lag smoothing problem is converted into an H2 estimation problem for an associated system with instantaneous and delayed measurements. The existence condition and estimator are derived by using the re-organized innovation analysis.

6.2 Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems In this section we study the optimal estimation problem for systems with multiple measurement delays. The standard Kalman filtering formulation is not applicable to such problem due to the delays in measurements. We shall extend the reorganized innovation analysis developed in the previous chapters to continuoustime systems. 6.2.1

Problem Statement

We consider the linear time-invariant system described by ˙ x(t) = Φx(t) + Γ e(t),

(6.1)

where x(t) ∈ Rn is the state, and e(t) ∈ Rr is the system noise. The state is observed by l + 1 channels with delays given by y(i) (t) = H(i) x(t − di ) + v(i) (t), i = 0, 1, · · · , l,

(6.2)

where y(i) (t) ∈ Rpi is the delayed measurement, v(i) (t) ∈ Rpi is the measurement noise. The initial state x(0), and noises e(t) and v(i) (t), i = 0, 1, · · · , l, are uncorrelated white noises with zero means and covariance matrices as E [x(0)x (0)] = x(0), x(0) = P0 , E [e(t)e (s)] = Qe δt−s , and E v(i) (t)v(i) (s) = Qv(i) δt−s ,

respectively, where E(·) denotes the expectation. Without loss of generality, the time delays di are assumed to be in a strictly increasing order: 0 = d0 < d1 < · · · < dl . In (6.2), y(i) (t) means the observation of the state x(t − di ) at time t, with delay di . Let y(t) be the observation of the system (6.2) at time t, then we have ⎤ ⎡ y(0) (t) ⎥ ⎢ .. (6.3) y(t) = ⎣ ⎦ , di−1 ≤ t < di , . y(i−1) (t) and for t ≥ dl ,

⎡

⎤ y(0) (t) ⎢ ⎥ y(t) = ⎣ ... ⎦ . y(l) (t)

(6.4)

The problem to be studied is stated as: Given the observation {y(s)|0≤s≤t }, ˆ (t | t) of x(t). find a linear least mean square error estimator x

6.2 Linear Minimum Mean Square Error Estimation

117

Remark 6.2.1. For the convenience of discussion, we shall assume that the time instant t ≥ dl throughout the section. When the measurement (6.2) is delay free, i.e., di = 0 for 1 ≤ i ≤ l, the above problem can be solved by using the standard Kalman filtering formulation. However, in the case when the system (6.2) is of measurement delays, the standard Kalman filtering is not applicable. One possible way to such problem is the PDE approach [13, 52], which however leads to difficult computation. Our aim is to present an analytical solution in terms of Riccati Differential Equations (RDEs). The key is the re-organization of the measurement and innovation sequences. Throughout the section we denote that

ti = t − di , i = 0, 1, · · · , l. 6.2.2

Re-organized Measurement Sequence

In this subsection, the instantaneous and l-delayed measurements will be reorganized as delay free measurements so that the Kalman filtering is applicable. As is well known, given the measurement sequence {y(s)|0≤s≤t }, the optiˆ (t | t) is the projection of x(t) onto the linear space mal state estimator x L {y(s)|0≤s≤t } [38, 3]. Note that the linear space L {y(s)|0≤s≤t } is equivalent to the following linear space (see Remark 2.2.2 for explanation): L{yl+1 (s)|0≤s≤tl ; · · · ; yi (s)|ti <s≤ti−1 ; · · · ; y1 (s)|t1 <s≤t }, (6.5) where

⎡

⎤ y(0) (s) ⎢ ⎥ .. yi (s) = ⎣ ⎦ . y(i−1) (s + di−1 )

(6.6)

is a new measurement obtained by re-organizing the delayed measurement y(t). It is clear that the new measurement yi (t) satisfies that yi (t) = Hi x(t) + vi (t), where

i = 1, · · · , l + 1,

⎤ ⎡ ⎤ H(0) v(0) (t) ⎥ ⎢ ⎢ ⎥ .. Hi = ⎣ ... ⎦ , vi (t) = ⎣ ⎦, . H(i−1) v(i−1) (t + di−1 )

(6.7)

⎡

(6.8)

and vi (t) is a white noise of zero mean and covariance matrix Qvi = diag{Qv(0) , · · · , Qv(i−1) }, i = 1, · · · , l + 1.

(6.9)

Note that the measurement sequence in (6.5) are no longer with any delay, which is termed as the re-organized measurement sequence of {y(s)|0≤s≤t }. ˆ (t | t) becomes the projection of x(t) onto the linear The optimal estimate x space (6.5).

118


6.2.3

Re-organized Innovation Sequence

In this subsection we shall define the innovation associated with the re-organized measurement sequence (6.5). To this end, we first define the projection in the linear space spanned by the re-organized measurement sequence. Definition 6.2.1. Given the re-organized observation yi (s) with ti ≤ s ≤ ti−1 . ˆ i (s) denotes the projection of yi (s) onto the linear space: For s > ti , y L{yl+1 (τ )|0≤τ ≤tl ;

· · · ; yk (τ )|tk hl − hi , following from (8.196), the optimal controller is given by (8.191). This completes the proof of the theorem. ∇ hl −hi , vi∗ (t)

In Theorem 8.4.1, vi∗ (τ ) is given in terms of the initial state x(0) rather than the current state x(τ ). This problem can be addressed by shifting time interval from [0, hl ] to [τ, τ + hl ]. To this end, we first introduce the following notations. For any given τ ≥ 0, denote: ⎧ ⎡ ⎤ v0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ .. ⎪ ⎪ ⎣ ⎦ , hi ≤ t < hi+1 , . ⎪ ⎪ ⎨ ⎡ vi (t + τ − hi ) ⎤ v τ (t) = (8.206) v0 (t + τ − h0 ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ .. ⎪ ⎪ ⎦ , t ≥ hl ⎣ . ⎪ ⎪ ⎩ vl (t + τ − hl ) 1 l j=i+1 Γ(j) vj (t + τ − hj ), hi ≤ t < hi+1 , v˜τ (t) = (8.207) 0, t ≥ hl .

194

8. Optimal and H∞ Control of Continuous-Time Systems with I/O Delays

Using the notations of (8.206)-(8.207), for a given τ > 0, the system (8.144) and the cost (8.149) can be rewritten respectively as 1 Φx(t + τ ) + Γ (t)v τ (t) + v¯τ (t), hi ≤ t < hi+1 , (8.208) x(t ˙ + τ) = t ≥ hl Φx(t + τ ) + Γ (t)v τ (t), and Jtf

=

Jtτf +

l 9 i=0

τ

vi (t)R(i) vi (t)dt +

9

0

τ

x (t)Qx(t)dt,

(8.209)

0

where 9 Jtτf

tf −τ

=

τ

9 τ

tf −τ

[v (t)] R(t)v (t)dt + 0

x (t + τ )Qx(t + τ )dt. (8.210)

0

Following a similar discussion and the well known dynamic programming arguments, the H∞ control for system (8.208) associated with the cost index (8.210) is given in the theorem below. Theorem 8.4.2. Given a scalar γ, suppose that the RDE (8.184) has a bounded solution P (t), 0 ≤ t ≤ tf and for any τ > 0, the following RDE with initial value P τ (t) |t=hl = P (τ +hl ) for hl +τ ≤ tf or P τ (t) |t=tf −τ = P (tf ) = 0 for hl +τ > tf admits a bounded solution P τ (t), 0 ≤ t ≤ min{hl , tf − τ }, −

dP τ (t) dt

=

Φ P τ (t) + P τ (t)Φ + Q − Ktτ R(t) (Ktτ )

(8.211)

where Ktτ = P τ (t)Γ (t)R(t)−1 .

(8.212)

Let v ∗ be decomposed as in (8.178), i.e. v ∗ = {v ∗ (t), 0 ≤ t ≤ tf }, where v ∗ (τ ) as in (8.189). Then vi∗ (τ ) in (8.189) is calculated by −1 vi∗ (τ ) = −R(i) Γ(i) × P τ (hi )[Ψ¯ τ (0, hi )] x(τ )+ 9 hi P τ (hi ) [Ψ¯ τ (s, hi )] {In − Gτ (s)P τ (s)} v˜τ ∗ (s)ds+ h0 , 9 hl τ τ τ τ τ∗ ¯ [In − P (hi )G (hi )] Ψ (hi , s)P (s)˜ v (s)ds , (8.213) hi+1

while v˜τ ∗ (·) is as in (8.207) with vi (·) replaced by vi∗ (·) for i = 0, · · · , l, Σ τ (·, ·) and Gτ (·) is given by 9 s ! Ψ¯ τ (r, s) Γ (r)R−1 (r)Γ (r)Ψ¯ τ (r, s)dr. Gτ (s) = (8.214) 0

8.4 H∞ Full-Information Control

195

In the above, R(r) and Γ (r) are respectively as in (8.155) and (8.154) and Ψ¯ τ (s, ·) is the transition matrix of −Φτ (s) = K τ (s)Γ (s) − Φ . The results of Theorems 8.4.1 and 8.4.2 give an explicit calculation of the quadratic form (8.177). Note, however, that the exogenous inputs play a contradictory role with the control inputs, namely the former aims to maximize the cost whereas the latter minimize the cost. Since the second term of (8.177) is not definite and v involves both the control inputs and the exogenous inputs, it is not clear from (8.177) that under what conditions a minimax solution exists. Thus, some further simplification of (8.177) is needed, which will be given in the next section. 8.4.4

H∞ Control

In this subsection, we shall discuss conditions under which a maximizing solution of Jtf with respect to the exogenous inputs exists and then derive a suitable H∞ controller. To this end, we recall the Krein space stochastic model (8.159)-(8.160) and decompose the observation y(t) and the noise v(t) as follows: ⎧ ⎡ ⎤ y0 (t) ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎪ ⎣ . ⎦ , hi ≤ t < hi+1 , ⎪ ⎪ ⎨ ⎡ yi (t) ⎤ y(t) = (8.215) y0 (t) ⎪ ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎣ . ⎦ , t ≥ hl ⎪ ⎪ ⎩ yl (t) ⎧ ⎡ ⎤ v0 (t) ⎪ ⎪ ⎪ ⎢ .. ⎥ ⎪ ⎪ ⎪ ⎣ . ⎦ , hi ≤ t < hi+1 , ⎪ ⎪ ⎨ ⎡ vi (t) ⎤ (8.216) v(t) = v0 (t) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ . ⎪ ⎪ ⎪ ⎣ .. ⎦ , t ≥ hl ⎪ ⎩ vl (t) where yi (t) ∈ Rm+r and vi (t) ∈ Rm , i = 0, · · · , l, satisfy yi (t)

= Γ(i) x(t) + vi (t), 0 ≤ t ≤ tf − hi ,

(8.217)

and vi (t), vi (s) = R(i) δt,s . In view of the input delays of the original system, we re-organize (8.215)-(8.216) as follows: 1 col{y0 (t + h0 ), · · · , yl (t + hl )}, 0 ≤ t ≤ tf − hl , ¯ (t) = y (8.218) col{y0 (t + h0 ), · · · , yi (t + hi )}, tf − hi+1 < t ≤ tf − hi

196


and 1

¯ (t) = v

col{v0 (t + h0 ), · · · , vl (t + hl )}, 0 ≤ t ≤ tf − hl , (8.219) col{v0 (t + h0 ), · · · , vi (t + hi )}, tf − hi+1 < t ≤ tf − hi .

¯ (t) obeys It is easy to know that for 0 ≤ t ≤ tf − hl , y ⎡ ¯ (t) = y

⎤ Γ(0) x(t + h0 ) ⎢ ⎥ .. ¯ (t), ⎣ ⎦+v . Γ(l) x(t + hl )

(8.220)

and for tf − hi+1 < t ≤ tf − hi , ⎡ ¯ (t) = y

⎤ Γ(0) x(t + h0 ) ⎢ ⎥ .. ¯ (t). ⎣ ⎦+v . Γ(i) x(t + hi )

¯ v (t), is calculated by ¯ (t), denoted by R The covariance matrix of v 1 diag{R(0) , · · · , R(l) }, 0 ≤ t ≤ tf − hl , ¯ v (t) = R diag{R(0) , · · · , R(i) }, tf − hi+1 < t ≤ tf − hi .

(8.221)

(8.222)

Further, it is not difficult to verify that the linear space generated by {¯ y(t), 0 ≤ t ≤ tf } is the same as the one generated by {y(t), 0 ≤ t ≤ tf }. We denote that v¯ =

{¯ v (t), 0 ≤ t ≤ tf },

(8.223)

{¯ y(t), 0 ≤ t ≤ tf },

(8.224)

¯ y

=

¯ (·) is as in (8.218) and where y 1 col{v0 (t), · · · , vl (t)}, 0 ≤ t ≤ tf − hl , v¯(t) = col{v0 (t), · · · , vi (t)}, tf − hi+1 < t ≤ tf − hi

(8.225)

with vi (t) as defined in (8.151) which is different from the Krein space element vi (t). Then we have the following result. Lemma 8.4.3. Under the conditions of Theorem 8.4.2, Jtf of (8.177) can be rewritten as: Jtf = x (0)P (0)x(0) + (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ),

(8.226)

¯ and v¯∗ is where P (0) is the solution of RDE (8.184) at t = 0, Ry¯ = ¯ y, y ∗ obtained from v¯ with vi (t) replaced by vi (t) for i = 0, · · · , l and 0 ≤ t ≤ tf − hi .


197

Proof: Recall from (8.177) that Jtf = ξ Pξ + (v − v ∗ ) Ry (v − v ∗ ).

(8.227)

Since for t < 0, wi (t) = 0 and ui (t) = 0, i.e. vi (t) = 0, it follows from (8.153) that ˆ (0), x(0) − x ˆ (0) = v˜(t) = 0, 0 ≤ t ≤ hl , which implies that v˜ = 0. Since x(0) − x P (0), we have ξ Pξ = x (0)P (0)x(0). Further, it is straightforward to verify that (v − v ∗ ) Ry (v − v ∗ ) is equal to (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ). Hence, the result follows. ∇ ¯ Now we introduce the innovation sequence w(t) associated with the new obser¯ (t) as vation y ˆ ˆ ¯ (tf | tf ) = 0, ¯ (t | t), y ¯ ¯ (t) − y w(t) =y

(8.228)

ˆ ¯ (t | t) is the projection of y ¯ (t) onto the linear space L{¯ where y y(τ ), t < τ ≤ ¯ ), t < τ ≤ tf }. It is easy to observe that tf } = L{w(τ ¯ w (t)δ(t − τ ) = R ¯ v (t)δ(t − τ ), ¯ ¯ ) = R w(t), w(τ ¯ v (t) is as in (8.222). The observation y ¯ = {¯ where R y(t), 0 ≤ t ≤ tf } and the ¯ = {w(t), ¯ innovation w 0 ≤ t ≤ tf } have the following relationship, ¯ = Lw w, ¯ y

(8.229)

where Lw = I + kw is a causal operator, i.e. the kernel of kw (t, τ ) is zero for ¯ τ ≥ t. By using the innovation sequence w(t), we have Ry¯

¯ w L , ¯ = Lw R = ¯ y, y w

(8.230)

¯ w = w, ¯ w ¯ is a diagonal operator. where R Lemma 8.4.4. Under the conditions of Theorem 8.4.2, the linear quadratic form Jtf of (8.177) can be further rewritten as Jtf

= x (0)P (0)x(0) +

l 9 i=0

tf −hi

{vi (τ ) − vi∗ (τ )} R(i) {vi (τ ) − vi∗ (τ )} dτ,

0

(8.231) where vi∗ (τ ) is as in (8.213). Proof: From (8.226) and (8.230), it follows that Jtf

= =

ξ Pξ + (¯ v − v¯∗ ) Ry¯ (¯ v − v¯∗ ) ¯ w (τ )L (¯ v − v¯∗ ) Lw R ¯∗ ). ξ Pξ + (¯ w v−v

Similar to the discussion in [30], (8.231) follows immediately. Now we are in the position to give the main result of this section.

(8.232) ∇

198


Theorem 8.4.3. Consider the system (8.144)-(8.145) and the performance criterion (8.146). For a given γ > 0, suppose the RDE (8.184) has a bounded solution P (t), 0 ≤ t ≤ tf and for any τ > 0, the RDE (8.211) admits a bounded solution P τ (t), 0 ≤ t ≤ min{hl , tf − τ }, with initial value P τ (t) |t=hl = P (τ + hl ) for hl + τ ≤ tf or P τ (t) |t=tf −τ = P (tf ) = 0 for hl + τ > tf , where P (τ + hl ) is calculated by RDE (8.184). Then, there exists an H∞ controller that solves the H∞ full-information control problem if and only if Π −1 − γ −2 P (0) >

0,

(8.233)

where P (0) is the terminal value of P (τ ) of (8.184). In this case, a suitable H∞ controller (central controller) u∗i (τ ) is given by ui (τ ) = [Im , 0] × vi∗ (τ ) where vi∗ (τ ) is as in (8.213), i.e. −1 vi∗ (τ ) = −R(i) Γ(i) × P τ (hi )[Ψ¯ τ (0, hi )] x(τ )+ 9 hi P τ (hi ) [Ψ¯ τ (s, hi )] {In − Gτ (s)P τ (s)} v˜τ ∗ (s)ds+ h0 , 9 hl τ τ τ τ τ ∗ [In − P (hi )G (hi )] Ψ¯ (hi , s)P (s)˜ v (s)ds , hi+1

(8.234) while v˜τ ∗ (·) is as in (8.207) with vi (·) replaced by vi∗ (·) for i = 0, · · · , l, i.e. ⎧ l ⎪ ⎨ Γ v ∗ (t + τ − h ), h ≤ t < h , i = 0, 1, · · · , l − 1 j i i+1 (j) j τ∗ v˜ (t) = j=i+1 ⎪ ⎩ 0, t ≥ hl (8.235) and Gτ (s) is as defined in (8.214), i.e., 9 s ! Ψ¯ τ (r, s) Γ (r)R−1 (r)Γ (r)Ψ¯ τ (r, s)dr. Gτ (s) =

(8.236)

0

Proof: Substituting (8.231) into (8.148) yields Jt∞ f

=

x (0)Π −1 x(0) − γ −2 Jtf

x (0)[Π −1 − γ −2 P (0)]x(0) l 9 tf −hi {vi (τ ) − vi∗ (τ )} R(i) {vi (τ ) − vi∗ (τ )} dτ, −γ −2 =

i=0

0

where vi∗ (τ ) is as in (8.234). Note R(i) = diag{D(i) D(i) , − γ 2 Ir },

(8.237)


and vi (t) =

199

ui (t) , wi (t)

can be further written as then Jt∞ f Jt∞ f

=

x (0)[Π −1 − γ −2 P (0)]x(0) l 9 tf −hi −γ −2 {ui (τ ) − u∗i (τ )} D(i) D(i) {ui (τ ) − u∗i (τ )} dτ i=0

+

0

l 9 tf −hi i=0

{wi (τ ) − wi∗ (τ )} {wi (τ ) − wi∗ (τ )} dτ.

(8.238)

0

Our aim is to find the controller ui (t) such that Jt∞ is positive for all x(0) f has minimum over and exogenous input wi (t), which is equivalent to that Jt∞ f {x(0); wi (t), 0 ≤ t ≤ tf − hi , i = 0, · · · , l}, and ui (t) can be chosen such that the minimum of Jt∞ is positive. Thus, the necessary and sufficient condition for f the existence of the H∞ controller is that Π −1 − γ −2 P (0) >

0.

(8.239)

In view of (8.238), a suitable controller (central controller) can be chosen such that ui (τ ) = u∗i (τ ) = [Im , 0] × vi∗ (τ ). Therefore, we have established the theorem.

(8.240) ∇

Remark 8.4.3. For delay-free systems, i.e. h1 = · · · = hl = 0, it is obvious that a suitable H∞ controller is given from (8.234) as u∗0 (τ )

= [Im , 0] × vi∗ (τ ) = −(D(0) D(0) )−1 B(0) P τ (0)[Ψ¯ τ (0, 0)] x(τ ) D(0) )−1 B(0) P (τ )x(τ ), = −(D(0)

(8.241)

where Ψ¯ τ (0, 0) = I has been used in the second equality. Note that P τ (0) = P (τ ) obeys the standard Riccati equation (8.184). Thus the obtained H∞ controller u∗0 (τ ) is the same as the well-known result for H∞ control [30]. Remark 8.4.4. Observe that a similar H∞ full-information control for systems with multiple input delays has been investigated in [44] using an operator Riccati equation approach. In the present work, we give an explicit solution based on a duality between the H∞ control and a smoothing problem. 8.4.5

Special Cases

In this subsection, we shall discuss some special cases of the H∞ control problem tackled earlier.

200


H∞ control for single input delay systems. We consider the following time invariant system for the H∞ control problem. = Ax(t) + B1 w(t) + B2 u(t − h) % & Cx(t) z(t) = , D D = Im . Du(t − h)

x(t) ˙

(8.242) (8.243)

It is clear that (8.242)-(8.243) is a special case of the system (8.144)-(8.145) with l = 1, G(0) = B1 , G(1) = 0, B(0) = 0, B(1) = B2 , D(0) = 0, D(1) = D and Q = C C. Associated with the system (8.242)-(8.243) and performance index (8.146), we introduce the following notations as in the last section, R(0) = −γ 2 Ir , R(1) = Im , v0 (t) = w(t), v1 (t) = u(t), Γ(0) = B1 , Γ(1) = B2 , and

v τ (t) =

v˜τ (t) =

Γ (t) =

⎧ ⎨ v0 (t + τ ), 0 ≤ t < h, (t + τ ) v 0 ⎩ , t≥h v1 (t + τ − h) 1 B2 u(t + τ − h), 0 ≤ t < h, 0, t≥h 1 0≤t 0, assume that: 1) RDE −

dP (t) dt

=

A P (t) + P (t)A + C C − P (t)Γ (t)R(t)−1 Γ (t) P (t), P (tf ) = 0, (8.248)

has a bounded solution P (t) for 0 ≤ t ≤ tf . 2) RDE −

dP τ (t) dt

=

A P τ (t) + P τ (t)A + C C +

1 τ P (t)B1 B1 P τ (t), (8.249) γ2

has a bounded solution for 0 ≤ τ < tf , 0 ≤ t < min{h, tf − τ }, where for τ + h < tf the initial value P τ (t) |t=h = P (τ + h) which is calculated by (8.248), while for the case of τ + h ≥ tf the initial value is as P τ (t) |t=tf −τ =P (tf ) = 0.


201

Then, there exists an H∞ controller that solves the H∞ control problem if and only if Π −1 − γ −2 P (0) >

0,

(8.250)

where P (0) is the terminal value of P (t) of (8.248). In this situation, a suitable H∞ controller is given by u(τ ) =

−B2 P τ (h)[Ψ¯ τ (0, h)] x(τ ) 9 h τ −B2 P (h) [Ψ¯ τ (s, h)] {[In − Gτ (s)P τ (s)} B2 u(s + τ − h)ds, 0

(8.251) where

9

s

[Ψ¯ τ (r, s)] Γ (r)R(r)−1 Γ (r) Ψ¯ τ (r, s)dr 0 9 s 1 = − 2 [Ψ¯ τ (r, s)] B1 B1 Ψ¯ τ (r, s)dr, (8.252) γ 0 ¯τr = − 12 P τ (r)B1 B1 + A . and Ψ¯ τ (r, .) is the transition matrix of −Φ γ Gτ (s) =

Remark 8.4.5. We note that when tf → ∞, (8.248) and (8.249) will be replaced respectively by ; < 1 B1 B1 − B2 B2 P + C C = 0 (8.253) A P + PA + P γ2 and − P˙ τ (t)

= A P τ (t) + P τ (t)A + C C +

1 τ P (t)B1 B1 P τ (t), (8.254) γ2

where 0 ≤ t ≤ h and the terminal condition P τ (h) = P . In this case, it is known from [87], where the infinite horizon H∞ control of systems with single input delay is considered, that an H∞ controller exists if (8.253) admits a positive semi-definite stabilizing solution P and (8.254) admits a bounded non-negative definite solution P τ (t), 0 ≤ t ≤ h. Noting that P τ (h) = P , it follows from (8.251) that u(τ ) =

−B2 P [Ψ¯ τ (0, h)] x(τ ) 9 h ¯ τ (s)P τ (s) B2 u(s + τ − h)ds,(8.255) −B2 P [Ψ¯ τ (s, h)] I + G 0

¯ τ (s) is where G ¯ τ (s) = G =

−Gτ (s) 9 s 1 [Ψ¯ τ (r, s)] B1 B1 Ψ¯ τ (r, s)dr. γ2 0

(8.256)

Note that Ψ¯ τ (r, s) is the transition matrix of −Φ¯τr = − γ12 (P τ (r)B1 B1 + A ), ! thus Ψ¯ τ (r, s) is the transition matrix of 12 (B1 B1 P τ (s) + A) [77]. Let h = 1, γ

202


then the controller (8.255) is the same as the controller in [87]. We note that there is a typing mistake in G(s) of [87]. The correct form should be (8.256). H∞ control with preview. H∞ control with preview is concerned with the following system x(t + 1) = z(t) =

Ax(t) + B1 w(t − h) + B2 u(t), & % Cx(t) , D D = Im . Du(t)

(8.257) (8.258)

It is clear that (8.257)-(8.258) is a special case of the system (8.144)-(8.145) with l = 1, Φ = A, G(0) = 0, G(1) = B1 , B(0) = B2 , B(1) = 0, D(0) = D, D(1) = 0, and Q = C C. Denote R(0) = Im , R(1) = −γ 2 Ir , v0 (t) = u(t), v1 (t) = w(t), Γ(0) = B2 , Γ(1) = B1 , and

v τ (t) =

v˜τ (t) =

Γ (t) =

⎧ ⎨ v0 (t + τ ), 0 ≤ t < h, (t + τ ) v 0 ⎩ , t≥h v1 (t + τ − h) 1 B1 w(t + τ − h), 0 ≤ t < h, 0, t≥h 1 0≤t 0, assume that the RDE −

dP (t) dt

= A P (t) + P (t)A + C C − P (t)Γ (t)R(t)−1 Γ (t) P (t),(8.263)

with P (tf ) = 0 has a bounded solution P (t) for 0 ≤ t ≤ tf . Then, there exists a controller that solves the H∞ control problem with preview if and only if Π −1 − γ −2 P (0) >

0,

(8.264)

where P (0) is the terminal value of P (τ ) of (8.263). In this situation, a suitable H∞ preview controller is given by 9 h τ τ ¯ Ψ¯ τ (0, s)P τ (s)B1 w(s + τ − h)ds, u(τ ) = −B2 P (0)[Ψ (0, 0)] x(τ ) − B2 0

(8.265)

8.5 Conclusion

203

where Ψ¯ τ (s, ·) is the transition matrix of − Φ¯τs = K τ (s)Γ (s) − Φ = P τ (s)B2 B2 − A

(8.266)

and P τ (t) for t ≤ min{h, tf − τ } is a solution to the RDE: −

dP τ (t) dt

= A P τ (t) + P τ (t)A + C C − P τ (t)B2 B2 P τ (t), (8.267)

for any 0 ≤ τ < tf and t ≤ min{h, tf − τ }, where for τ + h < tf the initial value P τ (t) |t=h = P (τ + h) which is calculated by (8.263), while for the case of τ + h ≥ tf the initial value is as P τ (t) |t=tf −τ =P (tf ) = 0. Remark 8.4.6. We note from [26] that if the RDE (8.263) has a bounded solution P (t), then P (t) ≥ 0, which is in fact a necessary and sufficiency solution for the solvability of the standard H∞ control (h = 0). Also, since (8.267) is a LQ-type of RDE, its existence of a bounded solution is guaranteed. Remark 8.4.7. When tf → ∞, (8.263) and (8.267) will be replaced respectively by ; < 1 A P + P A + P B B − B B (8.268) 1 2 1 2 P +C C = 0 γ2 and − P˙ τ (t) =

A P τ (t) + P τ (t)A + C C − P τ (t)B2 B2 P τ (t),

(8.269)

where 0 ≤ t ≤ h and the terminal condition P τ (h) = P . We note that the H∞ preview control problem in the infinite horizon case has been studied in [45, 88]. In [45], a sufficient solvability condition of the H∞ preview control is related to the existence of a stabilizing solution to (8.268) and the non-singularity of a matrix. The latter is shown to be equivalent to the existence of a stabilizing solution of an operator Riccati equation. An interesting necessary and sufficient condition for the infinite horizon H∞ control with preview has been given in [88].

8.5 Conclusion In this chapter we have studied the LQR problem for systems with multiple input delays. Firstly, an analytical solution to the LQR problem for input delayed systems has been derived by establishing a duality between the LQR and a smoothing estimation and applying standard projections. Next, the LQG control problem has been approached by applying a separation principle and the Kalman filter for systems with measurement delays discussed in Chapter 7. Finally, the H∞ full information control problem for multiple input delay systems has been solved by extending the result for an indefinite LQR problem.

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Index

ARMA, 46 ABR, 46 ATM, 46 augmented system, 67 augmentation, 8 augmentation approach, 26 auto-regressive(AR), 48, 111

differential equation, 129, 138 discrete-time systems, 17, 36, 44, 53 disturbance inputs, 100 downstream, 186 duality, 27, 29, 33, 34, 36, 163 duality principle, 27 dynamic programming, 39, 51, 194

backward delay, 48 backward iteration, 22, 121, 130 backward RDE, 38 backward stochastic state-space model, 30, 90, 188 boundary conditions, 145, 154 buffer, 47 buffer length, 48

energy gain, 53 estimation error, 53 Euclidean, 55 exogenous input, 88, 96 exogenous noise, 87

causal, 185 CBR(constant bit rate), 46 central estimator, 154 central smoother, 63 congested switch, 47 congestion control, 47, 51 continuous-time systems, 115, 130, 132, 141 cost function, 42, 109 covariance matrix, 12, 13, 19, 20, 48, 57, 59, 83 cross-covariance matrices, 128 cross-Gramian, 3

fictitious observation, 78, 124 finite dimensional approach, 145 finite-horizon, 108, 134 finite element method, 157 fixed-interval smoothing, 145 gain matrix, 34 game theoretic theory, 162 Gaussian, 42, 48, 131 Gaussian sequence, 47 Gramian, 2 matrix, 2, 3 operator, 189 Hamlitonian matrix, 163 H∞ :

delay free system, 53, 179 delayed inputs, 43 delayed measurements, 7, 18, 26, 43 difference equation, 60

control, 1 estimation, 15 filter, 80 filtering, 53, 77

212

Index

fixed-lag smoothing, 15, 26, 53, 63, 115, 132, 140 full-information control, 87, 104 multiple step-ahead prediction, 4, 53 performance, 115 prediction, 143, 161 H2 fixed-lag smoothing, 67 H2 estimation, 83, 141 Hilbert spaces,1, 2, 15, 70, 134 indefinite quadratic form, 87, 185 indefinite quadratic optimization, 143 indefinite space, 85 inertia, 64, 72, 80 infinite-dimensional systems, 143, 163 infinite horizon, 203 initial condition, 44, 75 initial estimate, 54 initial state, 8, 30, 39 innovation, 73, 81 innovation covariance, 61, 63, 147 innovation sequence, 14, 19 input delays, 27 input noises, 53, 69 integral operators, 166 i/o, 51 isotropic vector, 4 J-factorization,

54, 163

Kalman: filter, 4, 14, 42, 44 filtering, 6, 7, 10, 15, 13-17, 20, 22, 27, 34, 43, 51 Kernels, 167, 189 Krein spaces, 1, 2, 5, 15, 53, 55, 71, 134, 145 LDU factorization, 103 linear combination, 146 linear least mean square error, 18 linear estimation theory, 33 linear optimal estimation problem, 18 linear quadratic form, 100 linear quadratic Gaussian (LQG), 28, 43, 44, 47, 51, 163 linear quadratic regulation(LQR), 28, 29, 34, 36, 41, 42

linear space, 35, 74 link capacity, 48 measurement delay, 77, 141 measurement feedback control, 178 measurement noises, 42, 53 measurement output, 30, 42, 54, 69 mean square error, 7 minimax, 54, 95, 185 minimizing solution, 33 minimum, 72, 125 Minkowski, 1 multiple input delays, 87, 105, 163, 164 multiple measurement delays, 17, 84 multiple state delays, 143 multi-step prediction, 26 negative definite, 146 networked congestion control, networked control, 27 non-negative definite, 28, 42 null space, 4

27, 87

operator Riccati equation, 115 observation, 10 observation sequence, 10 one-step ahead prediction, 4, 13, 59 operator approach, 162 operator Riccati equations, 143 operator Riccati differential equations, 165 optimal control, 33, 36, 38 optimal controller, 37, 39, 41, 45, 50 optimal estimate, 9 optimal estimator, 12, 24 optimal estimation, 19 optimal filter, 13, 16, 22 optimal filtering, 26, 34 optimal filtering problem, 22 optimal smoothing estimate, 35 optimal state estimator, 18 optimization, 29 orthogonality, 150 partial differential equations, 115 performance criterion, 54, 63 performance index, 28 predictor, 162 preview, 88, 108, 202 priority sources, 46

Index projection, 9, 14, 18, 23 projections, 1, 3 projection formula, 13, 14, 21, 22, 35 quadratic function, 5, 6 quadratic performance index, quality of service(QoS), 46 queue buffer, 111 queue length, 46, 47, 49

28

re-organized innovation, 1, 11, 20, 22 23, 26, 66, 85 re-organized innovation analysis, 7, 9 15, 53 re-organized innovation sequence, 137 re-organized measurements, 136 Riccati difference equation (RDE), 7, 12, 27, 28, 44, 136 Riccati difference recursions, 15 Riccati equation, 15, 20, 24, 60 Riccati Partial Differential Equation, 115 RM cells, 47 separation principle, 28, 164 single input delay, 88, 106 single measurement delay, 84 smoothing estimation, 27 smoothing gain matrix, 169 smoothing problem, 33 state augmentation, 53

213

state estimation error, 20 state feedback, 27, 87 state prediction error, 119 static output feedback control, 27, 87 source rates, 47, 111 source sharing, 48 stationary point, 5 stochastic backward state-space model, 35 stochastic optimization, 145 stochastic sequence, 19 stochastic system, 30, 55, 70 system augmentation, 15 target queue length, 111 time delay, 15, 17 time-horizon, 54, 133 time-varying, 130 traffic, 46 transmission capacity, 46 transition matrix, 121, 169, 171 uncertainty, 77 unilateral delay, 186 unitary matrices, 102 upstream, 186 VBR(variable bit rate), white noises, wind-tunnel,

46

10, 12, 17, 30, 48, 66, 70 186

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