Multivariable Computer-Controlled Systems: A Transfer Function Approach (Communications and Control Engineering)

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Author: Efim N. Rosenwasser | Bernhard P. Lampe

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Communications and Control Engineering

Published titles include: Stability and Stabilization of Infinite Dimensional Systems with Applications Zheng-Hua Luo, Bao-Zhu Guo and Omer Morgul Nonsmooth Mechanics (Second edition) Bernard Brogliato Nonlinear Control Systems II Alberto Isidori L2 -Gain and Passivity Techniques in Nonlinear Control Arjan van der Schaft Control of Linear Systems with Regulation and Input Constraints Ali Saberi, Anton A. Stoorvogel and Peddapullaiah Sannuti Robust and H∞ Control Ben M. Chen Computer Controlled Systems Efim N. Rosenwasser and Bernhard P. Lampe Dissipative Systems Analysis and Control Rogelio Lozano, Bernard Brogliato, Olav Egeland and Bernhard Maschke Control of Complex and Uncertain Systems Stanislav V. Emelyanov and Sergey K. Korovin Robust Control Design Using H∞ Methods Ian R. Petersen, Valery A. Ugrinovski and Andrey V. Savkin Model Reduction for Control System Design Goro Obinata and Brian D.O. Anderson Control Theory for Linear Systems Harry L. Trentelman, Anton Stoorvogel and Malo Hautus Functional Adaptive Control Simon G. Fabri and Visakan Kadirkamanathan Positive 1D and 2D Systems Tadeusz Kaczorek Identification and Control Using Volterra Models Francis J. Doyle III, Ronald K. Pearson and Bobatunde A. Ogunnaike Non-linear Control for Underactuated Mechanical Systems Isabelle Fantoni and Rogelio Lozano Robust Control (Second edition) Jürgen Ackermann Flow Control by Feedback Ole Morten Aamo and Miroslav Krstić Learning and Generalization (Second edition) Mathukumalli Vidyasagar Constrained Control and Estimation Graham C. Goodwin, María M. Seron and José A. De Doná Randomized Algorithms for Analysis and Control of Uncertain Systems Roberto Tempo, Giuseppe Calafiore and Fabrizio Dabbene Switched Linear Systems Zhendong Sun and Shuzhi S. Ge Subspace Methods for System Identification Tohru Katayama Digital Control Systems Ioan D. Landau and Gianluca Zito

Efim N. Rosenwasser and Bernhard P. Lampe

Multivariable Computer-controlled Systems A Transfer Function Approach

With 27 Figures

123

Efim N. Rosenwasser, Dr. rer. nat. Dr. Eng. State Marine Technical University Lozmanskaya str. 3 190008 Saint Petersburg Russia

Bernhard P. Lampe, Dr. rer. nat. Dr. Eng. University of Rostock Institute of Automation 18051 Rostock Germany

Series Editors E.D. Sontag · M. Thoma · A. Isidori · J.H. van Schuppen

British Library Cataloguing in Publication Data Rosenwasser, Efim Multivariable computer-controlled systems : a transfer function approach. - (Communications and control engineering) 1.Automatic control I.Title II.Lampe, Bernhard P. 629.8 ISBN-13: 9781846284311 ISBN-10: 1846284317 Library of Congress Control Number: 2006926886 Communications and Control Engineering Series ISSN 0178-5354 ISBN-10: 1-84628-431-7 e-ISBN 1-84628-432-5 ISBN-13: 978-1-84628-431-1

Printed on acid-free paper

© Springer-Verlag London Limited 2006 MATLAB® is a registered trademark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098, U.S.A. http://www.mathworks.com Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed in Germany 987654321 Springer Science+Business Media springer.com

To Elena and Bärbel

Preface

Classical control theory comprehends two principal approaches for continuoustime and discrete-time linear time-invariant (LTI) systems. The first, constituted of frequency-domain methods, is based on the concepts of the transfer function and the frequency response. The second approach arises from the state space concept and uses either differential or difference equations for describing dynamical systems. Although these approaches were originally separate, it was finally accepted that rather than hindering each other, they are, in fact, complementary; therefore more constructive and comprehensive methods of investigation could be developed by applying and combining frequency-domain techniques with state-space ones [68, 55, 53, 40, 49]. A different situation exists in the theory of linear computer-controlled systems, which are a subclass of sampled-data (SD) systems, because they are built of both continuous- and discrete-time components. Traditionally, approximation methods, where the problem is reduced to a complete investigation of either continuous- or discrete-time LTI models, predominate in this theory. This assertion can easily be corroborated by studying the leading monograph in this field [14]. However, to obtain rigorous results, a unified and accurate description of discrete- as well as continuous-time elements in continuous time is needed. Unfortunately, as a consequence of this approach, the models become variable in time. Over the last few years, a series of methods has been developed for this more complicated problem; a lot of them are cited in [30, 158]. An analysis of those references, however, shows that there are no frequency-domain methods for the analysis and design of SD systems that could be applied analogously to those used in the theory of LTI systems. The reason for this deficiency seems to be the lack of a transfer function concept for this wider class of systems that would parallel the classical transfer function for LTI systems [7]. Difficulties in introducing such a concept are caused by the fact that linear computer-controlled systems are non-stationary and have periodically varying coefficients.

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Preface

In [148] the authors demonstrated that these difficulties could be conquered by the concept of the parametric transfer function (PTF) w(s, t), which, in contrast with the ordinary transfer function for LTI systems, depends on an additional parameter: the time t. Applying the PTF permits the development of frequency methods for the analysis and design of SD systems after the pattern of classical methods and by doing so provides important additional results, practical methods and solutions for a number of new problems. Last but not least, the PTF yields deeper insight into the structure and nature of SD systems. Though, for the most part, [148] handles single-input, single-output (SISO) systems, it pays attention to practical constraints and makes clear the broad potential of the PTF approach. Since its publication, the authors have taken forward a number of investigations which extend these methods to multiinput, multi-output (MIMO) systems. The results of these investigations are summarized in the present monograph. In place of the PTF, we now make use of the parametric transfer matrix (PTM) w(s, t). In making this extension, obstacles arise because, in contrast with the transfer matrix for LTI systems, the PTM w(s, t) is not a rational function of the argument s. Fortunately, these obstacles turn out to be surmountable and we have developed investigation methods that use only polynomial and rational matrices. Though the final results are stated in a fairly general form, they open new possibilities for solving important classes of applied multivariable problems for which other methods fail. The theory presented in Multivariable Computer-controlled Systems is conceptually based on the work of J.S. Tsypkin [177], who proposed a general frequency-domain description of SD systems in the complex s-plane, and L.A. Zadeh [198, 199], who introduced the PTF concept into automatic control theory. Other significant results in this field are due to J.R. Raggazini, J.T. Tou and S.S.L. Chang, [136, 175, 29]. A version of the well-known WienerHopf method by D. Youla et al. [196] was a useful tool. The main body of the book consists of ten chapters and is divided into three parts. Part I (Chapters 1–3) contains preliminary algebraic material. Chapters 1 and 2 handle the fundamentals of polynomial and rational matrices that are necessary for understanding ideas explained later. Chapter 3 describes a class of rational matrices that are termed “normal” in the text. At first sight, these matrices seem to have a number of exotic properties because their entries are bounded by a multitude of algebraic conditions; however, it follows from the results of Chapters 1, 2 and 3 that, in practical applications, it is with these matrices that we mostly have to deal. Chapters 4 and 5 form Part II of the book, dedicated to some control problems which are also necessary to further investigations but which are of additional, independent, importance. Chapter 4 handles the eigenvalue assignment problem and the structure of the characteristic matrix of the closed

Preface

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systems, where the processes are given by polynomial pairs or by polynomial matrix description (PMD), from a general standpoint. Chapter 5 looks more deeply into the question whether the z- and ζ-transforms are applicable for the investigation of normal and anomalous discrete systems. In this connection it considers the construction of controllable forward and backward models of such systems. We would emphasize that Chapters 1, 2, 4 and 5 use many results that are known from the fundamental literature: see, for instance, [51, 133, 69, 114, 27, 80, 68, 206, 111, 167, 164] and others. We have decided to include this material in the main body of the current book because of the following considerations: 1. Their inclusion makes the book more readable because it reduces the reader’s need for additional literature to a minimum. 2. The representation of the information is adapted to be more suited to achieving the objectives of this particular book. 3. Chapters 1, 2, 4 and 5 contain a number of new results that, in our opinion, will be interesting for readers who are not directly engaged with SD systems. Among the latter results are the concept of the simple polynomial matrix, its property of structural stability and the analysis of rational matrices on basis of their dominance and subordination, all of which appear in Chapters 1 and 2. Chapter 2 also details investigations into the reducibility of rational transfer matrices. Chapter 4 covers the theorems on eigenvalue and eigenstructure assignment for control systems with PMD processes. In Chapter 5, the investigation of the applicability of the z- and ζ- (or Taylor-) transformations to the mathematical description of anomalous discrete systems is obviously new, as is the generation of controllable forward and backward models for such systems. Part III (Chapters 6–10) is mainly concerned with frequency methods for the investigation of MIMO SD systems. Chapter 6 presents a frequency approach for parametrically discretized continuous MIMO processes and makes clear the mutual algebraic properties of the continuous process and the discrete model. Chapter 7 is dedicated to the mathematical description of the standard SD system. It is here that we introduce the PTM, among other substantial methods of description, and make careful investigation of its properties. Stability and stabilizing problems for closed-loop systems, in which the polynomial solution of the stabilizing problem obtained has very general character, are studied. Particularly, cases with pathological sampling periods are included. Chapter 8 deals with the analysis of the response of the standard SD system to stationary stochastic excitation and with the solution of the H2 optimization problem on the basis of the PTM concept and the Wiener-Hopf method. The method presented is extremely general and, in addition to finding the optimal control program, it permits us to state a number of fundamental

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Preface

properties of the optimal system: its structure and the set of its poles, for instance. Chapter 9 describes the methods of the preceding three chapters in greater detail for the special case of single-loop MIMO SD systems. This is done with the supposition that the transfer matrices of all continuous parts are normal and that the sampling period is non-pathological. When theses suppositions hold, important special cancellations take place; thus, the critical case, in which the transfer matrices of continuous elements contain poles on the imaginary axis, is considered. In this way, the fact, important for applications, that the solvability of the connected H2 problem in the critical case depends on the location of the critical elements inside the control loop with respect to the input and output of the system is stated. In this case there may be situations in which the H2 problem has no solution. Chapter 10 is devoted to the L2 problem for the standard SD system; it contains, as special cases, the design of optimal tracking systems and the redesign problem. In our opinion, this case constitutes a splendid example for demonstrating the possibilities of frequency methods. This chapter demonstrates that in the multidimensional case the solution of the L2 problem always leads to a singular quadratic functional for that a set of minimizing control programs exists. Applying Laplace transforms during the evaluation by the Wiener-Hopf method allows us to find the complete set of optimal solutions; by doing this, input signals of finite duration and constant signals are included. We know of no alternative methods for constructing the general solution to this problem. The book closes with four appendices. Appendix A gives a short introduction to the ζ-transformation (Taylor transformation), and its relationship to other operator transformations for discrete sequences. In Appendix B some auxiliary formulae are derived.Appendix C, written by Dr. K. Polyakov, presents the MATLAB DirectSDM Toolbox. Using this toolbox, various H2 and L2 problems for single-loop MIMO systems can be solved numerically.Appendix D, composed by Dr. V. Rybinskii, describes a design method for control with guaranteed performance. These controllers guarantee a required performance for arbitrary members of certain classes of stochastic disturbances. The MATLAB GarSD Toolbox, used for the numerical solution of such problems, is also presented. In our opinion, the best way to get well acquainted with the content of the book is, of course, the thorough reading of all the chapters in sequence, starting with Chapter 1. We recognize, however, that this requires effort and staying-power of the reader and an expert, interested only in SD systems, can start directly with Chapter 6, looking into the preceding chapters only when necessary. The book is written in a mathematical style. We do not include elementary introductory material on the functioning or the physical and technological characteristics of computer-controlled systems; likewise, there

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is no relation of the theory and practice of such systems in an historical context. For material of this sort, we refer the reader to the extensive literature in those fields, the above mentioned reference [14] by ˚ Aström and Wittenmark, for example. From this viewpoint, our book and its predecessor [148] can be seen as extensions of [14] the titles of both being inspired by it. Multivariable Computer-controlled Systems is addressed to engineers and scientific workers involved in the investigation and design of computercontrolled systems. It can also be used as a complementary textbook on process-oriented methods in computer-controlled systems by students on courses in control theory, communications engineering and related fields. Practically oriented mathematicians and engineers working in systems theory will find interesting insight in the following pages. The mathematical tools used in this book are, in general, included in basic mathematics syllabuses for engineers at technical universities. Necessary additional material is given directly in the text. The References section is by no means a complete bibliography as it contains only those works we used directly in the preparation of the book. The authors gratefully acknowledge the financial support by the German science foundation (Deutsche Forschungsgemeinschaft), especially Dr. Andreas Engelke for his engagement and helpful hints. Due to Mrs. Hannelore Gellert from the University of Rostock and Mrs. Ludmila Patrashewa from the Saint Petersburg University of Ocean Technology, additional support by the Euler program of the German academic exchange service (Deutscher Akademischer Austauschdienst) was possible, which is thankfully mentioned. We are especially indebted to the professors B.D.O. Anderson, K.J. ˚ Aström, P.M. Frank, G.C. Goodwin, M. Grimble, T. Kaczorek, V. Kuˇcera, J. Lunze, B. ˇ Lohmann, M. Sebek, A. Weinmann and a great number of unnamed colleagues for many helpful discussions and valuable remarks. The engaged work of Oliver Jackson from Springer helped us to overcome various editorial problems - we appreciate his careful work. We thank Sri Ramoju Ravi for comments after reading the draft version. The MATLAB-Toolbox DirectSDM by K.Y. Polyakov is available as free download from http://www.iat.uni-rostock.de/blampe/matlab toolbox.html In case of any problems, please contact [email protected].

Rostock, May 17, 2006

Efim Rosenwasser Bernhard Lampe

Contents

Part I Algebraic Preliminaries 1

Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Concepts of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Matrices over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Left and Right Equivalence of Polynomial Matrices . . . . . . . . . . 1.6 Row and Column Reduced Matrices . . . . . . . . . . . . . . . . . . . . . . . 1.7 Equivalence of Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Normal Rank of Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . 1.9 Invariant Polynomials and Elementary Divisors . . . . . . . . . . . . . . 1.10 Latent Equations and Latent Numbers . . . . . . . . . . . . . . . . . . . . . 1.11 Simple Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Pairs of Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Polynomial Matrices of First Degree (Pencils) . . . . . . . . . . . . . . . 1.14 Cyclic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Simple Realisations and Their Structural Stability . . . . . . . . . . .

3 3 5 7 10 12 15 20 21 23 26 29 34 38 44 49

2

Fractional Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rational Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 McMillan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Matrix Fraction Description (MFD) . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Double-sided MFD (DMFD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Index of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Strictly Proper Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Separation of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Inverses of Square Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . 2.10 Transfer Matrices of Polynomial Pairs . . . . . . . . . . . . . . . . . . . . . . 2.11 Transfer Matrices of PMDs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 59 60 63 73 74 77 82 85 87 90

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Contents

2.12 Subordination of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . 94 2.13 Dominance of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3

Normal Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.1 Normal Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Algebraic Properties of Normal Matrices . . . . . . . . . . . . . . . . . . . 110 3.3 Normal Matrices and Simple Realisations . . . . . . . . . . . . . . . . . . . 114 3.4 Structural Stable Representation of Normal Matrices . . . . . . . . . 116 3.5 Inverses of Characteristic Matrices of Jordan and Frobenius Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.6 Construction of Simple Jordan Realisations . . . . . . . . . . . . . . . . . 126 3.7 Construction of Simple Frobenius Realisations . . . . . . . . . . . . . . 132 3.8 Construction of S-representations from Simple Realisations. General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.9 Construction of Complete MFDs for Normal Matrices . . . . . . . . 138 3.10 Normalisation of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . 141

Part II General MIMO Control Problems 4

Assignment of Eigenvalues and Eigenstructures by Polynomial Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2 Basic Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.3 Recursive Construction of Basic Controllers . . . . . . . . . . . . . . . . . 154 4.4 Dual Models and Dual Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.5 Eigenvalue Assignment for Polynomial Pairs . . . . . . . . . . . . . . . . 165 4.6 Eigenvalue Assignment by Transfer Matrices . . . . . . . . . . . . . . . . 169 4.7 Structural Eigenvalue Assignment for Polynomial Pairs . . . . . . . 172 4.8 Eigenvalue and Eigenstructure Assignment for PMD Processes 174

5

Fundamentals for Control of Causal Discrete-time LTI Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.1 Finite-dimensional Discrete-time LTI Processes . . . . . . . . . . . . . . 183 5.2 Transfer Matrices and Causality of LTI Processes . . . . . . . . . . . . 189 5.3 Normal LTI Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.4 Anomalous LTI Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.5 Forward and Backward Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 5.6 Stability of Discrete-time LTI Systems . . . . . . . . . . . . . . . . . . . . . 222 5.7 Closed-loop LTI Systems of Finite Dimension . . . . . . . . . . . . . . . 225 5.8 Stability and Stabilisation of the Closed Loop . . . . . . . . . . . . . . . 230

Contents

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Part III Frequency Methods for MIMO SD Systems 6

Parametric Discrete-time Models of Continuous-time Multivariable Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.1 Response of Linear Continuous-time Processes to Exponential-periodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.2 Response of Open SD Systems to Exp.per. Inputs . . . . . . . . . . . 245 6.3 Functions of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.4 Matrix Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 6.5 DPFR and DLT of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . 258 6.6 DPFR and DLT for Modulated Processes . . . . . . . . . . . . . . . . . . . 261 6.7 Parametric Discrete Models of Continuous Processes . . . . . . . . . 266 6.8 Parametric Discrete Models of Modulated Processes . . . . . . . . . 271 6.9 Reducibility of Parametric Discrete Models . . . . . . . . . . . . . . . . . 275

7

Description and Stability of SD Systems . . . . . . . . . . . . . . . . . . . 279 7.1 The Standard Sampled-data System . . . . . . . . . . . . . . . . . . . . . . . 279 7.2 Equation Discretisation for the Standard SD System . . . . . . . . . 280 7.3 Parametric Transfer Matrix (PTM) . . . . . . . . . . . . . . . . . . . . . . . . 283 7.4 PTM as Function of the Argument s . . . . . . . . . . . . . . . . . . . . . . . 289 7.5 Internal Stability of the Standard SD System . . . . . . . . . . . . . . . 295 7.6 Polynomial Stabilisation of the Standard SD System . . . . . . . . . 298 7.7 Modal Controllability and the Set of Stabilising Controllers . . . 304

8

Analysis and Synthesis of SD Systems Under Stochastic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.1 Quasi-stationary Stochastic Processes in the Standard SD System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 8.2 Mean Variance and H2 -norm of the Standard SD System . . . . . 312 8.3 Representing the PTM in Terms of the System Function . . . . . 315 8.4 Representing the H2 -norm in Terms of the System Function . . 325 8.5 Wiener-Hopf Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.6 Algorithm for Realisation of Wiener-Hopf Method . . . . . . . . . . . 332 8.7 Modified Optimisation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 336 8.8 Transformation to Forward Model . . . . . . . . . . . . . . . . . . . . . . . . . 340

9

H2 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Optimisation of a Single-loop System . . . . . . . . . . . . . . . . . . 347 Single-loop Multivariable SD System . . . . . . . . . . . . . . . . . . . . . . . 347 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Stabilisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 Wiener-Hopf Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Factorisation of Quasi-polynomials of Type 1 . . . . . . . . . . . . . . . 355 Factorisation of Quasi-polynomials of Type 2 . . . . . . . . . . . . . . . 364 Characteristic Properties of Solution for Single-loop System . . . 373

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9.8 Simplified Method for Elementary System . . . . . . . . . . . . . . . . . . 374 10 L2 -Design of SD Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 10.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 10.2 Pseudo-rational Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . 383 10.3 Laplace Transforms of Standard SD System Output . . . . . . . . . . 387 10.4 Investigation of Poles of the Image Z(s) . . . . . . . . . . . . . . . . . . . . 392 10.5 Representing the Output Image in Terms of the System Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 10.6 Representing the L2 -norm in Terms of the System Function . . . 399 10.7 Wiener-Hopf Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 10.8 General Properties of Optimal Systems . . . . . . . . . . . . . . . . . . . . . 407 10.9 Modified Optimisation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 409 10.10Single-loop Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 10.11Wiener-Hopf Method for Single-loop Tracking System . . . . . . . . 412 10.12L2 Redesign of Continuous-time LTI Systems under Persistent Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 10.13L2 Redesign of a Single-loop LTI System . . . . . . . . . . . . . . . . . . . 426 Appendices A

Operator Transformations of Taylor Sequences . . . . . . . . . . . . 431

B

Sums of Certain Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

C

DirectSDM – A Toolbox for Optimal Design of Multivariable SD Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 C.2 Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 C.3 Operations with Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . 438 C.4 Auxiliary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 C.5 H2 -optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 C.5.1 Extended Single-loop System . . . . . . . . . . . . . . . . . . . . . . . 440 C.5.2 Function sdh2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 C.6 L2 -optimal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 C.6.1 Extended Single-loop System . . . . . . . . . . . . . . . . . . . . . . . 443 C.6.2 Function sdl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

D

Design of SD Systems with Guaranteed Performance . . . . . . 447 D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 D.2 Design for Guaranteed Performance . . . . . . . . . . . . . . . . . . . . . . . . 448 D.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 D.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 D.2.3 Calculation of Performance Criterion . . . . . . . . . . . . . . . . 451

Contents

xvii

D.2.4 Minimisation of Performance Criterion Estimate for SD Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 D.3 MATLAB -Toolbox GarSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 D.3.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 D.3.2 Setting Properties of External Excitations . . . . . . . . . . . . 454 D.3.3 Investigation of SD Systems . . . . . . . . . . . . . . . . . . . . . . . . 455 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Part I

Algebraic Preliminaries

1 Polynomial Matrices

1.1 Basic Concepts of Algebra 1. Let a certain set A with elements a, b, c, d, . . . be given. Assume that over the set A an algebraic operation is defined which relates every pair of elements (a, b) to a third element c ∈ A that is called the result of the operation. If the named operation is designated by the symbol ‘∗’, then the result is symbolically written as a ∗ b = c. In general, we have a ∗ b = b ∗ a. However, if for any two elements a, b in A the equality a ∗ b = b ∗ a holds, then the operation ‘∗’ is called commutative. The operation ‘∗’ is named associative, if for any a, b, c ∈ A the relation (a ∗ b) ∗ c = a ∗ (b ∗ c) is true. The set A is called a semigroup, if an associative operation ‘∗’ is defined in it. A semigroup A is called a group , if it contains a neutral element e, such that for every a ∈ A a∗e=e∗a=a is correct, and furthermore, for any a ∈ A there exists a uniquely determined element a−1 ∈ A, such that a ∗ a−1 = a−1 ∗ a = e .

(1.1)

The element a−1 is called the inverse element of a. A group, where the operation ‘∗’ is commutative, is called a commutative group or Abelian group. In many cases the operation ‘∗’ in an Abelian group is called addition, and it is designated by the symbol ‘+’. This notation is called additive. In additive notation the neutral element is called the zero element , and it is denoted by the symbol ‘0’ (zero).

4


In other cases the operation ‘∗’ is called multiplication, and it is written in the same way as the ordinary multiplication of numbers. This notation is named multiplicative. The neutral element in the multiplicative notation is designated by the symbol ‘1’ (one). For the inverse element in multiplicative notation is used a−1 , and in additive notation we write −a. In the last case the inverse element −a is also named the opposite element to a. 2. The set A is called an (associative) ring, if the two operations ‘addition’ and ‘multiplication’ are defined on A. Hereby, the set A forms an Abelian group with respect to the ‘addition’, and a semigroup with respect to the ‘multiplication’. From the membership to an Abelian group it follows (a + b) + c = a + (b + c) and a + b = b + a. Moreover, there exists a zero element 0, such that for an arbitrary a ∈ A a + 0 = 0 + a = a. The element 0 is always uniquely determined. Between the operations ‘addition’ and ‘multiplication’ of a ring the relations (a + b)c = ac + bc ,

c(a + b) = ca + cb

(left and right distributivity) are valid. In many cases rings are considered, which possess a number of further properties. If for any two a, b always ab = ba is true, then the ring is called commutative. If a unit element exists with 1a = a1 for all a ∈ A, then the ring is named as a ring with unit element. The element 1 in such a ring is always uniquely determined. The non-zero elements a, b of a ring, satisfying ab = 0, are named (left resp. right) zero divisor. A ring is called integrity region , if it has no zero divisor. 3. A commutative associative ring with unit element, where every non-zero element a has an inverse a−1 that satisfies Equation (1.1), is called a field. In others words, a field is a ring, where all elements different from zero with respect to multiplication form a commutative group. It can be shown that an arbitrary field is an integrity region. The set of complex or the set of real numbers with the ordinary addition and multiplication as operations are important examples for fields. In the following, these fields will be designated by C and R, respectively.

1.2 Polynomials

5

1.2 Polynomials 1. Let N be a certain commutative associative ring with unit element, especially it can be a field. Let us consider the infinite sequence (a0 , a1 , . . . , ak ; 0, . . .), where ak = 0, and all elements starting from ak+1 are equal to zero. Furthermore, we write (a0 , a1 , . . . , ak ; 0, . . .) = (b0 , b1 , . . . , bk ; 0, . . .) , if and only if ai = bi (i = 0, . . . , k). Over the set of elements of the above form, the operations addition and multiplication are introduced in the following way. The sum is defined by the relation (a0 , a1 , . . . , ak ; 0, . . .)+(b0 , b1 , . . . , bk ; 0, . . .) = (a0 +b0 , a1 +b1 , . . . , ak +bk ; 0, . . .) and the product of the sequences is given by (a0 , a1 , . . . , ak ; 0, . . .)(b0 , b1 , . . . , bk ; 0, . . .)

(1.2) = (a0 b0 , a0 b1 + a1 b0 , . . . , a0 bk + a1 bk−1 + . . . + ak b0 , . . . , ak bk ; 0, . . .) .

It is easily proven that the above explained operations addition and multiplication are commutative and associative. Moreover, these operations are distributive too. Any element a ∈ N is identified with the sequence (a; 0, . . .). Furthermore, let λ be the sequence λ = (0, 1; 0, . . .) . Then using (1.2), we get λ2 = (0, 0, 1; 0, . . .) ,

λ3 = (0, 0, 0, 1; 0, . . .) ,

etc.

Herewith, we can write (a0 , a1 , . . . , ak ; 0, . . .) = = (a0 ; 0, . . .) + (0, a1 ; 0, . . .) + . . . + (0, . . . , 0, ak ; 0, . . .) = a0 + a1 (0, 1; 0, . . .) + . . . + ak (0, . . . , 0, 1; 0, . . .) = a0 + a1 λ + a2 λ2 + . . . + ak λk . The expression on the right side of the last equation is called a polynomial in λ with coefficients in N . It is easily shown that this definition of a polynomial is equivalent to other definitions in elementary algebra. For ak = 0 the polynomial ak λk is called the term of the polynomial f (λ) = a0 + a1 λ + . . . + ak λk

(1.3)

with the highest power. The number k is called the degree of the polynomial (1.3), and it is designed by deg f (λ). If we have in (1.3) a0 = a1 = . . . = ak = 0,

6


then the polynomial (1.3) is named the zero polynomial . A polynomial with ak = 1 is called monic. If for two polynomials f1 (λ), f2 (λ) the relation f1 (λ) = af2 (λ) with a ∈ N is valid, then these polynomials are called equivalent. In what follows, we will use the notation f1 λ) ≈ f2 (λ) for the fact that the polynomials f1 (λ) and f2 (λ) are equivalent. Inside this book we only consider polynomials with coefficients from the real number field R or the complex number field C. Following [206] we use the notation F for a field that is either R or C. The set of polynomials over these fields are designated by R[λ], C[λ] or F[λ] respectively. The sets R[λ] and C[λ] are commutative rings without zero divisor. In what follows, the elements in R[λ] are called real polynomials. 2.

Some general properties of polynomials are listed below:

1. Any polynomial f (λ) ∈ C[λ] with deg f (λ) = n can be written in the form f (λ) = an (λ − λ1 ) · · · (λ − λn ) .

(1.4)

This representation is unique up to permutation of the factors. Some of the numbers λ1 , . . . , λn that are the roots of the polynomial f (λ), could be equal. In that case the product (1.4) is represented by f (λ) = an (λ − λ1 )µ1 · · · (λ − λq )µq ,

µ1 + . . . + µq = n,

(1.5)

where all λi , (i = 1, . . . , q) are different. The number µi , (i = 1, . . . , q) is called the multiplicity of the root λi . If f (λ) ∈ R[λ] then an is a real number, and in the products (1.4), (1.5) for every complex root λi there exists the conjugate complex root with equal multiplicity. 2. For given polynomials f (λ), d(λ) ∈ F[λ] there exists a uniquely determined pair of polynomials q(λ), r(λ) ∈ F[λ], such that f (λ) = q(λ)d(λ) + r(λ) ,

(1.6)

where deg r(λ) < deg d(λ) . Hereby, the polynomial q(λ) is called the entire part, and the polynomial r(λ) is the remainder from the division of f (λ) by d(λ). 3. Let us have f (λ), g(λ) ∈ F[λ]. It is said, that the polynomial g(λ) is a divisor of f (λ), and we write g(λ)|f (λ), if f (λ) = q(λ)g(λ) is true, where q(λ) is a certain polynomial. The greatest common divisor (GCD) of the polynomials f1 (λ) and f2 (λ) should be designated by p(λ). At the same time the GCD is a divisor of f1 (λ) and f2 (λ), and it possesses the greatest possible degree . Up to

1.3 Matrices over Rings

7

equivalence, the GCD is uniquely determined. Any GCD p(λ) permits a representation of the form p(λ) = f1 (λ)m1 (λ) + f2 (λ)m2 (λ) , where m1 (λ), m2 (λ) are certain polynomials in F[λ]. 4. The two polynomials f1 (λ) and f2 (λ) are called coprime if their monic GCD is equal to one, that means, up to constants, these polynomials possess no common divisors. For the polynomials f1 (λ) and f2 (λ) to be coprime, it is necessary and sufficient that there exist polynomials m1 (λ) and m2 (λ) with f1 (λ)m1 (λ) + f2 (λ)m2 (λ) = 1 . 5. If f1 (λ) = p(λ)f˜1 (λ) ,

f2 (λ) = p(λ)f˜2 (λ) ,

where p(λ) is a GCD of f1 (λ) and f2 (λ), then the polynomials f˜1 (λ) and f˜2 (λ) are coprime.

1.3 Matrices over Rings 1. Let N be a commutative ring with unit element forming an integrity region, such that ab = 0 implies a or b equal to zero, where 0 is the zero element of the ring N . Then from ab = 0, a = 0 it always follows b = 0. 2.

The rectangular scheme ⎡

a11 ⎢ .. A=⎣ . an1

⎤ . . . a1m .. .. ⎥ . . ⎦ . . . anm

(1.7)

is named a rectangular matrix over the ring N , where the aik , (i = 1, . . . , n; k = 1, . . . , m) are elements of the ring N . In what follows, the set of matrices is designated by Nnm . The integers n and m are called the dimension of the matrix. In case of m = n we speak of a quadratic matrix A, for m < n of a vertical and for m > n of a horizontal matrix A. For matrices over rings the operations addition, (scalar) multiplication with elements of the ring N , multiplication of matrices by matrices and transposition are defined. All these operations are defined in the same way as for matrices over numbers [51, 44]. 3. Every quadratic matrix A ∈ Nnn is related to its determinant det A which is calculated in the same way as for number matrices. However, in the given case the value of det A is an element of the ring N . A matrix A with det A = 0N is called regular or non-singular, for det A = 0N it is called singular.

8

4.


For any matrix A ∈ Nnn there uniquely exists a matrix adj A of the form ⎡ ⎤ A11 . . . An1 ⎢ ⎥ adj A = ⎣ ... ... ... ⎦ , (1.8) A1n . . . Ann

where Aik is the algebraic complement (the adjoint) of the element aik of the matrix A, which is received as the determinant of those matrix that remains by cutting the i−th row and k−th column multiplied by the sign-factor (−1)i+k . The matrix adj A is called the adjoint of the matrix A. The matrices A and adj A are connected by the relation A(adj A) = (adj A)A = (det A)In ,

(1.9)

where the identity matrix In is defined by ⎡ ⎤ 1N 0N . . . 0N ⎢ 0N 1N . . . 0N ⎥ ⎢ ⎥ In = ⎢ . . . . ⎥ = diag{1N , . . . , 1N } ⎣ .. .. . . .. ⎦ 0N 0N . . . 1N with the unit element 1N of the ring N , and diag means the diagonal matrix. 5. In the following, matrices of dimension n × 1 are called as columns and matrices of dimension 1 × m as rows, and both are referred to as vectors . The number n is named the height of the column, and the number m the width of the row, and both are the length of the vector. Let u1 , u2 , . . . , uk be rows of N1m . As a linear combination of the rows u1 , . . . , uk , we term the row u ˜ = c1 u1 + . . . + ck uk , where the ci , (i = 1, . . . , k) are elements of the ring N . The set of rows {u1 , . . . , uk } is named linear dependent, if there exist coefficients c1 , . . . , ck , that are not all equal to zero, such that u ˜ = O1m . Here and furthermore, Onm designates the zero matrix, i.e. that matrix in Nnm having all its elements equal to the zero element 0N . If the equation cu = c1 u1 + . . . + ck uk is valid with a c = 0N , then we say that the column u depends linearly on the columns u1 , . . . , uk . For the set {u1 , . . . , uk } of columns to be linear dependent, it is necessary and sufficient that one column depends linearly on the others in the sense of the above definition. For rows over the ring N the important statement is true: Any set of rows of the width m with more than m elements is linear dependent. In analogy, any set of columns of height n with more than n elements is also linear dependent.

1.3 Matrices over Rings

9

6. Let a finite or infinite set U of rows of width m be given. Furthermore, let r be the maximal number of linear independent elements of U, where due to the above statement r ≤ m is valid. An arbitrary subset of r linear independent rows of U is called a basis of the set U, , the number r itself is called the normal rank of U. All that is said above can be directly transferred to sets of columns. 7. Let a matrix A ∈ Nnm be given, and U should be the set of rows of A, and V the set of its columns. Then the following important statements take place: 1. The normal rank of the set U of the rows of the matrix A is equal to the normal rank of the set V of its columns. The common value of these ranks is called the normal rank of the matrix A, and it is designated by rank A. 2. The normal rank of the matrix A is equal to the highest order of its subdeterminants (minors) different from zero. (Here zero again means the zero element of the ring N .) 3. For the linear independence of all rows (columns) of a quadratic matrix, it is necessary and sufficient that it is non-singular. For arbitrary matrices A ∈ Nnm , the above statements imply

rank V = rank U ≤ min(n, m) = γA .

Hereinafter, the symbol ‘=’ stands for equality by definition. In the following, we say that the matrix A has maximal or full normal rank, if rank A = γA , or that it is non-degenerated. In the following the symbol ‘rank’ also denotes the rank of an ordinary number matrix. This notation does not lead to contradictions, because for matrices over the fields of real or complex numbers the normal rank coincides with the ordinary rank. 8.

For Matrix (1.7), the expression A

i1 i2 . . . ip k1 k2 . . . kp

⎡

ai1 k1 ⎢ .. = det ⎣ . aip k1

⎤ . . . ai1 kp .. .. ⎥ . . ⎦ . . . aip kp

denotes the minor of the matrix A, which is calculated by the elements, that are at the same time members of the rows with the numbers i1 , . . . , ip , and of the columns with the numbers k1 , . . . , kp . Let C = AB be given with C ∈ Nnm , A ∈ Nn , B ∈ Nm . Then if n = m, the matrix C is quadratic, and for n ≤ we have

10


det C =

···

A

1≤k1 m the vertical n × m matrix A(λ) is alatent if and only if

Im A(λ) = ϕ(λ) On−m,m becomes true with a certain unimodular matrix ϕ(λ). 5. A non-degenerated matrix A(λ) ∈ Fnm [λ] is said to be latent, if it has latent roots. Due to (1.40)–(1.42) it is clear that for n < m a latent matrix A(λ) allows the representation A(λ) = a(λ)b(λ) ,

(1.58)

where a(λ) = p(λ) diag{a1 (λ), . . . , an (λ)} ,

b(λ) = In On,m−n q(λ) and p(λ), q(λ) are unimodular matrices. Obviously, det a(λ) ≈ a1 (λ) · · · an (λ) is valid. The matrix b(λ) proves to ˜ A corresponding be alatent, i.e. its rank is equal to n = ρ for all λ = λ. representation for n > m is also possible.

1.11 Simple Matrices

29

6. Theorem 1.32. Suppose the n × m matrix A(λ) to be alatent. Then every submatrix generated from any of its rows is also alatent. Proof. Take a positive integer p < n and present the matrix A(λ) in the form ⎡

⎤ a11 (λ) . . . a1m (λ) .. .. ⎢ ⎥ ⎢ ⎥ . . . . . ⎢ ⎥ ⎢ ap1 (λ) . . . apm (λ) ⎥ Ap (λ) ⎢ ⎥ A(λ) = ⎢ . ⎥= ⎢ ap+1,1 (λ) . . . ap+1,m (λ) ⎥ A1 (λ) ⎢ ⎥ ⎢ ⎥ .. .. ⎣ ⎦ . ... . an1 (λ) . . . anm (λ)

(1.59)

It is indirectly shown that the submatrix Ap (λ) over the line turns out to be alatent. Suppose the contrary. Then owing to (1.58), we get Ap (λ) = ap (λ)bp (λ) , where the matrix ap (λ) is latent, and ord ap (λ) > 0. Applying this result from (1.59),

ap (λ) Op,n−p bp (λ) A(λ) = On−p,p In−p A1 (λ) ˜ be an eigenvalue of the matrix ap (λ), so is acquired. Let λ

˜ ˜ bp (λ) ˜ = ap (λ) Op,n−p A(λ) ˜ On−p,p In−p A1 (λ) is valid.Because the rank of the first factors on the right side is smaller than n, ˜ < n, which is in contradiction to the supposed alatency this implies rank A(λ) of A(λ). Remark 1.33. In the same way, it is shown that any submatrix of an alatent matrix A(λ) built from any of its columns also becomes alatent. Corollary 1.34. Every submatrix built from any rows or columns of a unimodular matrix is alatent.

1.11 Simple Matrices 1. A non-degenerated latent n × m matrix A(λ) of full rank ρ = γA is called simple, if Dρ (λ) = aρ (λ),

Dρ−1 (λ) = Dρ−2 (λ) = . . . = D1 (λ) = 1 .

30


In dependence on the dimension for a simple matrix A(λ) from (1.40)–(1.42), we derive the representations γA = n = m : A(λ) = ϕ(λ) diag{1, . . . , 1, an (λ)}ψ(λ) , ⎡

γA

1 ... ⎢ .. . . ⎢ = n < m : A(λ) = ϕ(λ) ⎢ . . ⎣0 ... 0 ... ⎡

γA

1 ... ⎢ .. . . ⎢. . ⎢ ⎢0 ... ⎢ = m < n : A(λ) = ϕ(λ) ⎢ ⎢0 ... ⎢0 ... ⎢ ⎢. ⎣ .. . . .

0 .. .

0 .. .

0 ... .. . ··· 1 0 0 ... 0 an (λ) 0 . . .

0 .. .

0 .. .

1 0 0 .. .

0 ... 0

⎤ 0 .. ⎥ .⎥ ⎥ ψ(λ) , 0⎦ 0

⎤

⎥ ⎥ ⎥ 0 ⎥ ⎥ am (λ) ⎥ ⎥ ψ(λ) , 0 ⎥ ⎥ .. ⎥ . ⎦ 0

where ϕ(λ) and ψ(λ) are unimodular matrices. 2.

From the last relations, we directly deduce the following statements:

a) For a non-degenerated matrix A(λ) to be simple, it is necessary and sufficient that every latent root λi is configured to only one elementary divisor. b) Let the non-degenerated n × m matrix A(λ) of rank γA have the latent roots λ1 , . . . , λq . Then for the simplicity of A(λ), the relation rank A(λi ) = γA − 1,

(i = 1, . . . , q)

or, equivalently the condition def A(λi ) = 1,

(i = 1, . . . , q)

(1.60)

is necessary and sufficient. c) Another criterion for the membership of a matrix A(λ) to the class of simple matrices yields the following theorem. Theorem 1.35. A necessary and sufficient condition for the simplicity of the n × n matrix A(λ) is, thatthere exists a n × 1 column B(λ), such that the

matrix L(λ) = A(λ) B(λ) becomes alatent.

Proof. Sufficiency: Let the matrix A(λ) B(λ) be alatent and λi , (i = 1, . . . , q) are the eigenvalues of A(λ). Hence it follows

rank A(λi ) B(λi ) = n, (i = 1, . . . , q) .


31

Hereby, we deduce from Theorem 1.28, that we need Condition (1.60) to be satisfied if the last conditions should be fulfilled, i.e. the matrix A(λ) has to be simple. Necessity: It is shown that for a simple matrix A(λ), there exists a col umn B(λ), such that the matrix A(λ) B(λ) becomes alatent. Let us have det A(λ) = d(λ) and ∆(λ) ≈ d(λ) as the equivalent monic polynomial. Then the matrix A(λ) can be written in the form A(λ) = ϕ(λ) diag{1, . . . , 1, ∆(λ)}ψ(λ) , where ϕ(λ), ψ(λ) are unimodular n × n matrices. The matrix Q(λ) of the shape

In−1 On−1,1 On−1,1 Q(λ) = O1,n−1 ∆(λ) 1 is obviously alatent, because it has a minor of n-th order that is equal to one. ˜ The matrix ψ(λ) with

ψ(λ) On1 ˜ ψ(λ) = = diag{ψ(λ), 1} O1n 1 is unimodular. Applying the last two equations, we get

˜ ϕ(λ)Q(λ)ψ(λ) = A(λ) B(λ) = L(λ) with

⎡ ⎤ 0 ⎢ .. ⎥ ⎢ ⎥ B(λ) = ϕ(λ) ⎢ . ⎥ . ⎣0⎦ 1

(1.61)

The matrix L(λ) is alatent per construction. Remark 1.36. If the matrix ϕ(λ) is written in the form

ϕ(λ) = ϕ1 (λ) . . . ϕn (λ) , where ϕ1 (λ), . . . , ϕn (λ) are the corresponding columns, then from (1.61), we gain B(λ) = ϕn (λ) . 3. Square simple matrices possess the property of structural stability, which will be explained by the next theorem. Theorem 1.37. Let the matrices A(λ) ∈ Fnn (λ), B(λ) ∈ Fnn [λ] be given, where the matrix A(λ) is simple, but the matrix B(λ) is of any structure. Furthermore, let us have det A(λ) = d(λ) and

32


det[A(λ) + B(λ)] = d(λ) + d1 (λ, ) ,

(1.62)

where d1 (λ, ) is a polynomial, satisfying the condition deg d1 (λ, ) < deg d(λ) .

(1.63)

Then there exists a positive number 0 , such that for || < 0 all matrices A(λ) + B(λ) are simple. Proof. The proof splits into several stages. Lemma 1.38. Let · be a certain norm for finite-dimensional number matrices. Then for any matrix B = [ bik ] ∈ Fnn the estimation max |bik | ≤ βB

1≤i,k≤n

(1.64)

is true, where β > 0 is a constant, independent of B. Proof. Let · 1 and · 2 be any two norms in the space Cnn . Due to the finite dimension of Cnn , any two norms are equivalent, that means, for an arbitrary matrix B, we have α1 B1 ≤ B2 ≤ α2 B1 , where α1 , α2 are positive constants not depending on the choice of B. Take n B1 = max |bik | , 1≤i≤n

k=1

then under the assumption · 2 = · , we win |bik | ≤ B1 ≤ α1−1 B , which is adequate to (1.64) with β = α1−1 . Lemma 1.39. Let the matrix A ∈ Fnn be non-singular and · be a certain norm in Fnn . Then, there exists a positive constant α0 , such that for B < α0 , all matrices A + B become non-singular. Proof. Assume |bik | ≤ βB, where β > 0 is the constant configured in (1.64). Then we expand det(A + B) = det A + ϕ(A, B) , where ϕ(A, B) is a scalar function of the elements in A and B. For it, an estimation |ϕ(A, B)| < µ1 βB + µ2 β 2 B2 + . . . + µn β n Bn is true, where µi , (i = 1, . . . , n) are constants, that do not depend on B. Hence there exists a number α0 > 0, such that B < α0 always implies |ϕ(A, B)| < | det A| . That’s why for B < α0 , the desired relation det(A + B) = 0 holds.


33

Lemma 1.40. For the matrix A ∈ Fnn , we assume rank A = ρ, and let · be a certain norm in Fnn . Then there exists a positive constant α0 , such that for B ∈ Fnn with B < α0 always rank(A + B) ≥ ρ . Proof. Let Aρ be a non-zero minor of order ρ of A. Lemma 1.39 delivers the existence of a number α0 > 0, such that for B < α0 , the minor of the matrix A + B corresponding to Aρ is different from zero. However, this means that the rank will not reduce after addition of B, but that was claimed by the lemma. Let

Proof of Theorem 1.37

d(λ) = d0 λk + . . . + dk = 0,

d0 = 0

be the characteristic polynomial of the matrix A(λ). Then we obtain from (1.62) and (1.63)

det[A(λ) + B(λ)] = d(λ, ) = d0 λk + d1 ()λk−1 + . . . + dk () , where di () = di + di1 + di2 2 + . . .

, (i = 1, . . . , k)

˜ be a root of the are polynomials in the variable with di (0) = di . Let λ equation d(λ, 0) = d(λ) = 0 with multiplicity ν, i.e. an eigenvalue of the matrix A(λ) with multiplicity ν. Since the matrix A(λ) is simple, we obtain ˜ = n − 1. rank A(λ) Hereby, due to Lemma 1.40 it follows the existence of a constant α ˜ , such that for every matrix G ∈ Cnn with G < α ˜ the relation ˜ + G] ≥ n − 1 rank[A(λ)

(1.65)

is fulfilled. Now consider the equation d(λ, ) = 0 . As known from [188], for || < δ, where δ > 0 is sufficiently small, there exist ˜ i (), (i = 1, . . . , ν), such that ν continuous functions λ ˜ i (), ) = det[A(λ ˜ i ()) + B(λ ˜ i ())] = 0 , d(λ ˜ i () may coincide. Thereby, the limits where some of the functions λ

(1.66)

34


˜i , ˜ i () = λ lim λ

→0

(i = 1, . . . , ν)

exist, and we can write ˜ i + ψ˜i () , ˜ i () = λ λ ˜ where ψ˜i () are continuous functions with ψ(0) = 0. Consequently, we get ˜ i ()) + B(λ ˜ i ()) = A(λ ˜ i + ψ˜i ()) + B(λ ˜ i + ψ˜i ()) A(λ ˜ i ) + Gi () = A(λ with ˜i) + L ˜ i () Gi () = B(λ ˜ i () for || < δ depend continuously on , and L ˜ i (0) = Onn and the matrices L holds. Next choose a constant ˜ > 0 with the property that for || < ˜ and all i = 1, . . . , ν, the relation ˜i) + L ˜ i () < α Gi () = B(λ ˜ is true. Therefore, we receive for || < ˜ from (1.65) ˜ i ()) + B(λ ˜ i ())] ≥ n − 1 . rank[A(λ On the other side, it follows from (1.66), that for || < δ, we have ˜ i ()) + B(λ ˜ i ())] ≤ n − 1 . rank[A(λ Comparing the last two inequalities, we find for || < min{˜ , δ} ˜ i ()) + B(λ ˜ i ())] = n − 1 . rank[A(λ The above considerations can be made for all eigenvalues of the matrix A(λ), therefore, Theorem 1.37 is proved by (1.60).

1.12 Pairs of Polynomial Matrices 1. Let us have a(λ) ∈ Fnn [λ], b(λ) ∈ Fnm [λ]. The entirety of both matrices is called a horizontal pair, and it is designated by (a(λ), b(λ)). On the other side, if we have a(λ) ∈ Fmm [λ] and c(λ) ∈ Fnm [λ], then we speak about a vertical pair and we write [a(λ), c(λ)]. The pairs (a(λ), b(λ)) and [a(λ), c(λ)] may be configured to the rectangular matrices

a(λ) , (1.67) Rh (λ) = a(λ) b(λ) , Rv (λ) = c(λ) where the first one is horizontal, and the second one is vertical. Due to

1.12 Pairs of Polynomial Matrices

35

Rv (λ) = a (λ) c (λ) , the properties of vertical pairs can immediately deduced from the properties of horizontal pairs. Therefore, we will consider now only horizontal pairs. The pairs (a(λ), b(λ)), [a(λ), c(λ)] are called non-degenerated if the matrices (1.67) are non-degenerated. If not supposed explicitly otherwise, we will always consider non-degenerated pairs. 2.

Let for the pair (a(λ), b(λ)) exist a polynomial matrix g(λ), such that a(λ) = g(λ)a1 (λ) ,

b(λ) = g(λ)b1 (λ)

(1.68)

with polynomial matrices a1 (λ), b1 (λ). Then the matrix g(λ) is called a common left divisor of the pair (a(λ), b(λ)). The common left divisor g(λ) is named as a greatest common left divisor (GCLD) of the pair (a(λ), b(λ)), if for any left common divisor g1 (λ) g(λ) = g1 (λ)α(λ) with a polynomial matrix α(λ) is true. As known any two GCLD are rightequivalent [69]. 3. If the pair (a(λ), b(λ)) is non-degenerated, then from Theorem 1.4, it follows the existence of a unimodular matrix

n

m

r (λ) r12 (λ) r(λ) = 11 r21 (λ) r22 (λ) for which

a(λ) b(λ)

(1.69)

n m

m r11 (λ) r12 (λ)

n = N (λ) O n r21 (λ) r22 (λ)

(1.70)

holds. As known [69], the matrix N (λ) is a GCLD of the pair (a(λ), b(λ)) . 4. The pair (a(λ), b(λ)) is called irreducible, if the matrix Rh (λ) in (1.67) is alatent. From the above considerations, it follows that the pair (a(λ), b(λ)) is irreducible, if and only if there exists a unimodular matrix r(λ) according to (1.69) with

a(λ) b(λ) r(λ) = In Onm . 5.

Let s(λ) = r

−1

s (λ) s12 (λ) (λ) = 11 s21 (λ) s22 (λ)

be a unimodular polynomial matrix. Then we get from (1.70)

36


s11 (λ) s12 (λ) a(λ) b(λ) = N (λ) Onm . s21 (λ) s22 (λ)

Hence it follows immediately a(λ) = N (λ)s11 (λ) ,

b(λ) = N (λ)s12 (λ) ,

that can be written in the form

a(λ) b(λ) = N (λ) s11 (λ) s12 (λ) . Due to Corollary 1.34, the pair (s11 (λ), s12 (λ)) is irreducible. Therefore, the next statement is true: If Relation (1.68) is true, and g(λ) is a GCLD of the pair (a(λ), b(λ)), then the pair (a1 (λ), b1 (λ)) is irreducible. The reverse statement is also true: If Relation (1.68) is valid, and the pair (a1 (λ), b1 (λ)) is irreducible, then the matrix g(λ) is a GCLD of the pair (a(λ), b(λ)). 6. A necessary and sufficient condition for the irreducibility of the pair (a(λ), b(λ)) with the n × n polynomial matrix a(λ) and the n × m polynomial matrix b(λ) is the existence of an n × n polynomial matrix X(λ) and an m × n polynomial matrix Y (λ), such that the relation a(λ)X(λ) + b(λ)Y (λ) = In

(1.71)

becomes true [69]. 7. All what is said up to now, can be transferred practically without change to vertical pairs [a(λ), c(λ)]. In this case, instead of the concepts common left divisor and GCLD we introduce the concepts common right divisor and greatest common right divisor (GCRD) . Hereby, if

m a(λ) L(λ) m p(λ) = c(λ) Onm n is valid with a unimodular matrix p(λ), then L(λ) is a GCRD of the corresponding pair [a(λ), c(λ)]. If L(λ) and L1 (λ) are two GCRD, then they are related by L(λ) = f (λ)L1 (λ) where f (λ) is a unimodular matrix. The vertical pair [a(λ), c(λ)] is called irreducible, if the matrix Rv (λ) in (1.67) is alatent. The pair [a(λ), c(λ)] turns out to be irreducible, if and only if, there exists a unimodular matrix p(λ) with

1.12 Pairs of Polynomial Matrices

p(λ)

37

a(λ) Im = . c(λ) Onm

Immediately, it is seen that the pair [a(λ), c(λ)] is exactly irreducible, when there exist polynomial matrices U (λ), V (λ), for which U (λ)a(λ) + V (λ)c(λ) = Im . 8.

The above stated irreducibility criteria will be formulated alternatively.

Theorem 1.41. A necessary and sufficient condition for the pair (a(λ), b(λ)) to be irreducible, is the existence of a pair (αl (λ), βl (λ)), such that the matrix

a(λ) b(λ) Ql (λ) = βl (λ) αl (λ) becomes unimodular. For the pair [a(λ), c(λ)] to be irreducible, it is necessary and sufficient that there exists a pair [αr (λ), βr (λ)], such that the matrix

αr (λ) c(λ) Qr (λ) = βr (λ) a(λ) becomes unimodular.

9. Lemma 1.42. Necessary and sufficient for the irreducibility of the pair (a(λ), b(λ)), with the n × n and n × m polynomial matrices a(λ) and b(λ), is the condition

(i = 1, . . . , q) , (1.72) rank Rh (λi ) = rank a(λi ) b(λi ) = n , where the λi are the different eigenvalues of the matrix a(λ). ˜ = λi , (i = 1, . . . , q), we have rank a(λ) ˜ = n. Proof. Sufficiency: For λ = λ Therefore, together with (1.72) the relation rank Rh (λ) = n is true for all finite λ. This means, however, the pair (a(λ), b(λ)) is irreducible. The necessity of Condition (1.72) is obvious.

10. Lemma 1.43. Let the pair (a(λ), b(λ)) be given with the n × n and n × m polynomial matrices a(λ), b(λ). Then for the pair (a(λ), b(λ)) to be irreducible, it is necessary that the matrix a(λ) has not more than m invariant polynomials different from 1.

38


Proof. Assume the number of invariant polynomials different from 1 of the matrix a(λ) be κ > m. Then it follows from (1.57), that there exists an eigenvalue λ0 of the matrix

a(λ) with rank a(λ0 ) = n − κ. Applying Inequality a(λ0 ) b(λ0 ) ≤ n − κ + m < n that means, the ma(1.48), we gain rank

trix a(λ) b(λ) is not alatent and, consequently, the pair (a(λ), b(λ)) is not irreducible. Remark 1.44. Obviously, we could formulate adequate statements as in Lemmata 1.42 and 1.43 for vertical pairs too.

1.13 Polynomial Matrices of First Degree (Pencils) 1.

For q = 1, n = m the polynomial matrix (1.10) takes the form A(λ) = Aλ + B

(1.73)

with constant n × n matrices A, B. This special structure is also called a pencil. The pencil A(λ) is non-singular if det(Aλ + B) ≡ / 0. According to the general definition, the non-singular matrix (1.73) is called regular for det A = 0 and anomalous for det A = 0. Regular pencils arise in connection with state space representations, while anomalous pencils are configured to descriptor systems [109, 34, 182]. All introduced concepts and statements that were developed for polynomial matrices of general structure are also valid for pencils (1.73). At the same time, these matrices possess a number of important additional properties that will be investigated in this section. In what follows, we only consider non-singular pencils. 2. In accordance with the general definition, the two matrices of equal dimension (1.74) A(λ) = Aλ + B, A1 (λ) = A1 λ + B1 are called left(right)-equivalent, if there exists a unimodular matrix p(λ) (q(λ)), such that A(λ) = p(λ)A1 (λ),

(A(λ) = A1 (λ)q(λ)) .

The matrices (1.74) are equivalent, if they satisfy an equation A(λ) = p(λ)A1 (λ)q(λ) with unimodular matrices p(λ), q(λ). As follows from the above disclosures, the matrices (1.74) are exactly left(right)-equivalent, if their Hermitian canonical forms coincide. For the equivalence of the matrices (1.74), it is necessary and sufficient that their Smith-canonical forms coincide.

1.13 Polynomial Matrices of First Degree (Pencils)

39

3. The matrices (1.74) are named strictly equivalent, if there exist constant matrices P , Q with (1.75) A(λ) = P A1 (λ)Q . If in (1.74) the conditions det A = 0, det A1 = 0 are valid, i.e. the matrices are regular, then the matrices A(λ), B(λ) are only in that case equivalent, when they are strictly equivalent. If det A = 0 or det A1 = 0, i.e. the matrices (1.74) are anomalous, then the conditions for equivalence and strict equivalence do not coincide. 4. In order to formulate a criterion for the strict equivalence of anomalous matrices (1.74), following [51], we consider the n × n Jordan block ⎤ ⎡ a 1 0 ... 0 0 ⎢0 a 1 ... 0 0⎥ ⎥ ⎢ .. ⎢ . 0 0⎥ ⎢0 0 a ⎥ (1.76) Jn (a) = ⎢ . . . . ⎥, ⎢ .. .. .. . . . . . ... ⎥ ⎥ ⎢ ⎣0 0 0 ... a 1⎦ 0 0 0 ... 0 a where a is a constant. Theorem 1.45 ([51]). Let det A(λ) = det(Aλ + B) = 0 be given with det A = 0 and 0 < ord A(λ) = deg det A(λ) = η < n .

(1.77)

Furthermore, let (λ − λ1 )η1 , . . . , (λ − λq )ηq ,

η1 + . . . + ηq = η

(1.78)

be the entity of elementary divisors of A(λ) in the field C. In what follows, the elementary divisors (1.78) will be called finite elementary divisors. Then the matrix A(λ) is strictly equivalent to the matrix ˜ A(λ) = diag{λIη + Aη , In−η + λAν }

(1.79)

with Aη = diag{Jη1 (λ1 ), . . . , Jηq (λq )} ,

(1.80)

Aν = diag{Jp1 (0), . . . , Jp (0)} , where p1 , . . . , p are positive integers with p1 + . . . + p = n − η. The matrix Aν ∈ Cn−η,n−η is nilpotent, that means, there exists an integer κ with Aκν = On−η,n−η .

40


Remark 1.46. The above defined numbers p1 , . . . , p are determined by the infinite elementary divisors of the matrix A(λ), [51]. Thereby, the matrices (1.74) are strictly equivalent, if their finite and infinite elementary divisors coincide. Remark 1.47. Matrix (1.79) can be represented as ˜ A(λ) = Uλ + V ,

(1.81)

where U = diag{Iη , Aν } ,

V = diag{Aη , In−η } .

(1.82)

As is seen from (1.76) and (1.80)–(1.82) for η < n, we always obtain ˜ det U = 0 and the matrix A(λ) is generally spoken not row reduced. 5. As any non-singular matrix, also an anomalous matrix (1.73) can be brought into row reduced form by left equivalence transformations. Hereby, we obtain for matrices of first degree some further results. Theorem 1.48. Let Relation (1.77) be true for the non-singular anomalous matrix (1.73). Then there exists a unimodular matrix P (λ), such that ˜ ˜ +B ˜ P (λ)(Aλ + B) = A(λ) = Aλ

(1.83)

is true with constant matrices A˜ = Moreover

n

A˜1 On−η,n

η n−η

n ˜ ˜ = B1 B ˜2 B

,

A˜ det ˜1 B2

η

.

(1.84)

n−η

= 0

(1.85)

is true together with deg P (λ) ≤ n − η .

(1.86)

Proof. We apply the row transformation algorithm of Theorem 1.9 to the matrix Aλ + B. Then after a finite number of steps, we reach at a row reduced ˜ matrix A(λ). Due to the fact, that the degree of the transformed matrix does ˜ ˜ not increase, we conclude deg A(λ) ≤ 1. The case deg A(λ) = 0 is excluded, otherwise the matrix A(λ) would be unimodular in contradiction to (1.77). ˜ Therefore, only deg A(λ) = 1 is possible. Moreover, we prove ˜ ˜ +B ˜ = diag{λα1 , . . . , λαn } A˜0 + A˜1 λ−1 (1.87) A(λ) = Aλ with det A˜0 = 0, where each of the numbers αi , (i = 1, . . . , n) is either 0 or 1. Due to


41

α1 + . . . + αn = η , among the numbers α1 , . . . , αn are exactly η ones with the value one, and the other n − η numbers are zero. Without loss of generality, we assume the succession α1 = α2 = . . . = αη = 1,

αη+1 = αη+2 = . . . = αn = 0 .

Then the matrix A˜ in (1.83) takes the shape (1.84). Furthermore, if the matrix ˜ A(λ) is represented in the form (1.87), then with respect to (1.79) and (1.84), we get

A˜ A˜0 = ˜1 . B2 ˜ Since the matrix A(λ) is row reduced, Relation (1.85) arises. It remains to show Relation (1.86). As follows from (1.36), each step decreases the degree of one of the rows of the transformed matrices at least by one. Hence each row of the matrix A(λ) cannot be transformed more than once. Therefore, the number of transformation steps is at most n − η. Since however, in every step the transformation matrix P (λ) is either constant or with degree one, Relation (1.86) holds. Corollary 1.49. In the row-reduced form (1.83), n − η rows of the matrix ˜ A(λ) are constant. Moreover, the rank of the matrix built from these rows is equal to n − η, i.e., these rows are linearly independent. Example 1.50. Consider the anomalous matrix ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 112 213 λ + 2 λ + 1 2λ + 3 A(λ) = Aλ + B = ⎣ 1 1 2 ⎦ λ + ⎣ 3 2 5 ⎦ = ⎣ λ + 3 λ + 2 2λ + 5 ⎦ 113 326 λ + 3 λ + 2 3λ + 6 appering in [51], that is represented in the form A(λ) = diag{λ, λ, λ} A0 + A1 λ−1 with A0 = A,

A1 = B .

In the first transformation step (1.30), we obtain ν1 + ν 2 + ν 3 = 0 2ν1 + 2ν2 + 3ν3 = 0 . Now, we can choose ν1 = 1, ν2 = −1, ν3 = 0, and the matrices (1.32) and (1.35) take the form ⎡ ⎤ 1 −1 0 P1 (λ) = D1 = ⎣ 0 1 0 ⎦ 0 0 1

42


hence ⎞ ⎤ ⎡ ⎤ −1 −1 −2 000 A1 (λ) = P1 (λ)A(λ) = diag{1, λ, λ} ⎝⎣ 1 1 2 ⎦ + ⎣ 3 2 5 ⎦ λ−1 ⎠ . 1 1 3 326 ⎛⎡

By appropriate manipulations, these matrices are transformed into ⎡ ⎤ ⎡ ⎤ 100 100 P2 (λ) = ⎣ λ 1 0 ⎦ , D2 = ⎣ 1 1 0 ⎦ . 001 001 Finally, we receive over the product A2 (λ) = P2 (λ)A1 (λ) = P2 (λ)P1 (λ)A(λ) the row-reduced matrix ⎡

⎤ −1 −1 −2 2 5 ⎦. A2 (λ) = ⎣ 3 λ + 3 λ + 2 3λ + 6 6. Let B be a constant n × n matrix. We assign to this matrix a matrix Bλ of degree one by Bλ = λIn − B , which is called the characteristic matrix of B. For polynomial matrices of this form, all above introduced concepts and statements for polynomial matrices of general form remain valid. Hereby, the characteristic polynomial of the matrix Bλ det Bλ = det(λIn − B) = dB (λ) usually is named the characteristic polynomial of the matrix B. In the same way, we deal with the terminology of minimal polynomials, invariant polynomials, elementary divisor etc. Obviously, ord Bλ = deg det Bλ = n . As a consequence from Relation (1.75) for A1 = A = In we formulate: Theorem 1.51 ([51]). For two characteristic matrices Bλ = λIn − B and B1λ = λIn − B1 to be equivalent, it is necessary and sufficient, that the matrices B and B1 are similar, i.e. the relation B1 = LBL−1 is true with a certain non-singular constant matrix L.


43

Remark 1.52. Theorem 1.51 implies the following property. If the matrix B (the matrix λIn − B) has the entirety of elementary divisors (λ − λ1 )ν1 · · · (λ − λq )νq ,

ν1 + . . . + νq = n ,

then the matrix B is similar to the matrix J of the form J = diag{Jν1 (λ1 ), . . . , Jνq (λq )} .

(1.88)

The matrix J is said to be the Jordan (canonical) form or shortly, Jordan matrix of the corresponding matrix B. For any n × n matrix B, the Jordan matrix is uniquely determined, except the succession of the diagonal blocks. 7. Let the horizontal pair of constant matrices (A, B) with A, n × n, and B, n×m be given. The pair (A, B) is called controllable, if the polynomial pair (λIn − A, B) is irreducible. This means, that the pair (A, B) is controllable if and only if the matrix

Rc (λ) = λIn − A B is alatent. It is known, see for instance [72, 69], that the pair (A, B) is controllable, if and only if rank Qc (A, B) = n , where the matrix Qc (A, B) is determined by

Qc (A, B) = B AB . . . An−1 B .

(1.89)

The matrix Qc (A, B) is named controllability matrix of the pair (A, B). Some statements regarding the controllability of pairs are listed now: a) If the pair (A, B) is controllable, and the n × n matrix R is non-singular, then also the pair (A1 , B1 ) with A1 = RAR−1 , B1 = RB is controllable. Indeed, from (1.89) we obtain

Qc (A1 , B1 ) = RB RAB . . . RAn−1 B == RQc (A, B) , from which follows rank Qc (A1 , B1 ) = rank Qc (A, B) = n, because R is non-singular. b) Theorem 1.53. Let the pair (A, B) with the n×n matrix A and the n×m matrix B be given, and moreover, an n×n matrix L, which is commutative with A, i.e. AL = LA. Then the following statements are true: 1. If the pair (A, B) is not controllable, then the pair (A, LB) is also not controllable. 2. If the pair (A, B) is controllable and the matrix L is non-singular, then the pair (A, LB) is controllable. 3. If the matrix L is singular, then the pair (A, LB) is not controllable.

44


Proof. The controllability matrix of the pair (A, LB) has the shape

Qc (A, LB) = LB ALB . . . An−1 LB (1.90)

= L B AB . . . An−1 B = LQc (A, B) where Qc (A, B) is the controllability matrix (1.89). If the pair (A, B) is not controllable, then we have rank Qc (A, B) < n, and therefore, rank Qc (A, LB) < n. Thus the 1st statement is proved. If the pair (A, B) is controllable and the matrix L is non-singular, then we have rank Qc (A, B) = n, rank L = n and from (1.90) it follows rank Qc (A, LB) = n. Hence the 2nd statement is shown. Finally, if the matrix L is singular, then rank L < n and rank Qc (A, LB) < n are true, which proves 3. c) Controllable pairs are structural stable - this is stated in the next theorem. Theorem 1.54. Let the pair (A, B) be controllable, and (A1 , B1 ) be an arbitrary pair of the same dimension. Then there exists a positive number 0 , such that the pair (A + A1 , B + B1 ) is controllable for all || < 0 . Proof. Using (1.89) we obtain Qc (A + A1 , B + B1 ) = Qc (A, B) + Q1 + . . . + n Qn ,

(1.91)

where the Qi , (i = 1, . . . , n) are constant matrices, that do not depend on . Since the pair (A, B) is controllable, the matrix Qc (A, B) contains a non-zero minor of n-th order. Then due to Lemma 1.39 for sufficiently small ||, the corresponding minor of the matrix (1.91) also remains different from zero. Remark 1.55. Non-controllable pairs do not possess the property of structural stability. If the pair (A, B) is not controllable, then there exists a pair (A1 , B1 ) of equal dimension, such that the pair (A + A1 , B + B1 ) for arbitrary small || > 0 becomes controllable. 8. The vertical pair [A, C] built from the constant m × m matrix A and n × m matrix C is called observable, if the vertical pair of polynomial matrices [λIm − A, C] is irreducible. Obviously, the pair [A, C] is observable, if and only if the horizontal pair (A , C ) is controllable, where the prime means the transposition operation. Due to this reason, observable pairs possess all the properties that have been derived above for controllable pairs. Especially, observable pairs are structural stable.

1.14 Cyclic Matrices 1. The constant n × n matrix A is said to be cyclic, if the assigned characteristic matrix Aλ = λIn − A is simple in the sense of the definition in Section 1.11, see [69, 78, 191].

1.14 Cyclic Matrices

45

Cyclic matrices are provided with the important property of structural stability, as is substantiated by the next theorem. Theorem 1.56. Let the cyclic n × n matrix A, and an arbitrary n × n matrix B be given. Then there exists a positive number 0 > 0, such that for || < 0 all matrices A + B become cyclic. Proof. Let det(λIn − A) = dA (λ),

deg dA (λ) = n .

Then we obtain det(λIn − A − B) = dA (λ) + d1 (λ, ) with deg d1 (λ, ) < n for all . Therefore, by virtue of Theorem 1.37, there exists an 0 , such that for || < 0 the matrix λIn − A − B remains simple, i.e. the matrix A + B is cyclic. 2. Square constant matrices that are not cyclic, will be called in future composed. Composed matrices are not equipped with the property of structural stability in the above defined sense. For any composed matrix A, we can find a matrix B, such that the sum A + B becomes cyclic, as small even || > 0 is chosen. Moreover, the sum A + B will become composed only in some special cases. This fact is illustrated by a 2 × 2 matrix in the next example. Example 1.57. As follows from Theorem 1.51, any composed 2 × 2 matrix A is similar to the matrix

a0 (1.92) B= = aI2 , 0a where a = const., so we have A = LBL−1 = B . Therefore, the set of all composed matrices in C22 is determined by Formula (1.92) for any a. Assume now the matrix Q = A + F to be composed. Then

q0 Q= 0q is true, and hence

a−q 0 F =B−Q= 0 a−q

becomes an composed matrix. When the 2 × 2 matrix A is composed, then the sum A + F still becomes onerous, if and only if the matrix F is composed too.

46


3. The property of structural stability of cyclic matrices allows a probabilitytheoretic interpretation. For instance, the following statement is true: Let A ∈ Fnn be a composed matrix and B ∈ Fnn any random matrix with independent entries that are equally distributed in a certain interval α ≤ bik ≤ β. Then the sum A + B with probability 1 becomes a cyclic matrix. 4. The property of structural stability has great practical importance. Indeed, let for instance the differential equation of a certain linear process be given in the form dx = Ax + Bu , dt

A = A0 + ∆A ,

where x is the state vector, and A0 , ∆A are constant matrices, where A0 is cyclic. The matrix ∆A manifests the unavoidable errors during the set up and calculation of the matrix A. From Theorem 1.56 we conclude that the matrix A remains cyclic, if the deviation ∆A satisfies the conditions of Theorem 1.56. If however, the matrix A is composed, then this property can be lost due to the imprecision characterised by the matrix ∆A, as tiny this ever has been with respect to the norm. 5.

Assume d(λ) = λn + d1 λn−1 + . . . + dn

to be a monic polynomial. Then the n × n matrix AF of the form ⎤ ⎡ 0 1 0 ... 0 0 ⎢ 0 0 1 ... 0 0 ⎥ ⎥ ⎢ ⎢ .. .. .. ⎥ .. . . .. AF = ⎢ . . . . . . ⎥ ⎥ ⎢ ⎣ 0 0 0 ... 0 1 ⎦ −dn −dn−1 −dn−2 . . . −d2 −d1

(1.93)

(1.94)

is called its accompanying (horizontal) Frobenius matrix with respect to the polynomial d(λ). Moreover, we consider the vertical accompanying Frobenius matrix ⎤ ⎡ 0 0 . . . 0 −dn ⎢ 1 0 . . . 0 −dn−1 ⎥ ⎥ ⎢ ⎢. . . .. .. ⎥ . . . ¯ ⎢ AF = ⎢ . . . . (1.95) . ⎥ ⎥. ⎥ ⎢ .. ⎣ 0 0 . 0 −d2 ⎦ 0 0 . . . 1 −d1 The properties of the matrices (1.94) and (1.95) are analogue, so that we could restrict ourself to the investigation of (1.94). The characteristic matrix of AF has the form

1.14 Cyclic Matrices

⎡ ⎢ ⎢ ⎢ λIn − AF = ⎢ ⎢ ⎣

47

⎤

λ −1 0 . . . 0 0 0 λ −1 . . . 0 0 ⎥ ⎥ .. .. .. ⎥ . . . . . .. . . . . . . ⎥ ⎥ 0 0 0 . . . λ −1 ⎦ dn dn−1 dn−2 . . . d2 λ + d1

(1.96)

Appending Matrix (1.96) with the column b = 0 . . . 0 1 , we receive the extended matrix ⎡ ⎤ λ −1 0 . . . 0 0 0 ⎢ 0 λ −1 . . . 0 0 0⎥ ⎢ ⎥ ⎢ .. .. . .. .. ⎥ . . . . . . ⎢ . . . . . . .⎥ ⎢ ⎥ ⎣ 0 0 0 . . . λ −1 0 ⎦ dn dn−1 dn−2 . . . d2 λ + d1 1 This matrix is alatent, because it has a minor of n-th order that is equal to (−1)n−1 . Strength to Theorem 1.35, Matrix (1.96) is simple, and therefore, the matrix AF is cyclic. 6.

By direct calculation we recognise det(λIn − AF ) = λn + d1 λn−1 + . . . + dn = d(λ) .

According to the properties of simple matrices, we conclude that the whole of invariant polynomials corresponding to the matrix AF is presented by a1 (λ) = a2 (λ) = . . . = an−1 (λ) = 1,

an (λ) = f (λ) .

Let A be any cyclic n × n matrix. Then the accompanying matrix Aλ = λIn − A is simple. Therefore, by applying equivalence transformations, Aλ might be brought into the form λIn − A = p(λ) diag{1, 1, . . . , d(λ)}q(λ) , where the matrices p(λ), q(λ) are unimodular, and d(λ) is the characteristic polynomial of the matrix A. From the last equation, we conclude that the set of invariant polynomials of the cyclic matrix A coincides with the set of invariant polynomials of the accompanying Frobenius matrix of its characteristic polynomial d(λ). Hereby, the matrices λIn − A and λIn − AF are equivalent, hence the matrices A and AF are similar, i.e. A = LAF L−1 is true with a certain non-singular matrix L. It can be shown that in case of a real matrix A, also the matrix L could be chosen real.

48


7.

As just defined in (1.76), let ⎡

a10 ⎢0 a 1 ⎢ ⎢ ⎢0 0 a Jn (a) = ⎢ ⎢. . . ⎢ .. .. .. ⎢ ⎣0 0 0 000

⎤ ... 0 0 ... 0 0⎥ ⎥ ⎥ .. . 0 0⎥ ⎥ . . . . .. ⎥ . . .⎥ ⎥ ... a 1⎦ ... 0 a

be a Jordan block. The matrix Jn (a) turns out to be cyclic, because the matrix ⎤ ⎡ λ − a −1 0 ... 0 0 0 ⎢ 0 λ − a −1 . . . 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢ 0 . 0 λ − a 0 0 0 ⎥ ⎢ ⎥ ⎢ . . . . . . . .. .. .. .. .. .. ⎥ ⎢ .. ⎥ ⎢ ⎣ 0 0 0 . . . λ − a −1 0 ⎦ 0 0 0 . . . 0 λ − a −1 is alatent. Let us represent the polynomial (1.93) in the form d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq , where all numbers λi are different. Consider the matrix J = diag{Jµ1 (λ1 ), . . . , Jµq (λq )}

(1.97)

and its accompanying characteristic matrix λIn − J = diag{λIµ1 − Jµ1 (λ1 ), . . . , λIµq − Jµq (λq )} , where the corresponding diagonal blocks take the shape ⎡ ⎤ λ − λi −1 0 ... 0 0 ⎢ 0 λ − λi −1 . . . 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ . . ⎢ 0 0 ⎥ 0 λ − λi . 0 ⎢ ⎥. λIµi − Jµi (λi ) = ⎢ . .. .. .. ⎥ .. .. ⎢ .. . . . . . ⎥ ⎢ ⎥ ⎣ 0 0 0 . . . λ − λi −1 ⎦ 0 0 0 . . . 0 λ − λi Obviously, we have det[λIµi − Jµi (λi )] = (λ − λi )µi so that from (1.98), we obtain

(1.98)

(1.99)

1.15 Simple Realisations and Their Structural Stability

49

det(λIn − J) = (λ − λ1 )µ1 · · · (λ − λq )µq = d(λ) . At the same time, using (1.98) and (1.99), we find rank(λi In − J) = n − 1,

(i = 1, . . . , q)

that means, Matrix (1.98) is cyclic. Therefore, Matrix (1.97) is similar to the accompanying Frobenius matrix of the polynomial (1.93), thus J = LAF L−1 , where L in general is a complex non-singular matrix.

1.15 Simple Realisations and Their Structural Stability 1. The triple of matrices a(λ), b(λ), c(λ) of dimensions p × p, p × m, n × p, according to [69] and others, is called a polynomial matrix description (PMD) τ (λ) = (a(λ), b(λ), c(λ)) .

(1.100)

The integers n, p, m are the dimension of the PMD. In dependence on the membership of the entries of the matrices a(λ), b(λ), c(λ) to the sets F[λ], R[λ], C[λ], the sets of all PMDs with dimension n, p, m are denoted by Fnpm [λ], Rnpm [λ], Cnpm [λ], respectively. A PMD (1.100) is called minimal, if the pairs (a(λ), b(λ)), [a(λ), c(λ)] are irreducible. 2.

A PMD of the form τ (λ) = (λIp − A, B, C) ,

(1.101)

where A, B, C are constant matrices, is said to be an elementary. Every elementary PMD (1.101) is characterised by a triple of constant matrices A, (p×p); B, (p×m); C, (n×p). The triple (A, B, C) is called a realisation of the linear process in state space, or shortly realisation. The numbers n, p, m are named the dimension of the elementary realisation. The set of all realisations with given dimension is denoted by Fnpm , Rnpm , Cnpm , respectively. Suppose the p × p matrix Q to be non-singular. Then the realisations (A, B, C) and (QAQ−1 , QB, CQ−1 ) are called similar. 3. The realisation (A, B, C) is called minimal, if the pair (A, B) is controllable and the pair [A, C] is observable, i.e. the elementary PMD (1.101) is minimal. A minimal realisation with a cyclic matrix A is called a simple realisation. The set of all minimal realisations of a given dimension will be ¯ npm , C ¯ npm respectively, and the set of all simple re¯ npm , R symbolised by F s s s alisations by Fnpm , Rnpm Cnpm . For a simple realisation (A, B, C) ∈ Rsnpm

50


−1 s always exists a similar realisation (QJ AQ−1 J , QJ B, CQJ ) ∈ Cnpm , where the −1 matrix QJ AQJ is of Jordan canonical form. Such a simple realisation is called a Jordan realisation. Moreover, for this realisation, there exists a sim−1 −1 s ilar realisation (QF AQ−1 F , QF B, CQF ) ∈ Rnpm , where the matrix QF AQF is a Frobenius matrix of the form (1.94). Such a simple realisation is called a Frobenius realisation.

4. Simple realisations possess the important property of structural stability, as the next theorem states. Theorem 1.58. Let the realisation (A, B, C) of dimension n, p, m be simple, and (A1 , B1 , C1 ) be an arbitrary realisation of the same dimension. Then there exists an 0 > 0, such that the realisation (A + A1 , B + B1 , C + C1 ) for all || < 0 remains simple. Proof. Since the pair (A, B) is controllable and the pair [A, C] is observable, there exists, owing to Theorem 1.54, an 1 > 0, such that the pair (A + A1 , B + B1 ) becomes controllable and the pair [A + A1 , C + C1 ] observable for all || < 1 . Furthermore, due to Theorem 1.56, there exists an 2 > 0, such that the matrix A + A1 becomes cyclic for all || < 2 . Consequently, for || < min(1 , 2 ) = 0 all realisations (A + A1 , B + B1 , C + C1 ) are simple. Remark 1.59. Realisations that are not simple, are not provided by the property of structural stability. For instance, from the above considerations we come to the following conclusion: Let the realisation (A, B, C) be not simple, and (A1 , B1 , C1 ) be a random realisation of equal dimension, where the entries of the matrices A1 , B1 , C1 are in the whole statistically independent and equally distributed in a certain interval [α, β]. Then the realisation (A + A1 , B + B1 , C + C1 ) will be simple with probability 1. 5. Theorem 1.58 has fundamental importance for developing methods on base of a mathematical description of linear time-invariant multivariable systems. The dynamics of such systems are described in continuous time by state-space equations of the form y = Cx,

dx = Ax + Bu , dt

(1.102)

corresponding to the realisation (A, B, C). In practical investigations, we always will meet A = A0 +∆A, B = B0 +∆B, C = C0 +∆C, where (A0 , B0 , C0 ) is the nominal realisation and the realisation (∆A, ∆B, ∆C) characterises inaccuracies due to finite word length etc. Now, if the nominal realisation is simple, then at least for sufficiently small deviations (∆A, ∆B, ∆C), the simplicity is preserved. Analogue considerations are possible for the description of the dynamics of discrete-time systems, where

1.15 Simple Realisations and Their Structural Stability

yk = Cxk ,

xk+1 = Axk + Buk

51

(1.103)

is used. If however, the nominal realisation (A, B, C) is not simple, then the structural properties will, roughly spoken, not be preserved even for tiny deviations. 6. In principle in many cases, we can find suitable bounds of disturbances for which a simple realisation remains simple. For instance, let the matrices A1 , B1 , C1 depend continuously on a scalar parameter α, such that A1 = A1 (α), B1 = B1 (α), C1 = C1 (α) with A1 (0) = Opp , B1 (0) = Opm , C1 (0) = Onp is valid. Now, if the parameter α increases from zero to positive values, then the realisation (A + A1 (α), B + B1 (α), C + C1 (α) for 0 ≤ α < α0 remains simple, where α0 is the smallest positive number, for which at least one of the following conditions takes place: a) The pair (A + A1 (α0 ), B + B1 (α0 )) is not controllable. b) The pair [A + A1 (α0 ), C + C1 (α0 )] is not observable. c) The matrix A + A1 (α0 ) is not cyclic.

2 Fractional Rational Matrices

2.1 Rational Fractions 1. A fractional rational (rat.) function, or shortly rational fraction means the relation of two polynomials (λ) =

m(λ) ˜ , ˜ d(λ)

(2.1)

˜ where d(λ) = 0. In dependence on the coefficient sets for the numerator and denominator polynomials in (2.1), the corresponding set of rational fractions is designated by F(λ), C(λ) or R(λ), respectively. Hereby, rational fractions in R(λ) are named real. Over the set of rational functions, various algebraic operations can be explained. 2.

Two rational fractions 1 (λ) =

m1 (λ) , d1 (λ)

2 (λ) =

m2 (λ) d2 (λ)

(2.2)

are considered as equal, and we write 1 (λ) = 2 (λ) for that, when m1 (λ)d2 (λ) − m2 (λ)d1 (λ) = 0 .

(2.3)

Let in particular m2 (λ) = a(λ)m1 (λ) ,

d2 (λ) = a(λ)d1 (λ)

with a polynomial a(λ), then (2.3) is fulfilled, and the fractions 1 (λ) =

m1 (λ) , d1 (λ)

2 (λ) =

a(λ)m1 (λ) a(λ)d1 (λ)

are equal in the sense of the above definition. Immediately, it follows that the rational fraction (2.1) does not change if the numerator and denominator are

54


cancelled by the same factor. Any polynomial f (λ) can be represented in the form f (λ) . f (λ) = 1 Therefore, for polynomial rings the relations F[λ] ⊂ F(λ), C[λ] ⊂ C(λ), R[λ] ⊂ R(λ) are true. 3. Let the fraction (2.1) be given, and g(λ) is the GCD of the numerator ˜ m(λ) ˜ and the denominator d(λ), such that ˜ = g(λ)d1 (λ) d(λ)

m(λ) ˜ = g(λ)m1 (λ),

with coprime m1 (λ), d1 (λ). Then we have (λ) =

m1 (λ) . d1 (λ)

(2.4)

Notation (2.4) is called an irreducible form of the rational fraction. Furthermore, assume d1 (λ) = d0 λn + d1 λn−1 + . . . + dn ,

d0 = 0 .

Then, the numerator and denominator of (2.4) can be divided by d0 , yielding (λ) =

m(λ) . d(λ)

Herein the numerator and denominator are coprime polynomials, and besides the polynomial d(λ) is monic. This representation of a rational fraction will be called its standard form. The standard form of a rational fraction is unique. 4.

The sum of rational fractions (2.2) is defined by the formula 1 (λ) + 2 (λ) =

5.

m1 (λ)d2 (λ) + m2 (λ)d1 (λ) m1 (λ) m2 (λ) + = . d1 (λ) d2 (λ) d1 (λ)d2 (λ)

The product of rational fractions (2.2) is explained by the relation 1 (λ)2 (λ) =

m1 (λ)m2 (λ) . d1 (λ)d2 (λ)

6. In algebra, it is proved that the sets of rational fractions C(λ), R(λ) with the above explained rules for addition and multiplication form fields. The zero element of those fields proves to be the fraction 0/1, the unit element is the rational fraction 1/1. If we have in (2.1) m(λ) ˜ = 0, then the inverse element −1 (λ) is determined by the formula −1 (λ) =

˜ d(λ) . m(λ) ˜

2.1 Rational Fractions

7.

55

The integer ind , for which the finite limit lim (λ)λind = 0 = 0

λ→∞

exists, is called the index of the rational fraction (2.1). In case of ind = 0 (ind > 0), the fraction is called proper (strictly proper). In case of ind ≥ 0, the fraction is said to be at least proper. In case of ind < 0 the fraction (λ) is named improper. If the rational fraction (λ) is represented in the form (2.1) ˜ and we introduce deg m(λ) ˜ = α, deg d(λ) = β, then the fraction is proper, strictly proper or at least proper, if the corresponding relation α = β, α < β or α ≤ β is true. The zero rational fraction is defined as strictly proper. 8.

Any fraction (2.1) can be written in the shape (λ) =

r(λ) + q(λ) d(λ)

(2.5)

with polynomials r(λ), q(λ), where deg r(λ) < deg d(λ), such that the first summand at the right side of (2.5) is strictly proper. The representation (2.5) is unique. Practically, the polynomials r(λ) and q(λ) could be found in the following way. Using (1.6), we uniquely receive m(λ) = d(λ)q(λ) + r(λ) with deg r(λ) < deg d(λ). Inserting the last relation into (2.1), we get (2.5). 9. The sum, the difference and the product of strictly proper fractions are also strictly proper. The totality of strictly proper fractions builds a commutative ring without unit element. 10.

Let the strictly proper rational fraction (λ) =

m(λ) d1 (λ)d2 (λ)

be given, where the polynomials d1 (λ) and d2 (λ) are coprime. Then we can find a separation m1 (λ) m2 (λ) + , (2.6) (λ) = d1 (λ) d2 (λ) where both fractions on the right side are strictly proper. Hereby, the polynomial m1 (λ) and m2 (λ) are determined uniquely.

56


11. A separation of the form (2.6) can be generalised as follows. Let the strictly proper fraction (λ) possess the shape (λ) =

m(λ) , d1 (λ)d2 (λ) · · · dn (λ)

where all polynomials in the denominator are two and two coprime, then there exists a unique representation of the form (λ) =

mn (λ) m1 (λ) m1 (λ) + + ... + , d1 (λ) d1 (λ) dn (λ)

(2.7)

where all fractions on the right side are strictly proper. In particular, let the strictly proper irreducible fraction (λ) =

m(λ) d(λ)

with d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq be given, where all λi are different. Introduce (λ − λi )µi = di (λ) and apply (2.7), then we obtain a representation of the form (λ) =

q i=1

mi (λ) , (λ − λi )µi

deg mi (λ) < µi .

(2.8)

The representation (2.8) is unique. 12.

Furthermore, we can show that the fractions of the form i (λ) =

mi (λ) , (λ − λi )µi

deg mi (λ) < µi

can be uniquely presented in the form i (λ) =

mi1 mi2 miµi + + ... + , (λ − λi )µi (λ − λi )µi −1 λ − λi

where the mij are certain constants. Inserting this relation into (2.8), we get (λ) =

q i=1

mi1 mi2 miµi + + ... + µ µ −1 i i (λ − λi ) (λ − λi ) λ − λi

,

(2.9)

which is named a partial fraction expansion . A representation of the form (2.9) is unique.

2.1 Rational Fractions

57

13. For calculating the coefficients mik of the partial fraction expansion (2.9) the formula ∂ k−1 m(λ)(λ − λi )µi 1 mik = (2.10) (k − 1)! ∂λk−1 d(λ) λ=λi can be used. The coefficients (2.10) are closely connected to the expansion of the function m(λ)(λ − λi )µi ˜i (λ) = d(λ) into a Taylor series in powers of (λ − λi ), that exists because the function ˜i (λ) is analytical in the point λ = λi . Assume for instance ˜i (λ) = i1 + i2 (λ − λi ) + . . . + iµi (λ − λi )µi + . . . with ik =

dk−1 ˜ 1 (λ) . i k−1 (k − 1)! dλ λ=λi

Comparing this expression with (2.10) yields mik = ik . 14.

Let (λ) =

m(λ) d1 (λ)d2 (λ)

(2.11)

be any rational fraction, where the polynomials d1 (λ) and d2 (λ) are coprime. Moreover, we assume (λ) =

m(λ) ˜ + q(λ) d1 (λ)d2 (λ)

for a representation of (λ) in the form (2.5), where m(λ), ˜ q(λ) are polynomials with deg m(λ) ˜ < deg d1 (λ) + deg d2 (λ). Since d1 (λ), d2 (λ) are coprime, there exists a unique decomposition m(λ) ˜ m1 (λ) m2 (λ) = + , d1 (λ)d2 (λ) d1 (λ) d2 (λ) where the fractions on the right side are strictly proper. Altogether, for (2.11) we get the unique representation m(λ) m1 (λ) m2 (λ) = + + q(λ) , d1 (λ)d2 (λ) d1 (λ) d2 (λ)

(2.12)

where deg m1 (λ) < deg d1 (λ) and deg m2 (λ) < deg d2 (λ) are valid. Let g(λ) be any polynomial. Then (2.12) can be written in the shape

58


m2 (λ) m1 (λ) + g(λ) + + q(λ) − g(λ) , (λ) = d1 (λ) d2 (λ)

(2.13)

which is equivalent to (λ) = 1 (λ) + 2 (λ) =

n1 (λ) n2 (λ) + , d1 (λ) d2 (λ)

(2.14)

where n1 (λ) = m1 (λ) + g(λ)d1 (λ) , n2 (λ) = m2 (λ) + [ q(λ) − g(λ)] d2 (λ) .

(2.15)

The representation of a rational fraction (λ) in the form (2.14) is called a separation with respect to the polynomials d1 (λ) and d2 (λ) . From (2.14), (2.15) it follows that a separation always is possible, but not uniquely determined. It is shown now that Formulae (2.14), (2.15) include all possible separations. Indeed, if (2.14) holds, then the division of the fractions on the right side of (2.14) by the denominator yields (λ) =

k2 (λ) k1 (λ) + q1 (λ) + + q2 (λ) d1 (λ) d2 (λ)

with deg k1 (λ) < deg d1 (λ), deg k2 (λ) < deg d2 (λ). Comparing the last equation with (2.12) and bear in mind the uniqueness of the representation (2.12), we get k1 (λ) = m1 (λ), k2 (λ) = m2 (λ) and q1 (λ) + q2 (λ) = q(λ). While assigning q1 (λ) = g(λ), q2 (λ) = q(λ) − g(λ), we realise that the representation (2.14) takes the form (2.12). Selecting in (2.13) g(λ) = 0, we obtain the separation 1 (λ) =

m1 (λ) , d1 (λ)

2 (λ) =

m2 (λ) + q(λ) , d2 (λ)

(2.16)

where the fraction 1 (λ) is strictly proper. Analogously, we find for g(λ) = q(λ) the separation 1 (λ) =

m1 (λ) + q(λ) , d1 (λ)

2 (λ) =

m2 (λ) , d2 (λ)

(2.17)

where the fraction 2 (λ) is strictly proper. Separation (2.16) is called minimal with respect to d1 (λ), and Separation (2.17) minimal with respect to d2 (λ). From the above exposition, it follows that the minimal separations are uniquely determined. An important special case arises for strictly proper fractions (2.11). Then we receive in (2.12) q(λ) = 0, and the minimal separations (2.16) and (2.17) coincide.

2.2 Rational Matrices

59

2.2 Rational Matrices 1.

A n × m matrix L(λ) of the form ⎡ 11 (λ) ⎢ .. L(λ) = ⎣ .

⎤ . . . 1m (λ) ⎥ .. .. ⎦ . . n1 (λ) . . . nm (λ)

(2.18)

is called a broken rational, or shortly rational matrix, if all its entries are broken rational functions of the form (2.1). If ik ∈ F( λ) (or C(λ), R(λ)), then the corresponding set of matrices (2.18) is denoted by Fnm (λ), (or Cnm (λ), Rnm (λ)), respectively. In the following considerations we optionally assume matrices in Rnm [λ] and Rnm (λ), that practically arise in all technological applications. But it is mentioned that most of the results derived below also hold for matrices in Fnm (λ), Cnm (λ). Rational matrices in Rnm (λ) are named real. By writing all elements of Matrix (2.18) on the principal denominator, the matrix can be denoted in the form L(λ) =

˜ (λ) N , ˜ d(λ)

(2.19)

˜ ˜ (λ) is a polynomial matrix, and d(λ) where the matrix N is a scalar polyno˜ mial. In description (2.19) the matrix N (λ) is called the numerator and the ˜ polynomial d(λ) the denominator of the matrix L(λ). Without loss of gener˜ ality, we will assume that the polynomial d(λ) in (2.19) is monic and has the linear factorisation ˜ = (λ − λ1 )µ1 · · · (λ − λq )µq . d(λ)

(2.20)

The fraction (2.19) is called irreducible, if with respect to the factorisation (2.20) ˜ (λi ) = Onm , (i = 1, . . . , q) . N However, if for at least one 1 ≤ i ≤ q ˜ (λi ) = Onm N becomes true, then the fraction (2.19) is named reducible. The last equation is fulfilled, if every element of the matrix L(λ) is divisible by λ − λi . After performing all possible cancellations, we always arrive at a representation of the form N (λ) , (2.21) L(λ) = d(λ) where the fraction on the right side is irreducible, and the polynomial d(λ) is monic. This representation of a rational matrix (2.18) is named its standard form. The standard form of a rational matrix is uniquely determined. In future, we will always assume rational matrices in standard form if nothing else is denied.

60


Example 2.1. Let the rational matrix ⎤ ⎡ 5λ + 3 1 ⎢ λ−2 λ−3⎥ ⎥ L(λ) = ⎢ ⎦ ⎣ 2λ + 1 λ+2 (λ − 2)(λ − 3) be given that can be represented in the form (2.19)

(5λ + 3)(λ − 3) λ−2 2λ + 1 (λ2 − 4)(λ − 3) L(λ) = , (λ − 2)(λ − 3)

(2.22)

(2.23)

where N (λ) =

(5λ + 3)(λ − 3) λ−2 , 2λ + 1 (λ2 − 4)(λ − 3) (2.24)

d(λ) = (λ − 2)(λ − 3) . According to

−13 0 N (2) = , 5 0

01 N (3) = , 70

the fraction (2.23) is irreducible. Since the polynomial d(λ) in (2.24) is monic, the expression (2.23) estabishes as the standard form of the rational matrix L(λ). If the denominator d(λ) in (2.21) has the shape (2.20), then the numbers λ1 , . . . , λq are called the poles of the matrix L(λ), and the numbers µ1 , . . . , µq are their multiplicities.

2.3 McMillan Canonical Form 1. Let the rational n × m matrix L(λ) be given in the standard form (2.21). The matrix N (λ) is written in Smith canonical form (1.41):

diag{a1 (λ), . . . , aρ (λ)} Oρ,m−ρ N (λ) = p(λ) q(λ) . (2.25) On−ρ,ρ On−ρ,m−ρ Then we produce from (2.21) L(λ) = p(λ)ML (λ)q(λ) ,

where ML (λ) =

Mρ (λ) Oρ,m−ρ On−ρ,ρ On−ρ,m−ρ

(2.26) (2.27)

2.3 McMillan Canonical Form

and Mρ (λ) = diag

a1 (λ) aρ (λ) ,..., d(λ) d(λ)

61

.

Executing all possible cancellations in (2.28), we arrive at α1 (λ) αρ (λ) ,..., , Mρ (λ) = diag ψ1 (λ) ψρ (λ)

(2.28)

(2.29)

where the αi (λ), ψi (λ), (i = 1, . . . , ρ) are coprime monic polynomials, such that αi+1 (λ) is divisible by αi (λ), and ψi (λ) is divisible by ψi+1 (λ). Matrix (2.27), where Mρ (λ) is represented in the form (2.29), is designated as the McMillan (canonical) form of the rational matrix L(λ). The McMillan form of an arbitrary rational matrix is uniquely determined. 2.

The polynomial ψL (λ) = ψ1 (λ) · · · ψρ (λ)

(2.30)

is said to be the McMillan denominator of the matrix L(λ), and the polynomial (2.31) αL (λ) = α1 (λ) · · · αρ (λ) its McMillan numerator. The non-negative number

Mdeg L(λ) = deg ψL (λ)

(2.32)

is called the McMillan degree of the matrix L(λ), or shortly its degree. 3. Lemma 2.2. For a rational matrix L(λ) in standard form (2.21) the fraction ∆(λ) =

ψL (λ) d(λ)

(2.33)

turns out to be a polynomial. Proof. It is shown that under the actual assumptions the polynomials a1 (λ) and d(λ) are coprime, such that the fraction a1 (λ)/d(λ) is irreducible. Indeed, let us assume the contrary such that a1 (λ) b1 (λ) = , d(λ) ψ1 (λ) where deg ψ1 (λ) < deg d(λ). Since the polynomial ψ1 (λ) is divisible by the polynomials ψ2 (λ), . . . , ψρ (λ), we obtain from (2.29) b1 (λ) bρ (λ) ,..., Mρ (λ) = diag , ψ1 (λ) ψ1 (λ)

62


where b1 (λ), . . . , bρ (λ) are polynomials. Inserting this relation and (2.27) into (2.26), we arrive at the representation L(λ) =

N1 (λ) , ψ1 (λ)

where N1 (λ) is a polynomial matrix, and deg ψ1 (λ) < deg d(λ). But this inequality contradicts our assumption on the irreducibility of the standard form (2.21). This conflict proves the correctness of ψ1 (λ) = d(λ), and from (2.30) arises (2.33). From Lemma 2.2, for a denominator d(λ) of the form (2.20), we deduce the relation ψL (λ) = (λ − λ1 )ν1 · · · (λ − λq )νq = d(λ)ψ2 (λ) · · · ψρ (λ) ,

(2.34)

where νi ≥ µi , (i = 1, . . . , q). The number νi is called the McMillan multiplicity of the pole λi . From (2.34) and (2.32) arise Mdeg L(λ) = ν1 + . . . + νq . 4. Lemma 2.3. For any matrix L(λ), assuming (2.26), (2.27), we obtain deg d(λ) ≤ Mdeg L(λ) ≤ ρ deg d(λ) . Proof. The left side of the claimed inequality establishes itself as a consequence of Lemma 2.2. The right side is seen immediately from (2.28), because under the assumption that all fractions ai (λ)/d(λ) are irreducible, we obtain ψL (λ) = [d(λ)]ρ .

5. Lemma 2.4. Let L(λ) in (2.21) be an n×n matrix with rank N (λ) = n. Then det L(λ) = κ

αL (λ) , ψL (λ)

κ = const. = 0 .

(2.35)

Proof. For n = m and rank N (λ) = n, from (2.26)–(2.29) it follows α1 (λ) α2 (λ) αn (λ) , ,..., L(λ) = p(λ) diag q(λ) . d(λ) ψ2 (λ) ψn (λ) Calculating the determinant on the right side of this equation according to (2.30) and (2.31) yields Formula (2.35) with κ = det p(λ) det q(λ).

2.4 Matrix Fraction Description (MFD)

63

2.4 Matrix Fraction Description (MFD) 1. Let the rational n × m matrix L(λ) be given in the standard form (2.21). We suppose the existence of a non-singular n × n polynomial matrix al (λ) with al (λ)N (λ) al (λ)L(λ) = = bl (λ) d(λ) and an n × m polynomial matrix bl (λ). In this case, we call the polynomial matrix al (λ) a left reducing polynomial of the matrix L(λ). Considering the last equation, we gain the representation L(λ) = a−1 l (λ)bl (λ) ,

(2.36)

which is called an LMFD (left matrix fraction description) of the matrix L(λ). Analogously, if there exists a non-singular m × m matrix ar (λ) with L(λ)ar (λ) =

N (λ)ar (λ) = br (λ) d(λ)

and a polynomial n × m matrix br (λ), we call the representation L(λ) = br (λ)a−1 r (λ)

(2.37)

a right MFD (RMFD) of the matrix L(λ), [69, 68], and the matrix ar (λ) is named its right reducing polynomial. 2. The polynomials al (λ) and bl (λ) in the LMFD (2.36) are called left denominator and right numerator, and the polynomials ar (λ), br (λ) of the RMFD (2.37) its right denominator and left numerator, respectively. Obviously, the set of left reducing polynomials of the matrix L(λ) coincides with the set of its left denominators, and the same is true for the set of right reducing polynomials and the set of right denominators. Example 2.5. Let the matrices L(λ) =

2λ λ2 + λ − 2 2 λ − 7λ + 18 −λ2 + 7λ − 2 (λ − 2)(λ − 3)

and al (λ) =

λ−4 1 λ−6 λ

be given. Then by direct calculation, we obtain

3 λ+1 al (λ)L(λ) = = bl (λ) , λ 2 such that L(λ) = a−1 l (λ)bl (λ).

64


3. For any matrix L(λ) (2.21), there always exist LMFDs and RMFDs. Indeed, take al (λ) = d(λ)In , bl (λ) = N (λ) , then the rational matrix (2.21) can be written in form of an LMFD (2.36), where det al (λ) = [d(λ)]n , and therefore deg det al (λ) = ord al (λ) = n deg d(λ) . In the same way, we see that ar (λ) = d(λ)Im ,

br (λ) = N (λ)

is an RMFD (2.37) of Matrix (2.21), where ord ar (λ) = m deg d(λ). However, as will be shown in future examples, in most cases we are interested in LMFDs or RMFDs with lowest possible ord al (λ) or ord ar (λ). 4. In connection with the above demand, the problem arise to construct an LMFD or RMFD, where det al (λ) or det ar (λ) have the minimal possible degrees. Those MFDs are called irreducible. In what follows, we speak about irreducible left MFDs (ILMFDs) and irreducible right MFDs (IRMFDs). The following statements are well known [69]. Statement 2.1 An LMFD (2.36) is an ILMFD, if and only if the pair

(al (λ), bl (λ)) is irreducible, i.e. the matrix al (λ) bl (λ) is alatent. Statement 2.2 An RMFD (2.37) is an IRMFD, if and only if the pair ar (λ) [ar (λ), br (λ)] is irreducible, i.e. the matrix is alatent. br (λ) Statement 2.3 If the n × m matrix A(λ) possesses the two LMFDs −1 L(λ) = a−1 l1 (λ)bl1 (λ) = al2 (λ)bl2 (λ)

and the pair (al1 (λ), bl1 (λ)) is irreducible, then there exists a non-singular n × n polynomial matrix g(λ) with al2 (λ) = g(λ)al1 (λ) ,

bl2 (λ) = g(λ)bl1 (λ) .

Furthermore, if the pair (al2 (λ), bl2 (λ)) is also irreducible, then the matrix g(λ) is unimodular. Remark 2.6. A corresponding statement is true for right MFDs.


65

5. The theoretical equipment for constructing ILMFDs and IRMFDs is founded on using the canonical form of McMillan. Indeed, from (2.27) and (2.29), we get ˜−1 a−1 (2.38) ML (λ) = a r (λ) l (λ)b(λ) = b(λ)˜ with a ˜l (λ) = diag{d(λ), ψ2 (λ), . . . , ψρ (λ), 1, . . . , 1} , a ˜r (λ) = diag{d(λ), ψ2 (λ), . . . , ψρ (λ), 1, . . . , 1} , diag{α1 (λ), , . . . , αρ (λ)} Oρ,m−ρ b(λ) = . On−ρ,ρ On−ρ,m−ρ

(2.39)

Inserting (2.38) and (2.39) in (2.26), we obtain an LMFD (2.36) and an RMFD (2.37) with al (λ) = a ˜l (λ)p−1 (λ), bl (λ) = b(λ)q(λ) , ar (λ), br (λ) = p(λ)b(λ) . ar (λ) = q −1 (λ)˜

(2.40)

In [69] is stated that the pairs (al (λ), bl (λ)) and [ar (λ), br (λ)] are irreducible, i.e. by using (2.40), Relations (2.36) and (2.37) generate ILMFDs and IRMFDs of the matrix L(λ). 6. If Relations (2.36) and (2.37) define ILMFDs and IRMFDs of the matrix L(λ), then it follows from (2.40) and Statement 2.3 that the matrices al (λ) and ar (λ) possess equal invariant polynomials different from one. Herein, det al (λ) ≈ det ar (λ) ≈ d(λ)ψ2 (λ) · · · ψρ (λ) = ψL (λ) , where ψL (λ) is the McMillan denominator of the matrix L(λ). Besides, the last relation together with (2.32) yields ord al (λ) = ord ar (λ) = Mdeg L(λ) .

(2.41)

Moreover, we recognise from (2.40) that the matrices bl (λ) and br (λ) in the ILMFD (2.36) and the IRMFD (2.37) are equivalent. 7. Lemma 2.7. Let a ˜l (λ) (˜ ar (λ)) be a left (right) reducing polynomial for the ˜r (λ) = κ). Then matrix L(λ) with ord a ˜l (λ) = κ (ord a Mdeg L(λ) ≤ κ . Proof. Let us have the ILMFD (2.36). Then due to Statement 2.3, we have a ˜l (λ) = g(λ)al (λ) ,

(2.42)

where the matrix g(λ) is non-singular, from which directly follows the claim.

66


8. A number of auxiliary statements about general properties of MFDs should be given now. Lemma 2.8. Let an LMFD L(λ) = a−1 l1 (λ)bl1 (λ) be given. Then there exists an RMFD L(λ) = br1 (λ)a−1 r1 (λ) with det al1 (λ) ≈ det ar1 (λ). The reverse statement is also true. Proof. Let the ILMFD and IRMFD −1 L(λ) = a−1 l (λ)bl (λ) = br (λ)ar (λ)

be given. Then with (2.42), we have al1 (λ) = gl (λ)al (λ) , where the matrix gl (λ) is non-singular. Let det g(λ) = h(λ) and choose the m × m matrix gr (λ) with det gr (λ) ≈ h(λ). Then using ar1 (λ) = ar (λ)gr (λ),

br1 (λ) = br (λ)gr (λ) ,

we obtain an RMFD of the desired form.

9. Lemma 2.9. Let the PMD of the dimension n, p, m τ (λ) = (a(λ), b(λ), c(λ)) be given, where the pair (a(λ), b(λ)) is irreducible. Then, if we have an ILMFD c(λ)a−1 (λ) = a−1 1 (λ)c1 (λ) ,

(2.43)

the pair (a1 (λ), c1 (λ)b(λ)) becomes irreducible. On the other side, if the pair [a(λ), c(λ)] is irreducible, and we have an IRMFD (2.44) a−1 (λ)b(λ) = b1 (λ)a−1 2 (λ) , then the pair [a2 (λ), c(λ)b1 (λ)] becomes irreducible. Proof. Since the pair (a(λ), b(λ)) is irreducible, owing to (1.71), there exist polynomial matrices X(λ), Y (λ) with a(λ)X(λ) + b(λ)Y (λ) = Ip .


67

In analogy, the irreducibility of the pair (a1 (λ), c1 (λ)) implies the existence of polynomial matrices U (λ) and V (λ) with a1 (λ)U (λ) + c1 (λ)V (λ) = In .

(2.45)

Using the last two equations, we find a1 (λ)U (λ) + c1 (λ)V (λ) = a1 (λ)U (λ) + c1 (λ)Ip V (λ) = a1 (λ)U (λ) + c1 (λ) [a(λ)X(λ) + b(λ)Y (λ)] V (λ) = In which, due to (2.43), may be written in the form a1 (λ) [U (λ) + c(λ)X(λ)V (λ)] + c1 (λ)b(λ) [Y (λ)V (λ)] = In . From this equation by virtue of (1.71), it is evident that the pair (a1 (λ), c1 (λ)b(λ)) is irreducible. In the same manner, it can be shown that the pair [a2 (λ), c(λ)b1 (λ)] is irreducible. Remark 2.10. The reader finds in [69] an equivalent statement to Lemma 2.9 in modified form. 10. Lemma 2.11. Let the pair (a1 (λ)a2 (λ), b(λ)) be irreducible. Then also the pair (a1 (λ), b(λ)) is irreducible. Analogously, we have: If the pair [a1 (λ)a2 (λ), c(λ)] is irreducible, then the pair [a2 (λ), c(λ)] is also irreducible. Proof. Produce −1 −1 b(λ) . L(λ) = a−1 2 (λ)a1 (λ)b(λ) = [a1 (λ)a2 (λ)]

Due to our supposition, the right side of this equation is an ILMFD. Therefore, regarding (2.41), we get Mdeg L(λ) = ord[a1 (λ)a2 (λ)] = ord a1 (λ) + ord a2 (λ) .

(2.46)

Suppose the pair (a1 (λ), b(λ)) to be reducible. Then there would exist an ILMFD −1 a−1 3 (λ)b1 (λ) = a1 (λ)b(λ) , where ord a3 (λ) < ord a1 (λ), and we obtain −1 −1 b1 (λ) . L(λ) = a−1 2 (λ)a3 (λ)b1 (λ) = [a3 (λ)a2 (λ)]

From this equation it follows that a3 (λ)a2 (λ) is a left reducing polynomial for L(λ). Therefore, Lemma 2.7 implies Mdeg L(λ) ≤ ord[a3 (λ)a2 (λ)] < ord a1 (λ) + ord a2 (λ) . This relation contradicts (2.46), that’s why the pair (a1 (λ), b(λ)) has to be irreducible. The second part of the Lemma is shown analogously.

68


11. The subsequent Lemmata state further properties of the denominator and the McMillan degree. Lemma 2.12. Let a matrix of the form L(λ) = c(λ)a−1 (λ)b(λ)

(2.47)

be given with polynomial matrices a(λ), b(λ), c(λ), where the pairs (a(λ), b(λ)) and [a(λ), c(λ)] are irreducible. Then ψL (λ) ≈ det a(λ) is true, and thus Mdeg L(λ) = ord a(λ) .

(2.48)

c(λ)a−1 (λ) = a−1 1 (λ)c1 (λ) .

(2.49)

Proof. Build the ILMFD

Since by supposition the left side of (2.49) is an IRMFD, we have det a(λ) ≈ det a1 (λ) .

(2.50)

Using (2.49), we obtain from (2.47) L(λ) = a−1 1 (λ)[c1 (λ)b(λ)] . Due to Lemma 2.9, the right side of this equation is an ILMFD and because of (2.50), we get ψL (λ) ≈ det a1 (λ) ≈ det a(λ) . Relation (2.48) now follows directly from (2.32). Lemma 2.13. Let L(λ) = L1 (λ)L2 (λ)

(2.51)

be given with rational matrices L1 (λ), L2 (λ), L(λ), and ψL1 (λ), ψL2 (λ), ψL (λ) should be their accompanying McMillan denominators. Then the expression χ(λ) =

ψL1 (λ)ψL2 (λ) ψL (λ)

realises as a polynomial. Proof. Let the ILMFD and in addition be given. Then

L(λ) = a−1 (λ)b(λ)

Li (λ) = a−1 i (λ)bi (λ) ,

(i = 1, 2)

(2.52) (2.53)


ψL (λ) ≈ det a(λ),

ψLi (λ) ≈ det ai (λ),

(i = 1, 2) .

69

(2.54)

Equation (2.51) with (2.53) implies −1 L(λ) = a−1 1 (λ)b1 (λ)a2 (λ)b2 (λ) .

(2.55)

Owing to Lemma 2.8, there exists an LMFD −1 a−1 3 (λ)b3 (λ) = b1 (λ)a2 (λ) ,

(2.56)

where det a3 (λ) ≈ det a2 (λ) ≈ ψL2 (λ) . Using (2.55) and (2.56), we find −1 −1 L(λ) = a−1 1 (λ)a3 (λ)b3 (λ)b2 (λ) = a4 (λ)b4 (λ) ,

(2.57)

where a4 (λ) = a3 (λ)a1 (λ),

b4 (λ) = b3 (λ)b2 (λ) .

Per construction, we get det a4 (λ) ≈ ψL1 (λ)ψL2 (λ) .

(2.58)

Relations (2.52) and (2.57) define LMFDs of the matrix L(λ), where (2.52) is an ILMFD. Therefore, the relation a4 (λ) = g(λ)a(λ) holds with an n × n polynomial matrix g(λ). From the last equation arises that the object det a4 (λ) = det g(λ) det a(λ) is a polynomial. Finally, this equation together with (2.54) and (2.58) yields the claim of the Lemma. Remark 2.14. From Lemma 2.13 under supposition (2.51), we get Mdeg[L1 (λ)L2 (λ)] ≤ Mdeg L1 (λ) + Mdeg L2 (λ) . In the following investigations, we will call the matrices L1 (λ) and L2 (λ) independent, when the equality sign takes place in the last relation. Lemma 2.15. Let L(λ) ∈ Fnm (λ), G(λ) ∈ Fnm [λ] and L1 (λ) = L(λ) + G(λ) be given. Then, we have ψL1 (λ) = ψL (λ)

(2.59)

Mdeg L1 (λ) = Mdeg L(λ) .

(2.60)

and therefore,

70


Proof. Start with the ILMFD (2.52). Then the matrix

Rh (λ) = a(λ) b(λ)

(2.61)

becomes alatent. By using (2.52), we build the LMFD L1 (λ) = a−1 (λ) [b(λ) + a(λ)G(λ)]

(2.62)

for the matrix L1 (λ), to which the horizontal matrix

R1h (λ) = a(λ) b(λ) + a(λ)G(λ) is configured. The identity

In G(λ) R1h (λ) = Rh (λ) Omn Im

is easily proved. The first factor on the right side is the alatent matrix Rh (λ) and the second factor is a unimodular matrix. Therefore, the product is also alatent and consequently, (2.62) is an ILMFD, which implies Mdeg L1 (λ) = ord a(λ) = Mdeg L(λ) and Equation (2.60) follows. Lemma 2.16. For the matrix L(λ) ∈ Fnm (λ), let an ILMFD (2.52) be given, and the matrix L1 (λ) is determined by L1 (λ) = L(λ)D(λ) , where the non-singular matrix D(λ) ∈ Fmm [λ] should be free of eigenvalues that coincide with eigenvalues of the matrix a(λ) in (2.52). Then the relation L1 (λ) = a−1 (λ)[b(λ)D(λ)]

(2.63)

defines an ILMFD of the matrix L1 (λ), and Equations (2.59), (2.60) are fulfilled. Proof. Let an ILMFD (2.52) and the set λ1 , . . . , λq of eigenvalues of a(λ) be given. Since Matrix (2.61) is alatent, we gain

rank Rh (λi ) = rank a(λi ) b(λi ) = n, (i = 1, . . . , q) . Consider the LMFD (2.63) and the accompanying matrix

R1h (λ) = a(λ) b(λ)D(λ) . The latent numbers of the matrix R1h (λ) belong to the set of numbers λ1 , . . . , λq . But for any 1 ≤ i ≤ q, we have


71

R1h (λi ) = Rh (λi )F (λi ) , where the matrix F (λ) has the form

In Onm F (λ) = . Omn D(λ) Under the supposed conditions, rank F (λi ) = n + m is valid, that means, the matrix F (λi ) is non-singular, which implies rank R1h (λi ) = n,

(i = 1, . . . , q) .

Therefore, the matrix R1h (λ) satisfies Condition (1.72), and Lemma 1.42 guarantees that Relation (2.63) delivers an ILMFD of the matrix L1 (λ). From this fact we conclude the validity of (2.59), (2.60). 12. Lemma 2.17. Let the irreducible rational matrix L(λ) =

N (λ) d1 (λ)d2 (λ)

(2.64)

be given, where N (λ) is an n × m polynomial matrix, and d1 (λ), d2 (λ) are coprime scalar polynomials. Moreover, let the ILMFDs ˜ 1 (λ) = N (λ) = a−1 (λ)b1 (λ) , L 1 d1 (λ)

˜ 2 (λ) = b1 (λ) = a−1 (λ)b2 (λ) L 2 d2 (λ)

exist. Then the expression L(λ) = [a2 (λ)a1 (λ)]−1 b2 (λ) turns out to be an ILMFD of Matrix (2.64). Proof. The proof immediately follows from Formulae (2.25)–(2.29), because the polynomials d1 (λ) and d2 (λ) are coprime. 13. Lemma 2.18. Let irreducible representations of the form (2.21) Li (λ) =

Ni (λ) , di (λ)

(i = 1, 2)

(2.65)

with n × m polynomial matrices Ni (λ) be given, where the polynomials d1 (λ) and d2 (λ) are coprime. Then we have Mdeg[L1 (λ) + L2 (λ)] = Mdeg L1 (λ) + Mdeg L2 (λ) .

(2.66)

72


Proof. Proceed from the ILMFDs ˜ Li (λ) = a ˜−1 i (λ)bi (λ),

(i = 1, 2) .

(2.67)

Then under the actual assumptions, the matrices a ˜1 (λ) and a ˜2 (λ) have no common eigenvalues, and they satisfy Mdeg Li (λ) = ord a ˜i (λ),

(i = 1, 2) .

(2.68)

Using (2.65), we arrive at L(λ) = L1 (λ) + L2 (λ) =

N1 (λ)d2 (λ) + N2 (λ)d1 (λ) , d1 (λ)d2 (λ)

where the fraction on the right side is irreducible. Consider the matrix ˜ 1 (λ) = L(λ)d2 (λ) = N1 (λ) d2 (λ) + N2 (λ) . L d1 (λ) Applying (2.67), we obtain

˜b1 (λ)d2 (λ) + a ˜ 1 (λ) = a L ˜−1 (λ) ˜ (λ)N (λ) . 1 2 1

From Lemmata 2.15–2.17, it follows that the right side of the last equation is an ILMFD, because the polynomials d1 (λ) and d2 (λ) are coprime. Now introduce the notation N2 (λ) ˜ ˜ ˜ 2 (λ) = ˜b1 (λ) + a L = b1 (λ) + a ˜1 (λ) ˜1 (λ)˜ a−1 (2.69) 2 (λ)b2 (λ) d2 (λ) and investigate the ILMFD −1 a ˜1 (λ)˜ a−1 2 (λ) = a1 (λ)a2 (λ) .

(2.70)

The left side of this equation is an IRMFD, because the matrices a ˜1 (λ) and a ˜2 (λ) have no common eigenvalues. Therefore, ord a ˜2 (λ) = ord a1 (λ) ,

(2.71)

and from Lemmata 2.9 and 2.15 we gather that the right side of the equation ˜ 2 (λ) = a−1 (λ) a1 (λ)˜b1 (λ) + a2 (λ)˜b2 (λ) = a−1 (λ)b2 (λ) L 1 1 is an ILMFD. This relation together with (2.69) implies L(λ) = [a1 (λ)˜ a1 (λ)]

−1

b2 (λ) .

Hereby, Lemma 2.17 yields that the right side of the last equation is an ILMFD, from which by means of (2.68) and (2.71), we conclude (2.66). Corollary 2.19. If we write with the help of (2.70) −1 ˜b1 (λ) + a2 (λ)˜b2 (λ) , (λ)a (λ) a (λ) L(λ) = a ˜−1 1 1 1 then the right side is an ILMFD.

2.5 Double-sided MFD (DMFD)

73

2.5 Double-sided MFD (DMFD) 1. Assume in (2.64) d1 (λ) and d2 (λ) to be monic and coprime polynomials, i.e. N (λ) L(λ) = d1 (λ)d2 (λ) is valid. Then applying (2.26)–(2.29) yields L(λ) =

⎡ ⎢ diag p(λ) ⎣

α1 (λ) α2 (λ) αρ (λ) , ,..., d1 (λ)d2 (λ) ϕ2 (λ)ξ2 (λ) ϕρ (λ)ξρ (λ) On−ρ,ρ

⎤ Oρ,m−ρ ⎥ ⎦ q(λ) , On−ρ,m−ρ

where all fractions are irreducible, and all polynomials ϕ2 (λ), . . . , ϕρ (λ) are divisors of the polynomial d1 (λ), and all polynomials ξ2 (λ), . . . , ξρ (λ) are divisor of the polynomial d2 (λ). Furthermore, every ϕi (λ) is divisible by ϕi+1 (λ), and ξi (λ) by ξi+1 (λ). 2.

Consider now the polynomial matrices a ˜l (λ) = diag{d1 (λ), ϕ2 (λ), . . . , ϕρ (λ), 1, . . . , 1}p−1 (λ) , diag{α1 (λ), . . . , αρ (λ)} Oρ,m−ρ ˜b(λ) = , On−ρ,ρ On−ρ,m−ρ

(2.72)

a ˜r (λ) = q −1 (λ) diag{d2 (λ), ξ2 (λ), . . . , ξρ (λ), 1, . . . , 1} with the dimensions n × n, n × m, m × m, respectively. So we can write ˜ a−1 (λ) . L(λ) = a ˜−1 r l (λ)b(λ)˜

(2.73)

A representation of the form (2.73) is called double-sided or bilateral MFD (DMFD). 3. Lemma 2.20. The pairs (˜ al (λ), ˜b(λ)) and [˜ ar (λ), ˜b(λ)] defined by Relations (2.72) are irreducible. Proof. Build the LMFD and RMFD N (λ)˜ ar (λ) ˜ =a ˜−1 l (λ)b(λ), d1 (λ)d2 (λ)

a ˜l (λ)N (λ) = ˜b(λ)˜ a−1 r (λ) . d1 (λ)d2 (λ)

With the help of (2.72), we immediately recognise that the right sides are ar (λ), ˜b(λ)] are irreILMFD resp. IRMFD. Therefore, the pairs (˜ al (λ), ˜b(λ)), [˜ ducible.

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Suppose (2.72), then under the conditions of Lemma 2.20, it follows that ˜r (λ) take their in the representation (2.73), the quantities ord a ˜l (λ) and ord a minimal values. A representation like (2.73) is named irreducible DMFD (IDMFD). The set of all IDMFD of the matrix L(λ) according to given polynomials d1 (λ), d2 (λ) has the form −1 L(λ) = a−1 l (λ)b(λ)ar (λ)

with b(λ) = p(λ)˜b(λ)q(λ),

al (λ), al (λ) = p(λ)˜

ar (λ) = a ˜r (λ)q(λ) ,

where p(λ), q(λ) are unimodular matrices of appropriate type. Example 2.21. Consider the rational matrix

5λ2 − 6λ − 12 −2λ2 + 3λ + 4 −2λ2 − 2λ + 18 λ2 − 7 L(λ) = . 2 (λ + λ + 2)(λ − 3) Assume d1 (λ) = λ2 + λ + 2, d2 (λ) = λ − 3, then we can write −1 L(λ) = a−1 l (λ)b(λ)ar (λ)

with al (λ) =

λ+1 λ , 2 λ+2

ar (λ) =

λ−1 1 2λ 3

,

b(λ) =

λ−2 0 . 2 1

The obtained DMFD is irreducible because of det al (λ) = d1 (λ), det ar (λ) = d2 (λ), and the quantities ord al (λ) and ord ar (λ) take their minimal possible values.

2.6 Index of Rational Matrices 1. As in the scalar case, we understand by the index of a rational n × m matrix L(λ) that integer ind L for which the finite limit lim L(λ)λind L = L0 = Onm

λ→∞

(2.74)

exists. For ind L = 0, ind L > 0 and ind L ≥ 0 the matrix L(λ) is called proper, strictly proper and at least proper, respectively. For rational matrices of the form (2.21), we have ind L = deg d(λ) − deg N (λ) .

2.6 Index of Rational Matrices

75

2. In a number of cases we also can receive the value of ind L from the LMFD or RMFD. Lemma 2.22. Suppose the matrix L(λ) in the standard form L(λ) =

N (λ) d(λ)

(2.75)

and the relations −1 L(λ) = a−1 l (λ)bl (λ) = br (λ)ar (λ)

(2.76)

should define LMFD resp. RMFD of the matrix L(λ). Then ind L satisfies the inequalities ind L = deg d(λ) − deg N (λ) ≤ deg al (λ) − deg bl (λ) ≤ deg ar (λ) − deg br (λ) .

(2.77)

Proof. From (2.75) and (2.76) we arrive at d(λ)bl (λ) = al (λ)N (λ) , which results in deg[d(λ)bl (λ)] = deg[al (λ)N (λ)] .

(2.78)

According to d(λ)bl (λ) = [d(λ)In ]bl (λ) and due to the regularity of the matrix d(λ)In , we get through (1.12) deg[d(λ)bl (λ)] = deg d(λ) + deg bl (λ) .

(2.79)

Moreover, using (1.11) we realise deg[al (λ)N (λ)] ≤ deg al (λ) + deg N (λ) .

(2.80)

Comparing (2.78)–(2.80), we obtain deg d(λ) + deg bl (λ) ≤ deg al (λ) + deg N (λ) , which is equivalent to the first inequality in (2.77). The second inequality can be shown analogously. Corollary 2.23. If the matrix L(λ) is proper, i.e. ind L = 0, then for any MFD (2.76) from (2.77) it follows deg bl (λ) ≤ deg al (λ),

deg br (λ) ≤ deg ar (λ).

If the matrix L(λ) is even strictly proper, i.e. ind L < 0 is true, then we have deg bl (λ) < deg al (λ),

deg br (λ) < deg ar (λ).

76


3. A complete information about the index of L(λ) is received in that case, where in the LMFD (2.36) [RMFD (2.37)] the matrix al (λ) is row reduced [ar (λ) is column reduced]. Theorem 2.24. Consider the LMFD L(λ) = a−1 l (λ)bl (λ)

(2.81)

with al (λ), bl (λ) of the dimensions n × n, n × m, where al (λ) is row reduced. Let αi be the degree of the i-th row of al (λ), and βi the degree of the i-th row of bl (λ), and denote δi = αi − βi ,

(i = 1, . . . , n)

and δL = min [δi ] . 1≤i≤n

Then the index of the matrix L(λ) is determined by ind L = δL . Proof. Using (1.22), we can write al (λ) = diag{λα1 , . . . , λαn } A˜0 + A˜1 λ−1 + A˜2 λ−2 + . . . ,

(2.82)

(2.83)

where the A˜i , (i = 0, 1, . . .) are constant matrices with det A˜0 = 0. Extracting from the rows of bl (λ) the corresponding factors, we obtain ˜0 λ−δL + B ˜1 λ−δL −1 + B ˜2 λ−δL −2 + . . . , bl (λ) = diag{λα1 , . . . , λαn } B ˜0 = ˜i , (i = 0, 1, . . .) are constant matrices, and B Onm . Inserting where the B this and (2.83) into (2.81), we find −1 ˜0 + B ˜1 λ−1 + . . . . B L(λ)λδL = A˜0 + A˜1 λ−1 + . . . Now, due to det A˜0 = 0, it follows ˜ lim L(λ)λδL = A˜−1 0 B0 = Onm ,

λ→∞

(2.84)

and by (2.74) we recognise the statement (2.82) to be true. Corollary 2.25. ([69], [68]) If in the LMFD (2.81) the matrix al (λ) is row reduced, then the matrix L(λ) is proper, strictly proper or at least proper, if and only if we have δL = 0, δL > 0 or δL ≥ 0, respectively. In the same way the corresponding statement for right MFD can be seen.

2.7 Strictly Proper Rational Matrices

77

Theorem 2.26. Consider the RMFD L(λ) = br (λ)a−1 r (λ) with ar (λ), br (λ) of the dimensions m × m, n × m, where ar (λ) is column reduced. Let α ˜ i be the degree of the i-th column of ar (λ) and β˜i the degree of the i-th column of br (λ), and denote δ˜i = α ˜ i − β˜i ,

(i = 1, . . . , m)

and δ˜L = min [δ˜i ] . 1≤i≤m

Then the index of the matrix L(λ) is determined by ind L = δ˜L . Example 2.27. Consider the matrices 2

2λ + 1 λ + 2 al (λ) = , 1 λ+1

bl (λ) =

2 3λ2 + 1 . 5 7

In this case the matrix al (λ) is row reduced, where α1 = 2, α1 = 1 and β1 = 2, β1 = 0. Consequently, we get δ1 = 0, δ2 = 1, thus δL = 0. Therefore, the matrix a−1 l (λ)bl (λ) becomes proper. Hereby, we obtain

20 ˜0 = 0 3 , A˜0 = , B 01 00 and from (2.84) it follows

0 1.5 −1 ˜ ˜ B (λ)b (λ) = A = . lim a−1 l 0 0 l 0 0 λ→∞

2.7 Strictly Proper Rational Matrices 1. According to the above definitions, Matrix (2.21) is strictly proper if ind L = deg d(λ) − deg N (λ) > 0. Strictly proper rational matrices possess many properties that are analogue to the properties of scalar strictly proper rational fractions, which have been considered in Section 2.1. In particular, the sum, the difference and the product of strictly proper rational matrices are strictly proper too.

78


2. For any strictly proper rational n × m matrix L(λ), there exists an indefinite set of elementary PMDs τ (λ) = (λIp − A, B, C)

(2.85)

i.e. realisations (A, B, C), such that L(λ) = C(λIp − A)−1 B .

(2.86)

The right side of (2.86) is called a standard representation of the matrix, or simply its representation. The number p, configured in (2.86), is called its dimension. A representation, where the dimension p takes its minimal possible value, is called minimal. A standard representation (2.86) is minimal, if and only if its elementary PMD is minimal, that means, if the pair (A, B) is controllable and the pair [A, C] is observable. The matrix L(λ) (2.86) is called the transfer function (transfer matrix) of the elementary PMD (2.85), resp. of the realisation (A, B, C). The elementary PMD (2.85) and the PMD τ1 (λ) = (λIq − A1 , B1 , C1 )

(2.87)

are called equivalent, if their transfer matrices coincide. 3. Now a number of statements on the properties of strictly proper rational matrices is formulated, which will be used later. Statement 2.4 (see [69, 68]) The minimal PMD (2.85) and (2.87) are equivalent, if and only if p = q. In this case, there exists a non-singular p × p matrix R with A1 = RAR−1 ,

B1 = RB,

C1 = CR−1 ,

(2.88)

i.e., the corresponding realisations are similar. Statement 2.5 Let the representation (2.86) be minimal and possess the ILMFD (2.89) C(λIp − A)−1 = a−1 l (λ)bl (λ) . Then, as follows from Lemma 2.9, the pair (al (λ), bl (λ)B) is irreducible. In analogy, if we have an IRMFD (λIp − A)−1 B = br (λ)a−1 r (λ) ,

(2.90)

then the pair [ar (λ), Cbr (λ)] is irreducible. Statement 2.6 If the representation (2.86) is minimal, then the matrices al (λ) in the ILMFD (2.89) and ar (λ) in the IRMFD (2.90) possess the same invariant polynomials different from 1 as the matrix λIp − A. Hereby, we have (2.91) ψL (λ) = det(λIp − A) ≈ det al (λ) ≈ det ar (λ) .


79

Particularly, it follows from (2.91) that Mdeg L(λ) of a strictly proper rational matrix L(λ) is equal to the dimension of its minimal standard representation. 4. Lemma 2.28. Assume n = m in the standard representation (2.86) and det L(λ) ≡ / 0. Then p ≥ n holds, and det L(λ) =

k(λ) det(λIp − A)

(2.92)

is valid, where k(λ) is a scalar polynomial with deg k(λ) ≤ p − n ind L .

(2.93)

The case p < n results in det L(λ) ≡ 0. Proof. In accordance with Lemma 2.8, there exists an LMFD C(λIp − A)−1 = a−1 1 (λ)b1 (λ) , where det a1 (λ) ≈ det(λIp − A), that’s why L(λ) = a−1 1 (λ)[b1 (λ)B] . Calculating the determinants of both sides yields (2.92) with k(λ) = det[b1 (λ)B]. To prove (2.93), we write L(λ) in the form (2.21) obtaining ind L = deg d(λ) − deg N (λ) > 0. Now calculating the determinant of the right side of (2.21), we gain det N (λ) det L(λ) = . [d(λ)]n Let deg d(λ) = q. Then deg N (λ) = q − ind L holds, where deg det N (λ) ≤ n(q − ind L) and deg[d(λ)]n = nq. From this we directly generate (2.93). For p < n, on account of the Binet-Cauchy formula, we come to det[C adj(λIp − A)B] ≡ 0. Corollary 2.29. Consider the strictly proper n × n matrix L(λ) and its McMillan denominator and numerator ψL (λ) and αL (λ), respectively. Then the following relation is true: deg αL (λ) ≤ deg ψL (λ) − n ind L . Proof. Let (2.86) be a minimal standard representation of the matrix L(λ). Then due to Lemma 2.4, we have det L(λ) = k

αL (λ) , ψL (λ)

k = const.

and the claim immediately results from (2.93).

80

5.


Let the strictly proper rational matrix L(λ) of the form (2.21) with L(λ) =

N (λ) d1 (λ)d2 (λ)

be given, where the polynomials d1 (λ) and d2 (λ) are coprime. Then there exists a separation N1 (λ) N2 (λ) + , (2.94) L(λ) = d1 (λ) d2 (λ) where N1 (λ) and N2 (λ) are polynomial matrices and both fractions in (2.94) are strictly proper. The matrices N1 (λ) and N2 (λ) in (2.94) are uniquely determined. In practice, the separation (2.94) can be produced by performing the separation (2.6) for every element of the matrix L(λ). Example 2.30. Let

λ+2 λ λ + 3 λ2 + 1 L(λ) = (λ − 2)2 (λ − 1)

be given. By choosing d1 (λ) = (λ − 2)2 , d2 (λ) = λ − 1, a separation (2.94) is found with

−3λ + 10 −λ + 4 31 , N2 (λ) = . N1 (λ) = −4λ + 13 −λ + 7 42 6. The separation (2.94) is extendable to a more general case. Let the strictly proper rational matrix have the form L(λ) =

N (λ) , d1 (λ)d2 (λ) · · · dκ (λ)

where all polynomials in the denominator are two-by-two coprime. Then there exists a unique representation of the form L(λ) =

N1 (λ) Nκ (λ) + ... + , d1 (λ) dκ (λ)

(2.95)

where all fractions on the right side are strictly proper. Particularly consider (2.21), and the polynomial d(λ) should have the form (2.20). Then under the assumption µ di (λ) = (λ − λi ) i , (i = 1, . . . , q) from (2.95), we obtain the unique representation L(λ) =

q i=1

where deg Ni (λ) < µi .

Ni (λ) , (λ − λi )µi

(2.96)


7.

81

By further transformations the fraction Li (λ) =

Ni (λ) (λ − λi )µi

could be written as Li (λ) =

Ni1 Ni2 Niµi + + ... + , (λ − λi )µi (λ − λi )µi −1 λ − λi

(2.97)

where the Nik , (k = 1, . . . , µi ) are constant matrices. Inserting (2.97) into (2.96), we arrive at the representation L(λ) =

q i=1

Ni1 Ni2 Niµi + + ... + (λ − λi )µi (λ − λi )µi −1 λ − λi

,

(2.98)

which is called partial fraction expansion of the matrix L(λ). 8. For calculating the matrices Nik in (2.97), we rely upon the analogous formula to (2.10) k−1

∂ N (λ)(λ − λi )µi 1 Nik = . (2.99) (k − 1)! ∂λk−1 d(λ) |λ=λi In practice, the coefficients in (2.99) will be determined by partial fraction expansion of the scalar entries of L(λ). Example 2.31. Assuming the conditions of Example 2.30, we get L(λ) =

where N11

42 = , 55

N21 N11 N12 + , + 2 (λ − 2) λ−2 λ−1

N12

−3 −1 = , −4 −1

N21

31 = 42

.

9. The partial fraction expansion (2.98) can be used in some cases for solving the question on reducibility of certain rational matrices. Indeed, it is easily shown that for the irreducibility of the strictly proper matrix (2.21), it is necessary and sufficient that in the expansion (2.98) Ni1 = Onm , must be true.

(i = 1, . . . , q)

(2.100)

82


2.8 Separation of Rational Matrices 1. Let the n × m matrix L(λ) in (2.21) be not strictly proper, that means ind L ≤ 0. Then for every element of the matrix L(λ), the representation (2.5) can be generated, yielding L(λ) =

R(λ) + G(λ) = L0 (λ) + G(λ) , d(λ)

(2.101)

where the fraction in the middle part is strictly proper, and G(λ) is a polynomial matrix. The representation (2.101) is unique. Practically, the dissection (2.101) is done in such a way that the dissection (2.5) is applied on each element of L(λ). Furthermore, the strictly proper matrix L0 (λ) on the right side of (2.101) is called the broken part of the matrix L(λ), and the matrix G(λ) its polynomial part . Example 2.32. For Matrix (2.22), we obtain

13(λ − 3) λ − 2

2λ + 1 0 5 0 , G(λ) = . L0 (λ) = 0 λ+2 (λ − 2)(λ − 3)

2.

Let us have in (2.101) L0 (λ) =

N0 (λ) , d1 (λ)d2 (λ)

(2.102)

where the polynomials d1 (λ) and d2 (λ) are coprime, and deg N0 (λ) < deg d1 (λ) + deg d2 (λ). Then as was shown above, there exists the unique separation N1 (λ) N2 (λ) N0 (λ) = + , d1 (λ)d2 (λ) d1 (λ) d2 (λ) where the fractions on the right side are strictly proper. Inserting this separation into (2.101), we find a unique representation of the form L(λ) =

N1 (λ) N2 (λ) + + G(λ) . d1 (λ) d2 (λ)

(2.103)

Example 2.33. For Matrix (2.22), we generate the separation (2.103) of the shape

13 0 01

−5 0 70 5 0 A(λ) = + + . 0 λ+2 λ−2 λ−3

2.8 Separation of Rational Matrices

3.

83

From (2.103) we learn that Matrix (2.101) can be presented in the form L(λ) =

Q1 (λ) Q2 (λ) + , d1 (λ) d2 (λ)

(2.104)

where Q2 (λ) = N2 (λ) + d2 (λ) [G(λ) − F (λ)] , (2.105) where the polynomial matrix F (λ) is arbitrary. The representation of the rational matrix L(λ) from (2.102) in the form (2.104), (2.105) is called its separation with respect to the polynomials d1 (λ) and d2 (λ). It is seen from (2.105) that for coprime polynomials d1 (λ) and d2 (λ), the separation (2.104) is always possible, but not uniquely determined. Nevertheless, the following theorem holds. Q1 (λ) = N1 (λ) + d1 (λ)F (λ) ,

Theorem 2.34. The totality of pairs Q1 (λ), Q2 (λ) satisfying the separation (2.104), is given by Formula (2.105). Proof. By P we denote the set of all polynomial pairs Q1 (λ), Q2 (λ) satisfying Relation (2.104), and by Ps the set of all polynomial pairs produced by (2.105) when we insert there any polynomial matrices F (λ). Since for any Pair (2.105), Relation (2.104) holds, Ps ⊂ P is true. On the other side, let the matrices ˜ 1 (λ), Q ˜ 2 (λ) fulfill Relation (2.104). Then we obtain Q ˜ i (λ) Q Ri (λ) = + Gi (λ) , di (λ) di (λ)

(i = 1, 2) ,

where the fractions on the right sides are strictly proper, and G1 (λ), G2 (λ) are polynomial matrices. Therefore, L(λ) =

R1 (λ) R2 (λ) + + G1 (λ) + G2 (λ) . d1 (λ) d2 (λ)

Comparing this with (2.103), then due to the uniqueness of the expansion (2.103), we get R1 (λ) = N1 (λ),

R2 (λ) = N2 (λ),

G(λ) = G1 (λ) + G2 (λ) .

˜ 1 (λ) and Q ˜ 2 (λ) Denoting G1 (λ) = F (λ), G2 (λ) = G(λ) − F (λ) we find Q satisfying Relation (2.105), i.e. P ⊂ Ps is true. Consequently, the sets P and Ps contain each other. Example 2.35. According to (2.104) and (2.105), we find for Matrix (2.22) the set of all separations with respect to the polynomials d1 (λ) = λ − 2, d2 (λ) = λ − 3. Using the results of Example 2.33, we obtain

84


Q1 (λ) = Q2 (λ) =

13 + (λ − 2)f11 (λ) (λ − 2)f12 (λ) 5 + (λ − 2)f21 (λ) (λ − 2)f22 (λ)

(λ − 3)[5 − f11 (λ)]

,

1 − (λ − 3)f12 (λ)

7 + (λ − 3)f21 (λ) (λ − 3)[(λ + 2) − f22 (λ)]

,

where the fik (λ), (i, k = 1, 2) are some polynomials.

4. Setting in (2.105) F (λ) = Onm , we arrive at the special solution of the form (2.106) Q1 (λ) = N1 (λ), Q2 (λ) = N2 (λ) + d2 (λ)G(λ) . Otherwise, taking F (λ) = G(λ) results in Q1 (λ) = N1 (λ) + d1 (λ)G(λ),

Q2 (λ) = N2 (λ) .

(2.107)

For the solution (2.106), the first summand in the separation (2.104) becomes a strictly proper rational matrix, and for the solution (2.107) the second one does. The particular separations defined by Formulae (2.106) and (2.107) are called minimal with respect to d1 (λ) resp. d2 (λ). Due to their construction, the minimal separations are uniquely determined. Example 2.36. The separation of Matrix (2.22), which is minimal with respect to d1 (λ) = λ − 2, is given by the matrices

13 0 5(λ − 3) 1 , Q2 (λ) = . Q1 (λ) = −5 0 7 (λ − 3)(λ + 2) With respect to d2 (λ) = λ − 3 the separation by the matrices

5λ + 3 0 01 , Q2 (λ) = Q1 (λ) = −5 λ2 − 4 70 is minimal. These minimal separations are unique per construction.

5. If in particular the original rational matrix (2.101) is strictly proper, then G(λ) = Onm becomes true, and the minimal separations (2.106) and (2.107) coincide. Example 2.37. For the strictly proper matrix in Example 2.30, we obtain a unique minimal separation with Q1 (λ) = N1 (λ), Q2 (λ) = N2 (λ), where the matrices N1 (λ) and N2 (λ) were already determined in Example 2.30.

2.9 Inverses of Square Polynomial Matrices

85

2.9 Inverses of Square Polynomial Matrices 1. Assume the n×n polynomial matrix L(λ) to be non-singular, and adj L(λ) be its adjoint matrix, that is determined by Equation (1.8). Then the matrix L−1 (λ) =

adj L(λ) det L(λ)

(2.108)

is said to be the inverse of the matrix L(λ). Equation (1.9) implies L(λ)L−1 (λ) = L−1 (λ)L(λ) = In .

(2.109)

2. The matrix L(λ) could be written with the help of (1.40), (1.49) in the form ⎡ ⎤ h1 (λ) 0 ... 0 ⎢ 0 h1 (λ)h2 (λ) . . . ⎥ 0 ⎢ ⎥ −1 L(λ) = p−1 (λ) ⎢ . ⎥ q (λ) , .. .. .. ⎣ .. ⎦ . . . 0 0 . . . h1 (λ)h2 (λ) · · · hn (λ) where p(λ) and q(λ) are unimodular matrices. How the inverse matrix L−1 (λ) can be calculated? For that purpose, the general Formula (2.108) is used. Denoting H(λ) = diag{h1 (λ), h1 (λ)h2 (λ), . . . , h1 (λ)h2 (λ) · · · hn (λ)} we can write

L−1 (λ) = q(λ)H −1 (λ)p(λ) .

Now, we have to calculate the matrix H polynomial of H(λ) amounts to

−1

(2.110)

(λ). Obviously, the characteristic

dH (λ) = det H(λ) = hn1 (λ)hn−1 (λ) · · · hn (λ) ≈ det L(λ) = dL (λ) . 2

(2.111)

Direct calculating the matrix of adjuncts adj H(λ) results in adj H(λ) = diag hn−1 (λ)hn−1 (λ) · · · hn (λ), hn−1 (λ)hn−2 (λ) · · · hn (λ), . . . 1 2 1 2 (2.112) n−2 . . . , hn−1 (λ)h (λ) · · · h (λ) , n−1 1 2 from which we gain H −1 (λ) =

adj H(λ) . dH (λ)

(2.113)

Herein, the numerator and denominator are constrained by Relations (2.112), (2.111). In general, the rational matrix on the right side of (2.113) is reducible, and that’s why we will write

86


H −1 (λ) =

adj H(λ) dL min (λ)

(2.114)

with adj H(λ) = diag {h2 (λ) · · · hn (λ), h3 (λ) · · · hn (λ), . . . , hn (λ), 1} dL min (λ) = h1 (λ)h2 (λ) · · · hn (λ) = an (λ) ,

(2.115) (2.116)

where an (λ) is the last invariant polynomial. Altogether, we receive by using (2.110) adj L(λ) , (2.117) L−1 (λ) = dL min (λ) where

adj L(λ) = q(λ)adj H(λ) · p(λ) .

(2.118)

Matrix (2.118) is called the monic adjoint matrix, and the polynomial dL min (λ) the minimal polynomial of the matrix L(λ). The rational matrix on the right side of (2.117) will be named monic inverse of the polynomial matrix L(λ). 3. Opposing (2.111) to (2.116) makes clear that among the roots of the minimal polynomial dLmin (λ) are all eigenvalues of the matrix L(λ), however, possibly with lower multiplicity. It is remarkable that the fraction (2.117) is irreducible. The reason for that lies in the fact that the matrix adj H(λ) for no value of λ becomes zero. The same can be said about Matrix (2.118), because the matrices q(λ) and p(λ) are unimodular. 4. Comparing (2.108) with (2.117), we find out that the fraction (2.108) is irreducible, if and only if h1 (λ) = h2 (λ) = . . . = hn−1 (λ) = 1,

hn (λ) = an (λ) = dLmin (λ) ≈ det L(λ)

holds, i.e. if the characteristic polynomial of the matrix L(λ) is equivalent to its minimal polynomial. If the last conditions are fulfilled, then the matrix L(λ) can be presented in the form L(λ) = p−1 (λ) diag{1, . . . , 1, an (λ)}q −1 (λ) that means, it is simple in the sense of Section 1.11, and the following theorem has been proved. Theorem 2.38. The inverse matrix (2.108) is irreducible, if and only if the matrix L(λ) is simple. 5.

From (2.117) we take the important equation adj L(λ)L(λ) = L(λ)adj L(λ) = dL min (λ)In .

(2.119)

2.10 Transfer Matrices of Polynomial Pairs

87

2.10 Transfer Matrices of Polynomial Pairs 1. The pairs (al (λ), bl (λ)), [ar (λ), br (λ)] are called non-singular, if / 0 resp. det ar (λ) ≡/ 0. For a non-singular pair (al (λ), bl (λ)), the det al (λ) ≡ rational matrix (2.120) wl (λ) = a−1 l (λ)bl (λ) can be explained, and for the non-singular pair [ar (λ), br (λ)], we build the rational matrix (2.121) wr (λ) = br (λ)a−1 r (λ) . Matrix (2.120) or (2.121) is called the transfer matrix (transfer function) of the corresponding pair. Applying the general Formula (2.108), we obtain wl (λ) =

adj al (λ) bl (λ) , dal (λ)

wr (λ) =

br (λ) adj ar (λ) dar (λ)

(2.122)

with the notation dal (λ) = det al (λ), dar (λ) = det ar (λ). 2. Definition 2.39. The transfer matrices wl (λ) and wr (λ) are called irreducible, if the rational matrices on the right side of (2.122) are irreducible. Now, we collect some facts on the reducibility of transfer matrices. Lemma 2.40. If the matrices al (λ), ar (λ) are not simple, then the transfer matrices (2.120), (2.121) are reducible. Proof. If the matrices al (λ), ar (λ) are not simple, then owing to (2.116), we −1 conclude that the matrices a−1 l (λ), ar (λ) are reducible, and therefore, also the fractions wl (λ) =

adj al (λ) bl (λ) , dal min (λ)

wr (λ) =

ar (λ) br (λ)adj . dar min (λ)

(2.123)

But this means, fractions (2.120), (2.121) are reducible. The matrices (2.123) are said to be the monic transfer matrices. 3. Lemma 2.41. If the pairs (al (λ), bl (λ)), [ar (λ), br (λ)] are reducible, i.e. the matrices

a (λ) (2.124) Rh (λ) = al (λ) bl (λ) , Rv (λ) = r br (λ) are latent, then the fractions (2.120), (2.121) are reducible.

88


Proof. If the pair (al (λ), bl (λ)) is reducible, then by virtue of the results in Section 1.12, we obtain al (λ) = g(λ)al1 (λ),

bl (λ) = g(λ)bl1 (λ)

(2.125)

with ord g(λ) > 0 and polynomial matrices al1 (λ), bl1 (λ), where due to det al (λ) = det g(λ) det al1 (λ) , the relation deg det al1 (λ) < deg det al (λ) holds. From (2.125), we gain wl (λ) = a−1 l1 (λ)bl1 (λ) =

adj al1 (λ) bl1 (λ) . det al1 (λ)

The denominator of this rational matrix possesses a lower degree than that of (2.122), what implies that the fraction wl (λ) in (2.122) is reducible. For the vertical pair [ar (λ), br (λ)], we carry out the proof in the same way. 4. Let the matrices al (λ) and ar (λ) be not simple. Then using (2.117), we receive the monic transfer matrix (2.123). Theorem 2.42. If the pairs (al (λ), bl (λ)), [ar (λ), br (λ)] are irreducible, then the monic transfer matrices (2.123) are irreducible. Proof. Let the pair (al (λ), bl (λ)) be irreducible. Then the matrix Rh (λ) in ˜ yields (2.124) is alatent. Therefore, an arbitrary fixed λ = λ

˜ = rank al (λ) ˜ bl (λ) ˜ = n. (2.126) rank Rh (λ) Multiplying the matrix Rh (λ) from left by the monic adjoint matrix adj al (λ), with benefit from (2.119), we find (2.127) adj al (λ)Rh (λ) = dal min (λ)In adj al (λ)bl (λ) . Now, let λ = λ0 be any root of the polynomial dal min (λ), then due to dal min (λ0 ) = 0 in (2.127) adj al (λ0 )Rh (λ0 ) = Onn adj al (λ0 )bl (λ0 ) is preserved. If we assume that the matrix wl (λ) in (2.123) is reducible, then ˜ 0 , we obtain for a certain root λ ˜ 0 )bl (λ ˜ 0 ) = Onm adj al (λ and therefore

2.10 Transfer Matrices of Polynomial Pairs

˜ 0 )Rh (λ ˜ 0 ) = On,n+m . adj al (λ

89

(2.128)

˜ 0 ) ] ≥ 1. Moreover, But from Relations (2.115), (2.118), we know rank [ adj al (λ ˜ from (2.126) we get rank Rh (λ0 ) = n, and owing to the Sylvester inequality (1.44), we conclude ˜ 0 )Rh (λ ˜0) ≥ 1 . rank adj al (λ Consequently, Equation (2.128) cannot be fulfilled and therefore, the fraction wl (λ) in (2.123) is irreducible. The proof for the irreducibility of the matrix wr (λ) in (2.123) runs analogously. Remark 2.43. The reverse statement of the just proven Theorem 2.42 is in general not true, as the next example illustrates. Example 2.44. Consider the pair (al (λ), bl (λ)) with

λ0 1 al (λ) = , bl (λ) = . 0λ 1 In this case, we have

a−1 l (λ) =

which means

So we arrive at

10 adj al (λ) = , 01

(2.129)

10 01 , λ dal min (λ) = λ .

1 1 adj al (λ)bl (λ) wl (λ) = = dal min (λ) λ

(2.130)

and the fraction on the right side is irreducible. Nevertheless, the pair (2.129) is not irreducible, because the matrix

λ01 Rh (λ) = 0λ1 for λ = 0 has only rank 1. On the other side, we immediately recognise that the pair

λ 0 1 al1 (λ) = , bl1 (λ) = −1 1 0 is an ILMFD of the transfer matrix (2.130), because the matrix

λ 01 Rh1 (λ) = −1 1 0 possesses rank 2 for all λ.

90


5. Theorem 2.45. For the transfer matrices (2.122) to be irreducible, it is necessary and sufficient that the pairs (al (λ), bl (λ)), [ar (λ), br (λ)] are irreducible and the matrices al (λ), ar (λ) are simple. Proof. The necessity follows from the above considerations. That the condition is also sufficient, we see by assuming al (λ) to be simple. Then dal min (λ) ≈ dal (λ), adj al (λ) ≈ adj al (λ) hold, and Theorem 2.42 yields that the fraction adj al (λ) bl (λ) wl (λ) = dal (λ) is irreducible. For the second fraction in (2.122), the statement is proven analogously.

2.11 Transfer Matrices of PMDs 1.

A PMD of the dimension n, p, m τ (λ) = (a(λ), b(λ), c(λ))

(2.131)

is called regular, if the matrix a(λ) is non-singular. All descriptor systems of interest belong to the set of regular PMDs. A regular PMD (2.131) is related to a rational transfer matrix wτ (λ) = c(λ)a−1 (λ)b(λ)

(2.132)

that is named the transfer function (-matrix) of the PMD (2.131). Using (2.108), the transfer matrix can be presented in the form wτ (λ) =

c(λ) adj a(λ) b(λ) . det a(λ)

(2.133)

When in the general case a(λ) is not simple, then by virtue of (2.117), we obtain c(λ)adj a(λ) b(λ) wτ (λ) = . (2.134) da min (λ) The rational matrix on the right side of (2.134) is called the monic transfer matrix of the PMD (2.131).

2.11 Transfer Matrices of PMDs

91

2. Theorem 2.46. For a minimal PMD (2.131), the monic transfer matrix (2.134) is irreducible. Proof. Construct the ILMFD c(λ)a−1 (λ) = a−1 1 (λ)c1 (λ) .

(2.135)

Since the left side of (2.135) is an IRMFD, det a(λ) ≈ det a1 (λ) .

(2.136)

Furthermore, also the minimal polynomials of the matrices a(λ) and a1 (λ) coincide, because they possess the same sequences of invariant polynomials different from one. Therefore, we have da min (λ) = da1 min (λ) .

(2.137)

Utilising (2.132) and (2.135), we can write wτ (λ) = a−1 1 (λ)[c1 (λ)b(λ)] .

(2.138)

The right side of (2.138) is an ILMFD, what follows from the minimality of the PMD (2.131) and Lemma 2.9. But then, employing Lemma 2.8 yields the fraction adj a1 (λ) c1 (λ)b(λ) wτ (λ) = da1 min (λ) to be irreducible, and this implies, owing to (2.137), the irreducibility of the right side of (2.134).

3. Theorem 2.47. For the right side of Relation (2.133) to be irreducible, it is necessary and sufficient that the PMD (2.131) is minimal and the matrix a(λ) is simple. Proof. The necessity results from Lemmata 2.40 and 2.41. Sufficiency: If the matrix a(λ) is simple, then det a(λ) ≈ da min (λ) , and the irreducibility of the right side of (2.133) follows from Theorem 2.46.

92


4. Let in addition to the PMD (2.131) be given a regular PMD of dimension n, q, m τ˜(λ) = (˜ a(λ), ˜b(λ), c˜(λ)) . (2.139) The PMD (2.131) and (2.139) are called equivalent, if their transfer functions coincide, that means c(λ)a−1 (λ)b(λ) = c˜(λ)˜ a−1 (λ)˜b(λ) .

(2.140)

Lemma 2.48. Assume the PMDs (2.131) and (2.139) be equivalent and the PMD (2.131) be minimal. Then the expression ∆(λ) =

det a ˜(λ) det a(λ)

(2.141)

turns out to be a polynomial. Proof. Lemma 2.8 implies the existence of the LMFD c˜(λ)˜ a−1 (λ) = a−1 2 (λ)c2 (λ) , where det a ˜(λ) ≈ det a2 (λ) .

(2.142)

Utilising (2.140), from this we gain the LMFD of the matrix wτ (λ) ˜ wτ (λ) = a−1 2 (λ)[c2 (λ)b(λ)] .

(2.143)

On the other side, the minimality of the PMD (2.131) allows to conclude that the right side of (2.138) is an ILMFD of the matrix wτ (λ). Comparing (2.138) with (2.143), we obtain a2 (λ) = g(λ)a(λ), where g(λ) is a polynomial matrix. Therefore, the expression ∆1 (λ) =

det a2 (λ) = det g(λ) det a1 (λ)

proves to be a polynomial. Taking into account (2.136) and (2.142), we realise that the right side of Equation (2.141) becomes a polynomial. Hereby, ∆(λ) ≈ ∆1 (λ) holds. Corollary 2.49. If the PMDs (2.131) and (2.139) are equivalent and minimal, then det a(λ) ≈ det a ˜(λ) . Proof. Lemma 2.48 offers under the given suppositions that det a(λ) , det a ˜(λ) are polynomials, this proves the claim.

det a ˜(λ) det a(λ)

2.11 Transfer Matrices of PMDs

93

5. Lemma 2.50. Consider a regular PMD (2.131) and its corresponding transfer matrix (2.132). Moreover, let the ILMFD and IRMFD −1 wτ (λ) = p−1 l (λ)ql (λ) = qr (λ)pr (λ)

(2.144)

exist. Then the expressions ∆l (λ) =

det a(λ) , det pl (λ)

∆r (λ) =

det a(λ) det pr (λ)

(2.145)

turn out to be polynomials. Besides, the sets of poles of each of the matrices w1 (λ) = pl (λ)c(λ)a−1 (λ),

w2 (λ) = a−1 (λ)b(λ)pr (λ)

are contained in the set of roots of the polynomial ∆l (λ) ≈ ∆r (λ). Proof. Consider the PMDs τ1 (λ) = (pl (λ), ql (λ), In ) ,

(2.146)

τ2 (λ) = (pr (λ), Im , qr (λ)) . Per construction, the PMDs (2.131) and (2.146) are equivalent, where the PMD (2.146) is minimal. Therefore, due to Lemma 2.48, the functions (2.145) are polynomials. Now we build the LMFD c(λ)a−1 (λ) = a−1 3 (λ)c3 (λ) ,

(2.147)

det a3 (λ) ≈ det a(λ) .

(2.148)

where As above, we have an LMFD of the transfer matrix wτ (λ) wτ (λ) = a−1 3 (λ)[c3 (λ)b(λ)] .

(2.149)

This relation together with (2.144) determines two LMFDs of the transfer matrix wτ (λ), where (2.144) is an ILMFD. Therefore, a3 (λ) = gl (λ)pl (λ)

(2.150)

holds with a non-singular n × n polynomial matrix gl (λ). Inversion of both sides of the last equation leads to −1 −1 a−1 3 (λ) = pl (λ)gl (λ) .

(2.151)

Moreover, from (2.150) through (2.148), we receive det gl (λ) =

det a(λ) det a3 (λ) ≈ = ∆l (λ) . det pl (λ) det pl (λ)

(2.152)

From (2.147) and (2.151), we earn pl (λ)c(λ)a−1 (λ) = gl−1 (λ)c3 (λ) =

adj gl (λ) c3 (λ) , det gl (λ)

and with the aid of (2.152), this yields the proof for a left MFD. The relation for a right MFD is proven analogously.

94


2.12 Subordination of Rational Matrices 1.

Let us have the rational n × m matrix w(λ) and the ILMFD w(λ) = p−1 l (λ)ql (λ) .

(2.153)

Furthermore, let the rational n × s matrix w1 (λ) be given. Definition 2.51. The matrix w1 (λ) is said to be subordinated from left to the matrix w(λ), and we write w1 (λ) ≺ w(λ)

(2.154)

l

for that, when the polynomial pl (λ) is a left-cancelling polynomial for w1 (λ), i.e. the product ql1 (λ) = pl (λ)w1 (λ) is a polynomial. 2.

In analogy, if the n × m matrix w(λ) has an IRMFD w(λ) = qr (λ)p−1 r (λ)

and the s × m matrix w2 (λ) is of such a kind, that the product

qr1 (λ) = w2 (λ)pr (λ) turns out as a polynomial matrix, then the matrix w2 (λ) is said to be subordinated from right to the matrix w(λ), and we denote this fact by w2 (λ) ≺ w(λ) . r

(2.155)

3. Lemma 2.52. Let the right side of (2.153) define an ILMFD of the matrix w(λ), and Condition (2.154) should be fulfilled. Let ψw (λ) and ψw1 (λ) be the McMillan denominators of the matrices w(λ) resp. w1 (λ). Then the fraction ∆(λ) =

ψw (λ) ψw1 (λ)

proves to be a polynomial. Proof. Take the ILMFD of the matrix w1 (λ): w1 (λ) = p−1 l1 (λ)ql1 (λ) . Since the polynomial pl (λ) is left cancelling for w1 (λ), a factorisation pl (λ) = g1 (λ)pl1 (λ) with an n × n polynomial matrix g1 (λ) is possible. Besides, we obtain ψw (λ) det pl (λ) ≈ = ∆(λ) . det g1 (λ) = det pl1 (λ) ψw1 (λ) Remark 2.53. A corresponding statement holds, when (2.155) is true.

2.12 Subordination of Rational Matrices

95

4. Lemma 2.54. Assume (2.154) be valid, and Q(λ), Q1 (λ) be any polynomial matrices of appropriate dimension. Then w1 (λ) + Q1 (λ) ≺ w(λ) + Q(λ) . l

Proof. Start with the ILMFD (2.153). Then the expression w(λ) + Q(λ) = p−1 l (λ) [ql (λ) + pl Q1 (λ)] , due to Lemma 2.15, is also an ILMFD. Hence owing to (2.154), the product pl (λ)[w1 (λ) + Q1 (λ)] turns out as a polynomial, that’s what the lemma claims. Remark 2.55. An analogous statement is true for subordination from right. Therefore, when the matrix w1 (λ) is subordinated to the matrix w(λ), then the broken part of w1 (λ) is subordinated to the broken part of w(λ). The reverse is also true.

5. Theorem 2.56. Consider the strictly proper n × m matrix w(λ), and its minimal realisation (2.156) w(λ) = C(λIp − A)−1 B . Then for holding the relation w1 (λ) ≺ w(λ) ,

(2.157)

l

where the rational n × s matrix w1 (λ) is strictly proper, it is necessary and sufficient, that there exists a constant p × s matrix B1 , which guarantees w1 (λ) = C(λIp − A)−1 B1 . Proof. Sufficiency: Build the ILMFD C(λIp − A)−1 = a−1 1 (λ)b1 (λ) . Then owing to Lemma 2.9, the expression w(λ) = a−1 1 (λ)[b1 (λ)B] defines an ILMFD of the matrix w(λ). Hence the product a1 (λ)w1 (λ) = b1 (λ)B1 proves to be a polynomial matrix, that’s what (2.157) declares. Necessity: Build the matrix w(λ) ˜ of the form

(2.158)

96

2 Fractional Rational Matrices s

m

w(λ) ˜ = [ w1 (λ) w(λ) ] n .

(2.159)

We will show Mdeg w(λ) ˜ = Mdeg w(λ) = p .

(2.160)

The equality Mdeg w(λ) = p immediately follows because the realisation (2.156) is minimal. It remains to show Mdeg w(λ) ˜ = Mdeg w(λ). For this purpose, we multiply (2.159) from left by the matrix a1 (λ). Then taking into account (2.157) and (2.158), we realise that

a1 (λ)w(λ) ˜ = a1 (λ)w1 (λ) a1 (λ)w(λ) is a polynomial matrix. Using ord a1 (λ) = p, Lemma 2.7 yields Mdeg w(λ) ˜ ≤ p. Now, we will prove that the inequality cannot happen. Indeed, assume Mdeg w(λ) ˜ = κ < p, then there exists a polynomial a ˜(λ) with ord a ˜(λ) = deg det a ˜(λ) = κ, and

˜(λ)w1 (λ) a ˜(λ)w(λ) a ˜(λ)w(λ) ˜ = a becomes a polynomial matrix. When this happens, also a ˜(λ)w(λ) becomes a polynomial matrix, and regarding to Lemma 2.7, we have Mdeg w(λ) ≤ κ < p. But this is impossible, due to our supposition Mdeg w(λ) = p, and the correctness of (2.160) is proven. Since the matrix w(λ) ˜ is strictly proper and Mdeg w(λ) ˜ = p holds, there exists a minimal realisation ˜ ˜ −1 B ˜ p − A) w(λ) ˜ = C(λI

(2.161)

˜ C, ˜ B. ˜ Bring the matrix with constant p × p, n × p and p × (s + m) matrices A, ˜ B into the form

s m ˜= B ˜1 B ˜2 p . B Then from (2.161), we gain

˜1 C(λI ˜ p − A) ˜2 . ˜ p − A) ˜ −1 B ˜ −1 B w(λ) ˜ = C(λI When we relate this and (2.159), we find ˜1 , ˜ p − A) ˜ −1 B w1 (λ) = C(λI

(2.162)

˜2 . ˜ p − A) ˜ −1 B w(λ) = C(λI

(2.163)

Expressions (2.156) and (2.163) define realisations of the matrix w(λ) of the same dimension p. However, since realisation (2.156) is minimal, also realisation (2.163) has to be minimal. According to (2.88), we can find a non-singular matrix R with

2.12 Subordination of Rational Matrices

A˜ = RAR−1 ,

˜2 = RB , B

97

C˜ = CR−1 .

From this and (2.162), it follows w1 (λ) = C(λIp − A)−1 B1 ,

˜1 , B1 = R−1 B

and the theorem is proven. Remark 2.57. A corresponding theorem can be proven for subordination from right. Theorem 2.58. Let (2.156) and the rational q × m matrix w1 (λ) be given. Then for holding the relation w1 (λ) ≺ w(λ) , r

it is necessary and sufficient, that there exists a constant q × p matrix C1 with w1 (λ) = C1 (λIp − A)−1 B .

6. Theorem 2.59. Consider the rational matrices F (λ),

G(λ),

H(λ) = F (λ)G(λ)

(2.164)

and the ILMFD F (λ) = a−1 1 (λ)b1 (λ),

G(λ) = a−1 2 (λ)b2 (λ) .

(2.165)

Furthermore, let us have the ILMFD −1 b1 (λ)a−1 2 (λ) = a3 (λ)b3 (λ) .

(2.166)

F (λ) ≺ H(λ)

(2.167)

Then the relation l

is true, if and only if the matrix

Rh (λ) = a3 (λ)a1 (λ) b3 (λ)b2 (λ)

(2.168)

is alatent, i.e. the pair (a3 (λ)a1 (λ), b3 (λ)b2 (λ)) is irreducible. Proof. Sufficiency: Start with the ILMFD H(λ) = a−1 (λ)b(λ) . Then from (2.165) and (2.166), we obtain

(2.169)

98

2 Fractional Rational Matrices −1 −1 −1 H(λ) = a−1 1 (λ)b1 (λ)a2 (λ)b2 (λ) = a1 (λ)a3 (λ)b3 (λ)b2 (λ)

= [a3 (λ)a1 (λ)]−1 b3 (λ)b2 (λ) .

(2.170)

Let Matrix (2.168) be alatent. Then the right side of (2.170) is an ILMFD and with the aid of (2.169), we get a(λ) = g(λ)a3 (λ)a1 (λ), where g(λ) is a unimodular matrix. Besides a(λ)F (λ) = g(λ)a3 (λ)b1 (λ) is a polynomial matrix, and hence (2.167) is true. Necessity: Assume (2.167), then we have a(λ) = h(λ)a1 (λ) ,

(2.171)

where h(λ) is a non-singular polynomial matrix. This relation leads us to −1 (λ)b(λ) . H(λ) = a−1 (λ)b(λ) = a−1 1 (λ)h

(2.172)

Comparing the expressions for H(λ) in (2.170) and (2.172), we find −1 (λ)b(λ) . (2.173) a−1 3 (λ)b3 (λ)b2 (λ) = h

But the matrix a3(λ) b3 (λ)b2 (λ) due to Lemma 2.9 is alatent, and the matrix h(λ) b(λ) with respect to (2.171) and owing to Lemma 2.11 is alatent. Therefore, the left as well as the right side of (2.173) present ILMFDs of the same rational matrix. Then from Statement 2.3 on page 64 arise

h(λ) = ϕ(λ)a3 (λ),

b(λ) = ϕ(λ)b3 (λ)b2 (λ) ,

(2.174)

where the matrix ϕ(λ) is unimodular. Applying (2.172) and (2.174), we arrive at the ILMFD H(λ) = [ϕ(λ)a3 (λ)a1 (λ)]−1 b(λ) . This expression and (2.170) define two LMFDs of the same matrix H(λ). Since the matrix ϕ(λ) is unimodular, we have ord[ϕ(λ)a3 (λ)a1 (λ)] = ord[a3 (λ)a1 (λ)] , and the right side of (2.170) is an ILMFD too. Therefore, Matrix (2.168) is alatent. A corresponding statement holds for subordination from right. Theorem 2.60. Consider the rational matrices (2.164) and the IRMFDs a−1 F (λ) = ˜b1 (λ)˜ 1 (λ),

G(λ) = ˜b2 (λ)˜ a−1 2 (λ) .

Moreover, let the IRMFD ˜ ˜ a−1 a ˜−1 1 (λ)b2 (λ) = b3 (λ)˜ 3 (λ) be given. Then the relation G(λ) ≺ H(λ) r

is true, if and only if the pair [˜ a2 (λ)˜ a3 (λ), ˜b1 (λ)˜b3 (λ)] is irreducible.

2.13 Dominance of Rational Matrices

99

7. The following theorem states an important special case, where the conditions of Theorems 2.59 and 2.60 are fulfilled. Theorem 2.61. If for the rational n × p and p × m matrices F (λ) and G(λ) the relation Mdeg[F (λ)G(λ)] = Mdeg F (λ) + Mdeg G(λ) (2.175) holds, i.e. the matrices F (λ) and G(λ) are independent, then the relations F (λ) ≺ F (λ)G(λ), l

G(λ) ≺ F (λ)G(λ) r

(2.176)

take place. Proof. Let us have the ILMFD (2.165), then Mdeg F (λ) = ord a1 (λ) and Mdeg G(λ) = ord a2 (λ). Besides, the pair [a2 (λ), b1 (λ)] is irreducible, that can be seen by assuming the contrary. In case of ord a3 (λ) < ord a2 (λ) in (2.166), we would obtain from (2.170) Mdeg H(λ) ≤ ord a1 (λ) + ord a3 (λ) < ord a1 (λ) + ord a2 (λ) which contradicts (2.175). The irreducibility of the pairs [a2 (λ), b1 (λ)] and (2.175) implies that the right part of (2.170) is an ILMFD. Owing to Theorem 2.59, the first relation in (2.176) is shown. The second relation in (2.137) is seen analogously. 8. Remark 2.62. Under the conditions of Theorem 2.59 using (2.171) and (2.174), we obtain a(λ)F (λ) = ϕ(λ)a3 (λ)b1 (λ) , that means, the factor ϕ(λ)a3 (λ) is a left divisor of the polynomial matrix a(λ)F (λ). Analogously, we conclude from the conditions of Theorem 2.60, when the IRMFD H(λ) = ˜b(λ)˜ a−1 (λ) is present, that a3 (λ)ψ(λ) G(λ)˜ a(λ) = ˜b2 (λ)˜ takes place with a unimodular matrix ψ(λ). We learn from this equation that under the conditions of Theorem 2.60, the polynomial matrix a ˜3 (λ)ψ(λ) is a right divisor of the polynomial matrix G(λ)˜ a(λ).

2.13 Dominance of Rational Matrices 1.

Consider the rational block matrix ⎡ w11 (λ) ⎢ .. w(λ) = ⎣ .

⎤ . . . w1m (λ) ⎥ .. .. ⎦, . . wn1 (λ) . . . wnm (λ)

(2.177)

100


where the wik (λ) are rational matrices of appropriate dimensions. Let ψ(λ) and ψik (λ), (i = 1, . . . , n; k = 1, . . . , m) be the McMillan denominators of the matrix w(λ) resp. of its blocks wik (λ). Hereinafter, we abbreviate McMillan denominator by MMD. Lemma 2.63. All expressions ψ(λ) ψik (λ)

dik (λ) =

(2.178)

turn out to be polynomials. Proof. At first assume only a block row

w(λ) = wz (λ) = w1 (λ) . . . wm (λ) ,

(2.179)

and we should have an ILMFD wz (λ) = a−1 (λ)b(λ) for it. Then per construction det a(λ) ≈ ψ z (λ) , where ψ z (λ) is the MMD of the row (2.179). Besides, the polynomial a(λ) is canceling from left for all matrices wi (λ), (i = 1, . . . , m), that means wi (λ) ≺ wz (λ),

(i = 1, . . . , m) .

l

Therefore, the relations

dzi (λ) =

ψ z (λ) , ψi (λ)

(i = 1, . . . , m)

where the ψi (λ) are the MMDs of the matrices wi (λ), owing to Lemma 2.52, become polynomials. In the same way can be seen that for a block column ⎤ ⎡ w ˜1 (λ) ⎥ ⎢ w(λ) = ws (λ) = ⎣ ... ⎦ (2.180) w ñ (λ) the expressions dsk (λ) =

ψ˜s (λ) , ψ˜k (λ)

(k = 1, . . . , n)

become polynomials, where ψ˜s (λ), ψ˜k (λ) are the MMDs of the column (2.180) and of its elements.


101

Denote by wiz (λ), (i = 1, . . . , n), w ˜ks (λ), (k = 1, . . . , m) all rows and z columns of Matrix (2.177), and by ψi (λ), (i = 1, . . . , n), ψ˜ks (λ), (k = 1, . . . , m) their MMDs. With respect to the above shown, the relations ψ(λ) , (i = 1, . . . , n); ψiz (λ)

ψ(λ) , (k = 1, . . . , m) ˜ ψks (λ)

are polynomials. Therefore, all relations dik (λ) =

ψ(λ) ψiz (λ) ψ(λ) = z ψik (λ) ψi (λ) ψik (λ)

become polynomials. Corollary 2.64. For any Matrix (2.177), the inequalities Mdeg w(λ) ≥ Mdeg wik (λ),

(i = 1, . . . , n; k = 1, . . . , m)

are true. 2. The element wik (λ) in Matrix (2.177) is said to be dominant, if the equality ψ(λ) = ψik (λ) takes place. Lemma 2.65. The element wik (λ) is dominant in Matrix (2.177), if and only if Mdeg w(λ) = Mdeg wik (λ) . Proof. The necessity of this condition is obvious. That it is also sufficient follows from the fact that expression (2.178) is a polynomial, and from the equations Mdeg w(λ) = deg ψ(λ),

Mdeg wik (λ) = deg ψik (λ) .

3. Theorem 2.66. A necessary and sufficient condition for the matrix w2 (λ) to be dominant in the block row

w(λ) = w1 (λ) w2 (λ) is that it meets the relation w1 (λ) ≺ w2 (λ) . l

102


A necessary and sufficient condition for the matrix w2 (λ) to be dominant in the block column

w1 (λ) w(λ) = w2 (λ) is that it meets the relation w1 (λ) ≺ w2 (λ) . r

Proof. The proof immediately arises from the proof of Theorem 2.56.

4. Theorem 2.67. Consider the strictly proper rational block matrix G(λ) of the shape

m

K(λ) L(λ) G(λ) = M (λ) N (λ)

i

(2.181)

n

and the minimal realisation N (λ) = C(λIp − A)−1 B .

(2.182)

Then the matrix N (λ) is dominant in G(λ), i.e. Mdeg N (λ) = Mdeg G(λ) = p ,

(2.183)

if and only if there exist constant i × p and p × matrices C1 and B1 with K(λ) = C1 (λIp − A)−1 B1 , L(λ) = C1 (λIp − A)−1 B ,

(2.184)

M (λ) = C(λIp − A)−1 B1 . Proof. Necessity: Let (2.183) be valid. Since the matrix G(λ) is strictly proper, there exists a minimal realisation ˜ ˜ p − A) ˜ −1 B G(λ) = C(λI

(2.185)

˜ B ˜ and C˜ of the dimensions p × p, p × ( + m) and with constant matrices A, (i + n) × m, respectively. Assume

p

C˜ C˜ = ˜1 C2

i n

m ˜ ˜1 B ˜2 p B= B

and substitute this expression in (2.185), then we obtain


G(λ) =

103

˜1 C˜1 (λIp − A) ˜2 ˜ −1 B ˜ −1 B C˜1 (λIp − A) . ˜1 C˜2 (λIp − A) ˜2 ˜ −1 B ˜ −1 B C˜2 (λIp − A)

(2.186)

Relating (2.181) to (2.186), we find ˜2 . ˜ −1 B N (λ) = C˜2 (λIp − A)

(2.187)

Both Equations (2.182) and (2.187) are realisations of N (λ) and possess the same dimension. Since (2.182) is a minimal realisation, so (2.187) has to be minimal too. Therefore, Relations (2.88) can be used that will lead us to A˜ = RAR−1 ,

C˜2 = CR−1

˜2 = RB, B

with a non-singular matrix R. Inserting this into (2.186), Relation (2.184) is achieved with ˜1 , ˜ −1 B C1 = C˜1 R , B1 = R that proves the necessity of the conditions of the theorem. Sufficiency: Suppose Conditions (2.182), (2.184) to be true. Then,

C1 G(λ) = (λIp − A)−1 B1 B C holds. Consider the matrix

G2 (λ) = M (λ) N (λ) = C(λIp − A)−1 B1 B .

(2.188)

The realisation on the right side of is minimal, because of Mdeg N (λ) =

(2.188) p. Therefore, the pair (λIp − A, B1 B ) is irreducible, and Mdeg G2 (λ) = p. Utilising (2.181) and (2.188), the matrix G(λ) can be written in the form

G1 (λ) G(λ) = G2 (λ) with

G1 (λ) = C1 (λIp − A)−1 B1 B ,

where the pair [λIp − A, C1 ] is, roughly said, non-irreducible. Therefore, according to Theorem 2.58, we obtain G1 (λ) ≺ G2 (λ) , r

and with account of Theorem 2.66 Mdeg G(λ) = Mdeg G2 (λ) = p . Corollary 2.68. Under Conditions (2.181)–(2.184), the relations K(λ) ≺ L(λ), l

are true.

K(λ) ≺ M (λ) r

(2.189)

104


Proof. Assume the ILMFD C1 (λIp − A)−1 = αl−1 (λ)βl (λ) . Owing to Lemma 2.9, the right side of C1 (λIp − A)−1 B2 = αl−1 (λ)[βl (λ)B2 ] is an ILMFD of the matrix L(λ). Therefore, the product αl (λ)K(λ) = βl (λ)B1 becomes a polynomial matrix. Herewith the first relation in (2.189) is shown. The second part can be proven analogously.

3 Normal Rational Matrices

3.1 Normal Rational Matrices 1.

Consider the rational n × m matrix A(λ) in the standard form (2.21) A(λ) =

N (λ) , d(λ)

deg d(λ) = p

(3.1)

and, furthermore, let be given certain ILMFD and IRMFD −1 A(λ) = a−1 l (λ)bl (λ) = br (λ)ar (λ) ,

(3.2)

where, due to the irreducibility of the MFD, we have ord al (λ) = ord ar (λ) = Mdeg A(λ) , and Mdeg A(λ) is the degree of the McMillan denominator of the matrix A(λ). At first, Relation (2.34) implies Mdeg A(λ) ≥ p. In the following disclosure, matrices will play an important role for which Mdeg A(λ) = p .

(3.3)

Since det al (λ) ≈ det ar (λ) is valid and both polynomials are divisible by d(λ), Relation (3.3) is equivalent to d(λ) = ψA (λ) ,

(3.4)

where ψA (λ) is the McMillan denominator of A(λ). Further on, rational matrices satisfying (3.3), (3.4) will be called normal matrices . 2.

For a normal matrix (3.1), it is possible to build IMFDs (3.2), such that det al (λ) ≈ d(λ),

det ar (λ) ≈ d(λ) .

106


If both ILMFD and IRMFD satisfy such conditions, the pair is called a complete MFD. Thus, normal rational matrices are rational matrices that possess a complete MFD. It is emphasised that a complete MFD is always irreducible. Indeed, from (2.34) is seen that for any matrix A(λ) in form (3.1) it always follows deg ψA (λ) ≥ deg d(λ). Therefore, if we have any matrix A(λ) satisfying (3.3), then the polynomials det al (λ) and det ar (λ) possess the minimal possible degree, and hence the complete MFD is irreducible. 3. A general characterisation of the set of normal rational matrices yields the next theorem. Theorem 3.1. Let in (3.1) be min(n, m) ≥ 2. Then for the fact that the irreducible rational matrix (3.1) becomes normal, it is necessary and sufficient, that every minor of second order of the polynomial matrix N (λ) is divisible without remainder by the denominator d(λ). Moreover, if Relations (3.2) define a complete MFD, then the matrices al (λ) and ar (λ) are simple. Proof. Necessity: To consider a concrete case, assume a left MFD. Let A(λ) =

N (λ) = a−1 l (λ)bl (λ) d(λ)

be part of a complete MFD. Then the matrix al (λ) is simple, because from (2.34) it follows that Equations (3.3), (3.4) can be fulfilled only for ψ2 (λ) = ψ3 (λ) = . . . = ψρ (λ) = 1. Therefore, (2.39) delivers the representation al (λ) = µ(λ) diag{1, . . . , 1, d(λ)}ν(λ)

(3.5)

with unimodular matrices µ(λ), ν(λ). We take from (3.5), that the matrix al (λ) is simple, and furthermore from (3.5), we obtain a−1 l (λ) =

ν −1 (λ) diag{d(λ), . . . , d(λ), 1}µ−1 (λ) Q(λ) . = d(λ) d(λ)

(3.6)

All minors of second order of the matrix diag{d(λ), . . . , d(λ), 1} are divisible by d(λ). Thus, by the Binet-Cauchy theorem this property passes to the numerator of the fraction on the right side of (3.6). But, due to the Binet-Cauchy theorem, the matrix N (λ) = Q(λ)bl (λ) possesses the shown property. Hence the necessity of the condition of the theorem is proven. Sufficiency: Assume that all minors of second order of the matrix N (λ) are divisible by d(λ). Then we learn from (1.40)–(1.42), that this matrix can be presented in the form N (λ) = p(λ)SN (λ)q(λ) ,

(3.7)

where p(λ), q(λ) are unimodular, and the matrix SN (λ) has the appropriate Smith canonical form. Thus, from (1.49) we receive

3.1 Normal Rational Matrices

⎡

g1 (λ) 0 ⎢ 0 g1 (λ)g2 (λ)d(λ) ⎢ .. ⎢ .. . SN (λ) = ⎢ ⎢ . ⎢ 0 0 ⎣

... ... .. .

⎤

0 0 .. .

. . . g1 (λ) · · · gρ (λ)d(λ)

On−ρ,ρ

107

⎥ Oρ,m−ρ ⎥ ⎥ ⎥ , (3.8) ⎥ ⎥ ⎦ On−ρ,m−ρ

where the polynomial g1 (λ) and the denominator d(λ) are coprime, because in the contrary the fraction (3.1) would be reducible. According to (3.7) and (3.8), the matrix A(λ) of (3.1) can be written in the shape ⎡ ⎤ g1 (λ) 0 ... 0 ⎢ d(λ) ⎥ ⎢ ⎥ ⎢ 0 g1 (λ)g2 (λ) . . . 0 Oρ,m−ρ ⎥ ⎢ ⎥ .. .. A(λ) = p(λ) ⎢ .. ⎥ q(λ) , .. ⎢ . ⎥ . . . ⎢ ⎥ ⎣ 0 ⎦ 0 . . . g1 (λ) · · · gρ (λ) On−ρ,ρ where the fraction

On−ρ,m−ρ

g1 (λ) is irreducible. Therefore, choosing d(λ)

al (λ) = diag{d(λ), 1, . . . , 1}p−1 (λ) ⎡

g1 (λ) 0 ⎢ 0 g1 (λ)g2 (λ) ⎢ .. ⎢ .. . . bl (λ) = ⎢ ⎢ ⎢ 0 0 ⎣

... ... .. .

0 0 .. .

. . . g1 (λ) · · · gρ (λ)

On−ρ,ρ we obtain the LMFD

⎤ ⎥ Oρ,m−ρ ⎥ ⎥ ⎥ q(λ) , ⎥ ⎥ ⎦ On−ρ,m−ρ

A(λ) = a−1 l (λ)bl (λ) ,

which is complete, because det al (λ) ≈ d(λ) is true. Corollary 3.2. It follows from (3.5), (3.6) that for a simple n × n matrix a(λ), the rational matrix a−1 (λ) is normal, and vice versa. Corollary 3.3. From Equations (3.7), (3.8) we learn that for k ≥ 2, all minors of k-th order of the numerator of a normal matrix N (λ) are divisible by dk−1 (λ). Remark 3.4. Irreducible rational matrix rows or columns are always normal. Let for instance the column ⎡ ⎤ a1 (λ) 1 ⎢ . ⎥ A(λ) = (3.9) ⎣ .. ⎦ d(λ) an (λ)

108


with polynomials ai (λ), (i = 1, . . . , n) be given. Then by applying left elementary operations, A(λ) can be brought into the form ⎡ ⎤ (λ) 1 ⎢ ⎥ A(λ) = c(λ) ⎣ ... ⎦ , d(λ) 0 where c(λ) is a unimodular n × n matrix, and (λ) is the GCD of the polynomials a1 (λ), . . . , an (λ). The polynomials (λ) and d(λ) are coprime, because in other case the rational matrix (3.9) could be cancelled. Choose ⎡ ⎤ (λ) ⎢ ⎥ al (λ) = diag{d(λ), 1, . . . , 1}c−1 (λ), bl (λ) = ⎣ ... ⎦ , 0 then obviously we have

A(λ) = a−1 l (λ)bl (λ) ,

and this LMFD is complete, because of det al (λ) ≈ d(λ). 4. A general criterion for calculating the normality of the rational matrix (3.1) directly from its elements yields the following theorem. Theorem 3.5. Let the fraction (3.1) be irreducible, and furthermore d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq ,

µ1 + . . . + µq = p .

(3.10)

Then a necessary and sufficient condition for the matrix A(λ) to be normal is the fact that each of its minors of second order possess poles in the points λ = λi (i = 1, . . . , q) with multiplicity not higher than µi .

Proof. Necessity: Let N (λ) = nij (λ) and A

i j k

⎡

nik (λ) ⎢ d(λ) = det ⎢ ⎣ njk (λ) d(λ)

⎤ ni (λ) d(λ) ⎥ ⎥ nj (λ) ⎦ d(λ)

(3.11)

be a minor of the matrix A(λ) that is generated by the elements of the rows with numbers i, j and columns with numbers k, . Obviously nik (λ)nj (λ) − njk (λ)ni (λ) i j (3.12) A = k d2 (λ) is true. If the matrix A(λ) is normal, then, due to Theorem 3.1, the numerator of the last fraction is divisible by d(λ). Thus we have

3.1 Normal Rational Matrices

A

i j k

=

aij k (λ) , d(λ)

109

(3.13)

where aij k (λ) is a certain polynomial. It is seen from (3.13) and (3.10) that the minor (3.11) possess in λ = λi poles of order µi or lower. Sufficiency: Conversely, if for every minor (3.11) the representation (3.13) is correct, then the numerator of each fraction (3.12) is divisible by d(λ), that means, every minor of second order of the matrix N (λ) is divisible by d(λ), or in other words, the matrix A(λ) is normal. 5. Theorem 3.6. If the matrix A(λ) (3.1) is normal, and (3.10) is assumed, then (3.14) rank N (λi ) = 1, (i = 1, . . . , q) . Thereby, if the polynomial (3.10) has only single roots, i.e. q = p, µ1 = µ2 = . . . = µp = 1, then Condition (3.14) is not only necessary but also sufficient for the normality the matrix A(λ). Proof. Equation (3.6) implies rank Q(λi ) = 1, (i = 1, . . . , q). Therefore, the matrix N (λi ) = Q(λi )bl (λi ) is either the zero matrix or it has rank 1. The first possibility is excluded, otherwise the fraction (3.1) would have been reducible. Hence we get (3.14). If all roots λi are simple, and (3.14) holds, then every minor of second order of the matrix N (λ) is divisible by (λ − λi ), (i = 1, . . . , q). Since in the present case d(λ) = (λ − λ1 )(λ − λ2 ) · · · (λ − λp ) is true, so every minor of second order of N (λ) is divisible by d(λ), it means, that the matrix A(λ) is normal. 6. We learn from Theorems 3.1–3.6 that the elements of a normal matrix A(λ) are constrained by a number of strict equations that consist between them, which ensure that all minors of second order are divisible by the denominator. Even small deviations, sometimes only in one element, cause that these equations are violated, and the matrix A(λ) is no longer normal, with the consequence that the order of the McMillan denominator of this matrix grows abruptly. As a whole, this leads to incorrect solutions during the construction of the IMFD and the corresponding realisations in state space. The above said gives evidence of the structural instability of normal matrices, and from that we conclude immediately the instability of the numeric operations with such matrices. On the other side, it is shown below, that in practical problems, the frequency domain models for real objects are described essentially by normal transfer matrices. Therefore, the methods for practical solution of control problems have to be supplied by additional tools, which help to overcome the mentioned structural and numeric instabilities to reach correct results.

110


Example 3.7. Consider the rational matrix A(λ) =

with N (λ) =

λ−1 1 , λ−2

N (λ) d(λ) d(λ) = (λ − 1)(λ − 2) ,

where is a constant. Due to det N (λ) = (λ − 1)(λ − 2) − , the matrix A(λ) proves to be normal if and only if = 0. It is easily checked that

λ−1 1 1 0 1 0 λ−1 1 = , λ−2 λ − 2 −1 0 λ2 − 3λ + 2 − 1 0 where the first and last matrix on the right side are unimodular. Thus for = 0, the matrix A(λ) has the McMillan canonical form ⎤ ⎡ 1 0 ⎥ ⎢ MA (λ) = ⎣ (λ − 1)(λ − 2) ⎦. 0 1− (λ − 1)(λ − 2) In the present case we have ψ1 (λ) = (λ−1)(λ−2) = d(λ), ψ2 (λ) = (λ−1)(λ− 2) = d(λ). Hence the McMillan denominator is ψA (λ) = (λ − 1)2 (λ − 2)2 and Mdeg A(λ) = 4. However, if = 0 is true, then we get ⎤ ⎡ 1 0 MA (λ) = ⎣ (λ − 1)(λ − 2) ⎦ . 0 1 In this case, we obtain ψ1 (λ) = (λ − 1)(λ − 2) = d(λ), ψ2 (λ) = 1 and the McMillan denominator ψA (λ) = (λ − 1)(λ − 2) which yields Mdeg A(λ) = 2.

3.2 Algebraic Properties of Normal Matrices 1. In this section we give some general algebraic properties of normal matrices that will be used further. Theorem 3.8. Let two normal matrices A1 (λ) =

N1 (λ) , d1 (λ)

A2 (λ) =

N2 (λ) d2 (λ)

(3.15)

of dimensions n × resp. × m be given. Then, if the fraction A(λ) = A1 (λ)A2 (λ) =

N1 (λ)N2 (λ) d1 (λ)d2 (λ)

is irreducible, the matrix A(λ) becomes normal.

(3.16)

3.2 Algebraic Properties of Normal Matrices

111

Proof. If n = 1 or m = 1 is true, then the statement follows from the remark after Theorem 3.1. Now, let min(n, m) ≥ 2 and assume N (λ) = N1 (λ)N2 (λ). Due to the theorem of Binet-Cauchy, every minor of second order of the matrix N (λ) is a bilinear form of the minors of second order of the matrices N1 (λ) and N2 (λ), and consequently divisible by the product d1 (λ)d2 (λ). Therefore, the fraction (3.16) is normal, because it is also irreducible. 2. Theorem 3.9. Let the matrices (3.15) have the same dimension, and the polynomials d1 (λ) and d2 (λ) be coprime. Then the matrix A(λ) = A1 (λ) + A2 (λ)

(3.17)

is normal. Proof. From (3.15) and (3.17), we generate A(λ) =

d2 (λ)N1 (λ) + d1 (λ)N2 (λ) . d1 (λ)d2 (λ)

(3.18)

The fraction (3.18) is irreducible, because the sum (3.17) has its poles at the zeros of d1 (λ) and d2 (λ) with the same multiplicity. Denote

αik (λ) βik (λ) A1 (λ) = , A2 (λ) = . d1 (λ) d2 (λ) Then the minor (3.11) for the matrix A(λ) ⎡ αik (λ) βik (λ) + ⎢ d1 (λ) d2 (λ) i j A = det ⎢ ⎣ αjk (λ) βjk (λ) k + d1 (λ) d2 (λ)

has the shape ⎤ αi (λ) βi (λ) + d1 (λ) d2 (λ) ⎥ ⎥. αj (λ) βj (λ) ⎦ + d1 (λ) d2 (λ)

(3.19)

Applying the summation theorem for determinants, and using the normality of A1 (λ), A2 (λ) after cancellation, we obtain the expression bij i j k (λ) A = k d1 (λ)d2 (λ) with certain polynomials bij k (λ). It follows from this expression that the poles of the minor (3.19) can be found under the roots of the denominators of Matrix (3.18), and they possess no higher multiplicity. Since this rational matrix is irreducible, Theorem 3.5 yields that the matrix A(λ) is normal. Corollary 3.10. If A(λ) is a normal n × m rational matrix, and G(λ) is an n × m polynomial matrix, then the rational matrix A1 (λ) = A(λ) + G(λ) is normal.

112

3.


For normal matrices the reverse to Theorem 2.42 is true.

Theorem 3.11. Let the polynomial n × n matrix al (λ) be simple and bl (λ) be any n × m polynomial matrix. If under this condition, the fraction A(λ) = a−1 l (λ)bl (λ) =

adj al (λ) bl (λ) det al (λ)

(3.20)

is irreducible, then the pair (al (λ), bl (λ)) is irreducible and the matrix A(λ) is normal. Proof. It is sufficient to consider the case min{n, m} ≥ 2. Since the matrix al (λ) is simple, with the help of Corollary 3.2, it follows that the matrix a−1 l (λ) is normal and all minors of second order of the matrix adj al (λ) are divisible by det al (λ). Due to the theorem of Binet-Cauchy, this property transfers to the numerator on the right side of (3.20), that’s why the matrix A(λ) is normal. By using (3.6), we also obtain A(λ) =

Q(λ)bl (λ) . d(λ)

Here per construction, we have det al (λ) ≈ d(λ). Comparing this equation with (3.20), we realise that the middle part of (3.20) proves to be a complete LMFD, and consequently the pair (al (λ), bl (λ)) is irreducible. Analogously, the following statement for right MFD can be shown: Corollary 3.12. If the polynomial m × m matrix ar (λ) is simple, det ar (λ) ≈ d(λ), br (λ) is any polynomial n × m matrix, and the fraction br (λ)a−1 r (λ) =

R(λ) = A(λ) d(λ)

is irreducible, then the pair [ar (λ), br (λ)] is irreducible and the left side defines a complete RMFD of the matrix A(λ). 4. Let us investigate some general properties of the MFD of the product of normal matrices. Consider some normal matrices (3.15), where their product (3.16) should exist and be irreducible. Moreover, let us have the complete LMFD A2 (λ) = a−1 A1 (λ) = a−1 1 (λ)b1 (λ), 2 (λ)b2 (λ) , where the matrices a1 (λ), a2 (λ) are simple, and det a1 (λ) ≈ d1 (λ), det a2 (λ) ≈ d2 (λ) are valid. Applying these representations, we can write −1 A(λ) = a−1 1 (λ)b1 (λ)a2 (λ)b2 (λ) .

(3.21)

Notice, that the fraction L(λ) = b1 (λ)a−1 2 (λ) owing to the irreducibility of A(λ) is also irreducible. Hence as a result of Corollary 3.12, the fraction L(λ) is normal, and there exists the complete LMFD

3.2 Algebraic Properties of Normal Matrices

113

−1 b1 (λ)a−1 2 (λ) = a3 (λ)b3 (λ) ,

where det a3 (λ) ≈ d2 (λ). From this and (3.21), we get A(λ) = a−1 l (λ)bl (λ) with al (λ) = a3 (λ)a1 (λ),

bl (λ) = b3 (λ)b2 (λ) .

Per construction, det al (λ) ≈ d1 (λ)d2 (λ) is valid, that’s why the last relations define a complete LMFD, the matrix al (λ) is simple and the pair (a3 (λ)a1 (λ), b3 (λ)b2 (λ)) is irreducible. Hereby, we still obtain Mdeg[A1 (λ)A2 (λ)] = Mdeg A1 (λ) + Mdeg A2 (λ) . Hence the following theorem has been proven: Theorem 3.13. If the matrices (3.15) are normal and the product (3.16) is irreducible, then the matrices A1 (λ) and A2 (λ) are independent in the sense of Section 2.4. 5. From Theorems 3.13 and 2.61 we conclude the following statement, which is formulated in the terminology of subordination of matrices in the sense of Section 2.12. Theorem 3.14. Let us have the normal matrices (3.15), and their product (3.16) should be irreducible. Then A1 (λ) ≺ A1 (λ)A2 (λ), l

A2 (λ) ≺ A1 (λ)A2 (λ) . r

6. Theorem 3.15. Let the separation A(λ) =

N1 (λ) N2 (λ) N (λ) = + d1 (λ)d2 (λ) d1 (λ) d2 (λ)

(3.22)

exist, where the matrix A(λ) is normal and the polynomials d1 (λ), d2 (λ) are coprime. Then each of the fractions on the right side of (3.22) is normal. Proof. At first we notice that the fractions on the right side of (3.22) are irreducible, otherwise the fraction A(λ) would be reducible. The fraction A1 (λ) =

N (λ) d1 (λ)

is also normal, because it is irreducible and the minors of second order of the numerator are divisible by the denominator d1 (λ). Therefore,

114


A1 (λ) = a−1 1 (λ)b1 (λ) ,

(3.23)

where the matrix a1 (λ) is simple and det a1 (λ) ≈ d1 (λ). Multiplying both sides of Equation (3.22) from left by a1 (λ) and considering (3.23), we get b1 (λ) N1 (λ) N2 (λ) = a1 (λ) + a1 (λ) , d2 (λ) d1 (λ) d2 (λ) this means

b1 (λ) N2 (λ) N1 (λ) − a1 (λ) = a1 (λ) . d2 (λ) d2 (λ) d1 (λ)

The left side of the last equation is analytical at the zeros of the polynomials d1 (λ), and the right side at the zeros of d2 (λ). Consequently a1 (λ)N1 (λ) = L(λ) d1 (λ) has to be a polynomial matrix L(λ) and N1 (λ) = a−1 1 (λ)L(λ) . d1 (λ) The fraction on the right side is irreducible, otherwise the fraction (3.22) has been irreducible. The matrix a1 (λ) is simple, and therefore the last fraction owing to Theorem 3.11 is normal. In analogy it may be shown that the matrix N2 (λ)/d2 (λ) is normal.

3.3 Normal Matrices and Simple Realisations 1. At the first sight, normal rational matrices seem to be quite artificial constructions, because their elements are bounded by a number of crisp equations. However, in this section we will demonstrate that even normal matrices for the most of real problems will give the correct description of multidimensional LTI objects in the frequency domain. 2. We will use the terminology and the notation of Section 1.15, and consider an arbitrary realisation (A, B, C) of dimension n, p, m. Doing so, the realisation (A, B, C) is called minimal, if the pair (A, B) is controllable and the pair [A, C] is observable. A minimal realisation is called simple if the matrix A is cyclic. As shown in Section 1.15, the property of simplicity of the realisation (A, B, C) is structural stable, and it is conserved at least for sufficiently small deviations in the matrices A, B, C. Realisations, that are not simple, are not supplied with the property of structural stability. Practically, this means that correct models of real linear objects in state space amounts to simple realisations.

3.3 Normal Matrices and Simple Realisations

115

3. As explained in chapter 2, every realisation (A, B, C) is assigned to a strictly proper rational n × m matrix w(λ) by the relation w(λ) = C(λIp − A)−1 B

(3.24)

equivalently expressed by w(λ) =

C adj(λIp − A)B , dA (λ)

(3.25)

where adj(λIp − A) is the adjoint matrix and dA (λ) = det(λIp − A). As is taken from (3.24), every realisation (A, B, C) is uniquely related to a transfer matrix. Conversely, every strictly proper rational n × m matrix w(λ) =

N (λ) d(λ)

(3.26)

is configured to an infinite set of realisations (A, B, C) of dimensions n, q, m with q ≥ Mdeg w(λ), where Mdeg w(λ) means the McMillan-degree of the matrix w(λ). Realisations, where the number q takes its minimal value, as before will be called minimal. The realisation (A, B, C) is minimal, if and only if the pair (A, B) is controllable and the pair [A, C] is observable. In general, minimal realisations of arbitrary matrices w(λ) are not simple, and therefore, they do not possess the property of structural stability. In this case small deviations in the coefficients of the linear objects (1.102), (1.103) lead to essential changes in their transfer functions. In this connection, the question arises, for which class of matrices the corresponding minimal realisations will be simple. The answer to this question lies in the following statement. 4. Theorem 3.16. The transfer matrix (3.25) of the realisation (A, B, C) is irreducible, if and only if the realisation (A, B, C) is simple. Proof. As follows from Theorem 2.45, for the irreducibility of the fractions (3.25), it is necessary and sufficient that the elementary PMD τ (λ) = (λIp − A, B, C)

(3.27)

is minimal and the matrix λIp −A is simple, which is equivalent to the demand for simplicity of the realisation (A, B, C). Theorem 3.17. If the realisation (A, B, C) of dimension (n, p, m) is simple, then the corresponding transfer matrix (3.24) is normal.

116


Proof. Assume that the realisation (A, B, C) is simple. Then the elementary PMD (3.27) is also simple, and the fraction on the right side of (3.25) is irreducible. Hereby, due to the simplicity of the matrix λIp − A, the rational matrix adj(λIp − A) (λIp − A)−1 = det(λIp − A) becomes normal, what means, it is irreducible and all minors of 2nd order of the matrix adj(λIp −A) are divisible by det(λIp −A). But then for min{m, n} ≥ 2 owing to the theorem of Binet-Cauchy, also the minors of 2nd order of the matrix Q(λ) = C adj(λIp − A)B possess this property, and this means that Matrix (3.24) is normal. Theorem 3.18. For a strictly proper rational matrix to possess a simple realisation, it is necessary and sufficient, that this matrix is normal. Proof. Necessity: When the rational matrix (3.26) allows a simple realisation (A, B, C), then it is normal by virtue of Theorem 3.17. Sufficiency: Let the irreducible matrix (3.26) be normal and deg d(λ) = p. Then there exists a complete LMFD w(λ) = a−1 (λ)b(λ) for it with ord a(λ) = p, and consequently Mdeg w(λ) = p. From this we conclude, that Matrix (3.26) allows a minimal realisation (A, B, C) of dimension (n, p, m). We now assume that the matrix A is not cyclic. Then the fraction C adj(λIp − A)B det(λIp − A) would be reducible. Hereby, Matrix (3.26) would permit the representation w(λ) =

N1 (λ) , d1 (λ)

where deg d1 (λ) < deg d(λ). But this contradicts the supposition on the irreducibility of Matrix (3.26). Therefore, the matrix A must be cyclic and the matrix λIp − A simple, hence the minimal realisation has to be simple.

3.4 Structural Stable Representation of Normal Matrices 1. The notation of normal matrices in the form (3.1) is structural unstable, because it looses for arbitrary small errors in its coefficients the property that its minors of 2nd order of the numerator are divisible by the denominator. In that case, the quantity Mdeg A(λ) will abruptly increase. Especially, if Matrix

3.4 Structural Stable Representation of Normal Matrices

117

(3.1) is strictly proper, then the dimensions of the matrices in its minimal realisation in state space will also abruptly increase, i.e. the dynamical properties with respect to the original system will change drastically. In this section, a structural stable representation (S-representation) of normal rational matrices will be introduced, . Regarding normality, the S-representation is invariant related to parameter deviations in the transfer matrix, originated for instance by modeling or rounding errors. 2. Theorem 3.19 ([144, 145]). The irreducible rational n × m matrix A(λ) =

N (λ) d(λ)

(3.28)

is normal, if and only if its numerator permits the representation N (λ) = P (λ)Q (λ) + d(λ)G(λ)

(3.29)

with an n × 1 polynomial column P (λ), an m × 1 polynomial column Q(λ), and an n × m polynomial matrix G(λ). Proof. Sufficiency: Let us have ⎡ ⎤ ⎤ ⎡ p1 (λ) q1 (λ) ⎢ ⎥ ⎥ ⎢ P (λ) = ⎣ ... ⎦ , Q(λ) = ⎣ ... ⎦ , pn (λ) qm (λ)

⎤ g11 (λ) . . . g1m (λ) ⎥ ⎢ .. G(λ) = ⎣ ... ⎦ ··· . gn1 (λ) . . . gnm (λ) ⎡

with scalar polynomials pi (λ), qi (λ), gik (λ). The minor (3.11) of Matrix (3.29) possesses the form

pi (λ)qk (λ) + d(λ)gik (λ) pi (λ)q (λ) + d(λ)gi (λ) i j N = det k pj (λ)qk (λ) + d(λ)gjk (λ) pj (λ)q (λ) + d(λ)gj (λ) = d(λ)nij k (λ) where nij k (λ) is a polynomial. Therefore, an arbitrary minor is divisible by d(λ), and thus Matrix (3.28) is normal. Necessity: Let Matrix (3.28) be normal. Then all minors of second order of its numerator N (λ) are divisible by the denominator d(λ). Applying (3.7) and (3.8), we find out that the matrix N (λ) allows the representation ⎤ ⎡ g1 (λ) 0 ... 0 ⎥ ⎢ 0 g1 (λ)g2 (λ)d(λ) . . . 0 ⎢ Oρ,m−ρ ⎥ .. .. ⎥ ⎢ .. .. ⎥ q(λ) . . . N (λ) = p(λ) ⎢ ⎥ ⎢ . ⎥ ⎢ 0 0 . . . g1 (λ) · · · gρ (λ)d(λ) ⎦ ⎣ On−ρ,ρ

On−ρ,m−ρ (3.30)

118


where the gi (λ), (i = 1, . . . , ρ) are monic polynomials and p(λ), q(λ) are unimodular matrices. Relation (3.30) can be arranged in the form N (λ) = N1 (λ) + d(λ)N2 (λ) ,

where N1 (λ) = g1 (λ)p(λ) ⎡

0 0 0 .. .

1 On−1,1

0 0 1 0 0 g3 (λ) .. .. . .

⎢ ⎢ ⎢ ⎢ N2 (λ) = g1 (λ)g2 (λ)p(λ) ⎢ ⎢ ⎢ ⎣00

0

... ... ... .. .

(3.31)

O1,m−1 q(λ) , On−1,m−1 0 0 0 .. .

Oρ,m−ρ

. . . g3 (λ) · · · gρ (λ) On−ρ,ρ On−ρ,m−ρ

(3.32) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ q(λ) . ⎥ ⎥ ⎦ (3.33)

Obviously, we have N1 (λ) = g1 (λ)P1 (λ)Q1 (λ) ,

(3.34)

where P1 (λ) is the first column of the matrix p(λ) and Q1 (λ) is the first row of q(λ). Inserting (3.32)–(3.34) into (3.31), we arrive at the representation (3.29), where for instance P (λ) = g1 (λ)P1 (λ),

Q(λ) = Q1 (λ),

G(λ) = N2 (λ)

can be used. 3.

Inserting (3.29) into (3.28) yields A(λ) =

P (λ)Q (λ) + G(λ) . d(λ)

(3.35)

The representation of a normal rational matrix in the form (3.35) is called its structural stable representation or S-representation. Notice, that the Srepresentation of a normal matrix is structural stable (invariant) according to variations of the vectors P (λ), Q(λ), the matrix G(λ) and of the polynomial d(λ), because the essential structural specialities of Matrix (3.35) still hold. 4.

Assume P (λ) = d(λ)L1 (λ) + P˜1 (λ) , ˜ 1 (λ) , Q(λ) = d(λ)L2 (λ) + Q

where deg P˜1 (λ) < deg d(λ), Then from (3.23), we obtain

˜ 1 (λ) < deg d(λ) . deg Q

(3.36)


119

˜ 1 (λ) + d(λ)G1 (λ) N (λ) = P˜1 (λ)Q with ˜ (λ) + P˜1 (λ)L (λ) + d(λ)L1 (λ)L (λ) + G(λ) . G1 (λ) = L1 (λ)Q 1 2 2 Altogether from (3.35), we get A(λ) =

˜ (λ) P˜1 (λ)Q 1 + G1 (λ) , d(λ)

(3.37)

˜ 1 (λ) satisfy Relation (3.36). Representation where the vectors P˜1 (λ) and Q (3.37) is named as minimal S-representation of a normal matrix. A minimal S-representation (3.34) also turns out to be structural stable, if the parameter variations do not violate Condition (3.36). 5. The proof of Theorem 3.19 has constructive character, so it yields a practical method for calculating the vectors P (λ), Q(λ) and the matrix G(λ), which appear in the S-representations (3.35) and (3.37). However, at first the matrix N (λ) has to be given the shape (3.30), which is normally connected with extensive calculations. In constructing the S-representation of a normal matrix, the next statement allows essential simplification in most practical cases. Theorem 3.20. Let the numerator N (λ) of a normal matrix (3.28) be given in the form

α(λ) 1 N (λ) = g(λ)φ(λ) ψ(λ) , (3.38) L(λ) β(λ) where g(λ) is a scalar polynomial, which is equal to the GCD of the elements of N (λ). The polynomial matrices φ(λ), ψ(λ) are unimodular and L(λ) is an (n − 1) × (m − 1) polynomial matrix. Furthermore, let

α(λ) = αm (λ) . . . α2 (λ) ,

β (λ) = β2 (λ) . . . βn (λ) be row vectors. Then L(λ) = β(λ)α(λ) + d(λ)G2 (λ) ,

(3.39)

where G2 (λ) is an (n − 1) × (m − 1) polynomial matrix. Hereby, we have N (λ) = g(λ)P (λ)Q (λ) + g(λ)d(λ)G(λ) ,

where P (λ) = φ(λ)

1 , β(λ)

Q (λ) = α(λ) 1 ψ(λ) ,

(3.40)

(3.41)

120


and moreover

O1,m−1 0 G(λ) = φ(λ) ψ(λ) . G2 (λ) On−1,1

(3.42)

Doing so, Matrix (3.28) takes the S-representation A(λ) =

g(λ)P (λ)Q (λ) + g(λ)G(λ) . d(λ)

Proof. Introduce the unimodular matrices

1 O1,n−1 Om−1,1 Im−1 Nβ (λ) = . , Nα (λ) = −β(λ) In−1 1 −α(λ) By direct calculation, we obtain

1 O1,n−1 , Nβ−1 (λ) = β(λ) In−1

Nα−1 (λ) =

α(λ) 1 . Im−1 Om−1,1

(3.43)

(3.44)

(3.45)

Easily,

1 O1,m−1 α(λ) 1 Nβ (λ) Nα (λ) = = B(λ) ˜ L(λ) β(λ) On−1,1 R(λ)

(3.46)

is established, where ˜ R(λ) = L(λ) − β(λ)α(λ) .

(3.47)

The polynomials g(λ), d(λ) are coprime, otherwise the fraction (3.28) would be reducible. Thereby, all minors of second order of the matrix

α(λ) 1 Ng (λ) = φ(λ) ψ(λ) (3.48) L(λ) β(λ) are divisible by d(λ). With regard to Ng (λ) = φ(λ)Nβ−1 (λ)B(λ)Nα−1 (λ)ψ(λ) and the observation that the matrices φ(λ)Nβ−1 (λ) and Nα−1 (λ)ψ(λ) are unimodular, we realise that the matrices Ng (λ) and B(λ) are equivalent. Since all minors of second order of the matrix B(λ) are divisible by d(λ), we get ˜ immediately that all elements of the matrix R(λ) are also divisible by d(λ), which runs into the equality ˜ R(λ) = L(λ) − β(λ)α(λ) = d(λ)G2 (λ) that is equivalent to (3.39). Inserting the relation L(λ) = β(λ)α(λ) + d(λ)G2 (λ)


into (3.38), we get N (λ) = g(λ)φ(λ) = g(λ)φ(λ)

121

α(λ) 1 ψ(λ) β(λ)α(λ) + d(λ)G2 (λ) β(λ)

(3.49)

α(λ) 1 O1,m−1 0 ψ(λ) + g(λ)d(λ)φ(λ) ψ(λ) . β(λ)α(λ) β(λ) G2 (λ) On−1,1

Using

α(λ) 1 1 α(λ) 1 , = β(λ)α(λ) β(λ) β(λ)

we generate from (3.49) Formulae (3.40)–(3.42). Relation (3.43) is held by substituting (3.49) into (3.28). Remark 3.21. Let

φ(λ) = φ1 (λ) . . . φn (λ) ,

(λ) , ψ (λ) = ψ1 (λ) . . . ψm where φi (λ), (i = 1, . . . , n), ψi (λ), (i = 1, . . . , m) are columns or rows, respectively. Then from (3.41), it follows P (λ) = φ1 (λ) + φ2 (λ)β2 (λ) + . . . + φn (λ)βn (λ) , Q(λ) = ψ1 (λ)αm (λ) + ψ2 (λ)αm−1 (λ) + . . . + ψm (λ) .

Remark 3.22. Equation (3.42) delivers rank G(λ) ≤ min{n − 1, m − 1} .

6. Example 3.23. Generate the S-representation of a normal matrix (3.28) with

−λ + 1 2 1 N (λ) = , d(λ) = (λ − 1)(λ − 2) . 0 (λ + 1)(λ − 2) λ − 2 In the present case, the matrix A(λ) possesses only the two single poles λ1 = 1 and λ2 = 2. Hereby, we have

0 2 1 , rank N (λ1 ) = 1 , N (λ1 ) = 0 −2 −1

−1 2 1 , rank N (λ2 ) = 1 , N (λ2 ) = 0 0 0

122


that’s why the matrix A(λ), owing to Theorem 3.6 is normal. For construction of the S-representation, Theorem 3.20 is used. In the present case, we have g(λ) = 1,

φ(λ) = I2 ,

β(λ) = λ − 2,

ψ(λ) = I3 , α(λ) = −λ + 1 2 ,

L(λ) = 0 (λ + 1)(λ − 2) .

Applying (3.39), we produce

L(λ) − β(λ)α(λ) = d(λ) d(λ) = d(λ) 1 1 ,

and therefore

G2 (λ) = 1 1 .

On the basis of (3.41), we find

1 P (λ) = , λ−2 Moreover, due to (3.42), we get

Q (λ) = −λ + 1 2 1 .

000 G(λ) = 110

.

With these results, we obtain

1

1−λ 2 1 λ−2 000 A(λ) = + . 110 (λ − 1)(λ − 2) Regarding deg P (λ) = deg Q(λ) = 1, the generated S-representation is minimal.

3.5 Inverses of Characteristic Matrices of Jordan and Frobenius Matrices 1. In this section S-representations for matrices of the shape (λIp − A)−1 = A−1 λ will be constructed, where A = Jp (a) is a Jordan block (1.76) or A = AF is a Frobenius matrix (1.94). In the first case, the matrix Aλ is called the characteristic Jordan matrix , and in the second case the characteristic Frobenius matrix. 2.

Consider the upper Jordan block with the eigenvalue a ⎤ ⎡ a 1 ... 0 0 ⎢0 a ... 0 0⎥ ⎥ ⎢ ⎥ ⎢ Jp (a) = ⎢ ... ... . . . ... ... ⎥ ⎥ ⎢ ⎣0 0 ... a 1⎦ 0 0 ... 0 a

(3.50)

3.5 Inverses of Characteristic Matrices of Jordan and Frobenius Matrices

123

and the corresponding characteristic matrix Jp (λ, a) = λIp − Jp (a) . Now, a direct calculation of the adjoint matrix yields ⎤ ⎡ (λ − a)p−1 (λ − a)p−2 . . . λ − a 1 ⎢ λ−a ⎥ 0 (λ − a)p−1 . . . (λ − a)2 ⎥ ⎢ ⎥ ⎢ . . . .. .. .. .. .. adj Jp (λ, a) = ⎢ ⎥ . (3.51) . . ⎥ ⎢ ⎣ 0 0 . . . (λ − a)p−1 (λ − a)p−2 ⎦ 0 0 ... 0 (λ − a)p−1 Matrix (3.51) has the shape (3.38) with g(λ) = 1, φ(λ) = ψ(λ) = Ip ,

α(λ) = (λ − a)p−1 . . . λ − a ,

β (λ) = (λ − a) . . . (λ − a)p−1 . Consistent with Theorem 3.20, we get adj Jp (λ, a) = P (λ)Q (λ) + d(λ)G(λ) , where

P (λ) = 1 β (λ) ,

(3.52)

Q (λ) = α(λ) 1

and d(λ) = det Jp (λ, a) = (λ − a)p . For determining the polynomial matrix G(λ), take care of ⎡ ⎤ 1 ⎢ λ − a ⎥

⎢ ⎥ P (λ)Q (λ) = ⎢ ⎥ (λ − a)p−1 . . . (λ − a) 1 .. ⎣ ⎦ . (λ − a)p−1 ⎡

(λ − a)p−1 (λ − a)p−2 . . .

⎢ ⎢ (λ − a)p (λ − a)p−1 ⎢ =⎢ .. .. ⎢ . . ⎣ 2p−2 (λ − a)2p−3 (λ − a) As a result, we obtain

... ..

.

1 λ−a .. .

. . . (λ − a)p−1

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎦

124


⎡

0

0

...

0

0

⎤

⎥ ⎢ ⎢ −(λ − a)p 0 ... 0 0⎥ ⎥ ⎢ ⎥ ⎢ 0 0⎥ −(λ − a)p+1 −(λ − a)p . . . adj Jp (λ, a) − P (λ)Q (λ) = ⎢ ⎥ ⎢ ⎢ .. .. .. .. ⎥ .. ⎢ . . . . .⎥ ⎦ ⎣ −(λ − a)2p−2 −(λ − a)2p−3 . . . −(λ − a)p 0 ⎤ ⎡ 0 0 ... 0 0 ⎥ ⎢ ⎢ 1 0 ... 0 0⎥ ⎥ ⎢ ⎥ ⎢ p⎢ λ − a 1 . . . 0 0 ⎥. = −(λ − a) ⎢ ⎥ ⎢ .. .. . . .. .. ⎥ ⎢ . . .⎥ . . ⎦ ⎣ (λ − a)p−2 (λ − a)p−3 . . . 1 0 Substituting this into (3.52) yields ⎡ 0 ⎢ ⎢ 1 ⎢ ⎢ λ−a G(λ) = − ⎢ ⎢ ⎢ .. ⎢ . ⎣

0

... 0 0

0 1 .. .

(λ − a)p−2 (λ − a)p−3

⎤

⎥ ... 0 0⎥ ⎥ ⎥ ... 0 0⎥, ⎥ . . .. .. ⎥ . . .⎥ ⎦ ... 1 0

and applying (3.43) delivers the S-representation Jp−1 (λ, a) =

P (λ)Q (λ) + G(λ) . (λ − a)p

Since deg P (λ) < p and deg Q(λ) < p, the produced S-representation is minimal. 3.

Now consider the problem to find the S-representation for −1 A−1 , λF = (λIp − AF )

where

⎡

0 0 .. .

1 0 .. .

0 1 .. .

⎢ ⎢ ⎢ AF = ⎢ ⎢ ⎣ 0 0 0 −dp −dp−1 −dp−2

⎤ ... 0 ... 0 ⎥ ⎥ . . .. ⎥ . . ⎥ ⎥ ... 1 ⎦ . . . −d1

(3.53)

is the lower Frobenius normal form of dimension p×p. Its characteristic matrix has obviously the shape

3.5 Inverses of Characteristic Matrices of Jordan and Frobenius Matrices

⎡

AλF

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

λ −1 0 λ .. .. . . 0 0 dp dp−1

125

⎤

0 ... 0 −1 . . . 0 ⎥ ⎥ .. ⎥ .. .. . . . ⎥ ⎥. ⎥ .. . −1 ⎦ 0 dp−2 . . . λ + d1

The adjoint matrix for (3.54) is calculated by ⎡ d1 (λ) d2 (λ) . . . dp−1 (λ) ⎢ ⎢ adj(λIp − AF ) = ⎢ ⎣ LF (λ)

(3.54)

1 λ .. .

⎤ ⎥ ⎥ ⎥. ⎦

(3.55)

λp−1 Here and in what follows, we denote d(λ) = λp + d1 λp−1 + . . . + dp−1 λ + dp , d1 (λ) = λp−1 + d1 λp−2 + . . . + dp−2 λ + dn−1 , d2 (λ) = λp−2 + d1 λp−3 + . . . + dp−2 , .. .. . .

(3.56)

dp−1 (λ) = λ + d1 , dp (λ) = 1 , and LF (λ) is a certain (p − 1) × (m − 1) polynomial matrix. Relation (3.55) with Theorem 3.20 implies adj(λIp − AF ) = PF (λ)QF (λ) + d(λ)GF (λ) ,

(3.57)

PF (λ) = 1 λ . . . λp−1 ,

QF (λ) = d1 (λ) . . . dp−1 (λ) 1 .

(3.58)

where

It remains to calculate the matrix GF (λ). Denote adj(λIp − AF ) = [aik (λ)], GF (λ) = [gik (λ)],

PF (λ)QF (λ) = [bik (λ)],

(i, k = 1, . . . , p) ,

(3.59)

then from (3.57), we obtain bik (λ) = −d(λ)gik (λ) + aik (λ) . Per construction, deg d(λ) = p, deg aik (λ) ≤ p − 1. Bringing this face to face with (1.6), we recognise that −gik (λ) is the integral part and aik (λ) is the

126


rest, when dividing the polynomial bik (λ) by d(λ). Utilising (3.58), we arrive at (3.60) bik (λ) = λi−1 dk (λ) . Due to deg dk (λ) = p − k, we obtain deg bik (λ) = p − k + i − 1. Thus, for k ≥ i, we get gik (λ) = 0. Substituting i = k + , ( = 1, . . . , p − k) and taking into account (3.60) and (3.56), we receive bk+,k (λ) = λ−1 d(λ) + dk (λ) , where deg dk (λ) < p. From this, we read gik (λ) = −λ−1 ,

(i = k + ; = 1, . . . , p − k) .

Altogether, this leads to ⎡

0 0 0 ⎢ −1 0 0 ⎢ ⎢ −λ −1 0 ⎢ GF (λ) = − ⎢ .. .. ⎢ .. ⎣ . . . p−2 p−3 −λ −λ −λp−4

... ... ... .. . −...

⎤ 0 0 0 0⎥ ⎥ ⎥ 0 0⎥ ⎥ .. .. ⎥ . .⎦ −1 0

(3.61)

and the wanted S-representation (λIp − AF )−1 =

PF (λ)QF (λ) + GF (λ) . d(λ)

(3.62)

Per construction, deg PF (λ) < p, deg QF (λ) < p is valid, so the produced S-representation (3.62) is minimal.

3.6 Construction of Simple Jordan Realisations 1.

Suppose the strictly proper normal rational matrix A(λ) =

N (λ) , d(λ)

deg d(λ) = p

(3.63)

and let (A0 , B0 , C0 ) be one of its simple realisations. Then any simple realisation of the matrix A(λ) has the form (QA0 Q−1 , QB0 , C0 Q−1 ) with a certain non-singular matrix Q. Keeping in mind that all simple matrices of the same dimension with the same characteristic polynomial are similar, the matrix Q can be selected in such a way that the equation QA0 Q−1 = A1 for a cyclic matrix A1 fulfills a prescribed form. Especially, we can achieve A1 = J, where J is a Jordan matrix (1.97), and every distinct root of the polynomials d(λ) is configured to exactly one Jordan block. The corresponding simple realisation (J, BJ , CJ ) is named a Jordan realisation. But, if we choose A1 = AF , where

3.6 Construction of Simple Jordan Realisations

127

AF is a Frobenius matrix (3.53), then the corresponding simple realisation (AF , BF , CF ) is called a Frobenius realisation. These two simple realisations are said to be canonical. In this section the question is considered, how to produce a Jordan realisation from a given normal rational matrix in S-representation. 2. Suppose the normal strictly proper rational matrix (3.63) in Srepresentation P (λ)Q (λ) + G(λ) . (3.64) A(λ) = d(λ) Then the following theorem gives the answer to the question, how to construct a simple Jordan realisation. Theorem 3.24. Suppose the normal n × m matrix A(λ) in S-representation (3.64) with d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq ,

µ1 + . . . + µq = p .

(3.65)

Then a simple Jordan realisation of the matrix A(λ) is attained by the following steps: 1) For each j, (j = 1, . . . , q) calculate the vectors dk−1 P (λ) 1 , Pjk = (k − 1)! dλk−1 λ=λj

Qjk

1 = (k − 1)!

(k = 1, . . . , µj )

dk−1 Q(λ)(λ − λj )µj λ=λ . dλk−1 d(λ) j

2) For each j, (j = 1, . . . , q) build the matrices

P˜j = Pj1 Pj2 . . . Pjµj (n × µj ) , ⎡ ⎤ Qjµj ⎢ Qj,µ −1 ⎥ j ⎢ ⎥ ˜ (µj × m) . Qj = ⎢ ⎥ .. ⎣ ⎦ . Qj1

(3.66)

(3.67)

3) Put together the matrices

PJ = P˜1 P˜2 . . . P˜q (n × p) , ⎡˜ ⎤ Q1 ˜ ⎥ ⎢Q ⎢ 2⎥ (p × m) . QJ = ⎢ . ⎥ ⎣ .. ⎦ ˜q Q

(3.68)

128


4) Build the simple Jordan matrix J (1.88) according to the polynomial (3.65). Then (3.69) A(λ) = PJ (λIp − J)−1 QJ , and the realisation (J, QJ , PJ ) is a simple Jordan realisation. The proof of Theorem 3.24 is prepared by two Lemmata. Lemma 3.25. Let Jµ (a) be an upper Jordan block (3.50), and Jµ (λ, a) be its corresponding characteristic matrix. Introduce for fixed µ the µ × µ matrices Hµi , (i = 0, . . . , µ − 1) of the following shape: ⎤ ⎡ 0 1 0 ... 0 ⎢0 0 1 ... 0⎥ ⎥ ⎢

⎢ .. .. . . . . .. ⎥ O1,µ−1 1 ⎢ , . . . , H = Hµ0 = Iµ , Hµ1 = ⎢ . . . . . ⎥ . µ,µ−1 ⎥ Oµ−1,µ−1 Oµ−1,1 ⎥ ⎢ . ⎣ 0 0 0 .. 1 ⎦ 0 0 0 ... 0 (3.70) Then, adj [λIµ − Jµ (a)]

(3.71) = (λ − a)µ−1 Hµ0 + (λ − a)µ−2 Hµ1 + . . . + (λ − a)Hµ,µ−2 + Hµ,µ−1 .

Proof. The proof deduces immediately from (3.51) and (3.70). Lemma 3.26. Assume the constant n × µ and µ × m matrices U and V with ⎡ ⎤ vµ

⎢ .. ⎥ U = u1 . . . uµ , V = ⎣ . ⎦ , v1

where ui , vi , (i = 1, . . . , µ) are columns or rows, respectively. Then the equation U adj [λIµ − Jµ (a)] V = L1 + (λ − a)L2 + . . . + (λ − a)µ−1 Lµ

(3.72)

is true, where L1 = u1 v1 , L2 = u1 v2 + u2 v1 , .. . Lµ =

u1 vµ

+

u2 vµ−1

(3.73) + ... +

uµ v1

= UV .

Proof. The proof follows directly by inserting (3.71) and (3.70) into the left side of (3.72).


129

Remark 3.27. Concluding in reverse direction, it comes out that, under assumption (3.73), the right side of Relation (3.72) is equal to the left one. Proof (of Theorem 3.24). From (3.64), we obtain A(λ) = where

N (λ) , d(λ)

(3.74)

N (λ) = P (λ)Q (λ) + d(λ)G(λ) .

(3.75)

Since Matrix (3.74) is strictly proper, it can be developed into partial fractions (2.98). Applying (3.65) and (2.96)–(2.97), this expansion can be expressed in the form q A(λ) = Aj (λ) , (3.76) j=1

where Aj (λ) =

Aj,µj Aj1 Aj2 , + +. . .+ (λ − λj )µj (λ − λj )µj −1 (λ − λj )

(j = 1, . . . , q) . (3.77)

The constant matrices Ajk , (k = 1, . . . , µj ) appearing in (3.77) are determined by the Taylor expansion at the point λ = λj : N (λ)(λ − λj )µj = Aj1 +(λ−λj )Aj2 +. . .+(λ−λj )µj −1 Aj,µj +(λ−λj )µj Rj (λ), d(λ) (3.78) where Rj (λ) is a rational matrix that is analytical in the point λ = λj . Utilising (3.74), (3.75), we can write N (λ)(λ − λj )µj = P (λ)Qj (λ) + (λ − λj )µj G(λ) , d(λ) where Qj (λ) =

Q(λ) , dj (λ)

dj (λ) =

(3.79)

d(λ) . (λ − λj )µj

Conformable with (3.78), for the determination of the matrices Ajk , (k = 1, . . . , µj ), we have to find the first µj terms of the separation on the right side of (3.79) in the Taylor series. Obviously, the matrices Ajk , (k = 1, . . . , µj ) do not depend on the matrix G(λ). Near the point λ = λj , suppose the developments P (λ) = Pj1 + (λ − λj )Pj2 + . . . + (λ − λj )µj −1 Pj,µj + . . . , Qj (λ) = Qj1 + (λ − λj )Qj2 + . . . + (λ − λj )µj −1 Qj,µj + . . . , where the vectors Pjk and Qjk are determined by (3.66). Then we get

130


P (λ)Qj (λ) = Pj1 Qj1 + (λ − λj )(Pj1 Qj2 + Pj2 Qj1 ) + + (λ − λj )2 (Pj1 Qj3 + Pj2 Qj2 + Pj3 Qj1 ) + . . . . Comparing this with (3.78) delivers Aj1 Aj2 .. .

= Pj1 Qj1 , = Pj1 Qj2 + Pj2 Qj1 , .. .. . .

Aj,µj = Pj1 Qj,µj + Pj2 Qj,µj −1 + . . . + Pj,µj Qj1 . Substituting this into (3.77) leads to

Aj (λ) = (λ − λj )−µj Aj1 + (λ − λj )Aj2 + . . . + (λ − λj )µj −1 Aj,µj = (λ − λj )−µj Pj1 Qj1 + (λ − λj )(Pj1 Qj2 + Pj2 Qj1 ) + . . . . . . + (λ − λj )µj −1 (Pj1 Qj,µj + . . . + Pj,µj Qj1 )

.

Taking into account Remark 3.27, we obtain from the last expression

˜j Aj (λ) = (λ − λj )−µj P˜j adj λIµj − Jµj (λj ) Q

−1 ˜j , Q = P˜j λIµj − Jµj (λj ) ˜ j are committed by (3.67). From the last equations where the matrices P˜j , Q and (3.76), it follows A(λ) =

q

−1 ˜ j = PJ (λIp − J)−1 QJ , P˜j λIµj − Jµj (λj ) Q

j=1

where PJ , QJ are the matrices (3.68) and

λIp − J = diag λIµ1 − Jµ1 (λ1 ), λIµ2 − Jµ2 (λ2 ), . . . , λIµq − Jµq (λq ) , which is equivalent to Formula (3.69). Since in the present case the p × p matrix J possesses the minimal possible dimension, the realisation (J, BJ , CJ ) is minimal. Therefore, the pair (J, BJ ) is controllable, and the pair [J, CJ ] is observable. Finally, per construction, the matrix J is cyclic. Hence (J, BJ , CJ ) is a simple Jordan realisation of the matrix A(λ).

3. Example 3.28. Find the the Jordan realisation of the strictly proper normal matrix


(λ − 1)2 1 0 λ−2 . A(λ) = 2 (λ − 1) (λ − 2) Using the notation in Section 3.4 g(λ) = 1,

β(λ) = λ − 2,

φ(λ) = ψ(λ) = I2 ,

d(λ) = (λ − 1)2 (λ − 2),

131

L(λ) = 0,

(3.80)

α(λ) = (λ − 1)2 ,

λ1 = 1, λ2 = 2

is performed, and applying (3.41) yields

1 P (λ) = , Q (λ) = (λ − 1)2 1 . λ−2 For constructing a simple Jordan realisation, we have to find the vectors (3.66). For the root λ1 = 1, we introduce the notation

Q (λ) 1 (λ − 1)2 1 = P1 (λ) = P (λ) = , Q1 (λ) = . λ−2 λ−2 λ−2 λ−2 Using (3.66), we obtain P11 = P (λ) |λ=λ1

1 = , −1

P12

dP1 (λ) 0 = = , 1 dλ λ=λ1

Q12

dQ1 (λ) = = 0 −1 . dλ λ=λ1

and furthermore Q11

=

Q1 (λ) |λ=λ1

= 0 −1 ,

Then (3.67) ensures

P˜1 =

1 0 , −1 1

0 −1 ˜ Q1 = . 0 −1

For the single root λ2 = 2, we denote

1 Q (λ) 1 1 P2 (λ) = P (λ) = , Q2 (λ) = = λ−2 (λ − 1)2 (λ − 1)2 and for λ = 2, we get P21 = P (λ) |λ=λ2

1 = , 0

Applying (3.68) yields

1 0 1 PJ = , −1 1 0

Q21 = Q2 (λ) |λ=λ2 = 1 1 . ⎡

⎤ 0 −1 QJ = ⎣ 0 −1 ⎦ . 1 1

Thus the simple Jordan realisation of Matrix (3.80) possesses the shape (J, BJ , CJ ) with ⎡ ⎤ 110 J = ⎣0 1 0⎦. 002

132


3.7 Construction of Simple Frobenius Realisations 1. For constructing a simple Jordan-realisation, the roots of the polynomials d(λ) have to be calculated, and this task can be connected with honest numerical problems. From this point of view, it is much easier to produce the realisation (AF , BF , CF ), where the matrix AF has the Frobenius normal form (3.53), that turns out to be the accompanying matrix for the polynomial d(λ) in (3.56). The assigned characteristic matrix to AF has the shape (3.54), and the S-representation of the matrix (λIp − AF )−1 is determined by Relations (3.62). For a given realisation (AF , BF , CF ), the transfer matrix A(λ) has the shape (3.63) with N (λ) = CF adj(λIp − AF )BF . Taking advantage from (3.57), the last equation can be represented in the form ˜ (3.81) N (λ) = P (λ)Q (λ) + d(λ)G(λ) with

⎡ ⎢ ⎢ P (λ) = CF ⎢ ⎣

1 λ .. . λn−1

⎤ ⎥ ⎥ ⎥, ⎦

Q (λ) = d1 (λ) . . . dn−1 (λ) 1 BF ,

(3.82)

G(λ) = CF GF (λ)BF .

Inserting (3.81) and (3.82) into (3.63), we get the wanted S-representation. Per construction, Relation (3.36) is fulfilled, i.e. the obtained S-representation is minimal. Therefore, Formulae (3.81), (3.82) forth the possibility of direct transfer from the Frobenius realisation to the corresponding minimal S-representation of its transfer matrix. 2. Example 3.29. Assume the Frobenius realisation with ⎤ ⎤ ⎡ ⎡

0 1 0 10 12 0 . AF = ⎣ 0 0 1 ⎦ , BF = ⎣ 2 3 ⎦ , CF = 3 1 −1 −2 −1 −1 01

(3.83)

Here AF is the accompanying matrix for the polynomial d(λ) = λ3 + λ2 + λ + 2 , so we get the coefficients d1 = 1, d2 = 1, d3 = 2. In this case, the polynomials (3.56) have the shape d1 (λ) = λ2 + λ + 1,

d2 (λ) = λ + 1 .

Hence recall (3.58) for the considered example, we receive

3.7 Construction of Simple Frobenius Realisations

PF (λ) = 1 λ λ2 ,

133

QF (λ) = λ2 + λ + 1 λ + 1 1 .

Then using (3.82), we obtain ⎡ ⎤

1

12 0 ⎣ ⎦ 2λ + 1 λ P (λ) = = , 3 1 −1 −λ2 + λ + 3 λ2 (3.84) ⎡ ⎤ 1 0

Q (λ) = λ2 + λ + 1 λ + 1 1 ⎣ 2 3 ⎦ = λ2 + 3λ + 3 3λ + 4 . 01 Moreover, a direct calculation with the help of (3.61) yields ⎡ ⎤⎡ ⎤

0 0 0

10 12 0 ⎣ −2 0 ⎦ ⎣ ⎦ −1 0 0 23 = G(λ) = , 3 1 −1 λ+1 3 −λ −1 0 01 which together with (3.84) gives the result P (λ)Q (λ) + G(λ) d(λ)

2 2λ + 1

λ + 3λ + 3 3λ + 4 2 −λ + λ + 3 −2 0 = + . λ+1 3 λ3 + λ2 + λ + 2

A(λ) =

3. It is remarkable that there exists a rather easy way from a minimal Srepresentation (3.64) to the matrix A(λ) of the simple Frobenius realisation. Theorem 3.30. Let for the strictly proper normal n×m matrix A(λ) be given the minimal S-representation (3.64), where d(λ) = λs + d1 λs−1 + . . . + ds .

(3.85)

Since the S-representation (3.64) is minimal, it follows P (λ) = N1 + N2 λ + . . . Ns λs−1 , Q(λ) = M1 + M2 λ + . . . Ms λs−1 ,

(3.86)

where the Ni , Mi , (i = 1, . . . , s) are constant vectors of dimensions n × 1 and m × 1, respectively. Introduce the columns B1 , . . . , Bs recursively by B1 = Ms , B2 = Ms−1 − d1 B1 , B3 = Ms−2 − d1 B2 − d2 B1 , .. .

.. .

Bs = M1 − d1 Bs−1 − d2 Bs−2 − . . . − ds−1 B1 .

(3.87)

134


With account to (3.86) and (3.87), build the matrices ⎡ ⎤ B1

⎢ .. ⎥ CF = N1 . . . Ns , BF = ⎣ . ⎦ .

(3.88)

Bs

Then the matrix

˜ A(λ) = CF (λIs − AF )−1 BF ,

(3.89)

where AF is the accompanying Frobenius matrix for the polynomial (3.85), defines the minimal standard realisation of the matrix A(λ), that means, the realisation (AF , BF , CF ) is the simple Frobenius realisation of the matrix A(λ). Proof. Using (3.62), we obtain from (3.89) CF PF (λ)QF (λ)BF ˜ + CF GF (λ)BF . A(λ) = d(λ)

(3.90)

From (3.88) and (3.58), we get CF PF (λ) = N1 + N2 λ + . . . + Ns λs−1 = P (λ) ,

(3.91)

and also QF (λ)BF = d1 (λ)B1 + d2 (λ)B2 + . . . + ds−1 (λ)Bs−1 + Bs ,

where d1 (λ), . . . , ds−1 (λ) are the polynomials (3.56). Substituting (3.56) into the last equation, we find QF (λ)BF = (λs−1 + d1 λs−2 + . . . + ds−1 )B1 + . . . + (λ + d1 )Bs−1 + Bs (3.92) = λs−1 B1 + λs−2 (d1 B1 + B2 ) + . . . + (ds−1 B1 + ds−2 B2 + . . . + Bs )

such that from (3.87), it follows Ms = B1 , Ms−1 = B2 + d1 B1 , Ms−2 = B3 + d1 B2 + d2 B1 , .. .

.. .

(3.93)

M1 = Bs + d1 Bs−1 + d2 Bs−2 + . . . + ds−1 B1 , and from (3.92) with (3.86), we find QF (λ)BF = λs−1 Ms + . . . + λM2 + M1 = Q (λ) . Finally, by virtue of this and (3.91), we produce from (3.90)

(3.94)

3.7 Construction of Simple Frobenius Realisations

P (λ)Q (λ) ˜ A(λ) = + CF GF (λ)BF . d(λ)

135

(3.95)

Comparing this expressions with (3.64) and paying attention to the fact that ˜ the matrices A(λ) and A(λ) are strictly proper, and the matrices G(λ) and CF GF (λ)BF are polynomial matrices, we obtain ˜ A(λ) = A(λ) . Example 3.31. Under the conditions of Example 3.29, we obtain

1 2 0 P (λ) = + λ+ λ2 , 3 1 −1 so that with regard to (3.86), we configure 1 2 N1 = , N2 = , 3 1

N3 =

0 , −1

that agrees with (3.84). Moreover, (3.84) yields 3 3 1 2 Q(λ) = + λ+ λ 4 3 0 and thus

3 , M1 = 4

3 M2 = , 3

1 M3 = . 0

Applying (3.93), we obtain

B1 = M3 = 1 0 ,

B2 = M2 − d1 B1 = 3 3 − 1 0 = 2 3 ,

B3 = M1 − d1 B2 − d2 B1 = 3 4 − 2 3 − 1 0 = 0 1 , that with respect to (3.88) can be written as ⎤ ⎡ ⎤ ⎡ 1 0 B1 BF = ⎣ B2 ⎦ = ⎣ 2 3 ⎦ . B3 0 1 This result is again consistent with (3.83).

It is referred to the fact that analogue formulae to (3.87), (3.93) for realisations with vertical Frobenius matrices were dedicated in different way in [165]. Remark 3.32. It is important that Formulae (3.87), (3.93) only depend on the coefficients of the characteristic polynomial (3.85), but not on its roots, that’s why the practical handling of these formulae is less critical.

136


3.8 Construction of S-representations from Simple Realisations. General Case 1. If the simple realisation (A, B, C) of a normal rational n × m transfer matrix A(λ) is known, then the corresponding S-representation of A(λ) can be build on basis of the general considerations in Section 3.4. Indeed, let the simple realisation (A, B, C) be given, so A(λ) =

N (λ) C adj(λIp − A)B = . det(λIp − A) dA (λ)

(3.96)

is valid, and by equivalence transformations, the representation

α(λ) 1 adj(λIp − A) = φ(λ) ψ(λ) L(λ) β(λ) can be generated with unimodular matrices φ(λ), ψ(λ). Then for constructing the S-representation, Theorem 3.20 is applicable. Using Theorem 3.20, the last equation yields adj(λIp − A) = Pa (λ)Qa (λ) + dA (λ)Ga (λ) that leads to C adj(λIp − A)B = P (λ)Q (λ) + dA (λ)G(λ) , where P (λ) = CPa (λ),

Q(λ) = B Qa (λ),

G(λ) = CGa (λ)B .

The last relations proves to be an S-representation of the matrix (3.96) A(λ) =

P (λ)Q (λ) + G(λ) dA (λ)

that could easily transformed into a minimal S-representation. 2. For calculating the adjoint matrix adj(λIp −A), we can benefit from some general relations in [51]. Assume dA (λ) = λp − q1 λp−1 − . . . − qp . Then the adjoint matrix adj(λIp − A) is determined by the formula adj(λIp − A) = λp−1 Ip + λp−2 F1 + . . . + Fp−1 , where F1 = A − q1 Ip ,

F2 = A2 − q1 A − q2 Ip , . . .

(3.97)

3.8 Construction of S-representations from Simple Realisations. General Case

137

or generally Fk = Ak − q1 Ak−1 − . . . − qk Ip . The matrices F1 , . . . , Fp−1 can be calculated successively by the recursion Fk = AFk−1 − qk Ip ,

(k = 1, 2, . . . , p − 1; F0 = Ip ) .

After this, the solution can be checked by the equation AFp−1 − qp Ip = Opp . 3. Example 3.33. Suppose the simple realisation (A, B, C) with

0 −1 0 1 1 0 A= , B= , C= . 0 1 1 −1 −1 1

(3.98)

In the present case, we have dA (λ) = λ2 − λ so we read p = 2 and q1 = 1, q2 = 0. Using (3.97), we find adj(λI2 − A) = λI2 + F1

−1 −1 . F1 = A − I2 = 0 0

with

Formula (3.97) delivers adj(λI2 − A) = −

−λ + 1 1 , 0 −λ

which is easily produced by direct calculation. Applying Theorem 3.20, we get − adj(λI2 − A) = P (λ)Q (λ) + dA G(λ) , where

P (λ) =

1 , −λ

Q(λ) =

−λ + 1 , 1

G(λ) =

0 0 . −1 0

Therefore, the S-representation of the matrix A(λ) for the realisation (3.98) takes the shape

−1

1 −λ λ+1 00 A(λ) = + . (3.99) 01 λ2 − λ The obtained S-representation is minimal.

138


3.9 Construction of Complete MFDs for Normal Matrices 1.

Let the normal matrix in standard form (2.21) A(λ) =

N (λ) d(λ)

(3.100)

be given with deg d(λ) = p. Then in accordance with Section 3.1, Matrix (3.100) allows the irreducible complete MFD −1 A(λ) = a−1 l (λ)bl (λ) = br (λ)ar (λ)

(3.101)

for which ord al (λ) = ord ar (λ) = Mdeg L(λ) = p . In principle, for building a complete MFD (3.101), the general methods from Section 2.4 can be applied. However, with respect to numerical effort and numeric stability, essentially more effective methods can be developed when we profit from the special structure of normal matrices while constructing complete MFDs. 2. Theorem 3.34. Let the numerator of Matrix (3.100) be brought into the form (3.38)

α(λ) 1 ψ(λ) . (3.102) N (λ) = g(λ)φ(λ) L(λ) β(λ) Then the pair of matrices

d(λ) O1,n−1 −1 al (λ) = φ (λ) , −β(λ) In−1

bl (λ) = al (λ)A(λ)

(3.103)

proves to be a complete LMFD, and the pair of matrices

Om−1,1 Im−1 ar (λ) = ψ −1 (λ) , br (λ) = A(λ)ar (λ) d(λ) −α(λ) is a complete RMFD of Matrix (3.100). Proof. Applying Relations (3.44)–(3.49), Matrix (3.102) is represented in the form

1 O1,n−1 α(λ) 1 1 O1,m−1 N (λ) = g(λ)φ(λ) ψ(λ) β(λ) In−1 On−1,1 d(λ)G2 (λ) Im−1 Om−1,1 and with respect to (3.100), we get

3.9 Construction of Complete MFDs for Normal Matrices

A(λ) = g(λ)φ(λ)

139

⎤

⎡

1 α(λ) 1 1 O1,n−1 ⎣ d(λ) O1,m−1 ⎦ ψ(λ) . Im−1 Om−1,1 β(λ) In−1 On−1,1 G2 (λ)

Multiplying this from left with the matrix al (λ) in (3.103), and considering

1 O1,n−1 d(λ) O1,n−1 d(λ) O1,n−1 = , −β(λ) In−1 β(λ) In−1 On−1,1 In−1 we find out that the product al (λ)A(λ) = g(λ)

O1,m−1 α(λ) 1 ψ(λ) On−1,1 G2 (λ) Im1 Om−1,1

α(λ) 1 = g(λ) ψ(λ) = bl (λ) G2 (λ) Om−1,1 1

(3.104)

proves to be a polynomial matrix. Per construction, we have det al (λ) ≈ d(λ) and ord al (λ) = deg d(λ), that’s why the LMFD is complete. For a right MFD the proof runs analogously.

3. Example 3.35. Let us have a normal matrix (3.100) with

λ 2λ − 3 N (λ) = , d(λ) = λ2 − 4λ + 3 . λ2 + λ + 1 2λ2 − 5 In this case, we find N (λ) = so we get

φ(λ) =

0 1 1 λ+1

0 1 , 1 λ+1

λ−2 1 2λ − 3 λ

ψ(λ) =

01 , 10

01 , 10

g(λ) = 1 ,

and with respect to (3.103), (3.104), we obtain immediately

d(λ) 0 −(λ + 1) 1 −d(λ)(λ + 1) d(λ) al (λ) = = , −λ 1 1 0 λ2 + λ + 1 −λ

1 λ−2 . bl (λ) = al (λ)A(λ) = 0 −1 In the present case, we have deg al (λ) = 3. The degree of the matrix al (λ) can be decreased, if we build the row-reduced form. The extended matrix according to the above pair has the shape

140


−λ3 + 3λ2 + λ − 3 λ2 − 4λ + 3 1 λ − 2 Rh (λ) = . −λ 0 −1 λ2 + λ + 1 Multiplying the matrix Rh (λ) from left with the unimodular matrix

1 λ−4 , φ(λ) = 0.5λ 0.5λ2 − 2λ + 1 we arrive at φ(λ)Rh (λ) =

−2λ − 7 3 1 2 −2.5λ + 1 0.5λ 0.5λ λ − 1

.

This matrix corresponds to the complete LMFD, where

−2λ − 7 3 1 2 , bl (λ) = al (λ) = −2.5λ + 1 0.5λ 0.5λ λ − 1 and the matrix al (λ) is row-reduced. Thus deg al (λ) = 1, and this degree cannot be decreased. 4.

For a known S-representation of the normal matrix A(λ) =

P (λ)Q (λ) + G(λ) , d(λ)

(3.105)

a complete MFD can be built by the following theorem. Theorem 3.36. Suppose the ILMFD and IRMFD P (λ) ˜ = a−1 l (λ)bl (λ), d(λ)

Q (λ) ˜ = br (λ)a−1 r (λ) . d(λ)

(3.106)

Then the expressions −1 ˜ A(λ) = a−1 l (λ) bl (λ)Q (λ) + al (λ)G(λ) = al (λ)bl (λ) , −1 A(λ) = P (λ)˜br (λ) + G(λ)ar (λ) a−1 r (λ) = br (λ)ar (λ)

(3.107)

define a complete MFD of Matrix (3.105). Proof. Due to Remark 3.4, the matrix P (λ)/d(λ) is normal. Therefore, for the ILMFD (3.106) det al (λ) ≈ det d(λ) is true, and the first row in (3.107) proves to be a complete LMFD of the matrix A(λ). In analogy, we realise that a complete RMFD stands in the second row.

3.10 Normalisation of Rational Matrices

141

5. Example 3.37. For Matrix (3.99) in Example 3.33,

−1

λ+1 1 −λ P (λ) Q (λ) = 2 , = 2 d(λ) λ −λ d(λ) λ −λ is performed. It is easily checked, that in this case, we can choose

−2λ −λ −1 al (λ) = , ˜bl (λ) = , λ+1 1 0

−λ λ ˜br (λ) = 1 0 , ar (λ) = −λ 1 and according to (3.107), we build the matrices

−1 0 −1 0 , br (λ) = . bl (λ) = 0 1 1 1

3.10 Normalisation of Rational Matrices 1. During the construction of complete LMFD, RMFD and simple realisations for normal rational matrices, we have to take into account the structural peculiarities, and the equations that exist between their elements. Indeed, even arbitrarily small inaccuracies during the calculation of the elements of a normal matrix (3.100), most likely will lead to a situation, where the divisibility of all minors of second order by the denominator is violated, and the result˜ ing matrix A(λ) is no longer normal. After that, also the irreducible MFD, ˜ built from the matrix A(λ) will not be complete, and the values of ord al (λ) and ord ar (λ) in the configured IMFDs will get too large. Also the matrix A˜ ˜ B, ˜ C) ˜ according to the matrix that is assigned by the minimal realisation (A, ˜ A(λ), would have too high dimension. As a consequence of these errors after transition to an IMFD or to corresponding minimal realisations, we would obtain linear models with totally different dynamic behavior than that of the original object, which is described by the transfer matrix (3.100). 2.

Let us illustrate the above remarks by a simple example.

Example 3.38. Consider the nominal transfer matrix

λ−a 0 0 λ−b , a = b A(λ) = (λ − a)(λ − b)

(3.108)

142


that proves to be normal. Assume that, due to practical calculations, the approximated matrix

λ−a+ 0 0 λ−b ˜ A(λ) = (3.109) (λ − a)(λ − b) is built, that is normal only for = 0. For the nominal matrix (3.108), there exists the simple realisation (A, B, C) with

01 01 a0 , B= , C= . (3.110) A= 0b 10 10 All other simple realisations of Matrix (3.108) are produced from (3.110) by similarity transformations. Realisation (3.110) corresponds to the system of differential equations of second order x˙ 1 = ax1 + u2 x˙ 2 = bx2 + u1 y1 = x2 ,

(3.111)

y2 = x1 .

For = 0, we find the minimal realisation (A , B , C ) for (3.109), where ⎡ ⎤ ⎡ ⎤ 0

a00 ⎢ a−b ⎥ ⎥ , C = 1 0 1 . 0 1 A = ⎣ 0 a 0 ⎦ , B = ⎢ (3.112) ⎣ ⎦ 010 00b 1− 0 a−b All other minimal realisations of Matrix (3.109) are held from (3.112) by similarity transformations. Realisation (3.112) is assigned to the differential equation of third order u1 a−b x˙ 2 = ax2 + u2 x˙ 3 = bx3 + 1 − u1 a−b y1 = x1 + x3 , y2 = x2 .

x˙ 1 = ax1 +

For = 0, these equations do not turn into (3.111), and the component x1 looses controllability. Moreover, for = 0 and a > 0, the object is no more stabilisable, though the nominal object (3.111) was stabilisable. In constructing the MFDs, similarly different solutions are held for = 0 and = 0. Indeed, if the numerator of the perturbed matrix (3.109) is written in Smith canonical form, then we obtain for b − a + = 0


143

λ−a+ 0 0 λ−b λ−a+ = b−λ

−1 b−a+ 1 b−a+

1 0 0 (λ − a + )(λ − b)

λ−a+

Thus, the McMillan canonical form of (3.109) becomes 1 −1 λ−a+ 0 λ − a + b−a+ (λ−a)(λ−b) b−a+ ˜ A(λ) = 1 λ−a+ b − λ b−a+ 0 1 λ−a

b−a+

λ−b b−a+

1

1

λ−b b−a+

.

1

. (3.113)

For = 0, the McMillan denominator ψA˜ (λ) of Matrix (3.109) results to ψA˜ (λ) = (λ − a)2 (λ − b) . Hence the irreducible left MFD is built with the matrices

1

1 (λ − a)(λ − b) 0 b−a+ b−a+ al (λ) = , 0 λ−a λ−b λ−a+ λ−a+

λ−b b−a+ b−a+ bl (λ) = . λ−a+ λ−a+ For = 0 the situation changes. Then from (3.113), it arises

λ−a λ−b 1 1 λ − a − b−a (λ−a)(λ−b) 0 b−a b−a A(λ) = 1 b − λ b−a 0 1 1 1 and we arrive at the LMFD al (λ) =

(λ−a)(λ−b) b−a

λ−b

λ−a bl (λ) =

(λ−a)(λ−b) b−a

b−a

λ−b b−a

1

1

λ−a

,

,

and for that, according to the general theory, we get ord al (λ) = 2.

3. In connection with the above considerations, the following problem arises. Suppose a simple realisation (A, B, C) of dimension n, p, m. Then its assigned (ideal) transfer matrix A(λ) =

N (λ) C adj(λIp − A)B = det(λIp − A) d(λ)

(3.114)

is normal. However, due to inevitable inaccuracies, we could have the real transfer matrix

144


˜ (λ) N ˜ A(λ) = , ˜ d(λ)

(3.115)

which practically always deviates from a normal matrix. Even more, if the random calculation errors are independent, then Matrix (3.115) with probability 1 is not normal. Hence the transition from Matrix (3.115) to its minimal ˜ B, ˜ C) ˜ of dimension n, q, m with q > p, that realisation leads to a realisation (A, means, to an object of higher order with non-predictable dynamic properties. 4. Analogue difficulties arise during the solution of identification problems for linear MIMO systems in the frequency domain [108, 4, 120]. Let for instance the real object be described by the simple realisation (A, B, C). Any identification procedure in the frequency domain will only give an approximate transfer matrix (3.115). Even perfect preparation of the identification conditions cannot avoid that the coefficients of the estimated transfer matrix (3.115) will slightly deviate from the coefficients of the exact matrix (3.114). But this deviation suffices that the probability for Matrix (3.115) to become normal turns to zero. Therefore, the formal transition from Matrix (3.115) to the corresponding minimal realisation will lead to a system of higher order, i.e. the identification problem is incorrectly solved. 5. Situations of this kind also arise during the application of frequency domain methods for design of linear MIMO systems [196, 6, 48, 206, 95], . . . . The algorithm of the optimal controller normally bases on the demand that it is described by a simple realisation (A0 , B0 , C0 ). The design method is usually supplied by numerical calculations, so that the transfer matrix of the optimal controller A˜0 (λ) will practically not be normal. Therefore, the really produced ˜0 , C˜0 ) of the optimal controller will have an increased order. realisation (A˜0 , B Due to this fact, the system with this controller may show a unintentional behavior, especially it might become (internally) unstable. 6. As a consequence of the outlined problems, the following general task is stated [144, 145]. Normalisation problem. Suppose a rational matrix (3.115), the coefficients of which deviate slightly from the coefficients of a certain normal matrix (3.114). Then, find a normal matrix Aν (λ) =

Nν (λ) dν (λ)

the coefficients of which differ only a bit from the coefficients of Matrix (3.115). A possible approach for the solution of the normalisation problem consists in the following reflection. By equivalence transformation, the numerator of the


145

˜ rational matrix A(λ) can be brought into the form (3.102) α ˜ (λ) 1 ˜ ˜ ˜ (λ) = g˜(λ)φ(λ) N ψ(λ) . ˜ ˜ L(λ) β(λ) ˜ Let d(λ) be the denominator of the approximated strictly proper matrix (3.115). Then owing to (3.40), from the above considerations, it follows immediately the representation Aν (λ) =

gˆ(λ)Pν (λ)Qν (λ) + Gν (λ) , dν (λ)

where ˜ dν (λ) = d(λ),

1 ˆ Pν (λ) = φ(λ) ˆ , β(λ)

ˆ ˆ (λ) 1 ψ(λ) Qν (λ) = α

(3.116)

and the matrix Gν (λ) is determined in such a way that the matrix Aν (λ) becomes strictly proper. Example 3.39. Apply the normalisation procedure to Matrix (3.109) for = 0 and b − a + = 0. Notice that

λ−b 1 11 λ−a+ 0 b − a + −1 b−a+ = 10 0 λ−b 0 1 λ − b −λ + b is true, i.e. in the present case, we can choose g˜(λ) = 1, and

˜ β(λ) = −λ + b,

b − a + −1 ˜ φ(λ) = , 0 1

α ˜ (λ) =

λ−b b−a+

11 ˜ ψ(λ) = . 10

Using (3.116), we finally find

b − a + −1 1 λ−a+ = , Pν (λ) = 0 1 −λ + b −λ + b

11

λ−b

λ−b λ−b + 1 b−a+ . Qν (λ) = b−a+ 1 = b−a+ 10 Selecting the denominator dν (λ) = (λ − a)(λ − b), we get

1 −1 −1 Gν (λ) = . b−a+ 1 1

Part II

General MIMO Control Problems

4 Assignment of Eigenvalues and Eigenstructures by Polynomial Methods

In this chapter, and later on if possible, the fundamental results are formulated for real polynomials or real rational matrices, because this case dominates in technical applications, and its handling is more comfortable.

4.1 Problem Statement 1. Suppose the horizontal pair (a(λ), b(λ)) with a(λ) ∈ Rnn [λ], b ∈ Rnm [λ]. For the theory and in many applications, the following problem is important. For a given pair (a(λ), b(λ)), find a pair (α(λ), β(λ)) with α(λ) ∈ Rmm [λ], β(λ) ∈ Rmn [λ] such that the set of eigenvalues of the matrix

a(λ) −b(λ) Q(λ, α, β) = (4.1) −β(λ) α(λ) takes predicted values λ1 , . . . , λq with the multiplicities µ1 , . . . , µq . In what follows, the pair (a(λ), b(λ)) is called the process to control, or shortly the process, and the pair (α(λ), β(λ)) the controller . Matrix (4.1) is designated as the characteristic matrix of the closed loop, or shortly the characteristic matrix. Denote (4.2) d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq , then the problem of eigenvalue assignment is formulated as follows. Eigenvalue assignment. For a given process (a(λ), b(λ)) and prescribed polynomial d(λ), find all controllers (α(λ), β(λ)) that ensure

a(λ) −b(λ) det Q(λ, α, β) = det ≈ d(λ) . (4.3) −β(λ) α(λ) In what follows, the polynomial det Q(λ, α, β) is designated as the characteristic polynomial of the closed loop. For a given process and polynomial

150


d(λ), Relation (4.3) can be seen as an equation depending on the controller (α(λ), β(λ)). 2. Let the just formulated task of eigenvalue assignment be solvable for a given process with a certain polynomial d(λ). Suppose Ωd to be the set of controllers satisfying Equation (4.3). Assume a1 (λ), . . . , an+m (λ) to be the sequence of invariant polynomials of the matrix Q(λ, α, β). In principle, for different controllers in the set Ωd , these sequences will be different, because such a sequence a1 (λ), . . . , an+m (λ) only has to meet the three demands: All polynomials ai (λ) are monic, each polynomial ai+1 (λ) is divisible by ai (λ), and a1 (λ) · · · an+m (λ) = d(λ) . Assume particularly a1 (λ) = a2 (λ) = . . . = an+m−1 (λ) = 1,

an+m (λ) = d(λ).

Then the matrix Q(λ, α, β) is simple. In connection with the above said the following task seems substantiated. Structural eigenvalue assignment. For a given process (a(λ), b(λ)) and scalar polynomial d(λ), the eigenvalue assignment ˜ d ⊂ Ωd , (4.3) should deliver the solution set Ωd . Find the subset Ω where the matrix Q(λ, α, β) possesses a prescribed sequence of invariant polynomials a1 (λ), . . . , an+m (λ). 3. In many cases, it is useful to formulate the control problem more general, when the process to control is described by a PMD τ (λ) = (a(λ), b(λ), c(λ)) ∈ Rnpm [λ] ,

(4.4)

which then is called as a PMD process. Introduce the matrix Qτ (λ, α, β) of the shape ⎤ ⎡ a(λ) Opn −b(λ) Onm ⎦ , (4.5) Qτ (λ, α, β) = ⎣ −c(λ) In Omp −β(λ) α(λ) which is called the characteristic matrix of the closed loop with PMD process. Then the eigenvalue assignment can be formulated as follows: Eigenvalue assignment for a PMD process. For a given PMD process (4.4) and polynomial d(λ) of the form (4.2), find the set of all controllers (α(λ), β(λ)) for which the relation det Qτ (λ, α, β) ≈ d(λ) is fulfilled.

(4.6)

4.2 Basic Controllers

151

4. Let the task of eigenvalue assignment for a PMD process be solvable, and Ωτ be the configured set of controllers (α(λ), β(λ)). For different controllers in Ωτ , the sequence of the invariant polynomials of Matrix (4.5) can be different. Therefore, also the next task is of interest. Structural eigenvalue assignment for a PMD process. For a given PMD process (4.4) and polynomial d(λ), the set of solutions (α(λ), β(λ)) of the eigenvalue assignment (4.6) is designated by Ωτ . ˜τ ⊂ Ωτ , where the matrix Qτ (λ, α, β) possesses a Find the subset Ω prescribed sequence of invariant polynomials. In the present chapter, the general solution of the eigenvalue assignment problem is derived, where the processes are given as polynomial pairs or as PMDs. Moreover, the structure of the set of invariant polynomials is stated, which can be prescribed for this task. Although, the following results are formulated for real matrices, they could be transferred practically without changes to the complex case. In the considerations below, the eigenvalue assignment problem is also called modal control problem, and the determination of the structured eigenvalues is also named structural modal control problem.

4.2 Basic Controllers 1. In this section, the important question is investigated, how to design the controller (α(λ), β(λ)) that Matrix (4.1) becomes unimodular, i.e. a1 (λ) = a2 (λ) = . . . = an+m (λ) = 1 . In the following, such controllers are called basic controllers. It follows directly from Theorem 1.41 that for the existence of a basic controller for the process (a(λ, b(λ)), it is necessary and sufficient that the pair (a(λ), b(λ)) is irreducible, i.e. the matrix

Rh (λ) = a(λ) b(λ) is alatent. If a process meets this condition, it is called irreducible. 2. The next theorem presents a general expression for the set of all basic controllers for a given irreducible pair. Theorem 4.1. Let (α0∗ (λ), β0∗ (λ)) be a certain basic controller for the process (a(λ), b(λ)). Then the set of all basic controllers (α0 (λ), β0 (λ)) is determined by the formula α0 (λ) = D(λ)α0∗ (λ) − M (λ)b(λ) , β0 (λ) = D(λ)β0∗ (λ) − M (λ)a(λ) ,

(4.7)

where M (λ) ∈ Rmn [λ] is an arbitrary, and D(λ) ∈ Rmm [λ] is an arbitrary, but unimodular matrix.

152


Proof. The set of all basic controllers is denoted by R0 , and the set of all pairs satisfying Condition (4.7) by Rp . At first, we will show R0 ⊂ Rp . Let (α0∗ (λ), β0∗ (λ)) be a certain basic controller, and

a(λ) −b(λ) ∗ ∗ (4.8) Ql (λ, α0 , β0 ) = −β0∗ (λ) α0∗ (λ) be its configured characteristic matrix, which is unimodular. Introduce

n

m

αr∗ (λ) br (λ) ∗ ∗ ∗ ∗ Q−1 l (λ, α0 , β0 ) = Qr (λ, α0 , β0 ) = β ∗ (λ) a (λ) r r

(4.9)

n m

.

Owing to the properties of the inverse matrix (2.109), we have the relations a(λ)αr∗ (λ) − b(λ)βr∗ (λ) = In , a(λ)br (λ) − b(λ)ar (λ) = Onm . Let (α0 (λ), β0 (λ)) be any other basic controller, and

a(λ) −b(λ) Ql (λ, α0 , β0 ) = −β0 (λ) α0 (λ) be its configured characteristic matrix. Then due to (4.10),

In Onm ∗ ∗ Ql (λ, α0 , β0 )Qr (λ, α0 , β0 ) = M (λ) D(λ)

(4.10)

(4.11)

(4.12)

where D(λ) = −β0 (λ)br (λ) + α0 (λ)ar (λ) , M (λ) = −β0 (λ)αr∗ (λ) + α0 (λ)βr∗ (λ) . From (4.12) with regard to (4.9), we receive

In Onm Ql (λ, α0∗ , β0∗ ) , Ql (λ, α0 , β0 ) = M (λ) D(λ) which directly delivers Formulae (4.7). Calculating (4.12), we get det D(λ) = det Ql (λ, α0 , β0 ) det Qr (λ, α0∗ , β0∗ ) = const. , i.e. the matrix D is unimodular. Therefore, every basic controller (α0 (λ), β0 (λ) permits a representation (4.7), that’s why R0 ⊂ Rp is true. On the other side, if (4.7) is valid, then from (4.11), it follows

4.2 Basic Controllers

Ql (λ, α0 , β0 ) =

a(λ)

−b(λ)

153

−D(λ)β0∗ (λ) + M (λ)a(λ) D(λ)α0∗ (λ) − M (λ)b(λ)

In Onm a(λ) −b(λ) = M (λ) D(λ) −β0∗ (λ) α0∗ (λ)

In Onm = Q(λ, α0∗ , β0∗ ) . M (λ) D(λ)

Since D(λ) is unimodular, also this matrix has to be unimodular, i.e. R0 ⊂ Rp is proven, and therefore the sets R0 and Rp coincide. 3. As emerges from (4.7), before constructing the set of all basic controllers, at first we have to find one sample of them. Usually, search procedures for such a controller found on the following considerations. Lemma 4.2. For the irreducible process (a(λ), b(λ)), there exist an m × m polynomial matrix ar (λ) and an n × m polynomial matrix br (λ), such that the equation (4.13) a(λ)br (λ) = b(λ)ar (λ) is fulfilled, where the pair [ar (λ), br (λ)] is irreducible. Proof. Since the process (a(λ), b(λ)) is irreducible, there exists a basic controllers (α0∗ (λ), β0∗ (λ)), such that the matrix

a(λ) −b(λ) Q(λ, α0∗ , β0∗ ) = −β0∗ (λ) α0∗ (λ) becomes unimodular. Thus, the inverse matrix ∗

α0r (λ) br (λ) −1 ∗ ∗ Q (λ, α0 , β0 ) = ∗ (λ) ar (λ) β0r is also unimodular. Then from (4.10), it follows Statement (4.13). Moreover, the pair [ar (λ), br (λ)] is irreducible thanks to Theorem 1.32. Remark 4.3. If the matrix a(λ) is non-singular, i.e. det a(λ) ≡/ 0, then there exists the transfer matrix of the processes w(λ) = a−1 (λ)b(λ) . The right side of this equation proves to be an ILMFD of the matrix w(λ). If we consider an arbitrary IRMFD w(λ) = br (λ)a−1 r (λ) , then Equation (4.13) holds, and the pair [ar (λ), br (λ)] is irreducible. Therefore, Lemma 4.2 is a generalisation of this property in case the matrix a(λ) is singular.

154


In what follows, the original pair (a(λ), b(λ)) is called left process model, and any pair [ar (λ), br (λ)] satisfying (4.13), is named right process model. If in this case, the pair [ar (λ), br (λ)] is irreducible, then the right process model should also be designated as irreducible. Lemma 4.4. Let [ar (λ), br (λ)] be an irreducible right process model. Then any pair (α0 (λ), β0 (λ) satisfying the Diophantine equation −β0 (λ)br (λ) + α0 (λ)ar (λ) = P (λ)

(4.14)

with a unimodular matrix P (λ), turns out to be a basic controller for the left process model (a(λ), b(λ)). Proof. Since the pair (a(λ), b(λ)) is irreducible, there exists a vertical pair [αr (λ), βr (λ)] with a(λ)αr (λ) − b(λ)βr (λ) = In . The pair [α0 (λ), β0 (λ)] should satisfy Condition (4.14). Then build the product

αr (λ) br (λ) In Onm a(λ) −b(λ) = , M (λ) P (λ) βr (λ) ar (λ) −β0 (λ) α0 (λ) where M (λ) is a polynomial matrix. Since the matrix P (λ) is unimodular, the matrix on the right side becomes unimodular. Thus, both matrices on the left side are unimodular, and (α0 (λ), β0 (λ)) proves to be a basic controller. Corollary 4.5. An arbitrary pair (α0 (λ), β0 (λ)) satisfying the Diophantine equation −β0 (λ)br (λ) + α0 (λ)ar (λ) = Im , proves to be a basic controller. Remark 4.6. It is easily shown that the set of all pairs (α0 (λ), β0 (λ)) satisfying Equation (4.14) for all possible unimodular matrices P (λ) generate the complete set of basic controllers.

4.3 Recursive Construction of Basic Controllers 1. As arises from Lemma 4.4, a basic controller (α0 (λ), β0 (λ)) can be found as solution of the Diophantine matrix equation (4.14). In the present section, an alternative method for finding a basic controller is described that leads to a recursive solution of simpler scalar Diophantine equations, and does not need the matrices ar (λ), br (λ), arising in (4.14).

4.3 Recursive Construction of Basic Controllers

2.

155

For finding this approach, the polynomial equation n

ai (λ)xi (λ) = c(λ)

(4.15)

i=1

is considered, where the ai (λ), (i = 1, . . . , n), c(λ) are known polynomials, and xi (λ), (i = 1, . . . , n) are unknown scalar polynomials. We will say, that the polynomials ai (λ) are in all coprime, if their monic GCD is equal to 1. The next lemma is a corollary from a more general statement in [79]. Lemma 4.7. A necessary and sufficient condition for the solvability of Equation (4.15) is, that the greatest common divisor of the polynomials ai (λ), (i = 1, . . . , n) is a divisor of the polynomial c(λ). Proof. Necessity: Suppose γ(λ) as a GCD of the polynomials ai (λ). Then ai (λ) = γ(λ)a1i (λ) ,

(i = 1, . . . , n)

(4.16)

where the polynomials a1i (λ), (i = 1, . . . , n) are in all coprime. Substituting (4.16) into (4.15), we obtain n γ(λ) a1i (λ)xi (λ) = c(λ) , i=1

from which it is clear that the polynomial c(λ) must be divisible by γ(λ). Sufficiency: The proof is done by complete induction. The statement should be valid for one n = k > 0, and then it is shown that it is also valid for n = k + 1. Consider the equation k+1

ai (λ)xi (λ) = c(λ) .

(4.17)

i=1

Without loss of generality assume that the coefficients ai (λ), (i = 1, . . . , k + 1) are in all coprime, otherwise both sides of Equation (4.17) could be divided by the common factor. Let κ(λ) be the GCD of the polynomials a1 (λ), . . . , ak (λ). Then (4.16) is true, where the polynomials ai1 (λ), (i = 1, . . . , k) are in all coprime. Herein, the polynomials κ(λ) and ak+1 (λ) are also coprime, otherwise the coefficients of Equation (4.17) would not be in all coprime. Hence the Diophantine equation (4.18) κ(λ)u(λ) + ak+1 (λ)xk+1 (λ) = c(λ) is solvable. Let u ˜(λ), x ˜k+1 (λ) be a certain solution of Equation (4.18). Investigate the equation k a1i (λ)xi (λ) = u ˜(λ) (4.19) i=1

156


which is solvable due to the induction supposition, since all coefficients a1i (λ) are in all coprime. Let x ˜i (λ), (i = 1, . . . , k) be any solution of Equation (4.19). Then applying (4.16) and (4.18), we get k+1

ai (λ)˜ xi (λ) = c(λ)

i=1

this means, the totality of polynomials x ˜i (λ), (i = 1, . . . , k + 1) presents a solution of Equation (4.15). Since the statement of the theorem holds for k = 2, we have proved by complete induction that it is also true for all k ≥ 2.

3. The idea of the proof consists in constructing a solution of (4.15) by reducing the problem to the case of two variables. It can be used to generate successively the solutions of Diophantine equations with several unknowns, where in every step a Diophantine equation with two unknowns is solved. Example 4.8. Find a solution of the equation (λ − 1)(λ − 2)x1 (λ) + (λ − 1)(λ − 3)x2 (λ) + (λ − 2)(λ − 3)x3 (λ) = 1 . Here, the coefficients are in all coprime, though they are not coprime by twos. In the present case, the auxiliary equation (4.18) could be given the shape (λ − 1)u(λ) + (λ − 2)(λ − 3)x3 (λ) = 1 . A special solution takes the form u ˜(λ) = 2 − 0.5λ ,

x ˜3 (λ) = 0.5 .

Equation (4.19) can be represented in the form (λ − 2)x1 (λ) + (λ − 3)x2 (λ) = 2 − 0.5λ . As a special solution for the last equation, we find x ˜1 (λ) = 0.5 ,

x ˜2 (λ) = −1 .

Thus, as a special solution of the original equation, we obtain x ˜1 (λ) = 0.5 ,

x ˜2 (λ) = −1 ,

x ˜3 (λ) = 0.5 .


157

4. Lemma 4.9. Suppose the n × (n + 1) polynomial matrix ⎡ ⎤ a11 (λ) . . . a1,n+1 (λ) ⎢ ⎥ .. A(λ) = ⎣ ... ⎦ ··· .

(4.20)

an1 (λ) . . . an,n+1 (λ) with rank A(λ) = n. Furthermore, denote DA (λ) as the monic GCD of the minors of n-th order of the matrix A(λ). Then there exist scalar polynomials d1 (λ), . . . , dn+1 (λ), such that the (n + 1) × (n + 1) polynomial matrix ⎤ ⎡ A1 (λ) = ⎣

A(λ)

⎦

d1 (λ) . . . dn+1 (λ) satisfies the relation det A1 (λ) = DA (λ) .

(4.21)

Proof. Denote Bi (λ) as that n × n polynomial matrix which is held from A(λ) by cutting its i-th column. Then the expansion of the determinant by the last row delivers n+1 (−1)n+1+i di (λ)∆i (λ) det A1 (λ) = i=1

with ∆i (λ) = det Bi (λ). Thus, Relation (4.21) is equivalent to the Diophantine equation n+1 (−1)n+1+i di (λ)∆i (λ) = DA (λ) . (4.22) i=1

By definition, the polynomial DA (λ) is the GCD of the polynomials ∆i (λ). Hence we can write ∆i (λ) = DA (λ)∆1i (λ) , where the polynomials ∆1i (λ), (i = 1, . . . , n + 1) are in all coprime. By virtue of this relation, Equation (4.22) can take the form n+1

(−1)n+1+i di (λ)∆1i (λ) = 1 .

i=1

Since the polynomials ∆1i (λ) are in all coprime, this equation is solvable thanks to Lemma 4.7. Multiplying both sides by DA (λ), we conclude that Equation (4.22) is solvable.

158


5. Theorem 4.10. [193] Suppose the non-degenerated n × m polynomial matrix ˜ A(λ), m > n + 1, where ⎡ ⎤ a11 (λ) . . . a1n (λ) a1,n+1 (λ) a1,n+2 (λ) . . . a1m (λ) ⎢ ⎥ .. .. .. .. ˜ A(λ) = ⎣ ... ⎦ ··· . . . ··· . an1 (λ) . . . ann (λ) an,n+1 (λ) an,n+2 (λ) . . . anm (λ) ⎤ a1,n+2 (λ) . . . a1m (λ) ⎥ .. .. = ⎣ A(λ) ⎦. . ··· . an,n+2 (λ) . . . anm (λ) ⎡

Assume that the submatrix A(λ) on the left of the line has the form (4.20). ˜ Let DA˜ (λ) be the monic GCD of the minors of n-th order of the matrix A(λ), and DA (λ) the monic GCD of the minors of n-th order of the matrix A(λ). The polynomials d1 (λ), . . . , dn+1 (λ) should be a solution of Equation (4.22). Then the monic GCD of the minors of n-th order of the matrix ⎡ ⎤ a11 (λ) . . . a1n (λ) a1,n+1 (λ) a1,n+2 (λ) . . . a1m (λ) ⎢ .. ⎥ .. .. .. .. ⎢ ⎥ ··· . . . ··· . (4.23) Ad (λ) = ⎢ . ⎥ ⎣ an1 (λ) . . . ann (λ) an,n+1 (λ) an,n+2 (λ) . . . anm (λ) ⎦ d1 (λ) . . . dn (λ) dn+1 (λ) dn+2 (λ) . . . dm (λ) satisfies the condition DAd (λ) = DA˜ (λ) for any polynomials dn+2 (λ), . . . , dn+m (λ). ˜ Corollary 4.11. If the matrix A(λ) is alatent, then also Matrix (4.23) is alatent. 6. Suppose an irreducible pair (a(λ), b(λ)), where a(λ) has the dimension n × n and b(λ) dimension n × m. Then by successive repeating the procedure explained in Theorem 4.10, the unimodular matrix ⎡ ⎤ a11 (λ) . . . a1n (λ) −b11 (λ) . . . −b1m (λ) ⎢ ⎥ .. .. .. .. ⎢ ⎥ . ··· . . ··· . ⎢ ⎥ ⎢ an1 (λ) . . . ann (λ) −bn1 (λ) . . . −bnm (λ) ⎥ ⎥ Ql (λ, α0 , β0 ) = ⎢ ⎢ −β11 (λ) . . . −β1n (λ) α11 (λ) . . . α1m (λ) ⎥ ⎢ ⎥ ⎢ ⎥ .. .. .. .. ⎣ ⎦ . ··· . . ··· . −βm1 (λ) . . . −βmn (λ) αm1 (λ) . . . αmm (λ)

is produced. The last m rows of this matrix present a certain basic controller


⎡ ⎢ α0 (λ) = ⎣

⎤

α11 (λ) . . . α1m (λ) ⎥ .. .. ⎦, . ··· . αm1 (λ) . . . αmm (λ)

⎡ ⎢ β0 (λ) = ⎣

159

⎤

β11 (λ) . . . β1n (λ) ⎥ .. .. ⎦. . ··· . βm1 (λ) . . . βmn (λ)

For a transition to the matrix Ql (λ, α0 , β0 ) in every step essentially a scalar Diophantine equation must be solved with several unknowns of the type (4.15). As shown above, the solution of such equations amounts to the successive solution of simple Diophantine equations of the form (4.18). 7. Example 4.12. Determine a basic controller for the process (a(λ), b(λ)) with

λ−1 λ+1 0 −λ a(λ) = , b(λ) = . (4.24) 0 1 −λ 0 The pair (a(λ), b(λ)) establishes to be irreducible, because the matrix

λ−1 λ+1 0 λ 0 1 λ 0 is alatent, which is easily checked. Hence the design problem for a basic controller is solvable. In the first step, we search the polynomials d1 (λ), d1 (λ), d3 (λ), so that the matrix ⎤ ⎡ λ−1 λ+1 0 1 λ ⎦ A1 (λ) = ⎣ 0 d1 (λ) d2 (λ) d3 (λ) becomes unimodular. Without loss in generality, we assume det A1 (λ) = 1. Thus we arrive at the Diophantine equation λ(λ + 1)d1 (λ) − λ(λ − 1)d2 (λ) + (λ − 1)d3 (λ) = 1 . A special solution of this equation is represented in the form λ 1 +1 . d1 (λ) = , d2 (λ) = 0, d3 (λ) = − 2 2 The not designated polynomial d4 (λ) can be chosen arbitrarily. Take for instance d4 (λ) = 0, so the alatent matrix becomes ⎡ ⎤ λ−1 λ+1 0 λ ⎢ ⎥ ⎢ 1 λ 0⎥ A2 (λ) = ⎢ 0 ⎥. ⎣ 1 ⎦ λ 0 − −1 0 2 2

160


It remains the task to complete A2 (λ) that it becomes a unimodular matrix. For that, we attempt ⎤ ⎡ λ−1 λ+1 0 λ ⎢ 0 1 λ 0 ⎥ ⎥ ⎢ ⎥ ⎢ A3 (λ) = ⎢ 1 ⎥ −λ ⎥ ⎢ 0 − 1 0 ⎦ ⎣ 2 2 d˜1 (λ) d˜2 (λ) d˜3 (λ) d˜4 (λ) with unknown polynomials d˜i (λ), (i = 1, . . . , 4). Assume det A3 (λ) = 1, so we obtain the Diophantine equation λ λ2 λ + 1 d˜1 (λ) − d˜2 (λ) + d˜3 (λ) + d˜4 (λ) = 1 λ 2 2 2 which has the particular solution d˜1 (λ) = d˜2 (λ) = d˜3 (λ) = 0,

d˜4 (λ) = 1 .

In summary, we obtain the unimodular matrix ⎡ λ−1 λ+1 0 ⎢ 0 1 λ ⎢ A3 (λ) = ⎢ λ ⎢ 1 0 − −1 ⎣ 2 2 0 0 0

⎤ λ 0⎥ ⎥ ⎥, ⎥ 0⎦ 1

where we read the basic controller (α0∗ (λ), β0∗ (λ)) with ⎤ ⎡ 1 λ − 0 − 1 0 − ∗ ∗ ⎦, α0 (λ) = ⎣ 2 β0 (λ) = . 2 0 0 0 1 Using this solution and Formula (4.7), we construct the set of all basic controllers for the process (4.24). Example 4.13. Find a basic controller for the process (a(λ), b(λ)) with

λ−1 λ+1 −1 a(λ) = , b(λ) = . (4.25) λ−1 λ+1 0 In the present case the matrix a(λ) is singular, nevertheless, a basic controller can be found because the matrix

λ−1 λ+1 1 Rh (λ) = a(λ) −b(λ) = λ−1 λ+1 0 is alatent. In accordance with the derived methods, we search for polynomials d1 (λ), d2 (λ), d3 (λ) such that the condition

4.4 Dual Models and Dual Bases

⎡

161

⎤

λ−1 λ+1 1 det ⎣ λ − 1 λ + 1 0 ⎦ = 1 d1 (λ) d2 (λ) d3 (λ) is satisfied, which is equivalent to the Diophantine equation −(λ + 1)d1 (λ) + (λ − 1)d2 (λ) = 1 which has the particular solution d1 (λ) = d2 (λ) = −0.5. Thus, we can take α0∗ (λ) = d3 (λ), β0∗ (λ) = 0.5 0.5 , where d3 (λ) is an arbitrary polynomial.

4.4 Dual Models and Dual Bases 1. For the further investigations, we want to modify the introduced notation. The initial irreducible process (a(λ), b(λ)) is written as (al (λ), bl (λ)), and is called, as before, a left process model. Any basic controller α0 (λ), β0 (λ)) is written in the form (α0l (λ), β0l (λ)), and it is called a left basic controller. Hereby, the assigned unimodular matrix (4.8) that presents itself in the form

al (λ) −bl (λ) (4.26) Ql (λ, α0l , β0l ) = −β0l (λ) α0l (λ) is named a left basic matrix. However, if the pair [ar (λ), br (λ)] is an irreducible right process model, then the vertical pair [α0r (λ), β0r (λ)], for which the matrix

α0r (λ) br (λ) Qr (λ, α0r , β0r ) = (4.27) β0r (λ) ar (λ) becomes unimodular, is said to be a right basic controller, and the configured matrix (4.27) is called a right basic matrix. . Using (4.27), we find

α0r (λ) β0r β0r (λ) α0r ar (λ) br (λ) (λ) (λ) Qr (λ, α0r , β0r ) = ≈ ≈ , (λ) α0r (λ) br (λ) ar (λ) ar (λ) br (λ) β0r

2.

where the symbol ≈ stands for the equivalence of the polynomial matri∗ ∗ (λ), β0r (λ)] is any right basic controller, then applying Theoces. Now, if [α0r rem 4.1 and the last relation, the set of all right basic controllers is expressed by the formula ∗ α0r (λ) = α0r (λ)Dr (λ) − br (λ)Mr (λ) , ∗ (λ)Dr (λ) − ar (λ)Mr (λ) , β0r (λ) = β0r

(4.28)

where the m × n polynomial matrix Mr (λ) is arbitrary, and Dr (λ) is any unimodular n × n polynomial matrix.

162

3.


The basic matrices (4.26) and (4.27) are called dual , if the equation Qr (λ, α0r , β0r ) = Q−1 l (λ, α0l , β0l )

(4.29)

holds, or equivalently, if Ql (λ, α0l , β0l )Qr (λ, α0r , β0r ) = Qr (λ, α0r , β0r )Ql (λ, α0l , β0l ) = In+m . (4.30) The processes (al (λ), bl (λ), [ar (λ), br (λ)] configured by Equations (4.29) and (4.30), as well as the basic controllers (α0l (λ), β0l (λ)), [α0r (λ), β0r (λ)] will also be named as dual. From (4.30) and (4.26), (4.27) emerge two groups of equations, respectively for left or right dual models as well as left and right dual basic controllers al (λ)α0r (λ) − bl (λ)β0r (λ) = In ,

al (λ)br (λ) − bl (λ)ar (λ) = Onm ,

−β0l (λ)α0r (λ) + α0l (λ)β0r (λ) = Onm ,

−β0l (λ)br (λ) + α0l (λ)ar (λ) = Im (4.31)

and α0r (λ)al (λ) − br (λ)β0l (λ) = In ,

−α0r (λ)bl (λ) + br (λ)α0l (λ) = Onm ,

β0r (λ)al (λ) − ar (λ)β0l (λ) = Onm ,

−β0r (λ)bl (λ) + ar (λ)α0l (λ) = Im . (4.32) Relations (4.31) and (4.32) are called direct and reverse Bezout identity, respectively [69]. Remark 4.14. The validity of the relations of anyone of the groups (4.31) or (4.32) is necessary and sufficient for the validity of Formulae (4.29), (4.30). Therefore, each of the groups of Relations (4.31) or (4.32) follows from the other one. 4.

Applying the new notation, Formula (4.7) can be expressed in the form

∗ ∗ −β0l (λ) α0l (λ) = Ml (λ) Dl (λ) Ql (λ, α0l , β0l )

al (λ) −bl (λ) = Ml (λ) Dl (λ) , ∗ ∗ (λ) α0l (λ) −β0l

which results in

∗ ∗ Ml (λ) Dl (λ) = −β0l (λ) α0l (λ) Q−1 l (λ, α0l , β0l )

∗ ∗ , β0l ) = −β0l (λ) α0l (λ) Qr (λ, α0l with the dual basic matrix ∗ ∗ Qr (λ, α0r , β0r )=

∗ α0r (λ) br (λ) . ∗ β0r (λ) ar (λ)

(4.33)

4.4 Dual Models and Dual Bases

163

Alternatively, (4.33) can be written in the form Dl (λ) = −β0l (λ)br (λ) + α0l (λ)ar (λ) , ∗ ∗ Ml (λ) = −β0l (λ)α0r (λ) + α0l (λ)β0r (λ) .

Analogously, Formula (4.28) can be presented in the form

α0r (λ) Dr (λ) ∗ ∗ , β0r ) = Qr (λ, α0r , β0r (λ) −Mr (λ) where we derive

α (λ) Dr (λ) ∗ ∗ , β0l ) 0r = Ql (λ, α0l −Mr (λ) β0r (λ)

or Dr (λ) = al (λ)α0r (λ) − bl (λ)β0r (λ) , ∗ ∗ Mr (λ) = β0l (λ)α0r (λ) − α0l (λ)β0r (λ) .

5. Theorem 4.15. Let the left and right irreducible models (al (λ), bl (λ)), [ar (λ), br (λ)] of an object be given, which satisfy the condition al (λ)br (λ) = bl (λ)ar (λ)

(4.34)

and, moreover, an arbitrary left basic controller (α0l (λ), β0l (λ)). Then a necessary and sufficient condition for the existence of a right basic controller [α0r (λ), β0r (λ)] that is dual to the controller (α0l (λ), β0l (λ)), is that the pair (α0l (λ), β0l (λ)) is a solution of the Diophantine equation α0l (λ)ar (λ) − β0l (λ)br (λ) = Im . Proof. The necessity follows from the Bezout identity (4.31). To prove the sufficiency, we notice that due to the irreducibility of the pair (al (λ), bl (λ)), there exists a pair [˜ α0r (λ), β˜0r (λ)], that fulfills the relation al (λ)˜ α0r (λ) − bl (λ)β˜0r (λ) = In .

(4.35)

Now, build the product

α ˜ 0r (λ) br (λ) Onm In al (λ) −bl (λ) = −β0l (λ) α0l (λ) β˜0r (λ) ar (λ) −β0l (λ)˜ α0r (λ) + α0l (λ)β˜0r (λ) Im from which we held

164


where

al (λ) −bl (λ) −β0l (λ) α0l (λ)

α0r (λ) br (λ) In Onm = , Omn Im β0r (λ) ar (λ)

(4.36)

˜ 0r (λ) + br (λ) β0l (λ)˜ α0r (λ) − α0l (λ)β˜0r (λ) , α0r (λ) = α β0r (λ) = β˜0r (λ) + ar (λ) β0l (λ)˜ α0r (λ) − α0l (λ)β˜0r (λ) .

It arises from (4.36), that the last pair is a right basic controller, which is dual to the left basic controller (α0l (λ), β0l (λ)). Remark 4.16. In analogy, it can be shown that for a right basic controller α0l (λ), β˜0l (λ)), if and [α0r (λ), β0r (λ)], there exists a dual left basic controller (˜ only if the relation al (λ)α0r (λ) − bl (λ)β0r (λ) = In

(4.37)

is fulfilled. Hereby, if the pair (˜ α0l (λ), β˜0l ) satisfies the condition α ˜ 0l (λ)ar (λ) − β˜0l (λ)br (λ) = Im , then the formulae

˜ 0l (λ) − β˜0l (λ)α0r (λ) − α ˜ 0l (λ)β0r (λ) bl (λ) , α0l (λ) = α β0l (λ) = β˜0l (λ) − β˜0l (λ)α0r (λ) − α ˜ 0l (λ)β0r (λ) al (λ)

(4.38)

define a left basic controller, that is dual to the controller [α0r (λ), β0r (λ)]. 6. The next theorem supplies a parametrisation of the set of all pairs of dual basic controllers. ∗ ∗ (λ), β0l (λ)) and Theorem 4.17. Suppose two dual basic controllers (α0l ∗ ∗ [α0r (λ), β0r (λ)]. Then the set of all pairs of dual basic controllers (α0l (λ), β0l (λ)), [α0r (λ), β0r (λ)] is determined by the relations ∗ (λ) − M (λ)bl (λ), α0l (λ) = α0l ∗ (λ) − br (λ)M (λ), α0r (λ) = α0r

∗ β0l (λ) = β0l (λ) − M (λ)al (λ) , ∗ β0r (λ) = β0r (λ) − ar (λ)M (λ) ,

(4.39)

where M (λ) is any polynomial matrix of appropriate dimension. Proof. In order to determine the set of all pairs of dual controllers, we at first notice that from (4.7) and (4.28) it follows that the relations ∗ α0l (λ) = Dl (λ)α0l (λ) − Ml (λ)bl (λ),

∗ β0l (λ) = Dl (λ)β0l (λ) − Ml (λ)al (λ) ,

∗ (λ)Dr (λ) − br (λ)Mr (λ), α0r (λ) = α0r

∗ β0r (λ) = β0r (λ)Dr (λ) − ar (λ)Mr (λ) (4.40)

4.5 Eigenvalue Assignment for Polynomial Pairs

165

hold, from which we get

In Onm ∗ ∗ , β0l ) Ql (λ, α0l Ml (λ) Dl (λ)

Dr (λ) Onm ∗ ∗ Qr (λ, α0r , β0r ) = Qr (λ, α0r , β0r ) . −Mr (λ) Im Ql (λ, α0l , β0l ) =

(4.41)

For the duality of the controllers (α0l (λ), β0l (λ)) and [α0r (λ), β0r (λ)], it is necessary and sufficient that the matrices (4.41) satisfy Relation (4.29). ∗ ∗ (λ), β0l (λ)) and But from (4.41), owing to the duality of the controllers (α0l ∗ ∗ [α0r (λ), β0r (λ)], we get

Dr (λ) Onm Ql (λ, α0l , β0l )Qr (λ, α0r , β0r ) = Ml (λ)Dr (λ) − Dl (λ)Mr (λ) Dl (λ) and Relation (4.29) is fulfilled, if and only if Dr (λ) = In ,

Dl (λ) = Im ,

Ml (λ) = Mr (λ) = M (λ) .

Corollary 4.18. Each solution of Equation (4.35) uniquely corresponds to a right dual controller, and each solution of Equation (4.36) uniquely corresponds to a left dual controller. Remark 4.19. Theorems 4.15 and 4.17 indicate that the pairs of left and right process models, used for building the dual basic controllers, may be chosen arbitrarily, as long as Condition (4.34) holds. If the pairs (al (λ), bl (λ)), [ar (λ), br (λ)] satisfy Condition (4.34), and the n × n polynomial matrix p(λ) and the m × m polynomial matrix q(λ) are unimodular, then the pairs (p(λ)al (λ), p(λ)bl (λ)), [ar (λ)q(λ), br (λ)q(λ)] fulfill this condition. Therefore, we can reach for instance that in (4.34), the matrix al (λ) is row reduced and the matrix ar (λ) is column reduced. Remark 4.20. From Theorems 4.15 and 4.17, it follows that as a first right basic controller any solution [α0r (λ), β0r (λ)] of the Diophantine Equation (4.37) can be used. Then the corresponding dual left basic controller is found by Formula (4.38). After that, the complete set of all pairs of dual basic controllers is constructed by Relations (4.40).

4.5 Eigenvalue Assignment for Polynomial Pairs 1. As stated in Section 4.1, the eigenvalue assignment problem for the pair (al (λ), bl (λ)) amounts to finding the set of controllers (αl (λ), βl (λ)) which satisfy the condition (4.42) det Ql (λ, αl , βl ) ≈ d(λ) ,

166


where d(λ) is a prescribed monic polynomial and

al (λ) −bl (λ) Ql (λ, αl , βl ) = . −βl (λ) αl (λ)

(4.43)

The general solution for the formulated problem in case of an irreducible process provides the following theorem. Theorem 4.21. Let the process (al (λ), bl (λ)) be irreducible. Then Equation (4.42) is solvable for any polynomial d(λ). Thereby, if (α0l (λ), β0l (λ)) is a certain basic controller for the process (al (λ), bl (λ)), then the set of all controllers (αl (λ), βl (λ)) satisfying (4.42) can be represented in the form αl (λ) = Dl (λ)α0l (λ) − Ml (λ)bl (λ) , βl (λ) = Dl (λ)β0l (λ) − Ml (λ)al (λ) ,

(4.44)

where the m × n polynomial matrix Ml (λ) is arbitrary, and for the m × m polynomial matrix Dl (λ) the condition det D(λ) ≈ d(λ) is valid. Besides, the pair (αl (λ), βl (λ)) is irreducible, if and only if the pair (Dl (λ), Ml (λ)) is irreducible. Proof. Denote the set of solutions of Equation (4.42) by N0 , and the set of pairs (4.44) by Np . Let (α0l (λ), β0l (λ)) be a certain basic controller. Then the matrices

al (λ) −bl (λ) Ql (λ, α0l , β0l ) = , −β0l (λ) α0l (λ) (4.45)

α0r (λ) br (λ) −1 Ql (λ, α0l , β0l ) = Qr (λ, α0r , β0r ) = β0r (λ) ar (λ) are unimodular, and the condition Ql (λ, α0l , β0l )Qr (λ, α0r , β0r ) = In+m holds. Let (αl (λ), βl (λ)) be a controller satisfying Equation (4.42). Then using (4.34), (4.43), (4.45) and the Bezout identity (4.31), we get

al (λ) −bl (λ) α0r (λ) br (λ) Ql (λ, αl , βl )Qr (λ, α0r , β0r ) = β0r (λ) ar (λ) −βl (λ) αl (λ) (4.46)

In Onm = Ml (λ) Dl (λ) with

4.5 Eigenvalue Assignment for Polynomial Pairs

167

Dl (λ) = −βl (λ)br (λ) + αl (λ)ar (λ) , Ml (λ) = −βl (λ)α0r (λ) + αl (λ)β0r (λ) .

(4.47)

Applying (4.46) and (4.47), we find Ql (λ, αl , βl ) = Nl (λ)Ql (λ, α0l , β0l ) ,

where Nl (λ) =

In Onm , Ml (λ) Dl (λ)

(4.48)

(4.49)

where we read (4.44). Calculating the determinant on both sides of (4.48) shows that det Ql (λ, αl , βl ) ≈ det Dl (λ) ≈ d(λ) . Thus N0 ⊂ Np was proven. By reversing the conclusions, we deduce as in Theorem 4.1 that also Np ⊂ N0 is true. Therefore, the sets N0 and Np coincide. Notice that Formulae (4.44) may be written in the shape

−βl (λ) αl (λ) = Ml (λ) Dl (λ) Ql (λ, α0l , β0l )

al (λ) −bl (λ) = Ml (λ) Dl (λ) . −β0l (λ) α0l (λ)

Since the matrix Ql (λ, α0l , β0l ) is unimodular, the matrices −βl (λ) αl (λ) and Ml (λ) Dl (λ) are right-equivalent, and that’s why the pair (αl (λ), βl (λ)) is irreducible, if and only if the pair (Dl (λ), Ml (λ)) is irreducible.

2. Example 4.22. For a prescribed polynomial d(λ), the solution set of the eigenvalue assignment problem for the process (4.24) in Example 4.12 has the form

d11 (λ) d12 (λ) −(0.5λ + 1) 0 m11 (λ) m12 (λ) 0 −λ − , αl (λ) = 0 1 −λ 0 d21 (λ) d22 (λ) m21 (λ) m22 (λ)

−0.5 0 m11 (λ) m12 (λ) λ−1 λ+1 d11 (λ) d12 (λ) − . βl (λ) = 0 0 0 1 d21 (λ) d22 (λ) m21 (λ) m22 (λ) Here the mik (λ) are arbitrary polynomials and dik (λ) are arbitrary polynomials bound by the condition d11 (λ)d22 (λ) − d21 (λ)d12 (λ) ≈ d(λ) .

168


Example 4.23. The set of solutions of Equation (4.42) for the process (4.25) in Example 4.13 has the form αl (λ) = kd(λ)d3 (λ) + m1 (λ)} ,

βl (λ) = 0.5kd(λ) 1 1 − m1 (λ) m2 (λ) λ − 1 λ + 1 , where k is a constant and d3 (λ), m1 (λ), m2 (λ) are any polynomials.

3. Now, consider the question, how the solution of Equation (4.42) looks like when the process (al (λ), bl (λ)) is reducible. In this case, with respect to the results in Section 1.12, there exists a latent square n × n polynomial matrix q(λ), such that al (λ) = q(λ)al1 (λ),

bl (λ) = q(λ)bl1 (λ)

(4.50)

is true with an irreducible pair (al1 (λ), bl1 (λ)). The solvability conditions for Equation (4.42) in case (4.50) states the following theorem. Theorem 4.24. Let (4.50) be valid and det q(λ) = γ(λ). Then a necessary and sufficient condition for the solvability of Equation (4.42) is, that the polynomial d(λ) is divisible by γ(λ). Thus, if (˜ α0l (λ), β˜0l (λ)) is a certain basic controller for the process (al1 (λ), bl1 (λ)), then the set of all controllers satisfying Equation (4.42) is bound by the relations ˜ l (λ)˜ ˜ l (λ)bl1 (λ) , αl (λ) = D α0l (λ) − M ˜ l (λ)β˜0l (λ) − M ˜ l (λ)al1 (λ) , βl (λ) = D

(4.51)

˜ l (λ) is arbitrary, and the m × m polywhere the m × n polynomial matrix M ˜ ˜ l (λ) ≈ d(λ). ˜ Here, the polynomial matrix Dl (λ) satisfies the condition det D ˜ nomial d(λ) is determined by ˜ = d(λ) . d(λ) γ(λ)

(4.52)

Proof. Let (4.50) be true. Then (4.42) can be presented in the shape

q(λ) Onm ˜ Ql (λ, αl , βl ) ≈ d(λ) , det (4.53) Omn Im where

˜ l (λ, αl , βl ) = Q

al1 (λ) −bl1 (λ) . −βl (λ) αl (λ)

Calculating the determinants, we find ˜ l (λ, αl , βl ) ≈ d(λ) , γ(λ) det Q

4.6 Eigenvalue Assignment by Transfer Matrices

169

i.e. for the solvability of Equation (4.53), it is necessary that the polynomial d(λ) is divisible by γ(λ). If this condition is ensured and (4.52) is used, then Equation (4.53) leads to ˜ . ˜ l (λ, α, β) ≈ d(λ) det Q Since the pair (al1 (λ), bl1 (λ)) is irreducible, Equation (4.53) is always solvable, thanks to Theorem 4.17, and its solution has the shape (4.51). 4. Let (a(λ), b(λ)) be an irreducible process and (αl (λ), βl (λ)) such a controller, that det Ql (λ, αl , βl ) = d(λ) ≡/ 0 becomes true. Furthermore, let (α0l (λ), β0l (λ)) be a certain basic controller. Then owing to Theorem 4.17, there exist m × m and m × n polynomial matrices Dl (λ) and Ml (λ), such that αl (λ) = Dl (λ)α0l (λ) − Ml (λ)bl (λ) , βl (λ) = Dl (λ)β0l (λ) − Ml (λ)al (λ) ,

(4.54)

where det Dl (λ) ≈ d(λ). Relations (4.54) are called the basic representation of the controllers (αl (λ), βl (λ)) with respect to the basis (α0l (λ), β0l (λ)). Theorem 4.25. The basic representation (4.54) is unique in the sense, that from the validity of (4.54) and the relation αl (λ) = Dl1 (λ)α0l (λ) − Ml1 (λ)bl (λ) , βl (λ) = Dl1 (λ)β0l (λ) − M1l (λ)al (λ) ,

(4.55)

we can conclude Dl1 (λ) = Dl (λ), Ml1 (λ) = Ml (λ). Proof. Suppose (4.54) and (4.55) are fulfilled at the same time. Subtracting (4.55) from (4.54), we get [Dl (λ) − Dl1 (λ)] α0l (λ) − [Ml (λ) − Ml1 (λ)] bl (λ) = Omm , [Dl (λ) − Dl1 (λ)] β0l (λ) − [Ml (λ) − Ml1 (λ)] al (λ) = Omn , which is equivalent to

Ml (λ) − Ml1 (λ) Dl (λ) − Dl1 (λ) Ql (λ, α0l , β0l ) = Om,m+n . From this, it follows immediately Ml1 (λ) = Ml (λ), Dl1 (λ) = Dl (λ), because the matrix Ql (λ, α0l , β0l ) is unimodular.

4.6 Eigenvalue Assignment by Transfer Matrices 1. In case of det αl (λ) ≡ / 0, it means, the pair (αl (λ), βl (λ)) is not singular, the transfer function of the controller

170


w (λ) = αl−1 (λ)βl (λ)

(4.56)

might be included into our considerations. Its standard form (2.21) can be written as M (λ) w (λ) = (4.57) d (λ) for which Relation (4.56) defines a certain LMFD. Conversely, if the transfer function of the controller is given in the standard form (4.57), then various LMFD (4.56) and the corresponding characteristic matrices

al (λ) −bl (λ) Ql (λ, αl , βl ) = (4.58) −βl (λ) αl (λ) can be investigated. Besides, every LMFD (4.56) is uniquely related to a characteristic polynomial ∆(λ) = det Ql (λ, αl , βl ). In future, we will say that the transfer matrix w (λ) is a solution of the eigenvalue assignment for the process (al (λ), bl (λ)), if it allows an LMFD (4.56) such that the corresponding pair (αl (λ), βl (λ)) satisfies Equation (4.42). 2. The set of transfer matrices (4.57) that supply the solution of the eigenvalue assignment is generally characterised by the next theorem. Theorem 4.26. Let the pair (al (λ), bl (λ)) be irreducible and (α0l (λ), β0l (λ)) be an appropriate left basic controller. Then for the fact that the transfer matrix (4.56) is a solution of Equation (4.42), it is necessary and sufficient that it allows a representation of the form w (λ) = [α0l (λ) − φ(λ)bl (λ)]−1 [β0l (λ) − φ(λ)al (λ)] ,

(4.59)

where φ(λ) is a broken rational m × n matrix, for which exists an LMFD φ(λ) = Dl−1 (λ)Ml (λ) ,

(4.60)

where det Dl (λ) ≈ d(λ) is true and the polynomial matrix Ml (λ) is arbitrary. Proof. Sufficiency: Suppose the LMFD (4.60). Then from (4.59) we get w (λ) = [Dl (λ)α0l (λ) − Ml (λ)bl (λ)]−1 [Dl (λ)β0l (λ) − Ml (λ)al (λ)] .

(4.61)

Thus, the set of equations αl (λ) = Dl (λ)α0l (λ) − Ml (λ)bl (λ) , βl (λ) = Dl (λ)β0l (λ) − Ml (λ)al (λ)

(4.62)

describes a controller satisfying Relation (4.42). Necessity: If (4.56) and det Ql (λ, αl , βl )) ≈ d(λ) are true, then for the matrices αl (λ) and βl (λ), we can find a basic representation (4.54), and under the invertability condition for the matrix αl (λ), we obtain (4.59), so the proof is carried out.

4.6 Eigenvalue Assignment by Transfer Matrices

171

Corollary 4.27. From (4.59) we learn that the transfer matrices w (λ), defined as the solution set of Equation (4.42), depend on a matrix parameter, namely the fractional rational matrix φ(λ). 3. Let the transfer function of the controller be given in the form (4.59). Then under Condition (4.60), it can be represented in form of the LMFD (4.56), where the matrices αl (λ), βl (λ) are determined by (4.62). For applications, the question of the irreducibility of the pair (4.62) is important. Theorem 4.28. The pair (4.62) is exactly then irreducible, when the pair [Dl (λ), Ml (λ)] is irreducible, i.e. the right side of (4.59) is an ILMFD. Proof. The proof follows directly from Theorem 4.17. 4. Let the process (al (λ), bl (λ)) and a certain fractional rational m×n matrix w (λ) be given, for which the expression (4.56) defines a certain LMFD. Thus, if

al (λ) −bl (λ) ≈ d(λ) , det Ql (λ, αl , βl )) = det −βl (λ) αl (λ) then, owing to Theorem 4.26, the matrix w (λ) can be represented in the form (4.59), (4.61), where (α0l (λ), β0l (λ)) is a certain basic controller. Under these circumstances, the notation (4.59) of the matrix w (λ) is called its basic representation with respect to the basis (α0l (λ), β0l (λ)). Theorem 4.29. For a fixed basic controller (α0l (λ), β0l (λ)), the basic representation (4.59) is unique in the sense, that the validity of (4.59) and w (λ) = [α0l (λ) − φ1 (λ)bl (λ)]−1 [β0l (λ) − φ1 (λ)al (λ)]

(4.63)

at the same time implies the equality φ(λ) = φ1 (λ). Proof. Without loss of generality, we suppose that the right side of (4.60) is an ILMFD. Then owing to Theorem 4.26, the right side of (4.61) is an ILMFD of the matrix w (λ). In addition let us have the LMFD φ1 (λ) = D1−1 (λ)M1 (λ) . Then from (4.63) for the matrix w (λ), we obtain the LMFD w (λ) = [D1 (λ)α0l (λ) − M1 (λ)bl (λ)]−1 [D1 (λ)β0l (λ) − M1 (λ)al (λ)] . (4.64) This relation and (4.61) define two different LMFDs of the matrix w (λ). By supposition the LMFD (4.61) is irreducible, so with respect to Statement 2.3 on page 64, we come out with D1 (λ)α0l (λ) − M1 (λ)bl (λ) = U (λ)[Dl (λ)α0l (λ) − Ml (λ)bl (λ)] , D1 (λ)β0l (λ) − M1 (λ)al (λ) = U (λ) [Dl (λ)β0l (λ) − Ml (λ)al (λ)] ,

172


where U (λ) is a non-singular m × m polynomial matrix. These relations can be written as

M1 (λ) − U (λ)Ml (λ) D1 (λ) − U (λ)Dl (λ) Ql (λ, α0l , β0l ) = Om,m+n , (4.65) where

al (λ) −bl (λ) . Ql (λ, α0l , β0l ) = −β0l (λ) α0l (λ) Since this matrix is designed unimodular, it follows from (4.65) M1 (λ) = U (λ)Ml (λ),

D1 (λ) = U (λ)Dl (λ) .

Thus we derive φ(λ) = Dl−1 (λ)Ml (λ) = D1−1 (λ)M1 (λ) = φ1 (λ) , which completes the proof.

4.7 Structural Eigenvalue Assignment for Polynomial Pairs 1. The solution of the structural eigenvalue assignment for an irreducible process (al (λ), bl (λ)) by the controller (αl (λ), βl (λ)) bases on the following statement. Theorem 4.30. Let the process (al (λ), bl (λ)) be irreducible and the controller (αl (λ), βl (λ)) should have the basic representation (4.54). Then the matrices

al (λ) −bl (λ) In Onm , S(λ) = (4.66) Ql (λ, αl , βl ) = Omn Dl (λ) −βl (λ) αl (λ) are equivalent, and this fact does not depend on the matrix Ml (λ). Proof. Notice

In Onm In Onm In Onm = , Ml (λ) Dl (λ) Ml (λ) Im Omn Dl (λ)

then Relations (4.48), (4.49) can be written in the form

In Onm S(λ)Ql (λ, α0l , β0l ) . Ql (λ, αl , βl ) = Ml (λ) Im The first and the last factor on the right side are unimodular matrices and therefore, the matrices (4.66) are equivalent.

4.7 Structural Eigenvalue Assignment for Polynomial Pairs

173

Theorem 4.31. Let a1 (λ), . . . , an+m (λ) and b1 (λ), . . . , bm (λ) be the sequences of invariant polynomials of the matrices Ql (λ, αl , βl ) and Dl (λ), respectively. Then the equations a1 (λ) = a2 (λ) = . . . = an (λ) = 1

(4.67)

and furthermore an+i (λ) = bi (λ),

(i = 1, . . . , m) .

(4.68)

Proof. Assume b1 (λ), . . . , bm (λ) be the sequence of invariant polynomials of Dl (λ). Then the sequence of invariant polynomials of S(λ) is equal to 1, . . . , 1, b1 (λ), . . . , bm (λ). But the matrices Dl (λ) and Ql (λ, αl , βl ) are equivalent, hence their sequences of invariant polynomials coincide, that means, Equations (4.67) and (4.68) are correct. Corollary 4.32. Theorem 4.31 supplies a constructive procedure for the design of closed systems with a prescribed sequence of invariant polynomials of the characteristic matrix. Indeed, let a sequence of monic polynomials b1 (λ), . . . , bm (λ) with b1 (λ) · · · bm (λ) ≈ d(λ) be given and for all i = 2, . . . , m, the polynomial bi (λ) is divisible by bi−1 (λ). Then we take Dl (λ) = p(λ) diag{b1 (λ), . . . , bm (λ)}q(λ) , where p(λ), q(λ) are unimodular matrices. After that, independently of the selection of Ml (λ) in (4.54), the sequence of the last m invariant polynomials of the matrix Ql (λ, αl , βl ) coincides with the sequence b1 (λ), . . . , bm (λ). Corollary 4.33. If the process (al (λ), bl (λ)) is irreducible, then there exists a set of controllers Ωs for which the matrix Ql (λ, αl , βl ) becomes simple. This happens exactly when the matrix Dl (λ) is simple, i.e. it allows the representation Dl (λ) = p(λ) diag{1, . . . , 1, d(λ)}q(λ) with unimodular matrices p(λ), q(λ). Corollary 4.34. Let irreducible left and right models of the process (al (λ), bl (λ)) and [ar (λ), br (λ)] be given. Then the sequence of invariant polynomials an+1 (λ), . . . , an+m (λ) of the characteristic matrix Ql (λ, αl , βl ) coincides with the sequence of invariant polynomials of the matrix Dl (λ) = −βl (λ)br (λ) + αl (λ)ar (λ) , which is a direct consequence of Theorem 4.30 and Equations (4.47).

174


4.8 Eigenvalue and Eigenstructure Assignment for PMD Processes 1. In the present section, the generale solution for the eigenvalue assignment (4.6) for non-singular PMD processes is developed. Moreover, the set of sequences of invariant polynomials is described for which the structural eigenvalue assignment is solvable. Theorem 4.35. Let the PMD (4.4) be non-singular and minimal, and wτ (λ) = c(λ)a−1 (λ)b(λ)

(4.69)

should be its corresponding transfer matrix. Furthermore, let us have the ILMFD wτ (λ) = a−1 (4.70) l (λ)bl (λ) . Then the eigenvalue assignment problem (4.5) is solvable for any polynomial d(λ). Besides, the set of pairs (αl (λ), βl (λ)) that are solutions of (4.6) coincides with the set of pairs that are determined as solutions of the eigenvalue assignment for the irreducible pair (al (λ), bl (λ)), and these may be produced on the base of Theorem 4.17. Preparing the proof, some auxiliary statements are given. Lemma 4.36. For the non-singular PMD (4.4), formula det Qτ (λ, α, β) = det a(λ) det [α(λ) − β(λ)wτ (λ)]

(4.71)

holds, where the matrix Qτ (λ, α, β) is established in (4.5). Proof. The matrix Qτ (λ, α, β) is brought into the form

A(λ) −B(λ) , Qτ (λ, α, β) = −C(λ) D(λ) where

a(λ) Opn , −c(λ) In

C(λ) = Omp β(λ) , A(λ) =

B(λ) =

b(λ) , Onm

D(λ) = α(λ) .

(4.72)

(4.73) (4.74)

Under the taken propositions, we have det A(λ) = det a(λ) ≡ / 0. Therefore, the well-known formula [51]

det Qτ (λ, α, β) = det A(λ) det D(λ) − C(λ)A−1 (λ)B(λ)

(4.75)

(4.76)

4.8 Eigenvalue and Eigenstructure Assignment for PMD Processes

175

is applicable. Observing −1

A

(λ) =

a−1 (λ)

Opn

c(λ)a−1 (λ) In

and (4.72)–(4.75), we obtain (4.71). Lemma 4.37. Let the non-singular PMD (4.4) and its corresponding transfer matrix (4.69) be given, for which Relation (4.70) defines an ILMFD. Consider the matrix

al (λ) −bl (λ) , (4.77) Ql (λ, α, β) = −β(λ) α(λ) where the matrices α(λ) and β(λ) are defined as in (4.5). If under this condition, the PMD (4.4) is minimal, then det Qτ (λ, α, β) ≈ det Ql (λ, α, β) .

(4.78)

Proof. Applying Formula (4.76) to Matrix (4.77), we find det Ql (λ, α, β) = det al (λ) [α(λ) − β(λ)wτ (λ)] .

(4.79)

Consider now the ILMFD c(λ)a−1 (λ) = a−1 1 (λ)c1 (λ) . Since the left side is an IRMFD, the relation det a1 (λ) ≈ det a(λ) holds. Thus due to Lemma 2.9, the expression wτ (λ) = a−1 (λ)[c1 (λ)b(λ)] defines an ILMFD of the matrix wτ (λ). This expression and (4.70) define at the same time ILMFDs of the matrix wτ (λ), so we have det a1 (λ) ≈ det a(λ) ≈ det al (λ) . Using this and (4.70) from (4.79), we obtain the statement (4.78). Proof of Theorem 4.35. The minimality of the PMD (4.4) and Lemma 4.37 imply that the sets of solutions of (4.6) and of the equation det Ql (λ, α, β) ≈ d(λ) coincide.

176


2. The next theorem supplies the solution of the eigenvalue assignment for the case, when the PMD (4.4) is not minimal. Theorem 4.38. Let the non-singular PMD (4.4) be not minimal, and Relation (4.70) should describe an ILMFD of the transfer matrix wτ (λ). Then the relation det a(λ) (4.80) χ(λ) = det al (λ) turns out to be a polynomial. Thereby, Equation (4.6) is exactly then solvable, when d(λ) = χ(λ)d1 (λ) , (4.81) where d1 (λ) is any polynomial. If (4.81) is true, then the set of controllers that are solutions of Equation (4.5) coincide with the set of solutions of the equation

al (λ) −bl (λ) det (4.82) ≈ d1 (λ) . −β(λ) α(λ) This solution set can be constructed with the help of Theorem 4.17. Proof. Owing to Lemma 2.48, Relation (4.80) is a polynomial. With the help of (4.71) and (4.80), we gain det Qτ (λ, α, β) = det a(λ) det[α(λ) − β(λ)wτ (λ)] = χ(λ) det al (λ) det[α(λ) − β(λ)wτ (λ)] . Using (4.79), we find out that Equation (4.5) leads to χ(λ) det Ql (λ, α, β) ≈ d(λ) .

(4.83)

From (4.83), it is immediately seen that Equation (4.6) needs Condition (4.81) be fulfilled for its solvability. Conversely, if (4.81) is fulfilled, Equation (4.83) leads to Equation (4.82). 3. The solution of the structural eigenvalue assignment for minimal PMD (4.4) supplies the following theorem. Theorem 4.39. Let the non-singular PMD (4.4) be minimal, and Relation (4.70) should define an ILMFD of the transfer matrix (4.69). Furthermore, let (α0 (λ), β0 (λ)) be a basic controller for the pair (al (λ), bl (λ)), and the set of pairs α(λ) = N (λ)α0 (λ) − M (λ)bl (λ) , β(λ) = N (λ)β0 (λ) − M (λ)al (λ)

(4.84)

should determine the set of solutions of the eigenvalue assignment (4.6). Moreover, let q1 (λ), . . . , qp+n+m (λ) be the sequence of invariant polynomials of the


177

polynomial matrix Qτ (λ, α, β), and ν1 (λ), . . . , νm (λ) be the sequence of invariant polynomials of the polynomial matrix N (λ). Then q1 (λ) = q2 (λ) = . . . = qp+n (λ) = 1 , qp+n+i (λ) = νi (λ),

(i = 1, . . . , m) .

(4.85)

Proof. a) It is shown that under the conditions of Theorem 4.39, the pair (A(λ), B(λ)) defined by Relation (4.73) is irreducible. Indeed, let (α0 (λ), β0 (λ)) be a basic controller for the pair (al (λ), bl (λ)) that is determined by the ILMFD (4.70). Then owing to Lemma 4.37, we have

A(λ) −B(λ) det = const. = 0 , −C0 (λ) D0 (λ) where

C0 (λ) = Omp β0 (λ) ,

D0 (λ) = α0 (λ) ,

and due to Theorem 1.41, the pair (A(λ), B(λ)) is irreducible. b) Equation (4.6) is written in the form

A(λ) −B(λ) det ≈ d(λ) . −C(λ) D(λ)

(4.86)

(4.87)

Since the pair A(λ), B(λ)) is irreducible, it follows from Theorem 4.17 that for any polynomial d(λ), Equation (4.87) is solvable and the set of solutions can be presented in the shape

b(λ) D(λ) = N1 (λ)D0 (λ) − M1 (λ) , Onm (4.88)

a(λ) Opn C(λ) = N1 (λ)C0 (λ) − M1 (λ) , −c(λ) In where the m×(p+n) polynomial matrix M1 (λ) is arbitrary, but the m×m polynomial matrix N1 (λ) has to fulfill the single condition det N1 (λ) ≈ d(λ) . c) On the other side, due to Theorem 4.38, the set of pairs (α(λ), β(λ)) satisfying Equation (4.87) coincides with the set of solutions of the equation

al (λ) −bl (λ) ≈ d(λ) det −β(λ) α(λ) that has the form (4.84), where the m × n polynomial matrix M (λ) is arbitrary, and the m × m polynomial matrix N (λ) has to satisfy the condition det N (λ) ≈ d(λ) .

178


Assume in (4.88) p n

˜ 1 (λ) M ˜ 2 (λ) m . M1 (λ) = M

Then with the help of (4.74), (4.86), Relation (4.88) can be presented in the shape

Omp

˜ 1 (λ)b(λ) , α(λ) = N1 (λ)α0 (λ) − M

˜ 2 (λ)c(λ) M ˜ 2 (λ) . ˜ 1 (λ)a(λ) − M β(λ) = N1 (λ) Omp β0 (λ) − M

In order to avoid a contradiction between these equations with (4.84), it is necessary and sufficient that the condition N1 (λ) = N (λ)

(4.89)

holds, and moreover, ˜ 1 (λ)b(λ) = M (λ)bl (λ), M

˜ 1 (λ)a(λ) − M ˜ 2 (λ)c(λ) = Omn , M

˜ 2 (λ) = M (λ)al (λ) M are fulfilled. Now we directly conclude that these relations are satisfied for ˜ 1 (λ) = M (λ)al (λ)c(λ)a−1 (λ), M

˜ 2 (λ) = M (λ)al (λ) . M

(4.90)

Besides, due to Lemma 2.9, the product al (λ)c(λ)a−1 (λ) is a polynomial matrix . Substituting the last relations and (4.89) into (4.88), we find

b(λ)

−1 D(λ) = N (λ)α0 (λ) − M (λ) al (λ)c(λ)a (λ) al (λ) , Onm

(4.91) a(λ) Opn

C(λ) = N (λ)C0 (λ) − M (λ) al (λ)c(λ)a−1 (λ) al (λ) . −c(λ) In From this and Theorem 4.8, Equations (4.85) emerge immediately. Corollary 4.40. In order to get a simple matrix Qτ (λ, α, β) under the conditions of Theorem 4.11, it is necessary and sufficient that the matrix N (λ) in Formula (4.91) is simple. 4. The structure of the characteristic matrix Qτ (λ, α, β) for the case, when the non-singular PMD (4.4) is not minimal, decides the following theorem. Theorem 4.41. For the non-singular PMD (4.4), the factorisations a(λ) = d1 (λ)a1 (λ),

b(λ) = d1 (λ)b1 (λ)

(4.92)

should be valid, where d1 (λ), a1 (λ) are p × p polynomial matrices, b1 (λ) is a p × m polynomial matrix and the pair (a1 (λ), b1 (λ)) is irreducible. Moreover, suppose c(λ) = c1 (λ)d2 (λ) (4.93) a1 (λ) = a2 (λ)d2 (λ), with p × p polynomial matrices d2 (λ), a2 (λ), the n × p polynomial matrix c1 (λ) and the irreducible pair [a2 (λ), c1 (λ)]. Then the following statements are true:


179

a) The PMD τ1 (λ) = (a2 (λ), b1 (λ), c1 (λ)) is equivalent to the PMD (4.4) and minimal. b) The relation det a(λ) (4.94) ξ(λ) = det a2 (λ) turns out to be a polynomial with ξ(λ) ≈ χ(λ) =

det a(λ) , det al (λ)

(4.95)

where χ(λ) is the polynomial (4.80). c) The relation Qτ (λ, α, β) = Gl (λ)Qτ1 (λ, α, β)Gr (λ)

(4.96)

is true with Gl (λ) = diag{d1 (λ), 1, . . . , 1} ,

Gr (λ) = diag{d2 (λ), 1, . . . , 1} , (4.97)

and the matrix Qτ1 (λ, α, β) has the shape ⎤ ⎡ a2 (λ) Opn −b1 (λ) Onm ⎦ . Qτ1 (λ, α, β) = ⎣ −c1 (λ) In Omp −β(λ) α(λ)

(4.98)

d) Formula ξ(λ) ≈ det d1 (λ) det d2 (λ)

(4.99)

is valid. e) Let q˜1 (λ), . . . , q˜p+n+m (λ) be the sequence of invariant polynomials of the matrix Qτ1 (λ, α, β) and ν1 (λ), . . . , νm (λ) be the sequence of invariant polynomials of the matrix N (λ) in the representation (4.91), where instead of al (λ), b(λ), c(λ) we have to write a2 (λ), b1 (λ), c1 (λ). Then q˜1 (λ) = q˜2 (λ) = . . . = q˜p+n (λ) = 1 , q˜p+n+i (λ) = νi (λ),

(4.100)

(i = 1, . . . , m) .

Proof. a) Using (4.92) and (4.93), we find wτ (λ) = c(λ)a−1 (λ)b(λ) = c1 (λ)a−1 2 (λ)b1 (λ) = wτ1 (λ) , where wτ1 (λ) is the transfer function of the PMD τ1 (λ), this means, the PMD τ (λ) and τ1 (λ) are equivalent. It is demonstrated that the PMD τ1 (λ) is minimal. Since the pair [a2 (λ), c1 (λ)] is irreducible per construction, it is sufficient to show that the pair (a2 (λ), b1 (λ)) is irreducible. Per construction, the pair (a1 (λ), b1 (λ)) = (a2 (λ)d2 (λ), b1 (λ)) is irreducible. Hence due to Lemma 2.11, also the pair (a2 (λ), b1 (λ)) is irreducible.

180


b) From (4.92) and (4.93), we recognise that Relation (4.94) is a polynomial. Since the PMD τ˜(λ) = (al (λ), bl (λ), In ) and τ1 (λ) are equivalent and minimal, Corollary 2.49 implies det al (λ) ≈ det a2 (λ) and this yields (4.95). c) Relations (4.96)–(4.98) can be taken immediately from (4.6), (4.92), (4.93). d) Applying Formula (4.71) to Matrix (4.98), we obtain det Qτ1 (λ, α, β) = det a2 (λ) det[α(λ) − β(λ)wτ (λ)] . Therefore, from (4.71) with the help of (4.94), we receive det Qτ (λ, α, β) = ξ(λ) det Qτ1 (λ, α, β) . On the other side from (4.96)–(4.98), it follows det Qτ (λ, α, β) = det Gl (λ) det Gr (λ) det Qτ1 (λ, α, β). Bringing face to face the last two equations proves Relation (4.99). e) Since the PMD τ1 (λ) is minimal and Relation (4.84) holds, Formula (4.100) follows from Theorem 4.39. Corollary 4.42. If one of the matrices d1 (λ) or d2 (λ) is not simple, then the matrix Qτ (λ, α, β) cannot be made simple with the help of any controller (α(λ), β(λ)). Proof. Let for instance the matrix d1 (λ) be not simple, then we learned from ˜ with the considerations in Section 1.11 that there exists an eigenvalue λ ˜ > 1. Hence considering (4.96), (4.97), we get def Qτ (λ, ˜ α, β) > 1, def Gl (λ) i.e., the matrix Qτ (λ, α, β) is not simple. If d2 (λ) is not simple, we conclude analogously. Corollary 4.43. Let the matrices d1 (λ) and d2 (λ) be simple and possess no eigenvalues in common. Then for the simplicity of the matrix Qτ (λ, α, β), it suffices that the matrix N (λ) in (4.84) is simple and has no common eigen˜ = d1 (λ)d2 (λ). values with the matrix d(λ) Proof. Let µ1 , . . . , µq and ν1 , . . . , νs be the different eigenvalues the matrices d1 (λ) and d2 (λ), respectively. Then the matrices Gl (λ) and Gr (λ) in (4.97) are also simple, where the eigenvalues of Gl (λ) are the numbers µ1 , . . . , µq , but the eigenvalues of the matrix Gr (λ) are the numbers ν1 , . . . , νs . Let the matrix N (λ) in (4.84) be simple and should possess the eigenvalues n1 , . . . , nk that are disjunct with all the values µi , (i = 1, . . . , q) and νj , (j = 1, . . . , s). Then from Corollary 4.40, it follows that the matrix Qτ1 (λ, α, β) is simple, and possesses the set of eigenvalues {n1 , . . . , nk }. From (4.96), we recall that


181

the set of eigenvalues of the matrix Qτ (λ, α, β) is built from the unification of the sets {µ1 , . . . , µq }, {ν1 , . . . , νs } and {n1 , . . . , nk }. Using (4.96), we find out that for all appropriate i def Qτ (µi , α, β) = 1,

def Qτ (νi , α, β) = 1,

def Qτ (ni , α, β) = 1 ,

and together with the results of Section 1.11, this yields that the matrix Qτ (λ, α, β) is simple.

5 Fundamentals for Control of Causal Discrete-time LTI Processes

5.1 Finite-dimensional Discrete-time LTI Processes 1. Generalised discrete linear processes can be represented by the abstract Fig. 5.1, where {u}, {y} are input and output vector sequences {u}

-

{y}

-

L

Fig. 5.1. Generalised discrete LTI process

⎤ {u}1 ⎥ ⎢ {u} = ⎣ ... ⎦ , {u}m ⎡

⎤ {y}1 ⎥ ⎢ {y} = ⎣ ... ⎦ ⎡

{y}n

and their components {u}i , {y}i are scalar sequences {u}i = {ui,0 , ui,1 , . . . },

{y}i = {yi,0 , yi,1 , . . . } .

The above vector sequences can also be represented in the form {u} = {u0 , u1 , . . . }, where

⎤ us,1 ⎥ ⎢ us = ⎣ ... ⎦ , us,m ⎡

{y} = {y0 , y1 , . . . } ,

(5.1)

⎤ ys,1 ⎥ ⎢ ys = ⎣ ... ⎦ .

(5.2)

⎡

ys,n

Furthermore, in Fig. 5.1, the letter L symbolises a certain system of linear equations that consists between the input and output sequences. If L stands for

184


a system with a finite number of linear difference equation with constant coefficients, then the corresponding process is called a finite-dimensional discretetime LTI object. In this section exclusively such objects will be considered and they will be called shortly as LTI objects. 2. Compatible with the introduced concepts, the LTI object in Fig. 5.1 is configured to a system of scalar difference equations n

(0)

aip yp,k+ + . . . +

p=1

n

()

aip yp,k =

p=1

m

(0)

bir ur,k+s + . . . +

r=1

m

(s)

bir ur,k

r=1

(i = 1, . . . , n; k = 0, 1, . . . ) , (j)

(5.3)

(j)

where the aip , bir are constant real coefficients. Introducing into the considerations the constant matrices (j) ˜bj = b(j) a ˜j = aip , ir and using the notation (5.1), the system of scalar Equations (5.3) can be written in form of the vector difference equation a ˜0 yk+ + . . . + a ˜ yk = ˜b0 uk+s + . . . + ˜bs uk ,

(k = 0, 1, . . . ) ,

(5.4)

which connects the components of the input and output sequences. Introduce the shifted vector sequences of the form {uj } = {uj , uj+1 , . . . },

{yj } = {yj , yj+1 , . . . } .

(5.5)

Then the system of Equations (5.4) can be written as equation connecting their shifted sequences: a ˜0 {y } + . . . + a ˜ {y} = ˜b0 {us } + . . . + ˜bs {u} .

(5.6)

It is easily checked that Equations (5.4) and (5.6) are equivalent. This can be done by substituting the expressions (5.5) into (5.6) and comparing the corresponding components of the sequences on the left and right side. 3. Next the right-shift (forward shift) operator q is introduced by the relations (5.7) qyk = yk+1 , quk = uk+1 . Herewith, (5.4) can be written in the form a ˜(q)yk = ˜b(q)uk ,

(5.8)

where a ˜(q) = a ˜0 q + . . . + a ˜ ,

˜b(q) = ˜b0 q s + . . . + ˜bs

(5.9)

5.1 Finite-dimensional Discrete-time LTI Processes

185

are polynomial matrices. However, if the operator q is defined by the relation q{yk } = {yk+1 },

q{uk } = {uk+1 } ,

then we come to equations that depend on the sequences a ˜(q){y} = ˜b(q){u} .

(5.10)

4. In what follows, Equations (5.10) or (5.8) will be called a forward model of the LTI process. The matrix a ˜(q) is named eigenoperator, and the matrix ˜b(q) inputoperator of the forward model. If not mentioned otherwise, we always suppose det a ˜(q) ≡ /0

(5.11)

i.e., the matrix a ˜(q) is non-singular. If (5.11) is valid, the LTI process is said to be non-singular. Moreover, we assume that in the equations of the non-singular processes (5.8), (5.9) always a ˜0 = Onn is true, and at least one of the matrices a ˜ or ˜bs is a nonzero matrix. If under the mentioned propositions, the relation det a ˜0 = 0

(5.12)

is valid, then the LTI process is called normal. If however, instead of (5.12) det a ˜0 = 0

(5.13)

is true, then the LTI process is named anomalous, [39]. For instance, descriptor processes can be modelled by anomalous systems [34]. 5. For a given input sequence {u}, Relations (5.4) can be regarded as a difference equation for the unknown output sequence {y}. If (5.13) is allowed, then in general the difference equation (5.4) cannot be written as recursion for yk , and we have to define, what we will understand by a solution. Solution of a not necessarily normal difference equation. For a known input sequence as a solution of Equation (5.4) {u}, we understand any sequence that is defined for all k ≥ 0, and the elements of which satisfy Relation (5.4) for all k ≥ 0. Suppose the non-singular matrix χ(q). Multiplying both sides of Equation (5.8) from left by χ(q), we obtain χ(q)˜ a(q)yk = χ(q)˜b(q)uk ,

(k = 0, 1, . . . )

(5.14)

and using (5.7), this can be written in an analogue form to (5.4). Equation (5.14) is said to be derived from the output Equation (5.4), and Equation (5.4) itself is called original.

186


Example 5.1. Consider the equations of the LTI processes in the form (5.3) y1,k+2 + y1,k + y2,k = uk+1

(5.15)

y1,k+1 + 2y2,k+1 = 2uk .

Denote yk = y1,k y2,k , then (5.15) is written in the form a ˜0 yk+2 + a ˜1 yk+1 + a ˜2 yk = ˜b0 uk+2 + ˜b1 uk+1 + ˜b2 uk ,

(5.16)

where

10 00 11 a ˜0 = , a ˜1 = , a ˜2 = , 00 11 00 ˜b0 = 0 , ˜b1 = 1 , ˜b2 = 0 , 0 0 2 so that we obtain (5.9) with

2 q +q 1 , a ˜(q) = q 2q

˜b(q) = q . 2

Since det a ˜(q) = 2q 3 + 2q 2 − q ≡/ 0, the LTI process (5.15) is non-singular. Besides, due to det a ˜0 = 0, the process is anomalous. Assume

q 1 χ(q) = , 1 −q then (5.14), that is derived from the original Equation (5.16), takes the form 2

3 q +2 q + q 2 + q 3q uk y = q 1 − 2q 2 k −q or, by means of (5.7), it is equivalently written as the system of equations y1,k+3 + y1,k+2 + y1,k+1 + 3y2,k+1 = uk+2 + 2uk y1,k+1 − 2y2,k+2 + y2,k = −uk+1 .

(k = 0, 1, . . . ) (5.17)

Hereby, Equations (5.15) are called original with respect to (5.17).

6. Lemma 5.2. For any matrix χ(q), all solutions of the original equation (5.8) are also solutions of the derived equation (5.14).

5.1 Finite-dimensional Discrete-time LTI Processes

Proof. The derived equation is written in the form χ(q) a ˜(q)yk − ˜b(q)uk = 0k ,

187

(5.18)

where 0k = 0 · · · 0 for all k ≥ 0. Obviously, the vectors uk , yk satisfy (5.18) for all k ≥ 0, when Equation (5.8) holds for all of them and all k ≥ 0. Remark 5.3. The inverse statement to Lemma 5.2 in general is not true. Indeed, let {v} be a solution of the equation χ(q)vk = 0k . Then any solution of the equation a ˜(q)yk = ˜b(q)uk + vk

(5.19)

for all possible vk presents a solution of the derived Equation (5.14), but only for vk = 0k it is a solution of the original equation. It is easy to show that Relation (5.19) contains all solutions of the derived equation. 7. Consider the important special case, when in (5.14) the matrix χ(q) is unimodular. In this case, the transition from the original equation (5.8) to the derived equation (5.14) means manipulating the system (5.3) by operations of the following types: a) Exchange the places of two equations. b) Multiply an equation by a non-zero constant. c) Add one equation to any other equation that was multiplied before by an arbitrary polynomial f (q). In what follows, Equations (5.8) and (5.14) are called equivalent by the unimodular matrix χ(q). The reasons for using this terminology arise from the next lemma. Lemma 5.4. The solution sets of the equivalent equations (5.8) and (5.14) coincide. Proof. Let Equations (5.8) and (5.14) be equivalent, and R, Rx are their solution sets. Lemma 5.2 implies R ⊂ Rx . On the other side, Equations (5.8) are gained from Equations (5.14) by multiplying them from left by χ−1 (q). Then also Lemma 5.2 implies Rx ⊂ R, thus R = Rx . 8.

Assume in (5.14) a unimodular matrix χ(q). Introduce the notation χ(q)˜ a(q) = a ˜ (q) ,

χ(q)˜b(q) = ˜b (q) .

Then the derived equation (5.14) can be written in the form

(5.20)

188


a ˜ (q)yk = ˜b (q)uk .

(5.21)

From Section 1.6, it is known that under supposition (5.11), the matrix χ(q) can always be selected in such a way that the matrix a ˜ (q) becomes row reduced. In this case, Equation (5.21) also is said to be row reduced. Let ñ (q) be the rows of the matrix a ˜ (q). As before, denote a ˜1 (q), . . . , a αi = deg a ˜i (q) ,

(i = 1, . . . , n) .

If under these conditions, Equation (5.21) is row reduced, then independently of the concrete shape of the matrix χ(q), the quantities αl =

n

αi ,

αmax = deg a ˜ (q) = max {αi } 1≤i≤n

i=1

take their minimal values in the set of equivalent equations to the original equation (5.8). Example 5.5. Consider the anomalous process y1,k+4 + 2y1,k+2 + y1,k+1 + 2y2,k+2 + y2,k = uk+3 + 2uk+1 y1,k+3 + y1,k+1 + y1,k + 2y2,k+1 = uk+2 + uk .

(5.22)

In the present case, we have 4

q + 2q 2 + q 2q 2 + 1 a ˜(q) = =a ˜0 q 4 + a ˜1 q 3 + a ˜2 q 2 + a ˜3 q + a ˜4 , 2q q3 + q + 1

3 ˜b(q) = q 2 + 2q = ˜b1 q 3 + ˜b2 q 2 + ˜b3 q + ˜b4 , q +1 where

10 00 22 10 01 a ˜0 = , a ˜1 = , a ˜2 = , a ˜3 = , a ˜4 = , 10 0 0 1 0 0 0 1 2 ˜b1 = 1 , ˜b2 = 0 , ˜b3 = 2 , ˜b4 = 0 . 0 1 0 1

Choose

1 −q χ(q) = . −q q 2 + 1

So, we generate the derived matrices 2

q 1 a ˜ (q) = χ(q)˜ a(q) = , q+1 q

˜b (q) = χ(q)˜b(q) = q , 1

where the matrix a ˜ (q) is row reduced. Applying this, Equations (5.22) might be expressed equivalently by y1,k+2 + y2,k = uk+1 y1,k+1 + y1,k + y2,k+1 = uk .

5.2 Transfer Matrices and Causality of LTI Processes

189

5.2 Transfer Matrices and Causality of LTI Processes 1. For non-singular processes (5.8) under Condition (5.11), the rational matrix (5.23) w(q) ˜ =a ˜−1 (q)˜b(q) is defined, which is called the transfer matrix (-function) of the forward model. The next lemma indicates an important property of the transfer matrix. Lemma 5.6. The transfer matrices of the original equation (5.8) and of the derived equation (5.21) coincide. Proof. Suppose

˜ ˜−1 w ˜ (q) = a (q)b (q) .

(5.24)

Then applying (5.20), we get ˜ ˜−1 ˜−1 (q)˜b(q) = w(q) ˜ . w ˜ (q) = a (q)b (q) = a Corollary 5.7. The transfer functions of equivalent forward models coincide. 2. From the above said emerges that any forward model (5.8) is uniquely assigned to a transfer matrix. The reverse statement is obviously wrong. Therefore the question arises, how is the set of forward models structured, that possess a given transfer matrix? The next theorem gives the answer. Theorem 5.8. Let the rational n × m matrix w(q) ˜ be given and ˜ w(q) ˜ =a ˜−1 0 b0 (q) be an ILMFD. Then the set of all forward models of LTI processes possessing this transfer matrix is determined by the relations a ˜(q) = ψ(q)˜ a0 (q),

˜b(q) = ψ(q)˜b0 (q) ,

(5.25)

where ψ(q) is any non-singular polynomial matrix. Proof. The right side of Relation (5.24) presents a certain LMFD of the rational matrix w(q). ˜ Hence by the properties of LMFDs considered in Section 2.4, we conclude that the set of all pairs (˜ a(q), ˜b(q)) according to the transfer matrix w(q) ˜ is determined by Relations (5.25). Corollary 5.9. A forward model of the LTI processes (5.8) is called controllable, if the pair (˜ a(q), ˜b(q)) is irreducible. Hence Theorem 5.8 is formulated in the following way: Let the forward model defined by the pair (˜ a(q), ˜b(q)) be controllable. Then the set of all forward models with transfer function (5.23) coincides with the set of all derived forward models.

190


3. The LTI process (5.8) is called weakly causal, strictly causal or causal, if its transfer matrix (5.23) is proper, strictly proper or at least proper, respectively. From the content of Section 2.6, it emerges that the LTI process (5.8), (5.9) is causal, if there exists the finite limit lim w(q) ˜ = w0 .

q→∞

(5.26)

Besides, when w0 = Onm holds, the process is strictly causal. When the limit (5.26) becomes infinite, the process is named non-causal. Theorem 5.10. For the process (5.8), (5.9) to be causal, the condition ≥s

(5.27)

is necessary. For strictly causality the inequality >s must be valid. Proof. Let us have the transfer matrix w(q) ˜ in the standard form (2.21) w(q) ˜ =

˜ (q) N . ˜ d(q)

˜ becomes true. ˜ (q) ≤ deg d(q) When this matrix is at least proper, then deg N Besides, Corollary 2.23 delivers for any LMFD (5.23) deg a ˜(q) ≥ deg ˜b(q), which is equivalent to (5.27). For strict causality, we conclude analogously. Remark 5.11. For further investigations, we optionally consider causal processes. Thus, in Equations (5.8), (5.9) always ≥ s is assumed. Remark 5.12. The conditions of Theorem 5.10 are in general not sufficient, as it is illustrated by the following example. Example 5.13. Assume the LTI process (5.8) with 3

q 1 1 ˜ a ˜(q) = , b(q) = 2 . q+1 q+2 q

(5.28)

In this case, we have = deg a ˜(q) = 3, s = deg ˜b(q) = 2. At the same time, we receive 2

−q + q + 2 q5 − q − 1 . w(q) ˜ = 4 q + 2q 3 − q − 1 Hence the process (5.28) is non-causal.

5.3 Normal LTI Processes

191

4. If Equation (5.8) is row reduced, the causality question for the processes (5.8) can be answered without constructing the transfer matrix. Theorem 5.14. Let Equation (5.8) be row reduced, and αi be the degree of the i-th row of the matrix a ˜(q) and βi be the degree of the i-th row of the matrix ˜b(q). Then the following statements are true: a) For the weak causality of the process, it is necessary and sufficient that the conditions (5.29) αi ≥ βi , (i = 1, . . . , n) are true, where at least for one 1 ≤ i ≤ n in (5.29) the equality sign has to be taken place. b) For the strict causality of the process, the fulfilment of the inequalities αi > βi ,

(i = 1, . . . , n)

(5.30)

is necessary and sufficient. c) When for at least one 1 ≤ i ≤ n αi < βi becomes true, then the process is non-causal. Proof. The proof emerges immediately from Theorem 2.24.

5.3 Normal LTI Processes 1.

This section considers LTI processes of the form a ˜0 yk+ + . . . + a ˜ yk = ˜b0 uk+ + . . . + ˜b uk ,

(k = 0, 1, . . . )

(5.31)

under the supposition det a ˜0 = 0 .

(5.32)

Some important properties of normal LTI processes will be formulated, which emerge from Relation (5.32). Theorem 5.15. For the weak causality of the normal processes (5.31), the fulfillment of ˜b0 = Onm (5.33) is necessary and sufficient. For the strict causality of the normal processes (5.31), the fulfillment of ˜b0 = Onm is necessary and sufficient.

(5.34)

192


Proof. From (5.32) it follows that the matrix a ˜(q) for a normal process is row reduced, and we have α1 = α2 = . . . = αn = . If (5.33) takes place, then in Condition (5.29) the equality sign stands for at least one 1 ≤ i ≤ n. Therefore, as a consequence of Theorem 5.14, the process is weakly causal. If however, (5.34) takes place, then Condition (5.30) is true and the process is strictly causal. 2.

Let the vector input sequence {u} = {u0 , u1 , . . . }

(5.35)

be given, and furthermore assume any ensemble of constant vectors of dimension n × 1 (5.36) y¯0 , y¯1 , . . . , y¯−1 . In what follows, the vectors (5.36) are called initial values. Theorem 5.16. For any input sequence (5.35) and any ensemble of initial values (5.36), there exists a unique solution of the normal equation (5.31) {y} = {y0 , y1 , . . . , y−1 , y , . . . } satisfying the initial conditions yi = y¯i ,

(i = 1, . . . , − 1) .

(5.37)

Proof. Assume that the vectors yi , (i = 0, 1, . . . , −1) satisfy Condition (5.37). Since Condition (5.32) is fulfilled, Equation (5.31) might be written in the shape ¯1 yk+−1 +. . .+ a ¯ yk + ¯b0 uk+ + ¯b1 uk+−1 +. . .+ ¯b uk , yk+ = a

(k = 0, 1, . . . ), (5.38)

where a−1 ˜i , a ¯i = −˜ 0 a

(i = 1, 2, . . . , );

¯bi = a ˜ ˜−1 0 bi , (i = 0, 1, . . . , ) .

For k = 0 from (5.38), we obtain ¯1 y−1 + . . . + a ¯ y0 + ¯b0 u + ¯b1 u−1 + . . . + ¯b u0 . y = a

(5.39)

Hence for a known input sequence (5.35) and given initial values (5.36), the vector y is uniquely determined. For k = 1 from (5.38), we derive ¯ 1 y + . . . + a ¯ y1 + ¯b0 u+1 + ¯b1 u + . . . + ¯b u1 . y+1 = a Thus with the help of (5.35), (5.36) and (5.39), the vector y+1 is uniquely calculated. Obviously, this procedure can be uniquely continued for all k > 0. As a result, in a unique way the sequence


193

{y} = {¯ y0 , . . . , y¯−1 , y , . . . } is generated, that is a solution of Equation (5.31) and fulfills the initial conditions (5.37). Remark 5.17. It follows from the proof of Theorem 5.16 that for weakly causal normal processes for given initial conditions, the vector yk of the solution {y} is determined by the values of the input sequence u0 , u1 , . . . , uk . If the process, however, is strictly causal, then the vector yk is determined by the vectors u0 , u1 , . . . , uk−1 . 3. Theorem 5.18. Let the input (5.2) be a Taylor sequence (see Appendix A). Then all solutions of Equation (5.31) are Taylor sequences. Proof. Using (5.8), (5.9), the polynomial matrices a(ζ) = ζ a ˜(ζ −1 ) = a ˜0 + a ˜1 ζ + . . . a ˜ ζ , b(ζ) = ζ ˜b(ζ −1 ) = ˜b0 + ˜b1 ζ + . . . ˜b ζ

(5.40)

are considered. Condition (5.32) implies det a(0) = det a ˜0 = 0 .

(5.41)

Under the assumed conditions, there exists the ζ-transform of the input sequence ∞ ui ζ i . (5.42) u0 (ζ) = i=0

Consider the vector

y 0 (ζ) = a−1 (ζ) a ˜0 y¯0 + ζ(˜ a0 y¯1 + a ˜1 y¯0 ) + . . .

a0 y¯−1 + a ˜1 y¯−2 + . . . + a ˜−1 y¯0 ) + ζ −1 (˜ (5.43)

−1 0 −1 ˜ + a (ζ) b0 u (ζ) − u0 − ζu1 − . . . − ζ u−1

+ ζ ˜b1 u0 (ζ) − u0 − ζu1 − . . . − ζ −2 u−2 + . . . + ζ ˜b u0 (ζ) ,

where u0 (ζ) is the convergent series (5.42). Since the vector u0 (ζ) is analytical in the point ζ = 0 and Condition (5.41) is valid, the right side of (5.43) is analytical in ζ = 0, and consequently y 0 (ζ) =

∞ i=0

yi ζ i

(5.44)

194


also defines a convergent series. For determining the coefficients of Expansion (5.44), substitute this equation on the left side of (5.43). Thus by taking advantage of (5.40), we come out with (˜ a0 + a ˜1 ζ + . . . + a ˜ ζ )

∞

yi ζ i = a ˜0 y¯0 + ζ(˜ a0 y¯1 + a ˜1 y¯0 ) + . . .

i=0

+ζ

−1

(˜ a0 y¯−1 + a ˜1 y¯−2 + . . . + a ˜−1 y¯0 ) (5.45)

0 + ˜b0 u (ζ) − u0 − ζu1 − . . . − ζ −1 u−1

+ ζ ˜b1 u0 (ζ) − u0 − ζu1 − . . . − ζ −2 u−2 + . . . + ζ ˜b u0 (ζ) ,

which holds for all sufficiently small |ζ|. Notice that the coefficients of the matrices ˜bi , (i = 0, . . . , ) on the right side of (5.45) are proportional to ζ i . Hence comparing the coefficients for ζ i , (i = 0, . . . , − 1) on both sides of (5.45) yields a ˜ 0 y0 = a ˜0 y¯0 a ˜ 1 y0 + a ˜ 0 y1 = a ˜1 y¯0 + a ˜0 y¯1 .. .

.. .

(5.46)

.. .

˜−1 y1 + . . . + a ˜ 0 y = a ˜−1 y¯0 + a ˜−2 y¯1 + . . . a ˜0 y¯−1 . a ˜−1 y0 + a With regard to (5.41), we generate from (5.46) yi = y¯i ,

(i = 0, 1, . . . , − 1) .

(5.47)

Using (5.47) and (5.46), Relation (5.45) is written as a ˜0

∞

i

yi ζ + ζ˜ a1

i=

∞ i=−1

= ˜b0

i

yi ζ .+ .. + ζ a ˜ ∞

∞

yi ζ i

i=0 ∞

ui ζ i + ζ ˜b1

i=

ui ζ i + . . . + ζ ˜b

i=−1

∞

ui ζ i .

i=0

Dividing both sides of the last equation by ζ yields the relation a ˜0

∞ i=0

yi+ ζ i + a ˜1

∞

yi+−1 ζ i.+ .. + a ˜

i=0

= ˜b0

∞ i=0

∞

yi ζ i

i=0

ui+ ζ + ˜b1 i

∞ i=0

ui+−1 ζ i + . . . + ˜b

∞

ui ζ i .

i=0

A comparison of the coefficients of equal powers of ζ on both sides produces


195

a ˜0 yk+ + a ˜1 yk+−1 + . . . + a ˜ yk = ˜b0 uk+ + ˜b1 uk+−1 + . . . + ˜b uk , (k = 0, 1, . . . ) . Bringing this face to face with (5.31) and taking advantage of (5.47), we conclude that the coefficients of the expansion (5.44) build a solution of Equation (5.31), the initial conditions of which satisfy (5.37) for any initial vectors (5.36). But owing to Theorem 5.16, every ensemble of initial values (5.36) uniquely corresponds to a solution. Hence we discover that for any initial vectors (5.36), the totality of coefficients of the expansion (5.44) exhaust the whole solution set of the normal equation (5.31). Thus in case of convergence of the ζ-transforms (5.42), all solutions of the normal equation (5.31) are Taylor sequences. Corollary 5.19. When the input is a Taylor sequence {u}, it emerges from the proof of Theorem 5.18 that the right side of Relation (5.43) defines the ζ-transform of the general solution of the normal equation (5.31). 4. From Theorem 5.18 and its Corollary, as well as from the relations between the z-transforms and ζ-transforms, it arises that for a Taylor input sequence {u}, any solution {y} of the normal equation (5.31) possesses the z-transform ∞ y ∗ (z) = yk z −k . k=0

Applying (A.8), after transition in (5.31) to the z-transforms, we arrive at

˜1 z −1 y ∗ (z) − z −1 y0 − . . . − zy−2 a ˜0 z y ∗ (z) − z y0 − . . . − zy−1 + a

+... + a ˜ y ∗ (z) = ˜b0 z u∗ (z) − z u0 − . . . − zu−1

+ ˜b1 z −1 u∗ (z) − z −1 u0 − . . . − zu−2 + . . . + ˜b u∗ (z) . Using (5.23) after rearrangement, this is represented in the form ∗ ˜ (z) + z (˜ a0 y0 − ˜b0 u0 ) + z −1 (˜ a0 y1 + a ˜1 y0 − ˜b0 u1 − ˜b1 u0 ) + . . . y ∗ (z) = w(z)u + z(˜ a0 y−1 + a ˜1 y−2 + . . . + a ˜−1 y0 − ˜b0 u−1 − ˜b1 u−2 − . . . − u0 ) . 0 have to be selected in such a way that the The initial vectors y00 , . . . , y−1 relations

a ˜0 y00 = ˜b0 u0 a ˜1 y00 + a ˜0 y10 = ˜b1 u0 + ˜b0 u1 .. .

.. .

.. .

0 ˜0 y−1 = ˜b−1 u0 + . . . + ˜b0 u−1 a ˜−1 y00 + . . . + a

(5.48)

196


hold. Owing to det a ˜0 = 0, the system (5.48) uniquely determines the to0 . Taking these vectors as initial values, tality of initial vectors y00 , . . . , y−1 we conclude that the solution {y 0 }, which is configured to the initial values 0 , possesses the z-transform y00 , . . . , y−1 ∗ ˜ (z) . y0∗ (z) = w(z)u

(5.49)

In what follows, those solution of Equation (5.31) having the transform (5.49) is called the solution with vanishing initial energy. As a result of the above considerations, the following theorem is formulated. Theorem 5.20. For the normal equation (5.31) and any Taylor input sequence {u}, there exists the solution with vanishing initial energy {y 0 }, which has the z-transform (5.49). The initial conditions of this solution are uniquely determined by the system of equations (5.48). 5.

The Taylor matrix sequence {H} = {H0 , H1 , . . . , },

(i = 0, 1, . . . )

for which the equation H ∗ (z) =

∞

Hi z −i = w(z) ˜

i=0

holds, is called the weighting sequence of the normal process (5.31). Based on the above reasons arising in the proof of Theorem 5.20, we are able to show that the weighting sequence {H} is the solution the matrix difference equation ˜ Hk = ˜b0 Uk+ + . . . + ˜b Uk , a ˜0 Hk+ + . . . + a

(k = 0, 1, . . . )

(5.50)

for the matrix input {U } = {Im , Omm , Omm , . . . }

(5.51)

with the solution of the equations a ˜0 H0 = ˜b0 a ˜ 1 H0 + a ˜ 0 H1 .. .

= ˜b1 .. .. . .

a ˜−1 H0 + . . . + a ˜0 H−1 = ˜b−1 as initial values H0 , . . . , H−1 . Notice that due to (5.51) for k > 0, Equation (5.50) converts into the homogeneous equation a ˜0 Hk+ + . . . + a ˜ Hk = Onm ,

(k = 0, 1, . . . ) .

5.4 Anomalous LTI Processes

197

5.4 Anomalous LTI Processes 1. Conformable with the above introduced concepts, the non-singular causal LTI process, described by the equations a ˜(q)yk = ˜b(q)uk

(5.52)

with a ˜(q) = a ˜0 q + a ˜1 q −1 + + . . . + a ˜ , ˜b(q) = ˜b0 q + ˜b1 q −1 + + . . . + ˜b

(5.53)

is called anomalous, when det a ˜0 = 0 .

(5.54)

From a mathematical point of view, anomalous processes, which include descriptor systems [34], are provided with a number of properties that distinguish them fundamentally from normal processes. Especially, notice the following: a) While a normal process, which is described by (5.52), (5.53), will always be causal, an anomalous process, described by (5.52)–(5.54), might be non-causal, as shown in Example 5.13. b) The successive procedure in Section 5.3 on the basis of (5.38), that was used for calculating the sequence is not applicable for anomalous processes. Indeed, denote a1 yk++i−1 − . . . − a ˜ yk+i + ˜b0 yk+i + . . . + ˜b uk+i , di = −˜ then Equation (5.52) is written as difference equation a ˜0 y+i = di ,

(i = 0, 1, . . . ) .

(5.55)

However due to (5.54), Equations (5.55) possess either no solution or infinitely many solutions. In both cases, a unique resolution of (5.55) is impossible and the successive calculation of the output sequence breaks down. c) For a normal equation (5.52), the totality of initial vectors (5.36), configured according to (5.37), can be prescribed arbitrarily. For causal anomalous processes (5.52)–(5.54), the solution in general exists only for certain initial conditions that are bound by additional equations, which also include values of the input sequence. Example 5.21. Consider the anomalous process y1,k+1 + 3y1,k + 2y2,k+1 = xk y1,k+1 + 2y2,k+1 + y2,k = 2xk .

(k = 0, 1, . . . )

(5.56)

198


In this case, we configure

a ˜(q) =

q + 3 2q , q 2q + 1

˜b(q) = 1 . 2

By means of (5.23), we find the transfer matrix

−2q + 1 q+6 . w(q) ˜ = 7q + 3 This matrix is proper and thus the process (5.56) is causal. For k = 0 from (5.56), we obtain y1,1 + 2y2,1 = x0 − 3y1,0 y1,1 + 2y2,1 = 2x0 − y2,0 .

(5.57)

Equations (5.57) are consistent under the condition x0 − 3y1,0 = 2x0 − y2,0

(5.58)

which makes the system (5.57) to y1,1 + 2y2,1 = x0 − 3y1,0 = 2x0 − 2y2,0 that has infinitely many solutions.

With respect to the above said, it is clear that dealing with anomalous processes (5.52)–(5.54) needs special attention. The present section presents adequate investigations. 2. Lemma 5.22. If the input of a causal anomalous process is a Taylor-sequence, then all solutions of Equation (5.52) are Taylor sequences. Proof. Without loss of generality, we assume that Equation (5.52) is row reduced, so that utilising (1.21) gives a ˜(q) = diag{q α1 , . . . , q αn }A˜0 + a ˜1 (q) , (5.59) where the degree of the i-th row of the matrix a ˜1 (q) is lower than αi and ˜(q) = αmax . Select det A˜0 = 0. Suppose deg a χ(q) = diag{q αmax −α1 , . . . , q αmax −αn } and consider the derived equations (5.14), which with the help of (5.59) takes the form A˜0 q αmax + χ(q)˜ a1 (q) yk = χ(q)˜b(q)uk . (5.60) As is easily seen, Equation (5.60) is normal under the given suppositions. Therefore, owing to Theorem 5.18 for Taylor input sequence {u}, all solutions of Equation (5.60) are Taylor sequences. But due to Lemma 5.2, all solutions of the original equation (5.52) are also solutions of the derived equation (5.60), thus Lemma 5.22 is proven.


199

3. Lemma 5.22 motivates a construction procedure for the solution set of Equation (5.52) according to its initial conditions. For this reason, the process equations (5.52) are written as a system of scalar equations of the shape (5.3): n

(0) aip yp,k+αi

+ ... +

p=1

n

(α ) aip i yp,k

=

p=1

m

(0) bir ur,k+αi

+ ... +

r=1

m

(α )

bir i ur,k

r=1

(5.61) (p = 1, . . . , n; r = 1, . . . , m; k = 0, 1, . . . ) , where due to the row reducibility, the condition (0) det aip = det A˜0 = 0 .

(5.62)

Without loss of generality, we suppose αi ≥ αi+1 ,

(i = 1, . . . , n − 1),

αn > 0 ,

because this can always be obtained by rearrangement. Passing formally from (5.61) to the z-transforms, we obtain n

(0)

aip z αi yp∗ (z) − z αi y¯p,0 − . . . − z y¯p,αi −1 +

p=1

+

n

(1)

aip z αi −1 yp∗ (z) − z αi −1 y¯p,0 − . . . − z y¯p,αi −2 + . . .

p=1

... +

n

˜i (z) , aip i yp∗ (z) = B (α )

p=1

˜i (z) is a polynomial in z, where y¯p,0 , . . . , y¯p,αi −1 are the initial values, and B (j) which depends on the coefficients bir and the excitation {u}. Substituting here ζ −1 for z, we obtain the equations for the ζ-transforms n

(0)

aip yp0 (ζ) − y¯p,0 − . . . − ζ αi −1 y¯p,αi −1 +

p=1

+

(5.63) n

(1)

aip yp0 (ζ) − y¯p,0 − . . . − z αi −2 y¯p,αi −2 + . . .

p=1

... + ζ

αi

n

(α )

aip i yp0 (z) = Bi (ζ) ,

p=1

where Bi (ζ) is a polynomial in ζ. The solution is put up as a set of power series ∞ 0 yp (ζ) = yp,k ζ k , (5.64) k=0

200


which exists due to Condition (5.62). Besides, the condition yp,k = y¯p,k

(5.65)

has to be fulfilled for all y¯p,k , that are practically configured by the left side of (5.63). Inserting (5.64) on the left side of (5.63), and comparing the coefficients of ζ k , (k = 0, 1, . . . ) on both sides, a system of successive linear equations for the quantities yp,k , (k = 0, 1, . . . ) is created, which due to (5.62) is always solvable. In order to meet Condition (5.65), we generate the totality of linear relations, that have to be fulfilled between the quantities y¯p,k and the first values of the input sequence {u}. These conditions determine the set of initial conditions y¯p,k , for which the wanted solution of Equation (5.61) exists. Since with respect to Lemma 5.22, all solutions of Equation (5.61) (whenever they exist) possess ζ-transforms, the suggested procedure always delivers the wanted result. Example 5.23. Investigate the row reduced anomalous system of equations y1,k+2 + y1,k + 2y2,k+1 = uk+1 y1,k + y2,k+1 = uk

(k = 0, 1, . . . )

(5.66)

of the form (5.61). In the present case, we have α1 = 2, α2 = 1 and

10 (0) aip = . 01 A formal pass to the z-transforms yields z 2 y1∗ (z) − z 2 y¯1,0 − z y¯1,1 + y1∗ (z) + 2zy2∗ (z) − 2z y¯2,0 = zu∗ (z) − zu0 y1∗ (z) + zy2∗ (z) − z y¯2,0 = u∗ (z) so, we gain

2

−1 2 y1∗ (z) z +1 2 z y¯1,0 + z y¯1,1 + 2z y¯2,0 + zu∗ (z) − zu0 = . (5.67) z y¯2,0 + u∗ (z) y2∗ (z) 1 z

Substituting ζ −1 for z gives

−1 y10 (ζ) 1 + ζ 2 2ζ y¯1,0 + ζ y¯1,1 + 2ζ y¯2,0 + ζu0 (ζ) − ζu0 = , (5.68) y¯2,0 + ζu0 (z) y20 (ζ) ζ 1

where the conditions y10 (ζ) = y1∗ (ζ −1 ),

y20 (ζ) = y2∗ (ζ −1 ),

u0 (ζ) = u∗ (ζ −1 )

were used. Since by supposition the input {u} is a Taylor sequence, the expansion


u0 (ζ) =

∞

201

uk ζ k

k=0

converges. Thus, the right side of (5.68) is analytical in the point ζ = 0. Hence the pair of convergent expansions y10 (ζ) =

∞

y1,k ζ k ,

y20 (ζ) =

k=0

∞

y2,k ζ k

(5.69)

k=0

exists uniquely. From (5.68), we obtain y10 (ζ) − y¯1,0 − ζ y¯1,1 + ζ 2 y10 (ζ) + 2ζy20 (ζ) − 2ζ y¯2,0 = ζu0 (ζ) − ζu0 ζy10 (ζ) + y20 (ζ) − y¯2,0 = ζu0 (z) .

(5.70)

Now we insert (5.69) into (5.70) and set equal those terms on both sides, which do not depend on ζ. Thus, we receive y1,0 = y¯1,0 ,

y2,0 = y¯2,0 ,

(5.71)

and for the term with ζ, we get y1,1 = y¯1,1 .

(5.72)

When (5.71) and (5.72) hold, then in the first row of (5.70) the terms of zero and first degree in ζ neutralise each other, respectively, and in the second equation the absolute terms cancel each other. Altogether, Equations (5.70) under Conditions (5.71), (5.72) might be written in the shape ∞

k

y1,k ζ + ζ

2

k=2

∞

k

y1,k ζ + 2ζ

k=0

ζ

∞

y1,k ζ k +

k=0

∞ k=1 ∞

k

y2,k ζ = ζ y2,k ζ k = ζ

k=1

∞ k=1 ∞

uk ζ k uk ζ k .

k=0

Canceling the first equation by ζ 2 , and the second by ζ, we find ∞ k=0

k

y1,k+2 ζ +

∞

k

y1,k ζ + 2

k=0 ∞ k=0

y1,k ζ k +

∞ k=0 ∞ k=0

k

y2,k+1 ζ = y2,k+1 ζ k =

∞ k=0 ∞

uk+1 ζ k uk ζ k .

k=0

Comparing the coefficients of the powers ζ k , (k = 0, 1, . . . ), we conclude that for any selection of the constants y1,0 = y¯1,0 , y2,0 = y¯2,0 , y1,1 = y¯1,1 the coefficients of the expansion (5.69) present a solution of Equation (5.66) which satisfies the initial conditions (5.71), (5.72). As result of the above analysis, the following facts are ascertained:

202


a) The general solution of Equation (5.66) is determined by the initial conditions y¯1,0 , y¯2,0 , y¯1,1 , which can be chosen arbitrarily. b) The right side of Relation (5.67) presents the z-transform of the general solution of Equation (5.66). c) The right side of Relation (5.68) presents the ζ-transform of the general solution of Equation (5.66). Example 5.24. Investigate the row reduced anomalous system of equations y1,k+2 + y1,k + 2y2,k+2 = uk+1 y1,k+1 + y2,k+1 = uk

(k = 0, 1, . . . )

of the form (5.61). In the present case, we have α1 = 2, α2 = 1 and

12 (0) (0) , det ai1 = 0 . ai1 = 11

(5.73)

(5.74)

By formal pass to z-transforms, we find ∗ y1 (z) = y2∗ (z) (5.75) 2

2

2 −1 2 ∗ z + 1 2z z y¯1,0 + z y¯1,1 + 2z y¯2,0 + 2z y¯2,1 + zu (z) − zu0 . z z z y¯1,0 + z y¯2,0 + u∗ (z) Although the values of the numbers α1 and α2 are the same as in Example 5.23, the right side of Relation (5.75) now depends on four values y¯1,0 , y¯2,0 , y¯1,1 and y¯2,1 . Substitute z = ζ −1 , so, as in the preceding example, the relations

−1 0 1 + ζ2 2 y2,0 + +2ζ y¯2,1 + ζu0 (z) − ζu0 y1 (ζ) y¯1,0 + ζ y¯1,1 + 2¯ = y¯1,0 + y¯2,0 + ζu0 (ζ) y20 (ζ) 1 1 (5.76) take place. The right side of (5.76) is analytical in the point ζ = 0. Hence there exists uniquely a pair of convergent expansions (5.69), which are the Taylor series of the right side of (5.76). Thus from (5.76), we obtain

y10 (ζ) − y¯1,0 − ζ y¯1,1 + ζ 2 y10 (ζ) + 2 y20 (ζ) − y¯2,0 − ζ y¯2,1 = ζu0 (ζ) − ζu0 (5.77) y10 (ζ) − y¯1,0 + y20 (ζ) − y¯2,0 = ζu0 (ζ) . Equating on both sides the terms not depending on ζ in (5.9), we find (y1,0 − y¯1,0 ) + 2(y2,0 − y¯2,0 ) = 0 (y1,0 − y¯1,0 ) + (y2,0 − y¯2,0 ) = 0 , so we gain with the help of (5.74)


y1,0 = y¯1,0 ,

y2,0 = y¯2,0 .

203

(5.78)

Comparing the coefficients for ζ, we find y1,1 − y¯1,1 + 2(y2,1 − y¯2,1 ) = 0 y1,1 + y2,1 = u0 which might be composed in the form (y1,1 − y¯1,1 ) + 2(y2,1 − y¯2,1 ) = 0 (y1,1 − y¯1,1 + (y2,1 − y¯2,1 = −¯ y1,1 − y¯2,1 + u0 . Recall (5.74) and recognise that the equations y1,1 = y¯1,1 ,

y2,1 = y¯2,1

(5.79)

hold, if and only if the condition y¯1,1 + y¯2,1 = u0

(5.80)

is satisfied. The last equation is a consequence of the second equation in (5.73) for k = 0. Suppose (5.80), then also Relations (5.78) and (5.79) are fulfilled, such that Relations (5.77) might be comprised to the equations ∞

y1,k ζ k + ζ 2

k=2

∞

y1,k ζ k + 2

k=0 ∞

y1,k ζ k +

k=1

∞ k=2 ∞

y2,k ζ k = ζ y2,k ζ k = ζ

k=1

∞ k=1 ∞

uk ζ k uk ζ k .

k=0

Reducing the first equation by ζ 2 , and the second one by ζ, we find ∞ k=0

y1,k+2 ζ k +

∞

y1,k ζ k + 2

k=0 ∞ k=0

y1,k+1 ζ k +

∞ k=0 ∞

y2,k+2 ζ k = y2,k+1 ζ k =

k=0

∞ k=0 ∞

uk+1 ζ k uk ζ k .

k=0

k

Comparing the coefficients for the powers ζ , (k = 0, 1, . . . ), we conclude that for any selection of the constants y¯1,0 , y¯2,0 and the quantities y¯1,1 , y¯2,1 , which are connected by Relation (5.80), there exists a solution of Equation (5.73) satisfying the initial conditions (5.78), (5.79). As result of the above analysis, the following facts are ascertained: a) The general solution of Equation (5.73) is determined by the quantities y¯1,0 , y¯2,0 , y¯1,1 and y¯2,1 , where the first two are free selectable and the other two are bound by Relation (5.80). b) When (5.80) holds, the right side of Relation (5.75) presents the z-transform of the general solution of Equation (5.73). c) When (5.80) holds, the right side of Relation (5.76) presents the ζ-transform of the general solution of Equation (5.73).

204


4. In order to find the solutions of the equation system (5.61) for admissible initial conditions, various procedures can be constructed. For this purpose, the discovered expressions for the z- and the ζ-transforms are very helpful. A further suitable approach consists in the direct solution of the derived normal equation (5.60). Using this approach, the method of successive approximation (5.38) is applicable. Example 5.25. Under the conditions of Example 5.23 substitute k by k + 1 in the second equation of (5.66). Then we obtain the derived normal system of equations y1,k+2 + y1,k + 2y2,k+1 = uk+1 y1,k+1 + y2,k+1 = uk+1 ,

(k = 0, 1, . . . )

that might be written in the form y1,k+2 = −y1,k − 2y2,k+1 + uk+1 y2,k+2 = −y1,k+1 + uk+1

(k = 0, 1, . . . ) .

(5.81)

Specify the initial conditions as y1,0 ,

y2,0 ,

y1,1 ,

y2,1 = u0 − y1,0 ,

then the wanted solution is generated directly from (5.81).

Example 5.26. A corresponding consideration of Equation (5.73) leads to the normal system y1,k+2 + 2y2,k+2 = −y1,k + uk+1 y1,k+2 + y2,k+2 =

uk+1

(k = 0, 1, . . . )

with the initial conditions y1,0 ,

y2,0 ,

y1,1 ,

y2,1 = u0 − y1,1 .

5. Although in general, the solution of a causal anomalous equation (5.52) does exist only over the set of admissible initial conditions, for such an anomalous system always exists the solution for vanishing initial energy. Theorem 5.27. For the causal anomalous process (5.52) with Taylor input sequence {u}, there always exists the solution {y0 }, the z-transform y0∗ (z) of which is determined by the relation ∗ y0∗ (z) = w(z)u ˜ (z) ,

where

w(z) ˜ =a ˜−1 (z)˜b(z)


205

is the assigned transfer matrix. The ζ-transform of the solution {y0 } has the view (5.82) y00 (ζ) = w(ζ)u0 (ζ) with

w(ζ) = w(ζ ˜ −1 ) = a ˜−1 (ζ −1 )˜b(ζ −1 ) .

(5.83)

Proof. Without loss of generality, we assume that Equation (5.52) is row reduced and in (5.59) det A˜0 = 0. In this case, the right side of (5.82) is analytical in the point ζ = 0. Thus, there uniquely exists the Taylor series expansion y00 (ζ)

∞

=

y˜k ζ k

(5.84)

k=0

that converges for sufficiently small |ζ|. From (5.83) and (5.53) using (5.40), we get (5.85) w(ζ) = a−1 (ζ)b(ζ) , where b(ζ) = ˜b0 + ˜b1 ζ + . . . + ˜b ζ .

a(ζ) = a ˜0 + a ˜1 ζ + . . . + a ˜ ζ ,

(5.86)

Applying (5.84)–(5.86) and (5.82), we obtain

a ˜0 + a ˜1 ζ + . . . + a ˜ ζ

∞

∞ ˜b0 + ˜b1 ζ + . . . + ˜b ζ u ˜k ζ k .

y˜k ζ k =

k=0

k=0

(5.87) By comparison of the coefficients for ζ k , (k = 0, 1, . . . , − 1), we find a ˜0 y˜0 = ˜b0 u0 a ˜1 y˜0 + a ˜0 y1 = ˜b1 u ˜0 + ˜b0 u1 .. .

.. .

(5.88)

.. .

˜0 y−1 = ˜b−1 u ˜0 + . . . + ˜b0 u−1 . a ˜−1 y˜0 + . . . + a With the aid of (5.88), Equation (5.87) is easily brought into the form a ˜0

∞ k=

∞

y˜k ζ k + ζ˜ a1

y˜k ζ k + . . . + ζ a ˜

k=−1 ∞

∞

k=

k=−1

= ˜b0

u ˜k ζ k + ζ ˜b1

∞

y˜k ζ k

k=0

Cancellation on both sides by ζ yields

u ˜k ζ k + . . . + ζ ˜b

∞ k=0

u ˜k ζ k .

206


a ˜0

∞

y˜k+ ζ k + a ˜1

k=0

∞

y˜k+−1 ζ k + . . . + a ˜

k=0

= ˜b0

∞

∞

y˜k ζ k

k=0

u ˜k+ ζ + ˜b1 k

k=0

∞

u ˜k+−1 ζ + . . . + ˜b k

k=0

∞

u ˜k ζ k .

k=0

Comparing the coefficients of the powers ζ k , (k = 0, 1, . . . ) on both sides, we realise that the coefficients of Expansion (5.84) satisfy Equation (5.52) for all k ≥ 0. Remark 5.28. Since in the anomalous case det a ˜0 = 0, Relations (5.88) do not allow to determine the initial conditions that are assigned to the solution with vanishing initial energy. For the determination of these initial conditions, the following procedure is possible. Using (5.59), we obtain w(ζ) = A−1 (ζ)B(ζ) ,

(5.89)

where a(ζ −1 ) = A˜0 + A˜1 ζ + . . . + A˜ ζ , A(ζ) = diag{ζ α1 , . . . ζ αn }˜ ˜0 + B ˜1 ζ + . . . + B ˜ ζ . B(ζ) = diag{ζ α1 , . . . ζ αn }˜b(ζ −1 ) = B

(5.90)

With the help of (5.82), (5.89) and (5.90), we derive

A˜0 + A˜1 ζ + . . . + A˜ ζ

∞

∞ ˜0 + B ˜1 ζ + . . . + B ˜ ζ yk ζ k = B uk ζ k .

k=0

k=0

By comparing the coefficients, we find ˜ 0 u0 A˜0 y˜0 = B ˜ 1 u0 + B ˜ 0 u1 A˜1 y˜0 + A˜0 y˜1 = B .. .

.. .

(5.91)

.. .

˜−1 u0 + . . . + B ˜0 u−1 . A˜−1 y˜0 + . . . + A˜0 y˜−1 = B Since per construction det A˜0 = 0, Equations (5.91) provide to determine the vectors y˜0 , . . . , y˜−1 . Example 5.29. Find the initial conditions for the solution with vanishing initial energy for Equations (5.73). Notice that in this example 2

z +1 2 z ˜ a ˜(z) = , b(z) = z z 1 is assigned. From this and (5.90), we obtain


A(ζ) = that means

1 + ζ 2 2ζ 1 1

12 ˜ , A0 = 11 ˜0 = 0 , B 0

2 ,

207

ζ B(ζ) = , ζ

00 10 ˜ ˜ A1 = , A2 = , 00 00 ˜0 = 1 , B ˜0 = 0 . B 1 0

(5.92)

Applying (5.92) and (5.91), we get ˜ 0 u0 A˜0 y˜0 = B i.e. y˜0 = O21 . Thus, the second equation in (5.91) takes the form ˜ 1 u0 A˜0 y˜1 = B with the consequence

u0 −1 ˜ ˜ y˜1 = A0 B1 u0 = . 0

Hence the solution with vanishing initial energy is determined by the initial conditions y¯1,0 = 0, y¯2,0 = 0, y¯1,1 = u0 , y¯2,1 = 0 . Here, Relation (5.80) is satisfied.

6. For anomalous causal LTI processes, in the same way as for normal processes, we introduce the concept of the weighting sequence {H} = {H0 , H1 , . . . } .

(5.93)

The weighting sequence is a matrix sequence, whose z-transform H ∗ (z) is determined by ˜ . (5.94) H ∗ (z) = w(z) From (5.94) it follows that the weighting sequence (5.93) might be seen as the matrix solution of Equation (5.52) under vanishing initial energy for the special input {U } = {Im , Omm , Omm , . . . } , because the z-transform of this input amounts to U ∗ (z) = Im . Passing in (5.94) to the variable ζ, we obtain an equation for the ζ-transform H 0 (ζ) = w(ζ) .

208


Since the right side is analytical in ζ = 0, there exists the convergent expansion H 0 (ζ) =

∞

Hk ζ k .

k=0

Applying Relations (5.85), (5.86), we obtain from the last two equations

a ˜0 + a ˜1 ζ + . . . + a ˜ ζ

∞

Hk ζ k = ˜b0 + ˜b1 ζ + . . . + ˜b ζ .

(5.95)

k=0

By comparison of the coefficients at ζ i , (i = 0, . . . , ) on both sides, we find a ˜0 H0 = ˜b0 a ˜ 1 H0 + a ˜0 H1 = ˜b1 .. .

(5.96)

.. .. . .

˜0 H = ˜b . a ˜ H0 + . . . + a When (5.96) is fulfilled, the terms with ζ i , (i = 0, . . . , ) on both sides of (5.95) neutralise each other, hence this equation might be written as a ˜0

∞

Hk ζ k + ζ˜ a1

k=+1

∞

Hk ζ k + . . . + ζ a ˜

k=

∞

Hk ζ k = Onm .

k=1

Canceling both sides by ζ +1 results in a ˜0

∞ k=0

Hk++1 ζ k + a ˜1

∞

Hk+ ζ k + . . . + a ˜

k=0

∞

Hk+1 ζ k = Onm .

k=0

If we make the coefficients at all powers of ζ on the left side equal to zero, then we find a ˜0 Hk++1 + a ˜1 Hk+ + . . . + a ˜ Hk+1 = Onm ,

(k = 0, 1, . . . ) ,

which is equivalently expressed by ˜1 Hk+−1 + . . . + a ˜ Hk = Onm , a ˜0 Hk+ + a

(k = 1, 2, . . . ) .

From this is seen that for k ≥ 1, the elements of the weighting sequence satisfy the homogeneous equation, which is derived from (5.52) for {u} = Om1 . Notice that for det a ˜0 = 0, the determination of the matrices Hi , (i = 0, 1, . . . , ) is not possible with the help of (5.96). To overcome this difficulty, Relation (5.89) is recruited. So instead of (5.95), we obtain the result

A˜0 + A˜1 ζ + . . . + A˜ ζ

∞ k=0

˜0 + B ˜1 ζ + . . . + B ˜ ζ , Hk ζ k = B

5.5 Forward and Backward Models

209

where a corresponding formula to (5.96) arises: ˜0 A˜0 H0 = B ˜1 A˜1 H0 + A˜0 H1 = B .. .

(5.97)

.. .. . .

˜ . A˜ H0 + . . . + A˜0 H = B Owing to det A˜0 = 0, the matrices Hi , (i = 0, 1, . . . , ) can be determined. If (5.97) is valid, we create the recursion formula A˜0 Hk++1 = −A˜1 Hk+ − . . . − A˜ Hk+1 ,

(k = 0, 1, . . . ) .

Thus with the help of the initial conditions (5.97), the weighting sequence {H} can be calculated. Example 5.30. Under the conditions of Example 5.29 and applying (5.97) and (5.92), we find 0 1 0 H0 = , H1 = , H2 = . 0 0 0 The further elements of the weighting sequence are calculated by means of the recursion formula A˜0 Hk+3 = −A˜2 Hk+1 ,

(k = 0, 1, . . . )

or Hk+3

1 0 = Hk+1 . −1 0

5.5 Forward and Backward Models 1.

Suppose the causal LTI process a ˜(q)yk = ˜b(q)uk ,

where

a ˜(q) = a ˜0 q + . . . + a ˜ , ˜b(q) = ˜b0 q + . . . + ˜b ,

(5.98) (n

× n) ,

(n

× m) .

(5.99)

As before, Equation (5.98) is designated as a forward model of the LTI process. a(q) in Select a unimodular matrix χ(q), such that the matrix a ˜ (q) = χ(q)˜ (5.20) becomes row reduced and consider the equivalent equation

210


a ˜ (q)yk = ˜b (q)uk ,

(5.100)

where ˜b (q) = χ(q)˜b(q). Let αi be the degree of the i-th row of the matrix a ˜ (q), then we have a ˜ (q) = diag {q α1 , . . . , q αn } A˜0 + A˜1 q −1 + . . . + A˜ q − , where det A˜0 = 0. Then the equation of the form

1

a(ζ)yk = b(ζ)uk

(5.101)

with a(ζ) = A0 + A1 ζ + . . . + A ζ = diag {ζ α1 , . . . , ζ αn } a ˜ (ζ −1 )

(5.102)

b(ζ) = B0 + B1 ζ + . . . + B ζ = diag {ζ α1 , . . . , ζ αn } ˜b (ζ −1 )

(5.103)

and

is called the associated backward model of the LTI process. From (5.29), we recognise that b(ζ) is a polynomial matrix. 2.

Hereinafter, the matrix ˜ w(q) ˜ =a ˜−1 (q)˜b(q) = a ˜−1 (q)b (q)

(5.104)

is called the transfer matrix of the forward model, and the matrix −1 ˜ w(ζ) = a−1 (ζ)b(ζ) = a ˜−1 (ζ −1 )˜b(ζ −1 ) = a ˜−1 )b (ζ −1 ) (ζ

(5.105)

is the transfer matrix of the backward model. From (5.104) and (5.105), we take the reciprocal relations w(ζ) = w(ζ ˜ −1 ),

w(q) ˜ = w(q −1 ) .

(5.106)

The matrix a ˜(q) is named as before the eigenoperator of the forward model and the matrix a(ζ) is the eigenoperator of the backward model. As seen from (5.102), the eigenoperator a(ζ) of the backward model is independent of the shape of the matrix ˜b(q) in (5.98). Obviously, the matrices a(ζ) and b(ζ) in (5.101) are not uniquely determined. Nevertheless, as we realise from (5.105), the transfer matrix w(ζ) is not affected. Moreover, later on we will prove that the structural properties of the matrix a(ζ) also do not depend on the special procedure for its construction. 1

In (5.101) for once ζ means the operator q −1 . A distinction from the complex variable of the ζ-transformation is not made, because the operator q −1 , due to the mentioned difficulties, will not be used later on.


211

Example 5.31. Consider the forward model 3y1,k+2 + y2,k+3 + y2,k = uk+2 + uk 2y1,k+1 + y2,k+2 = uk+1 . In the present case, we have 2 3

3q q + 1 a ˜(q) = , 2q q 2

2

˜b(q) = q + 1 . q

Thus, the transfer matrix of the forward model w(q) ˜ emerge as

q−1 q2 − 2 w(q) ˜ = q3 − 2 and the corresponding LTI process is strictly causal. Select the unimodular matrix

1 −q χ(q) = , 0 1 so we obtain a ˜ (q) = χ(q)˜ a(q) =

q2 1 , 2q q 2

˜b (q) = χ(q)˜b(q) = 1 . q

Thus, the matrices a(ζ), b(ζ) of the associated backward model take the shape

1 ζ2 10 00 01 2 a(ζ) = = + ζ+ ζ , 2ζ 1 01 20 00 2 0 1 2 ζ = ζ+ ζ . b(ζ) = ζ 1 0 3. Lemma 5.32. For the causality of the processes (5.98), it is necessary and sufficient that the transfer function w(ζ) is analytical in the point ζ = 0. For the strict causality of the process (5.98), the fulfillment of the equation w(0) = Onm is necessary and sufficient. Proof. Necessity: If the process (5.98) is causal, then the matrix w(q) ˜ is at least proper and thus analytical in the point q = ∞. Hence Matrix (5.105) is analytical in the point ζ = 0. If the process is strictly causal, then w(∞) ˜ = Onm is valid and we obtain the claim. Thus, the necessity is shown. We realise that the condition is also sufficient, when we reverse the steps of the proof.

212


Corollary 5.33. The process (5.98) is strictly causal, if and only if the equation w(ζ) = ζw1 (ζ) holds with a matrix w1 (ζ), which is analytical in the point ζ = 0. 4. It is shown that the concepts of forward and backward models are closely connected with the properties of the z- and ζ-transforms of the solution for Equation (5.98). Indeed, suppose a causal process, then it was shown above that for a Taylor input sequence {u}, independently of the fact whether the process is normal or anomalous, Equation (5.98) always possesses the solution with vanishing initial energy, and its z-transform y ∗ (z) satisfies the equation a ˜(z)y ∗ (z) = ˜b(z)u∗ (z)

(5.107)

that formally coincides with (5.98). In what follows, Relation (5.107) is also called a forward model of the process (5.98). From (5.107), we receive ∗ y ∗ (z) = w(z)u ˜ (z) = a ˜−1 (z)˜b(z)u∗ (z) .

Substituting here ζ −1 for z, we obtain y 0 (ζ) = w(ζ)u0 (ζ) , where y 0 (ζ), u0 (ζ) are the ζ-transforms of the process output for vanishing initial energy and the input sequence, respectively. Moreover, w(ζ) is the transfer matrix of the backward model (5.105). Owing to (5.105), the last equation might be presented in the form a(ζ)y 0 (ζ) = b(ζ)u0 (ζ)

(5.108)

which coincides with the associated backward model (5.101). 5. The forward model (5.107) is called controllable, if the pair (˜ a(z), ˜b(z)) is irreducible, i.e. for all finite z

˜ h (z) = rank a rank R ˜(z) ˜b(z) = n is true. Analogously, the backward model (5.108) is called controllable, if for all finite ζ

rank Rh = rank a(ζ) b(ζ) = n is true. We will derive some general properties according to the controllability of forward and backward models.


213

6. Lemma 5.34. If the forward model (5.107) is controllable, then the associated backward model (5.108) is also controllable. Proof. Let the model (5.107) be controllable. Then the row reduced model (5.100) is also controllable and hence for all finite z

rank a ˜ (z) ˜b (z) = n . Let z0 = 0, z1 , . . . , zq be the distinct eigenvalues of the matrix a ˜(z). Then the matrix a ˜ (z) has the same eigenvalues. From Formula (5.102), we gain with respect to det A˜0 = 0 that the set of eigenvalues of the matrix a(ζ) contains the quantities ζi = zi−1 , i = 1, . . . , q. From the above rank condition for all 1 ≤ i ≤ q, we obtain

rank a(ζi ) b(ζi ) = rank diag zi−αi , . . . , zi−αn a ˜ (zi ) ˜b (zi ) = n . Thus due to Lemma 1.42, the claim emerges. 7.

Let the forward model (5.107) be controllable and equation w(z) ˜ = C(zIp − A)−1 B + D

(5.109)

should describe a minimal standard realisation of the transfer matrix w(z). ˜ Then with the help of (5.106), we find w(ζ) = ζC(Ip − ζA)−1 B + D .

(5.110)

The expression on the right side is called a minimal standard realisation of the transfer matrix of the associated backward model. Besides, the rational matrix w0 (ζ) = ζC(Ip − ζA)−1 B might be seen as the transfer matrix of the PMD τ0 (ζ) = (Ip − ζA, ζB, C) . Lemma 5.35. Under the named suppositions, the PMD τ0 (ζ) is minimal, i.e. the pairs (Ip − ζA, ζB) and [Ip − ζA, C] are irreducible. Proof. Let z0 = 0, z1 , . . . , zq be the eigenvalues of the matrix A in (5.109). Then the eigenvalues of the matrix Ip − ζA turn out to be the numbers ζ1 = z1−1 , . . . , ζq = zq−1 . Since the representation (5.109) is minimal, for all finite z, it follows

zIp − A rank zIp − A B = n, rank = n. C Thus, for all i = 1, . . . , q

214


rank Ip − ζi A ζi B = rank zi Ip − A B = n ,

Ip − ζi A zi Ip − A Ip − ζi A rank = rank = rank = n. C ζi C C These conditions together with Lemma 1.42 imply the irreducibility of the pairs (Ip − ζA, ζB), [Ip − ζA, C] . Lemma 5.36. Let z0 = 0, z1 , . . . , zq be the different eigenvalues of the matrix A and a1 (z), . . . , aρ (z) be the totality of its invariant polynomials different from one, having the shape a1 (z) = z µ01 (z − z1 )µ11 · · · (z − zq )µq1 .. .. .. .. . . . . µ0ρ µ1ρ aρ (z) = z (z − z1 ) · · · (z − zq )µqρ ,

(5.111)

where µsi ≥ µs,i−1 , (s = 0, . . . q, i = 2, . . . , ρ),

q ρ

µsi = p .

s=0 i=1

Then the totality of invariant polynomials different from one of the matrix Ip − ζA consists of the ρ polynomials α1 (ζ), . . . , αρ (ζ) having the shape α1 (ζ) = (ζ − z1−1 )µ11 · · · (ζ − zq−1 )µq1 .. .. .. .. . . . . −1 µ1ρ αρ (ζ) = (ζ − z1 ) · · · (ζ − zq−1 )µqρ .

(5.112)

Proof. Firstly, we consider the matrix A as Jordan-Block (1.76) of dimension µ×µ ⎤ ⎡ a 1 ... 0 0 ⎥ ⎢ ⎢ 0 a ... 0 0 ⎥ ⎥ ⎢ ⎥ (5.113) A = Jµ (a) = ⎢ ⎢ ... ... . . . . . . ... ⎥ . ⎥ ⎢ ⎣0 0 ... a 1⎦ 0 0 ... 0 a For a = 0, we obtain

⎡

⎢ ⎢ ⎢ Iµ − ζJµ (a) = ⎢ ⎢ ⎢ ⎣

1 − ζa 0 .. . 0 0

⎤ ... 0 0 ⎥ . 1 − ζa . . 0 0 ⎥ ⎥ .. .. ⎥ .. .. . . . . . ⎥ ⎥ 0 . . . 1 − ζa −ζ ⎦ 0 . . . 0 1 − ζa −ζ

(5.114)


215

Besides, det[Iµ − ζJµ (a)] = (1 − ζa)µ

(5.115) −1

and Matrix (5.114) possesses only the eigenvalue ζ = a of multiplicity µ. For ζ = a−1 from (5.114), we receive ⎤ ⎡ 0 −a−1 0 . . . 0 ⎢ 0 0 −a−1 . . . 0 ⎥ ⎥ ⎢ ⎢ .. . . .. ⎥ . −1 (5.116) Iµ − a Jµ (a) = ⎢ ... ... . . ⎥ . ⎥ ⎢ −1 ⎣0 0 0 . . . −a ⎦ 0 0 0 ... 0 Obviously, rank[Iµ − a−1 Jµ (a)] = µ − 1 is true. Thus, owing to Theorem 1.28, Matrix (5.114) possesses an elementary divisor (ζ − a−1 )µ . For a = 0 from (5.114), we obtain ⎤ ⎡ 1 −ζ . . . 0 0 ⎥ ⎢ ⎢ 0 1 ... 0 0 ⎥ ⎥ ⎢ ⎥ (5.117) Iµ − ζJµ (0) = ⎢ ⎢ ... ... . . . . . . ... ⎥ . ⎥ ⎢ ⎣ 0 0 . . . 1 −ζ ⎦ 0 0 ... 0 1 Obviously det[Iµ − ζJµ (0)] = 1 , thus Matrix (5.117) is unimodular and has no elementary divisor. Now, consider the general case and A is expressed in the Jordan form A = U diag Jµ01 (0), Jµ11 (z1 ), . . . , Jµqρ (zq ) U −1 where U is a certain non-singular matrix. Hence Ip −ζA = U diag Iµ01 − ζJµ01 (0), Iµ11 − ζJµ11 (z1 ), . . . , Iµqρ − ζJµqρ (zq ) U −1 . (5.118) According to Lemma 1.25, the set of elementary divisors of the block-diagonal matrix (5.118) consists of the unification of the sets of elementary divisors of its diagonal blocks. As follows from (5.113)–(5.117), no elementary divisor of Matrix (5.118) is assigned to the eigenvalue zero and a non-zero eigenvalues zk corresponds to the totality of elementary divisors (ζ − zk−1 )µk1 , . . . , (ζ − zk−1 )µkρ , from which directly follows (5.112).

216


8. The next theorem establishes the connection between the eigenoperators of controllable forward and backward models. Theorem 5.37. Suppose the forward and backward models (5.107) and (5.108) be controllable, and the sequence of the invariant polynomials different from 1 of the matrix a ˜(z) should have the form (5.111). Then the sequence of invariant polynomials different from 1 of the matrix a(ζ) has the form (5.112). Proof. The proof is divided into several steps. a) When the right side of (5.109) defines a minimal standard realisation of the transfer matrix of a controllable forward model, then the sequences of the invariant polynomials different from 1 of the matrices a ˜(z) and zIp −A coincide. Thus, we get det a ˜(z) ≈ det(zIp − A) . This statement directly emerges from the content of Section 2.4. b) In the same way, we conclude that, when the right side of (5.110) presents a minimal standard realisation of the transfer matrix of the controllable backward model (5.108), then det a(ζ) ≈ det(Ip − ζA) . Thus, the sequences of invariant polynomials different from 1 of the matrices a(ζ) and Ip − ζA coincide, because the pairs (Ip − ζA, ζB) and [Ip − ζA, C] are irreducible. c) Owing to Lemma 5.36, the set of invariant polynomials of the matrices zIp − A and Ip − ζA are connected by Relations (5.111) and (5.112), such that with respect to a) and b) analogue connections also exist between the sets of invariant polynomials of the matrices a ˜(z) and a(ζ). Corollary 5.38. The last equivalence implies det a(0) = 0 . Hence the eigenoperator of a controllable backward models does not possess a zero eigenvalue. Corollary 5.39. Denote ˜ det a ˜(z) = ∆(z),

det a(ζ) = ∆(ζ)

(5.119)

˜ and let deg ∆(z) = p. Then ˜ −1 ) . ∆(ζ) ≈ ζ p ∆(ζ

(5.120)


217

Proof. Assume (5.111), then with µi =

ρ s=1

µis ,

q

µi = p

i=1

and taking advantage of (5.112), we can write ˜ ∆(z) ≈a ˜1 (ζ) · · · a ˜ρ (z) = z µ0 (z − z1 )µ1 · · · (z − zq )µq , ∆(ζ) ≈ α1 (ζ) · · · αρ (ζ) = (ζ − z1−1 )µ1 · · · (ζ − zq−1 )µq . Thus, (5.120) is ensured. In the following, the polynomials (5.119) are referred to as the characteristic polynomial of the forward or of the backward model, respectively. Remark 5.40. For an arbitrary polynomial f (ζ) with deg f (ζ) = p, the polynomial f ∗ (ζ) = ζ p f (ζ −1 ) usually is designated as the reciprocal to f (ζ), [14]. Thus Relation (5.120) might be written in the form ∆(ζ) ≈ ∆˜∗ (ζ) i.e. the characteristic polynomial of the controllable backward model is equivalent to the reciprocal characteristic polynomial of the controllable forward model. 9.

The next assertion can be interpreted as completion of Theorem 5.37.

Theorem 5.41. Let a ˜(z) be a non-singular n × n polynomial matrix and χ(z) a(z) becomes row reduced. is a unimodular matrix, such that a ˜ (z) = χ(z)˜ ˜ (z) and Furthermore, let αi be the degree of the i-th row of the matrix a build a(ζ) with the help of (5.102). Then for the matrix a(ζ) all assertions of Theorem 5.37 and its corollaries are true. Proof. Consider the controllable forward model a ˜(z)y ∗ (z) = χ−1 (z)u∗ (z) . Multiplying this from left by χ(z), we obtain the row reduced causal model a ˜ (z)y ∗ (z) = In u∗ (z) and hence both models are causal. Passing from the last model to the associated backward model by applying Relations (5.102), (5.103), we get a(ζ)y 0 (ζ) = b(ζ)u0 (ζ) and all assertions of Theorem 5.41 emerge from Theorem 5.37 and its corollaries.

218


10. As follows from the above shown, the set of eigenoperators of the associated backward model does not depend on the matrix ˜b(z) in (5.107) and it can be found by Formula (5.102). The reverse statement is in general not true. Therefore, the transition from a backward model to the associated forward model has to be considered separately. a) For a given controllable backward model a(ζ)y 0 (ζ) = b(ζ)u0 (ζ) ,

(5.121)

the transfer function of the associated controllable forward model can be designed with the help of the ILMFD a ˜−1 (z)˜b(z) = a−1 (z −1 )b(z −1 ) . b) The following lemma provides a numerical well posed method. Lemma 5.42. Let a controllable backward model (5.121) be given, where the matrix a(ζ) has the form (5.102) and det A0 = 0. Furthermore, let γi be the degree of the i-th row of the matrix

Rh (ζ) = a(ζ) b(ζ) . Introduce the polynomial matrices a ˜(z) = diag {z γ1 , . . . , z γn } a(z −1 ) , ˜b(z) = diag {z γ1 , . . . , z γn } b(z −1 ) .

(5.122)

Then under the condition

rank a ˜(0) ˜b(0) = n ,

(5.123)

the pair (˜ a(z), ˜b(z)) defines an associated controllable forward model. Proof. The proof follows the reasoning for Lemma 5.34. Example 5.43. Consider the controllable backward model (5.121) with

1 2ζ ζ . (5.124) a(ζ) = , b(ζ) = 2 1+ζ 1 ζ +1 In this case, we have γ1 = 1, γ2 = 2 and the matrices (5.122) take the form

z 2 1 ˜ a ˜(z) = 2 , b(z) = . z + z z2 1 + z2 Here Condition (5.123) is satisfied. Thus, the matrices define an associated controllable forward model.


219

11. As just shown for a known eigenoperator of the forward model a ˜(z), the set of all eigenoperators of the associated controllable backward models can be generated. When with the aid of Formula (5.102), one eigenoperator a0 (ζ) has been designed, then the set of all such operators is determined by the relation a(ζ) = ψ(ζ)a0 (ζ) , where ψ(ζ) is any unimodular matrix. The described procedure does not depend on the input operator ˜b(z). However, the reverse pass from an eigenoperator of a controllable backward model a(ζ) to the eigenoperator a ˜(z) in general requires additional information about the input operator b(ζ). In this connection, we ask for general rules for the transition from the matrix a(ζ) to the matrix a ˜(z). Theorem 5.44. Let the two controllable backward models a(ζ)y 0 (ζ) = b1 (ζ)u0 (ζ) , a(ζ)x0 (ζ) = b2 (ζ)v 0 (ζ)

(5.125)

be given, where a(ζ), b1 (ζ) and b2 (ζ) are polynomial matrices of dimensions ˜2 (z) be the n × n, n × m and n × , respectively. Furthermore, let a ˜1 (z) and a eigenoperators of the controllable forward models associated to (5.125). Then the relations a0 (z), a ˜2 (z) = β2 (z)˜ a0 (z) a ˜1 (z) = β1 (z)˜ are true, where β1 (z) and β2 (z) are nilpotent polynomial matrices, i.e. they only possess the eigenvalue zero, and the n × n polynomial matrix a ˜0 (z) can be chosen independently on the matrices b1 (ζ) and b2 (ζ), it is only committed by the matrix a(ζ). Proof. Consider the transfer matrices of the models (5.125) w1 (ζ) = a−1 (ζ)b(ζ),

w2 (ζ) = a−1 (ζ)b2 (ζ) .

(5.126)

Since the matrices (5.126), roughly speaking are not strictly proper, they could be written as w1 (ζ) = w ¯1 (ζ) + d1 (ζ),

w2 (ζ) = w ¯2 (ζ) + d2 (ζ) ,

(5.127)

where d1 (ζ) and d2 (ζ) are polynomial matrices, and the matrices w ¯1 (ζ) and w ¯2 (ζ) are strictly proper. But the right sides of Relations (5.126) are ILMFD, so Lemma 2.15 delivers that the relations w ¯1 (ζ) = a−1 (ζ)¯b1 (ζ),

w ¯2 (ζ) = a−1 (ζ)¯b2 (ζ) ,

where ¯b1 (ζ) = b1 (ζ) − a(ζ)d1 (ζ),

¯b2 (ζ) = b2 (ζ) − a(ζ)d2 (ζ)

(5.128)

220


determine an ILMFD of Matrix (5.128). Let us have the minimal standard realisation (5.129) w ¯1 (ζ) = C(ζIq − G)−1 B1 . In (5.129) the matrix G is non-singular, because of det a(0) = 0. Moreover, q = Mdeg w1 (ζ) is valid. Thus, we get from (5.128) ¯1 (ζ) w ¯2 (ζ) ≺ w l

and owing to Theorem 2.56, the matrix w ¯2 (ζ) allows the representation w ¯2 (ζ) = C(ζIq − G)−1 B2 .

(5.130)

Since the right sides of (5.128) are ILMFD, ¯2 (ζ) = deg det a(ζ) . Mdeg w ¯1 (ζ) = Mdeg w Hence the right side of (5.130) is a minimal standard realisation of the matrix w ¯2 (ζ). Inserting (5.129) and (5.130) into (5.127), we arrive at w1 (ζ) = C(ζIq − G)−1 B1 + d1 (ζ) , w2 (ζ) = C(ζIq − G)−1 B2 + d2 (ζ) , where the matrices C and G do not depend on the matrices b1 (ζ) and b2 (ζ) configured in (5.125). Substituting now z −1 for ζ, we obtain the transfer matrices of the forward models w ˜1 (z) = −C(zIq − G−1 )−1 G−2 B1 − CG−1 B1 + d1 (z −1 ) , w ˜2 (z) = −C(zIq − G−1 )−1 G−2 B2 − CG−1 B2 + d2 (z −1 ) ,

(5.131)

where the realisations (G−1 , G−2 B1 , C) and (G−1 , G−2 B2 , C) turn out to be minimal, because the realisations (G, B1 , C) and (G, B2 , C) are minimal. Build the ILMFD ˜0 (z)˜b0 (z) . (5.132) C(zIq − G−1 )−1 = a The matrix a ˜0 (z) does not depend on the matrices b1 (ζ) or b2 (z) in (5.126), because the matrices C and G do not. Besides, a ˜0 (z) has no eigenvalues equal to zero, because G−1 is regular. Using (5.132) from (5.131), we gain

−2 ˜−1 B1 − a ˜0 (z)CG−1 B1 + a ˜0 (z)d1 (z −1 ) , w ˜1 (z) = a 0 (z) −b0 (z)G

(5.133) −2 −1 −1 w ˜2 (z) = a ˜−1 (z) −b (z)G B − a ˜ (z)CG B + a ˜ (z)d (z ) . 0 2 0 2 0 2 0 The matrices in the brackets possess poles only in the point z = 0. Thus, in the ILMFDs ˜0 (z)CG−1 B1 + a ˜0 (z)d1 (z −1 ) = β1−1 (z)q1 (z) , −b0 (z)G−2 B1 − a −b0 (z)G−2 B2 − a ˜0 (z)CG−1 B2 + a ˜0 (z)d2 (z −1 ) = β2−1 (z)q2 (z)


221

the matrices β1 (z) and β2 (z) are nilpotent. Applying this and (5.133), as well as Corollary 2.19, we find out that the ILMFDs −1

w ˜1 (z) = [β1 (z)˜ a0 (z)]

q1 (z),

−1

w ˜2 (z) = [β2 (z)˜ a0 (z)]

q2 (z)

exist, from which all assertions of the Theorem may be read. 12. Sometimes in engineering literature, the pass from the original controllable forward model (5.98) to an associated backward model is made by procedures that are motivated by the SISO case. Then simply ˜(ζ −1 ), a(ζ) = ζ a

b(ζ) = ζ ˜b(ζ −1 )

(5.134)

is applied. It is easy to see that this procedure does not work, when det a ˜0 = 0 in (5.98). In this case, we would get det a(0) = 0, which is impossible for a controllable backward model. If however, in (5.99) det a ˜0 = 0 takes place, i.e. the original process is normal, then Formula (5.134) delivers a controllable associated backward model. 13. In recent literature [69, 80, 115], the backward model is usually written in the form (5.135) a(q −1 )yk = b(q −1 )uk , where q −1 is the right-shift operator that is inverse to the operator q. Per definition, we have q −1 yk = yk−1 ,

q −1 uk = uk−1 .

(5.136)

Example 5.45. The backward model corresponding to the matrices (5.124) is written with the notation (5.136) in the form y1,k + 2y2,k−1 = uk−1 y1,k + y1,k−1 + y2,k = uk−2 + uk .

(5.137)

As was demonstrated in [14], a strict foundation for using the operator q −1 for a correct description of discrete LTI processes is connected with honest difficulties. The reason arises from the fact, that the operator q is only invertible over the set of two-sided unlimited sequences. If however, the equations of the LTI process (5.4) are only considered for k ≥ 0, then the application of the operator q −1 needs special attention. From this point of view, the application of the ζ-transformation for investigating the properties of backward models seems more careful. Nevertheless, the description in the form (5.135) appears sometimes more comfortable, and it will also be used later on.

222


5.6 Stability of Discrete-time LTI Systems 1.

The vector sequence {y} = {y0 , y1 , . . . } is called stable, if the inequality yk < cθk ,

(k = 0, 1, . . . )

is true, where · is a certain norm for finite dimensional number vectors and c, θ are positive constants with 0 < θ < 1. If for the sequence {y}, such an estimate does not hold, then it is called unstable. The homogeneous vector difference equation ˜ yk = On1 a ˜0 yk+ + . . . + a

(5.138)

is called stable, if all of its solutions are stable sequences. Equations of the form (5.138), that are not stable, will be called unstable. 2. The next theorem establishes a criterion for the stability of Equation (5.138). Theorem 5.46. Suppose the n × n polynomial matrix a ˜(z) = a ˜0 (z)z + . . . + a ˜ be non-singular. Let zi , (i = 1, . . . , q) be the eigenvalues of the matrix a ˜(z), i.e. the roots of the equation ˜ ∆(z) = det a ˜(z) = 0 .

(5.139)

Then, for the stability of Equation (5.138), it is necessary and sufficient that |zj | < 1,

(j = 0, 1, . . . , q) .

(5.140)

Proof. Sufficiency: Let χ(z) be a unimodular matrix, such that the matrix a ˜ρ (z) = χ(z)˜ a(z) is row reduced. Thus, the equivalent equation a ˜ρ (z)yk = On1

(5.141)

at the same time with Equations (5.137) is stable or unstable, and the equation det a ˜ρ (z) = 0 possesses the same roots as Equation (5.139). Since the zero input is a Taylor sequence, owing to Lemma 5.22, all solutions of Equation (5.19) are Taylor sequences. Passing in Equation (5.141) to the z-transforms, we obtain the result that for any initial conditions, the transformed solution of Equation (5.141) has the shape y ∗ (z) =

R(z) , ˜ ∆(z)

5.6 Stability of Discrete-time LTI Systems

223

where R(z) is a polynomial vector. Besides under Condition (5.140), the inverse z-transformation formula [1, 123] ensures that all originals according to the transforms of (5.141) must be stable. Thus the sufficiency is shown. Necessity: It is shown that, if Equation (5.139) has one root z0 with |z0 | ≥ 1, then Equation (5.138) is unstable. Let d be a constant vector, which is a solution of the equation a ˜(z0 )d = On1 . Then, we directly verify that yk = z0k d,

(k = 0, 1, . . .)

is a solution of Equation (5.138). Besides due to |z0 | ≥ 1, this sequence is unstable and hence Equation (5.138) is unstable. 3. Let a(ζ) be the eigenoperator of the associated backward model designed by Formula (5.102). Then the homogeneous process equation might be written in form of the backward model a(ζ)y = a0 yk + a1 yk−1 + . . . + a yk− = On1

(5.142)

with det a0 = 0. Denote ∆(ζ) = det a(ζ) , then the stability condition of Equation (5.142) may be formulated as follows. Theorem 5.47. For the stability of Equation (5.142), it is necessary and sufficient that the characteristic polynomial det(a0 + a1 ζ + . . . + a ζ ) = det a(ζ) = ∆(ζ) = 0 has no roots inside the unit disc or on its border. Proof. The proof follows immediately from Theorems 5.41–5.46. Corollary 5.48. As a special case, Equation (5.142) is stable, if ∆(ζ) = const. = 0, i.e. if the matrix a(ζ) is unimodular. 4. Further on, the non-singular n × n polynomial matrices a ˜(z) and a(ζ) are called stable, if the conditions of Theorems 5.46, 5.47 are true for them. Matrices a ˜(z) and a(ζ) are named unstable, when they are not stable. The ˜ + [z] and set of real stable polynomial matrices a ˜(z) and a(ζ) is denoted by R nn + Rnn [ζ], respectively. For the sets of adequate scalar polynomials, we write ˜ + [z] and R+ [ζ], respectively. R

224


5. In the following considerations, the stability conditions for Equations (5.135) and (5.138) will be applied to explain the stability of the inverse matrices a ˜−1 (z) and a−1 (ζ). In what follows, the rational matrix w(z) ˜ ∈ Rnm (z) is called stable, if its poles z1 , . . . , zq satisfy Condition (5.140). The rational matrix w(ζ) is called stable, if it is free of poles inside or on the border of the unit disc. In the light of this definition, any polynomial matrix is a stable rational matrix. The sets ˜ + (z) and R+ (ζ), of real stable matrices w(z) ˜ and w(ζ) are denoted by R nm nm respectively. Rational matrices, which are not stable, are named unstable. Theorem 5.49. Equations (5.138) and (5.142) are stable, if and only if the rational matrices a ˜−1 (z) and a−1 (ζ) are stable. Proof. Applying Formula (2.114), we obtain the irreducible representation a ˜−1 (z) =

adj a ˜(z) , da˜ min (z)

(5.143)

where da˜ min (z) is the minimal polynomial of the matrix a ˜(z). Since the set of ˜(z), roots of the polynomial da˜ min (z) contains all roots of the polynomial det a the matrices a ˜(z) and a ˜−1 (z) are at the same time stable or unstable. The same can be said about the matrices a(ζ) and a−1 (ζ). 6. Hitherto, the forward model (5.107) and the backward model (5.108) are called stable, when the matrices a ˜(z) and a(ζ) are stable. For the considered class of systems, this definition is de facto equivalent to the asymptotic stability in the sense of Lyapunov. Theorem 5.50. Let the forward model (5.107) and the associated backward model (5.108) be controllable. Then for the stability of the corresponding models, it is necessary and sufficient that their transfer matrices w(z) ˜ resp. w(ζ) are stable. Proof. Using (5.104) and (5.143), we obtain w(z) ˜ =

adj a ˜(z) ˜b(z) . da˜ min (z)

Under the made suppositions, this matrix is irreducible, and this fact arises from Theorem 2.42. Thus, the matrices a ˜(z) and w(z) ˜ are either both stable or both unstable. This fact proves Theorem 5.50 for forward models. The proof for backward models runs analogously.

5.7 Closed-loop LTI Systems of Finite Dimension

225

5.7 Closed-loop LTI Systems of Finite Dimension 1. The input signal {u} of the process in Fig. 5.1 is now separated into two components. The first component is still denoted by {u} and contains the directly controllable quantities called as the control input. Besides the control input, additional quantities effect the process L, that depend on external factors. In Fig. 5.2, these quantities are assigned by the sequence {g} called as the disturbance input. The forward model of this process might be represented by the equation {gk }

-

{yk }

-

L

{uk } Fig. 5.2. Process with two inputs

a ˜(q)yk = ˜b(q)uk + f˜(q)gk ,

(k = 0, 1, . . . ) ,

(5.144)

where a ˜(q) ∈ Rnn [q], ˜b(z) ∈ Rnm [q], f˜(q) ∈ Rn [q]. In future, we will only consider non-singular processes, for which det a ˜(q) ≡/ 0 is true. When this condition is ensured, the rational matrices w(q) ˜ =a ˜−1 (q)˜b(q),

w ˜g (q) = a ˜−1 (q)f˜(q)

(5.145)

are explained, and they will be called the control and disturbance transfer matrix, respectively. For the further investigations, we always suppose the following assumptions: A1 The matrix w ˜g (q) is at least proper, i.e. the process is causal with respect to the input {g}. A2 The matrix w(q) ˜ is strictly proper, i.e. the process is strictly causal with respect to the input {u}. This assumption is motivated by the following reasons: a) In further considerations, only such kind of models will occur. b) This assumption enormously simplifies the answer to the question about the causality of the controller. c) It can be shown that, when the matrix w(q) ˜ is only proper, then the closed-loop system contains de facto algebraic loops, which cannot appear in real sampled-data control systems [19]. The process (5.144) is called controllable by the control input, if for all finite q

(5.146) rank Rh (q) = rank a ˜(q) ˜b(q) = n ,

226


and it is named controllable by the disturbance input, if for all finite q

rank Rg (q) = rank a ˜(q) f˜(q) = n . 2. To impart the process (5.144) appropriate dynamical properties, a controller R is fed back, what results in the structure shown in Fig. 5.3. The {gk }

-

{yk } L

•

-

{uk } R

Fig. 5.3. Controlled process

controller R itself is an at least causal discrete-time LTI object, which is given by the forward model ˜ α ˜ (q)uk = β(q)y k ˜ with α(q) ˜ ∈ Rmm [q], β(q) ∈ Rmn [q]. Together with Equation (5.144), this performs a model of the closed-loop system: a ˜(q)yk − ˜b(q)uk = f˜(q)gk (5.147) ˜ −β(q)y ˜ (q)uk = Om1 . k +α 3. The dynamical properties of the closed-loop system (5.147) are characterised by the polynomial matrix

a ˜(q) −˜b(q) ˜ ˜ Ql (q, α, ˜ β) = , (5.148) ˜ −β(q) α ˜ (q) which is named the (left) characteristic matrix of the forward model of the closed-loop system. A wide class of control problems might be expressed purely algebraic. Abstract control problem. For a given pair (˜ a(q), ˜b(q)) with ˜ strictly proper transfer matrix w(q), ˜ find the set of pairs (˜ α(q), β(q)) such that the matrix ˜ wd (q) = α(q) ˜ −1 β(q) is at least proper and the characteristic matrix (5.148) adopt certain prescribed properties. Besides, the closed-loop system (5.147), has to be causal.


227

4. For the solution of many control problems, it is suitable to use the associated backward model additionally to the forward model (5.144) of the process. We will give a general approach for the design of such models, which suppose the controllability of the process by the control input. For this reason, we write (5.144) in the form ˜ ˜g (q)gk . yk = w(q)u k +w Substituting here ζ −1 for q, we obtain yk = w(ζ)uk + wg (ζ)gk , where

w(ζ) = w(ζ ˜ −1 ) ,

(5.149)

wg (ζ) = w ˜g (ζ −1 ) .

When we have the ILMFD w(ζ) = a−1 (ζ)b0 (ζ) ,

(5.150)

then (5.149) might be written as a(ζ)yk = b0 (ζ)uk + a(ζ)wg (ζ)gk .

(5.151)

For the further arrangements the next property is necessary. Lemma 5.51. Let the process (5.144) be controllable by the control input. Then the matrix (5.152) bg (ζ) = a(ζ)wg (ζ) turns out to be a polynomial. Proof. Due to supposition (5.146), the first relation in (5.145) is an ILMFD of the matrix w(q). ˜ Thus from (5.145), we obtain ˜ , w ˜g (q) ≺ w(q)

(5.153)

l

because the polynomial a ˜(q) reduces the matrix w ˜g (q). Starting with the minimal standard realisation w(q) ˜ = C(qIp − A)−1 B + D , where A, B, C, D are constant matrices of appropriate dimension, we find from (5.153) with the help of Theorem 2.56, that the matrix w ˜g (q) allows the representation (5.154) w ˜g (q) = C(qIp − A)−1 Bg + Dg , where Bg and Dg are constant matrices of the dimensions p × and n × , respectively. Substituting ζ −1 for q, we obtain the standard realisation of the matrix w(ζ):

228


w(ζ) = ζC(Ip − ζA)−1 B + D , which is minimal due to Lemma 5.35, i.e. the pairs (Ip − ζA, ζB) and [Ip − ζA, C] are irreducible. Thus, when we build the ILMFD C(Ip − ζA)−1 = a−1 1 (ζ)b1 (ζ) ,

(5.155)

then the right side of the formula w(ζ) = a−1 (ζ) ζ b (ζ)B + a (ζ)D 1 1 1 turns out as an ILMFD of the matrix w(ζ). Hence the right side of (5.150) is also an ILMFD of the matrix w(ζ), such that a(ζ) = ψ(ζ)a1 (ζ) is valid with a unimodular matrix ψ(ζ). From (5.154), we find wg (ζ) = w ˜g (ζ −1 ) = ζC(Ip − ζA)−1 Bg + Dg . Therefore, using (5.155), we realise that a(ζ)wg (ζ) = ζψ(ζ) b1 (ζ)Bg + a1 (ζ)Dg = bg (ζ) is a polynomial matrix. Inserting (5.152) into (5.151), we obtain the wanted backward model of the form a(ζ)yk = b0 (ζ)uk + bg (ζ)gk . 4. Due to the supposed strict causality of the process with respect to the control input, the conditions det a(0) = 0,

b0 (0) = Onm

(5.156)

hold. That’s why for further considerations, the associated backward model of the process is denoted in the form a(ζ)yk = ζb(ζ)uk + bg (ζ)gk ,

(5.157)

where the first condition in (5.156) is ensured. Starting with the backward model of the process (5.157), the controller is attempted in the form α(ζ)uk = β(ζ)yk with det α(0) = 0 .

(5.158)


229

When we put this together with (5.157), we obtain the backward model of the closed-loop system a(ζ)yk − ζb(ζ)uk = bg (ζ)gk −β(ζ)yk + α(ζ)uk = Om1 .

(5.159)

Besides, the characteristic matrix of the backward model of the closed-loop system Ql (ζα, β) takes the form

a(ζ) −ζb(ζ) . (5.160) Ql (ζ, α, β) = −β(ζ) α(ζ) Introduce the extended output vector

yk , Υk = uk

so Equations (5.159) might be written in form of the backward model

bg (ζ) . Ql (ζ, α, β)Υk = B(ζ)gk , B(ζ) = Om 5. In analogy to the preceding investigations, consider the case when the process is described by a PMD of the form τ0 = (a(ζ), ζb(ζ), c(ζ)) ∈ Rnpm [ζ] with det a(0) = 0. Then the backward model of the closed-loop system might be presented in the shape a(ζ)xk = ζb(ζ)uk + bg (ζ)gk yk = c(ζ)xk

(5.161)

α(ζ)uk = β(ζ)yk . In this case, the characteristic matrix of the closed-loop system Qτ (ζ, α, β) takes the form ⎤ ⎡ a(ζ) Opn −ζb(ζ) ⎥ ⎢ Onm ⎦ . Qτ (ζ, α, β) = ⎣ −c(ζ) In (5.162) Omp −β(ζ) α(ζ) Introduce here the extended vector Υτ k

⎤ xk = ⎣ yk ⎦ , uk ⎡

so (5.161) might be written as backward model Qτ (ζ, α, β)Υτ k = Bτ (ζ)gk ,

⎤ bg (ζ) B(ζ) = ⎣ On ⎦ . Om ⎡

230


5.8 Stability and Stabilisation of the Closed Loop 1. The following investigations for stability and stabilisation refer to closedloop systems and will be done with the backward models (5.159) and (5.161), which are preferred in the whole book. In what follows, an arbitrary controller (α(ζ), β(ζ)) is said to be stabilising, if the closed-loop system with it is stable. The polynomial ∆(ζ) = det Ql (ζ, α, β)

(5.163)

is called the characteristic polynomial of the system (5.159), and the polynomial ∆τ (ζ) = det Qτ (ζ, α, β) the characteristic polynomial of the system (5.161). Theorem 5.52. For the stability of the systems (5.159) or (5.161), it is necessary and sufficient that the characteristic polynomials ∆(ζ) or ∆τ (ζ), respectively, are stable. Proof. The proof immediately follows from Theorems 5.46 and 5.47. Corollary 5.53. Any stabilising controller for the processes (5.157) or (5.161) is causal, i.e. det α(0) = 0 . (5.164) Proof. When the system (5.159) is stable, due to Theorem 5.46, we have det Ql (0, α, β) = ∆(0) = 0 , which with the aid of (5.160) yields det a(0) det α(0) = 0 , hence (5.164) is true. The proof for the system (5.161) runs analogously. Corollary 5.54. Any stabilising controller (α(ζ), β(ζ)) for the systems (5.159) or (5.161) possesses a transfer matrix wd (ζ) = α−1 (ζ)β(ζ) , because from (5.164) immediately emerge that the matrix α(ζ) is invertible. Corollary 5.55. The stable closed-loop systems (5.159) and (5.161) possess the transfer matrices w0 (ζ) = Q−1 l (ζ, α, β)B(ζ) , wτ (ζ) = Q−1 τ (ζ, α, β)Bτ (ζ) , which are analytical in the point ζ = 0.

5.8 Stability and Stabilisation of the Closed Loop

2.

Let the LMFD

w(ζ) = ζa−1 l (ζ)bl (ζ)

231

(5.165)

be given. Then using the terminology of Chapter 4, the pair (al (ζ), ζbl (ζ)) is called a left process model . Besides, if the pair (al (ζ), ζbl (ζ)) is irreducible, then the left process model is named controllable. If we have at the same time the RMFD (5.166) w(ζ) = ζbr (ζ)a−1 r (ζ) , then the pair [ar (ζ), ζbr (ζ)] is called a right process model. The right process model is named controllable, when the pair [ar (ζ), ζbr (ζ)] is irreducible. Related to the above, the concept of controllability for left and right models of the controllers might be introduced. If the LMFD and RMFD wd (ζ) = αl−1 (ζ)βl (ζ) = βr (ζ)αr−1 (ζ) exist, then the pairs (αl (ζ), βl (ζ)) and [αr (ζ), βr (ζ)] are left and right models of the controller, respectively. As above, we introduce the concepts of controllable left and right controller models. The matrices

al (ζ) −ζbl (ζ) Ql (ζ, αl , βl ) = , −βl (ζ) αl (ζ) (5.167)

αr (ζ) ζbr (ζ) Qr (ζ, αr , βr ) = βr (ζ) ar (ζ) are called the left and right characteristic matrices, respectively. Lemma 5.56. Let (al (ζ), ζbl (ζ)), [ar (ζ), ζbr (ζ)] as well as (αl (ζ), βl (ζ)), [αr (ζ), βr (ζ)] be irreducible left and right models of the process or controller, respectively. Then det Ql (ζ, αl , βl ) ≈ det Qr (ζ, αr , βr ) .

(5.168)

Proof. Applying the general formulae (4.76) and (5.167), we easily find

−1 det Ql (ζ, αl , βl ) = det al det αl det In − ζa−1 l (ζ)bl (ζ)αl (ζ)βl (ζ) ,

(5.169) −1 (ζ)β (ζ)α (ζ) . det Qr (ζ, αr , βr ) = det ar det αr det In − ζbr (ζ)a−1 r r r Due to the supposed irreducibility, we obtain for the left and right models det al (ζ) ≈ det ar (ζ),

det αl (ζ) ≈ det αr (ζ) .

Moreover, the expressions in the brackets of (5.169) coincide, that’s why (5.168) is true. From Lemma 5.56, it arises that the design problems for left and right models of stabilising controllers are in principal equivalent.

232


3. A number of statements is listed, concerning the stability of the closedloop system and the design of the set of stabilising controllers. Theorem 5.57. Let the process be controllable by the control input, and Relations (5.165) and (5.166) should determine controllable left and right IMFDs. Then a necessary and sufficient condition for the fact, that the pair (αl (ζ), βl (ζ)) is a left model of a stabilising controller, is that the matrices αl (ζ) and βl (ζ) satisfy the relation αl (ζ)ar (ζ) − ζβl (ζ)br (ζ) = Dl (ζ) ,

(5.170)

where Dl (ζ) is any stable polynomial matrix. For the pair [αr (ζ), βr (ζ)] to be a right model of a stabilising controller, it is necessary and sufficient that the matrices αr (ζ) and βr (ζ) fulfill the relation al (ζ)αr (ζ) − ζbl (ζ)βr (ζ) = Dr (ζ) ,

(5.171)

where Dr (ζ) is any stable polynomial matrix. Proof. Relation (5.170) will be shown. Necessity: Let the polynomial matrices α0r (ζ), β0r (ζ) satisfy the equation al (ζ)α0r (ζ) − ζbl (ζ)β0r (ζ) = In . Then owing to Lemma 4.4, the matrix Qr (ζ, α0r , β0r ) =

α0r (ζ) ζbr (ζ) β0r (ζ) ar (ζ)

is unimodular. Besides, we obtain Ql (ζ, αl , βl )Qr (ζ, α0r , β0r ) =

In Onm , Ml (ζ) Dl (ζ)

(5.172)

where Dl (ζ) and Ml (ζ) are polynomial matrices and in addition (5.170) is fulfilled. Per construction, the sets of eigenvalues of the matrices Ql (ζ, αl , βl ) and Dl (ζ) coincide. Thus, the stability of the matrix Ql (ζ, αl , βl ) implies the stability of the matrix Dl (ζ). Sufficiency: Take the steps of the proof in reverse order to realise that the conditions are sufficient. Relation (5.171) is shown analogously. Theorem 5.58. For the rational m × n matrix wd (ζ) to be the transfer matrix of a stabilising controller for the system (5.159), where the process is completely controllable, it is necessary and sufficient that it allows a representation of the form wd (ζ) = F1−1 (ζ)F2 (ζ) = G2 (ζ)G−1 1 (ζ) ,

(5.173)


233

where F1 (ζ), F2 (ζ) and G1 (ζ), G2 (ζ) are stable rational matrices satisfying F1 (ζ)ar (ζ) − ζF2 (ζ)br (ζ) = Im ,

(5.174)

al (ζ)G1 (ζ) − ζbl (ζ)G2 (ζ) = In .

Proof. The first statement in (5.174) will be shown. Necessity: Let (αl (ζ), βl (ζ)) be a left model of a stabilising controller. Then the matrices of this pair satisfy Relation (5.170) for a certain stable matrix Dl (ζ). Besides, we convince that the matrices F1 (ζ) = Dl−1 (ζ)αl (ζ),

F2 (ζ) = Dl−1 (ζ)βl (ζ)

are stable and satisfy Relations (5.173) and (5.174). Sufficiency: Let the matrices F1 (ζ) and F2 (ζ) be stable and Relation (5.174) be satisfied. Then the rational matrix

F (ζ) = F1 (ζ) F2 (ζ) is stable. Consider the ILMFD F (ζ) =

a−1 F (ζ)bF (ζ)

=

a−1 F (ζ)

m

n

d1 (ζ) d2 (ζ)

m,

where the matrix aF (ζ) is stable and d1 (ζ) = aF (ζ)F1 (ζ),

d2 (ζ) = aF (ζ)F2 (ζ)

are polynomial matrices. Due to d1 (ζ)ar (ζ) − ζd2 (ζ)br (ζ) = aF (ζ) , the pair (d1 (ζ), d2 (ζ)), owing to Theorem 5.57, is a stabilising controller with the transfer function −1 wd (ζ) = d−1 1 (ζ)d2 (ζ) = F1 (ζ)F2 (ζ) .

Thus, the first statement in (5.174) is proven. The second statement in (5.174) can be shown analogously. Theorem 5.59. Let the pair (al (ζ), ζbl (ζ)) be irreducible and (α0l (ζ), β0l (ζ)) should be an arbitrary basic controller, such that the matrix Ql (ζ, α0l , β0l ) becomes unimodular. Then the set of all stabilising left controllers (αl (ζ), βl (ζ)) for the system (5.159) is determined by the relations αl (ζ) = Dl (ζ)α0l (ζ) − ζMl (ζ)bl (ζ) , βl (ζ) = Dl (ζ)β0l (ζ) − Ml (ζ)al (ζ) , where Dl (ζ), Ml (ζ) are any polynomial matrices, but Dl (ζ) has to be stable.

234


Proof. The proof immediately emerges from Theorem 4.21. Theorem 5.60. Let the pairs (al (ζ), ζbl (ζ)) and (αl (ζ), βl (ζ)) be irreducible and the matrix Q−1 l (ζ, αl , βl ) be represented in the form

n

m

V1 (ζ) q12 (ζ) Q−1 l (ζ, αl , βl ) = V2 (ζ) q21 (ζ)

n

.

(5.175)

m

Then a necessary and sufficient condition for (αl (ζ), βl (ζ)) to be a stabilising controller is the fact that the matrices V1 (ζ) and V2 (ζ) are stable. Proof. The necessity of the conditions of the theorem emerges immediately from Theorem 5.49. Sufficiency: Let the ILMFD (5.165) and IRMFD (5.166) exist and (α0l (ζ), β0l (ζ)), (α0r (ζ), β0r (ζ)) should be dual left and right basic controllers. Then observing (5.172), we get Ql (ζ, αl , βl ) = Nl (ζ)Ql (ζ, α0l , β0l ) , where

(5.176)

In Onm Nl (ζ) = . Ml (ζ) Dl (ζ)

(5.177)

Inverting the matrices in Relation (5.176), we arrive at −1 −1 Q−1 l (ζ, αl , βl ) = Ql (ζ, α0l , β0l )Nl (ζ) .

With respect to the properties of dual controllers, we obtain

α0r (ζ) ζbr (ζ) −1 Ql (ζ, αl , βl ) = Qr (ζ, α0r , β0r ) = , β0r (ζ) ar (ζ)

(5.178)

where from (5.177), we find Nl−1 (ζ) =

In Onm . −Dl−1 (ζ)Ml (ζ) Dl−1 (ζ)

Applying this and (5.178), we obtain α0r (ζ) − ζbr (ζ)φ(ζ) ζbr (ζ)Dl−1 (ζ) −1 Ql (ζ) = β0r (ζ) − ar (ζ)φ(ζ) ar (ζ)Dl−1 (ζ) with the notation

φ(ζ) = Dl−1 (ζ)Ml (ζ) .

Comparing (5.175) with (5.179), we produce

(5.179)

(5.180)


V1 (ζ) = α0r (ζ) − ζbr (ζ)φ(ζ) ,

(5.181)

V2 (ζ) = β0r (ζ) − ar (ζ)φ(ζ) , or equivalently

235

In V1 (ζ) = Qr (ζ, α0r , β0r ) . −φ(ζ) V2 (ζ)

From this equation and (5.178), we generate

V1 (ζ) In = Ql (ζ, α0l , β0l ) , −φ(ζ) V2 (ζ) thus we read φ(ζ) = β0l (ζ)V1 (ζ) − α0l (ζ)V2 (ζ) . When the matrices V1 (ζ) and V2 (ζ) are stable, then the matrix φ(ζ) is also stable. Furthermore, notice that the pair (αl (ζ), βl (ζ)) is irreducible, because the pair (Dl (ζ), Ml (ζ)) is also irreducible. Hence Equation (5.180) defines an ILMFD of the matrix φ(ζ). But the matrix φ(ζ) is stable and therefore, Dl (ζ) is also stable. Since also the matrix Q−1 l (ζ, αl , βl ) is stable, the blocks in (5.179) must be stable. Hence owing to Theorem 5.49, it follows that the matrix Ql (ζ, αl , βl ) is stable and consequently, the controller (αl (ζ), βl (ζ)) is stabilising. Remark 5.61. In principle, we could understand the assertions of Theorem 5.60 as a corollary to Theorem 5.58. Nevertheless, in the proof we gain some important additional relations that will be used in the further disclosures. Besides (5.181), an additional representation of the matrices V1 (ζ), V2 (ζ) will be used. For this purpose, notice that from (5.167) and (5.175), it emerges al (ζ)V1 (ζ) − ζbl (ζ)V2 (ζ) = In , −βl (ζ)V1 (ζ) + αl (ζ)V2 (ζ) = Onm . Resolving these equations for the variables V1 (ζ) and V2 (ζ), we obtain −1

V1 (ζ) = [al (ζ) − ζbl (ζ)wd (ζ)]

, −1

V2 (ζ) = wd (ζ) [al (ζ) − ζbl (ζ)wd (ζ)]

(5.182) ,

where wd (ζ) is the transfer matrix of the controller. From (5.182), it follows directly (5.183) wd (ζ) = V2 (ζ)V1 (ζ)−1 .

236


4. On basis of Theorem 4.24, the stabilisation problem can be solved for the system (5.159) even in those cases, when the pair (a(ζ), ζb(ζ)) is reducible. Theorem 5.62. Suppose in (5.159) a(ζ) = λ(ζ)a1 (ζ),

ζb(ζ) = ζλ(ζ)b1 (ζ)

with a latent matrix λ(ζ) and the irreducible pair (a1 (ζ), ζb1 (ζ)). Then, if the matrix λ(ζ) is unstable, the system (5.159) never can be stabilised by a feedback of the form (5.158), i.e. the process (5.157) is not stabilisable. However, if the matrix λ(ζ) is stable, then there exists for this process a set of stabilising controllers, i.e. the process is stabilisable. The corresponding set of stabilising controllers coincides with the set of stabilising controllers of the irreducible pair (a1 (ζ), ζb1 (ζ)). 5. In analogy, following the reasoning of Section 4.8, the stabilisation problem for PMD processes is solved. Theorem 5.63. Suppose the strictly causal LTI process as a minimal PMD τ0 (ζ) = (a(ζ), ζb(ζ), c(ζ)) ,

(5.184)

where det a(0) ≡ / 0. For the transfer matrix wτ (ζ) = ζc(ζ)a−1 (ζ)b(ζ) , there should exist the ILMFD wτ (ζ) = ζp−1 (ζ)q(ζ) .

(5.185)

Then the stabilisation problem det Qτ (ζ, α, β) ≈ d+ (ζ) , where Qτ (ζ, α, β) is Matrix (5.162), is solvable for any stable polynomial d+ (ζ). Besides, the set of stabilising controllers coincides with the set of stabilising controllers of the irreducible pair (p(ζ), ζq(ζ)) and can be designed on basis of Theorems 5.57 and 5.59. Theorem 5.64. Let the strictly causal PMD process (5.184) be not minimal and the polynomial η(ζ) should be defined by the relation η(ζ) =

det a(ζ) , det p(ζ)

where the matrix p(ζ) is determined by an ILMFD (5.185). Then for the stabilisability of the PMD process (5.184), it is necessary and sufficient that the polynomial η(ζ) is stable. If this condition is fulfilled, the set stabilising controllers of the original system coincides with the set of stabilising controllers of the irreducible pair (p(ζ), ζq(ζ)) in the ILMFD (5.185).


237

6. The results in Sections 4.7 and 4.8, together with the design of the set of stabilising controllers allow to obtain at the same time information about the structure of the set of invariant polynomials for the characteristic matrices (5.160) and (5.162). For instance, in case of Theorem 5.57 or 5.63, the n invariant polynomials of the matrices (5.160) a1 (ζ), . . . , an (ζ) are equal to 1, and the set of the remaining invariant polynomials an+1 (ζ), . . . , an+m (ζ) coincides with the set of invariant polynomials of the matrix Dl (ζ). 7. For practical applications, the question on insensitivity of the obtained solution of the stabilisation problem plays a great role. The next theorem supplies the answer to this question. Theorem 5.65. Let (αl (ζ), βl (ζ)) be a stabilising controller for the strictly causal process (al (ζ), ζbl (ζ)), and instead of the process (al (ζ), ζbl (ζ)) there exists the disturbed strictly causal process (al (ζ) + al1 (ζ), ζbl (ζ) + ζbl1 (ζ)) with al1 (ζ) =

r

Ak ζ k ,

bl1 (ζ) =

k=0

r

(5.186)

Bk ζ k ,

k=0

where r ≥ 0 is an integer, and Ak , Bk , (k = 0, . . . , r) are constant matrices. Suppose · be a certain norm for finite-dimensional number matrices. Then there exists a positive constant , such that for Ak < ,

Bk < ,

(k = 0, . . . , r) ,

(5.187)

the closed-loop system with the disturbed process (5.186) and the controller (αl (ζ), βl (ζ)) remains stable. Proof. The characteristic matrix of the closed-loop system with the disturbed process has the form

al (ζ) + al1 (ζ) −ζ[bl (ζ) + bl1 (ζ)] . Ql1 (ζ, αl , βl )) = αl (ζ) −βl (ζ) Applying the sum theorem for determinants, we find det Ql1 (ζ, αl , βl ) = ∆1 (ζ) = ∆(ζ) + ∆2 (ζ) ,

where ∆(ζ) = det

al (ζ) −ζbl (ζ) −βl (ζ) αl (ζ)

(5.188)

is the characteristic polynomial of the undisturbed system, and ∆2 (ζ) is a polynomial, the coefficients of which tend to zero for → 0. Denote min |∆(ζ)| = δ .

|ζ|=1

(5.189)

238


Under our suppositions, δ > 0 is true, because the polynomial ∆(ζ) has no zeros on the unit circle. Attempt ∆2 (ζ) = d0 ζ µ + d1 ζ µ−1 + . . . + dµ , where the coefficients di , (i = 0, 1, . . . , µ) continuously depend on the elements of the matrices Ak , Bk , and all of them become zero, when for all κ = 0, 1, . . . , r Ak = Onn , Bk = Onm is valid. Thus, there exists an , such that the inequalities |di |
0 is a real constant. Here, s is designated as the exponent and T as the period of the exponential-periodic function x(t). Further on, all components of the vector xT (t) are supposed to be of bounded variation.

242

6 Parametric Discrete-time Models of Continuous-time Multivariable Processes

Assuming an exp.per. input signal x(t) (6.2), this section handles the existence problem for an exp.per. output signal of the processes (6.1), i.e. y(t) = est yT (t),

yT (t) = yT (t + T ) .

(6.3)

3. Lemma 6.1. Let the matrix w(p) be given in the standard form w(p) =

N (p) d(p)

with N (s) ∈ Rnm [p] and the scalar polynomial d(p) = (p − p1 )µ1 · · · (p − pq )µq ,

µ1 + . . . + µq = r .

(6.4)

Furthermore, suppose

x(t) = xs (t) = Xest ,

(6.5)

where X ∈ Cm1 is a constant vector and s is a complex number with s = pi ,

(i = 1, . . . , q) .

(6.6)

Then there exists a unique output of the form

y(t) = ys (t) = Y (s)est

(6.7)

with a constant vector Y (s) ∈ Cn1 . Besides, Y (s) = w(s)X and ys (t) = w(s)Xest .

(6.8)

Proof. Suppose a certain ILMFD w(s) = a−1 l (s)bl (s) with al (s) ∈ Rnn [s], bl (s) ∈ Rnm [s]. Then Relation (6.1) is equivalent to the differential equation d d al y = bl x. (6.9) dt dt Relations (6.5) and (6.7) should hold, and the vectors xs (t) and ys (t) should determine special solutions of Equation (6.9). Due to d al ys (t) = a(s)Y (s)est , dt d bl xs (t) = b(s)Xest , dt

6.1 Response of Linear Continuous-time Processes to Exponential-periodic Signals

the condition al (s)Y (s) = bl (s)X

(6.10)

must be satisfied. Owing to the properties of ILMFDs, the eigenvalues of the matrix al (s) turn out as the roots of the polynomial (6.4), but possibly with higher multiplicity. Thus, (6.6) implies det al (s) = 0 and from (6.10) we derive Y (s) = a−1 l (s)bl (s)X = w(s)X , i.e. Formula (6.8) really determines the wanted solution. Now, we prove that the found solution is unique. Beside of (6.5) and (6.7), let Equation (6.9) have an additional special solution of the form x(t) = Xest ,

ys1 (t) = Y1 (s)est ,

where Y1 (s) is a constant vector. Then the difference

εs (t) = ys (t) − ys1 (t) = [Y (s) − Y1 (s)]est must be a non-vanishing solution of the equation d al εs (t) = 0 . dt

(6.11)

(6.12)

Relation (6.12) represents a homogeneous system of linear difference equations with constant coefficients. This system may possess non-trivial solutions of the form (6.11) only when det al (s) = 0. But this case is excluded by (6.6). Thus, Y (s) = Y1 (s) is true, i.e. the solution of the form (6.7) is unique. 4. The question about the existence of an exp.per. output signal with the same exponent and the same period is investigated. Theorem 6.2. Let the transfer function of the processes (6.1) be strictly proper, the input signal should have the form (6.2), and for all k, (k = 0, ±1, . . .) the relations √ s + kjω = pi , (i = 1, . . . , q), ω = 2π/T, j = −1 (6.13) should be valid. Then there exists a unique exp.per. output of the form (6.3) with T yT (t) = ϕw (T, s, t − τ )xT (τ ) dτ , (6.14) 0

where ϕw (T, s, t) is defined by the series ϕw (T, s, t) =

∞ 1 w(s + kjω)ekjωt . T k=−∞

(6.15)

243

244


Proof. The function xT (t) is represented as Fourier series ∞

xT (t) =

xk ekjωt ,

k=−∞

where xk =

1 T

Then we obtain x(t) =

T

xT (τ )e−kjωτ dτ .

(6.16)

0 ∞

xk e(s+kjω)t .

k=−∞

According to the linearity of the operator (6.1) and Condition (6.13), Lemma 6.1 yields y(t) =

∞

w(s + kjω)xk e(s+kjω)t = est yT (t) ,

(6.17)

k=−∞

where yT (t) =

∞

w(s + kjω)xk ekjωt .

(6.18)

k=−∞

Using (6.16), the last expression sounds yT (t) =

T ∞ 1 w(s + kjω) xT (τ )e−kjωτ dτ ekjωt . T 0 k=−∞

Under our suppositions, series (6.15) converges. Hence due to the general properties of Fourier series [171], the order of summation and integration could be exchanged. Thus, we obtain Formula (6.14). It remains to show the uniqueness of the above generated exp.per. solution. Assume the existence of a second exp.per. output y1 (t) = est y1T (t),

y1T (t) = y1T (t + T )

in addition to the solution (6.3). Then the difference ε(t) = y(t) − y1 (t) is a solution of the homogeneous equation (6.12) with exponent s and period T . But, from (6.13) emerge that Equation (6.12) does not possess solutions different from zero. Thus, ε(t) = 0 and hence the exp.per. solutions (6.3) and (6.14) coincide. 5. In the following, the series (6.15) is called the displaced pulse frequency response, which is abbreviated as DPFR. This notation has a physical interpretation. Let δ(t) be the Dirac impulse and

6.2 Response of Open SD Systems to Exp.per. Inputs

δT (t) =

∞

245

δ(t − kT )

k=−∞

is a periodic pulse sequence. Then, it is well known [159] that the function δT (t) could be developed in a generalised Fourier series δT (t) =

∞ 1 kjωt e . T k=−∞

For the response of the process (6.1) to the exp.per. input x(t) = est δT (t)

(6.19)

recruit Formulae (6.17), (6.18) with xk = 1, (k = 0, ±1, . . . ). Thus, we obtain y(t) = est ϕw (T, s, t) . Hence the DPFR ϕw (T, s, t) is related to the response of the process (6.1) to an exponentially modulated sequence of unit impulses (6.19).

6.2 Response of Open SD Systems to Exp.per. Inputs 1. In this section and further on by a digital control unit DCU, we understand a system with a structure as shown in Fig. 6.2.1 If the digital control unit works as a controller, we also will call it a digital controller. Hereby, y = y(t) and DCU y

-

{ξ} ADC

-

{ψ} ALG

-

v

DAC

-

Fig. 6.2. Structure of a digital control unit

v = v(t) are vectors of dimensions m × 1 and n × 1, respectively. Furthermore, y(t) is assumed to be a continuous function of t. In Fig. 6.2 ADC is the analog to digital converter, which converts a continuous-time input signal y(t) into a discrete-time vector sequence {ξ} with the elements ξk , (k = 0, ±1, . . . ), i.e.

ξk = y(kT ) = yk , 1

(k = 0, ±1, . . . ) .

(6.20)

The concepts for the elements in a digital control system are not standardised in the literature.

246


The number T > 0, arising in (6.20), is named as the sampling period or the period of time quantisation. The block ALG in Fig. 6.2 stands for the control program or the control algorithm. If confusion is excluded, also the short name controller is used. It calculates from the sequence {ξ} a new sequence {ψ} with elements ψk , (k = 0, ±1, . . . ). The ALG is a causal discrete LTI object, which is described for instance by its forward model α ˜ 0 ψk+ + α ˜ 1 ψk+−1 + . . . + α ˜ ψk = β˜0 ξk+ + β˜1 ξk+−1 + . . . + β˜ ξk (6.21) or by the associated backward model α0 ψk + α1 ψk−1 + . . . + α ψk−ρ = β0 ξk + β1 ξk−1 + . . . + βρ ξk−ρ .

(6.22)

In (6.21) and (6.22) the α ˜ i , β˜i and αi , βi are constant real matrices of appropriate dimensions. Finally in Fig. 6.2, the block DAC is the digital to analog converter, which transforms a discrete sequence {ψ} into a continuous-time signal v(t) by the relation (6.23) v(t) = m(t − kT )ψk , kT < t < (k + 1)T . In (6.23), m(t) is a given function on the interval 0 < t < T , which is named as form function, because it establishes the shape of the control pulses [148]. In what follows, we always suppose that the function m(t) is of bounded variation on the interval 0 ≤ t ≤ T . 2. During the investigation of open and closed sampled-data systems, the transition of exp.per. signals through a digital control unit (6.20)-(6.23) plays an important role. Suppose the input of a digital control unit be the continuous-time signal y(t) = est yT (t),

yT (t) = yT (t + T )

(6.24)

with the exponent s and the period T , which coincides with the time quantisation period. We search for an exp.per. output of the form v(t) = est vT (s, t),

vT (s, t) = vT (s, t + T ) .

(6.25)

At first, notice a special feature, when an exp.per. signal (6.24) is sent through a digital control unit. If (6.24) and (6.20) is valid, we namely obtain ξk = eksT ξ0 ,

ξ0 = yT (0).

The result would be the same, if instead of the input y(t) the exponential signal ys (t) = est yT (0)

6.2 Response of Open SD Systems to Exp.per. Inputs

247

would be considered. The equivalence of the last two equations shows the so-called stroboscopic property of a digital control unit. The awareness of the stroboscopic property makes it possible to connect the response of the digital control unit to an exp.per. excitation with its response to an exponential signal. 3. In connection with the above said, consider the design task for a solution of Equations (6.20)–(6.23) under the conditions y(t) = est y0 ;

v(t) = est vT (t), vT (t) = vT (t + T ) .

(6.26)

Assume at first m(t) = 1,

0≤t 1 and f (λi ) = 0, the elementary divisor (λ − λi )φi of the matrix A corresponds to an elementary divisor φ (λ − f (ζi )) i of the matrix f (A). In case of f (λi ) = 0, φi > 1 for the elementary divisor (λ − λi )φi , there exist more than one elementary divisors of the matrix f (A). 12. Suppose again that the characteristic polynomial of the matrix A has the shape (6.58) and the sequence of its elementary divisors has the shape (6.59). It is said that the matrices A and f (A) have the same structure, if among the numbers f (λ1 ), . . . , f (λq ) are no equal ones and the sequence of elementary divisors of the matrix f (A) possesses the analogue form to (6.59) (λ − f (λ1 ))

φ1

, . . . , (λ − f (λr ))

φr

.

The above derived results are formulated in the next theorem. Theorem 6.7. The following two conditions are necessary and sufficient for the matrices A and f (A) to possess the same structure: a) f (λi ) = f (λk ),

(i = k; i, k = 1, . . . , q)

(6.60)

b) For all exponents φi ( = 1, . . . , ρ < r) with φi > 1 f (λi ) = 0,

( = 1, . . . , ρ) .

Corollary 6.8. Let the matrix A be cyclic, i.e. in (6.59) r = q and φi = µi are true. Then the matrix f (A) is also cyclic, if and only if Conditions a) and b) are true.

256


6.4 Matrix Exponential Function 1.

Consider the scalar function f (λ) = eλt ,

where t is a real parameter. This function is defined on the spectrum of every matrix. Thus for any matrix A, formula (6.56) is applicable and we obtain f (A) =

q teλi t t(νi −1) eλi t Mi2 + . . . + Mi,νi , eλi t Mi1 + 1! (νi − 1)! i=1

(6.61)

where f (λi ) = eλi t ,

f (λi ) = teλi t , . . . , f (νi −1) (λi ) = tνi −1 eλi t .

Matrix (6.61) is named the exponential function of the matrix A and it is denoted by f (A) = eAt . 2. A further important representation of the matrix eAt is obtained by applying the series expansion eλt = 1 + λt +

λ 2 t2 + ... , 2!

(6.62)

which converges for all λ, and consequently on any spectrum too. Inserting the matrix A instead of λ into (6.62), we receive eAt = Ip + At +

A2 t2 + ... . 2!

Particularly for t = 0, we get eAt |t=0 = Ip . 3.

Differentiating (6.63) by t, we obtain d At A2 t2 e + . . . = AeAt . = A Ip + At + dt 2!

4.

Substituting the parameter −τ for t in (6.62), we receive e−Aτ = Ip − Aτ +

A2 τ 2 − +... . 2!

By multiplying this expansion with (6.62), we prove

(6.63)

6.4 Matrix Exponential Function

eAt e−Aτ = e−Aτ eAt = Ip + A(t − τ ) + hence

257

A2 (t − τ )2 + ... 2!

eAt e−Aτ = e−Aτ eAt = eA(t−τ ) .

For τ = t, we find immediately eAt e−At = e−At eAt = Ip or

At −1 = e−At . e

5. Theorem 6.9. For a positive constant T , the matrices A and eAT possess the same structure, if the eigenvalues λ1 , . . . , λq of A satisfy the conditions eλi T = eλk T ,

(i = k; i, k = 1, . . . , q)

(6.64)

or equivalently λi − λk =

2nπj = njω , T

(i = k; i, k = 1, . . . , q) ,

(6.65)

where n is an arbitrary integer and ω = 2π/T . Proof. Owing to d(eλT )/ dT = T eλT = 0 for an exponential function, Condition b) in Theorem 6.7 is always ensured. Therefore, the matrices A and eAT have the same structure, if and only if Conditions (6.60) hold, which in the present case have the shape (6.64). Conditions (6.65) are obviously implications of (6.64). Corollary 6.10. Let the matrix A be cyclic. Then, a necessary and sufficient condition for the matrix eAT to become cyclic is the demand that Conditions (6.64) hold. 6.

The next theorem is fundamental for the future declarations.

Theorem 6.11 ([71]). Let the pair (A, B) controllable and the pair [A, C] observable. Then under Conditions (6.64), (6.65), the pair (eAT , B) is controllable and the pair [eAT , C] is observable. 7. Since for any λ, t always eλt = 0, the matrix eAt becomes non-singular for any finite t and any matrix A.

258


6.5 DPFR and DLT of Rational Matrices 1.

Let w(s) ∈ Rnm (s) be a strictly proper rational matrix and ϕw (T, s, t) =

∞ 1 w(s + kjω)ekjωt , T

ω=

k=−∞

2π T

(6.66)

be its displaced pulse frequency response. As the discrete Laplace transform (DLT) of the matrix w(s), we understand the sum of the series

Dw (T, s, t) =

∞ 1 w(s + kjω)e(s+kjω)t , T

−∞ < t < ∞ .

(6.67)

k=−∞

The transforms (6.66) and (6.67) are closely connected:

Dw (T, s, t) = est ϕw (T, s, t) .

(6.68)

Per definition, we have ϕw (T, s, t) = ϕw (T, s, t + T ) . In this section, we will derive closed formulae for the sums of the series (6.66) and (6.67). 2. Lemma 6.12. Let the matrix w(s) be strictly proper and possess the partial fraction expansion µi q wik w(s) = , (6.69) (s − si )k i=1 k=1

where the wik are constant matrices. Then the sum of the series (6.67) is determined by the formulae

Dw (T, s, t) = Dw (T, s, t),

0 0, k > 0 .

7.6 Polynomial Stabilisation of the Standard SD System x

-g 6

- N (p)

•

299

y

u DAC

{ψ}

ALG

{ξ}

ADC

Fig. 7.9. Closed loop of the standard sampled-data system

3. The following theorem presents a solution to the stabilisation problem for the closed loop, when the continuous-time plant is given by its minimal realisation. ˜2ν , C˜2ν ) be a minimal realisation of dimension Theorem 7.13. Let (A˜ν , B (n, λ, m), and the rational matrix wN (ζ) be determined by ˜ ˜ ˜2ν , wN (ζ) = DN µ (T, ζ, 0) = ζ C˜2ν eAν T µ(A˜ν )(Iλ − ζeAν T )−1 B

where

T

µ(A˜ν ) =

e−Aν τ m(τ ) dτ . ˜

0

Let us have an ILMFD wN (ζ) = ζa−1 N (ζ)bN (ζ)

(7.74)

with aN (ζ) ∈ Rnn [ζ] and bN (ζ) ∈ Rnm [ζ]. Then the relation

N (ζ) =

˜

det(Iη − ζeAν T ) det aN (ζ)

(7.75)

is a polynomial. For an arbitrary choice of the minimal realisation ˜2ν , C˜2ν ) and the matrices aN (ζ), bN (ζ), the polynomials (7.75) are (A˜ν , B equivalent, i.e., they are equal up to a constant factor. A necessary and sufficient condition for the set of minimal realisations of the matrix N (p) to be stabilisable is that the polynomial (7.75) is stable, i.e., is free of roots inside the closed unit disk. If the polynomial N (ζ) is stable and (aN (ζ), ζbN (ζ)) is an arbitrary pair forming the ILMFD (7.74), then the set of all controllers (α(ζ), β(ζ)) stabilising the minimal realisations of the matrix N (p) is defined as the set of all pairs ensuring the stability of the matrix

aN (ζ) −ζbN (ζ) . (7.76) QN (ζ, α, β) = −β(ζ) α(ζ) ˜2ν , C˜2ν ) and Proof. Using an arbitrary minimal realisation (A˜ν , B (7.62)–(7.64), we arrive at the problem of investigating the stability of the system

300

7 Description and Stability of SD Systems

y˜ = C˜2ν v˜ ,

d˜ v ˜2ν u = A˜ν v˜ + B dt

α(ζ)ψk = β(ζ)uk u(t) = m(t − kT )ψk ,

kT < t < (k + 1)T .

As follows from Theorem 7.9, a necessary and sufficient condition for the stability of this system is that the matrix ⎡ ⎤ ˜ ˜ ˜2ν Iλ − ζeAν T Oλn −ζeAν T µ(A˜ν )B ⎥ ˜ ν (ζ, α, β) = ⎢ Q (7.77) ⎣ −C˜2ν ⎦ In Onm Omλ −β(ζ) α(ζ) is stable. Hence the set of the pairs of stabilising polynomials (α(ζ), β(ζ)) coincides with the set of stabilising pairs for the nonsingular PMD ˜ ˜ ˜2ν , C˜2ν . (7.78) τÑ (ζ) = Iλ − ζeAν T , ζeAν T µ(A˜ν )B ˜2ν , C˜2ν ) and the Then the claim of the theorem for a given realisation (A˜ν , B pair (aN (ζ), ζbN (ζ)) follows from Theorem 5.64. It remains to prove that the set of stabilising controllers does not depend on the choice of the realisation ˜2ν , C˜2ν ) and the pair (aN (ζ), ζbN (ζ)). With this aim in view, we notice (A˜ν , B that from the formulae of Section 6.8, it follows that ∞ 1 N (s + kjω)µ(s + kjω) . (7.79) wN (ζ) = T e−sT =ζ k=−∞

Since all minimal realisations are equivalent, the matrix wN (ζ) is indepen˜2ν , C˜2ν ). Hence the set of pairs dent of the choice of the realisation (A˜ν , B (aN (ζ), ζbN (ζ)) is also independent of the choice of this realisation. The same proposition holds for the set of stabilising controllers. 4. A more complete result can be obtained under the assumption that the poles of the matrix N (p) satisfy the strict conditions for non-pathological behavior (6.124) and (6.125). Theorem 7.14. Let the eigenvalues s1 , . . . , sq of the matrix sIλ − A˜ν satisfy the strict conditions for non-pathological behavior si − sk =

2nπj , T

(i = k; i, k = 1, . . . , q; n = 0, ±1, . . . ) , µ(si ) = 0 ,

(i = 1, . . . , q) .

˜2ν , C˜2ν ) are stabilisable. Then all minimal realisations (A˜ν , B

(7.80) (7.81)

7.6 Polynomial Stabilisation of the Standard SD System

301

Proof. If (7.80) and (7.81) hold, due to Theorem 6.30, the PMD (7.78) is minimal. Then for the ILMFD ˜ C˜2ν Iλ − ζeAν T = a−1 1 (ζ)b1 (ζ) , we have

˜ det a1 (ζ) ≈ det Iλ − ζeAν T .

Hence from Lemma 2.9, it follows that for any ILMFD (7.74) aN (ζ) = φ(ζ)a1 (ζ) is true with a unimodular matrix φ(ζ). Therefore, in this case due to (7.75), the polynomial N (ζ) = const. = 0 is stable. Then the claim follows from Theorem 7.13. 5. A general criterion for the stabilisability of the closed loop is given by the following theorem. ˜ B ˜2 , C˜2 ) of dimension n, η, m be any realisation of Theorem 7.15. Let (A, ˜2ν , C˜2ν ) be one of its minimal realisation with ˜ the matrix N (p), and (Aν , B dimension n, λ, m such that η > λ. Then the function r(s) =

˜ det(sIη − A) det(sIλ − A˜ν )

(7.82)

˜2ν , C˜2ν ) is is a polynomial. If, in addition, the minimal realisation (A˜ν , B ˜ B ˜2 , C˜2 ). If the minimal realisation not stabilisable, so is the realisation (A, ˜2ν , C˜2ν ) is stabilisable, then for the stabilisability of the realisation (A˜ν , B ˜ B ˜2 , C˜2 ), it is necessary and sufficient that all roots of the polynomial (7.82) (A, be in the open left half-plane. Under this condition, the set of stabilising con˜ B ˜2 , C˜2 ) and is detertrollers (α(ζ), β(ζ)) is independent of the realisation (A, mined by the stability condition for the matrix (7.76). Proof. Under the given assumptions, the PMDs ˜ B ˜2 , C˜2 ) , τN (s) = (sIη − A, ˜2ν , C˜2ν ) τN ν (s) = (sIλ − A˜ν , B

(7.83)

are equivalent, i.e. their transfer matrices coincide. Moreover, since the PMD τN ν (s) is minimal, Relation (7.82) is a polynomial by Lemma 2.48. Let us have ˜ = (s − s1 )κ1 · · · (s − sq )κq , det(sIη − A)

κ1 + . . . + κq = η ,

det(sIλ − A˜ν ) = (s − s1 )1 · · · (s − sq )q ,

1 + . . . + q = λ ,

302


where κi ≥ i , (i = 1, . . . , q). Let i < κi for i = 1, . . . , γ and i = κi for i = γ + 1, . . . , q. Then from (7.82), we obtain r(s) = (s − s1 )m1 · · · (s − sγ )mγ ,

(7.84)

where mi = κi − i , (i = 1, . . . , γ). Moreover, since the PMDs (7.83) are equivalent, using (7.79), we receive −1 −1 ˜ ˜ ˜ ˜B ˜2 = ζ C˜2ν Iλ − ζeA˜ν T ˜2ν ζ C˜2 Iη − ζeAT eAT µ(A) eAν T µ(A˜ν )B (7.85) = wN (ζ) . From (7.85), it follows that the PMDs ˜ ˜ ˜B ˜2 , C˜2 , τd (ζ) = Iη − ζeAT , ζeAT µ(A)

˜

˜

˜2ν , C˜2ν τdν (ζ) = Iλ − ζeAν T , ζeAν T µ(A˜ν )B

(7.86) (7.87)

are equivalent. Then ˜

det(Iη − ζeAT ) = (1 − ζes1 T )κ1 · · · (1 − ζesq T )κq , ˜

det(Iλ − ζeAν T ) = (1 − ζes1 T )1 · · · (1 − ζesq T )q , and the relation ˜

1 (ζ) =

det(Iη − ζeAT ) det(Iλ − ζeA˜ν T )

= (1 − ζes1 T )m1 · · · (1 − ζesγ T )mγ

(7.88)

is a polynomial. Consider the characteristic matrix (7.77) for the PMD (7.86) ⎡ ⎤ ˜ ˜ ˜B ˜2 Iη − ζeAT Oηn −ζeAT µ(A) ⎢ ⎥ ˜ α, β) = ⎢ −C˜ ⎥. Q(ζ, In Onm 2 ⎣ ⎦ Omη −β(ζ) α(ζ) Using Equation (4.71) for this and Matrix (7.77), and taking account of (7.85), we find ˜ ˜ α, β) = det(Iη − ζeAT det Q(ζ, ) det[α(ζ) − β(ζ)w Ñ (ζ)] ,

˜ ν (ζ, α, β) = det(Iλ − ζeA˜ν T ) det[α(ζ) − β(ζ)w det Q Ñ (ζ)] . Hence with (7.88), it follows ˜ α, β) = 1 (ζ) det Q ˜ ν (ζ, α, β) . det Q(ζ, ˜2ν , C˜2ν ) is not stabilisable, then the matrix If the minimal realisation (A˜ν , B ˜ Qν (ζ, α, β) is unstable for any controller (α(ζ), β(ζ)). Due to the last equation,

7.6 Polynomial Stabilisation of the Standard SD System

303

˜ α, β) is also unstable. If the polynomial 1 (ζ) is not stable, the matrix Q(ζ, ˜ α, β) is also unstable, independently of the choice of the then the matrix Q(ζ ˜ α, β) controller. Finally, if the polynomial 1 (ζ) is stable, then the matrix Q(ζ, ˜ is stable or unstable together with the matrix Qν (ζ, α, β). As a conclusion, we note that from (7.88), it follows that the polynomial 1 (ζ) is stable iff in (7.84), we have Re si < 0,

(i = 1, . . . , γ) .

This completes the proof. 6. Using the above results, we can consider the stabilisation problem for the complete standard sampled-data system. Theorem 7.16. Let the continuous-time plant of the standard sampled-data system be given by the state equations (7.61) with a χ × χ matrix A. Let ˜2ν , C˜2ν ) with dimension (n, λ, m) be any minimal realisation of the also (A˜ν , B matrix N (p). Then, we have χ ≥ λ and the function r(s) =

det(sIχ − A) det(sIλ − A˜ν )

(7.89)

˜2ν , C˜2ν ) is not is a polynomial. Moreover, if the minimal realisation (A˜ν , B stabilisable, then also the standard sampled-data system with the plant (7.61) ˜2ν , C˜2ν ) is stabilisable, is not stabilisable. If the minimal realisation (A˜ν , B then for stabilisability of the standard sampled-data system, it is necessary and sufficient that all roots si of the polynomial (7.89) lie in the open left halfplane. Under this condition, the set of stabilising controllers for the standard sampled-data system coincides with the set of stabilising controllers for the ˜2ν , C˜2ν ). minimal realisation (A˜ν , B Proof. Using (7.61) and (7.62)–(7.64) and assuming x(t) = O1 , we can represent the standard sampled-data system in the form y = C2 v ,

dv = Av + B2 u dt

α(ζ)ψk = β(ζ)yk u(t) = m(t − kT )ψk ,

(7.90) kT < t < (k + 1)T .

that should be completed with the output equation z(t) = C1 y(t) + DL u(t) . Since

C2 (pIχ − A)−1 B2 = N (p) ,

(7.91)

304


due to (7.9), Equations (7.90) can be considered as equations of the closed loop, where the continuous-time plant N (p) is given in form of a realisation (A, B2 , C2 ) of dimension n, χ, m, which is not minimal in the general case. Obviously, a necessary and sufficient condition for the stability of the system (7.90) and (7.91) is that the system (7.90) is stable. Hence it follows the conclusion that the stabilisation problem for the standard sampled-data system with the plant (7.61) is equivalent to the stabilisation problem for the closed loop, where the continuous-time plant is given as a realisation (A, B2 , C2 ). Therefore, all claims of Theorem 7.16 are corollaries of Theorem 7.15.

7.7 Modal Controllability and the Set of Stabilising Controllers 1. Let the continuous-time plant of the standard sampled-data system be given by the state equations dv = Av + B1 x + B2 u dt z = C1 v ,

(7.92)

y = C2 v .

Then, as follows from Theorems 7.13–7.16, in case of a stabilisable plant (7.92), the characteristic polynomial of the closed-loop standard sampled-data system ∆(ζ) can be represented in the form ∆(ζ) = (ζ)∆d (ζ) ,

(7.93)

where (ζ) is a stable polynomial, which is independent of the choice of the controller. Moreover, in (7.93)

aN (ζ) − bN (ζ) (7.94) ∆d (ζ) ≈ det = det QN (ζ, α, β) , −β(ζ) α(ζ) where (α(ζ), β(ζ)) is a discrete controller and the matrices aN (ζ), ζbN (ζ) define an ILMFD wN (ζ) = ζC2 (I − ζeAT )−1 µ(A)eAT B2 = ζa−1 N (ζ)bN (ζ) .

(7.95)

From (7.93), it follows that the roots of the characteristic polynomial of the standard sampled-data system ∆(ζ) can be split up into two groups. The first group (roots of the polynomial ρ(ζ)) is determined only by the properties of the matrix w(p) and is independent of the properties of the discrete controller. Hereinafter, these roots will be called uncontrollable. The second group of roots consists of those roots of the polynomial (7.94), which are determined by the matrix w(p) and the controller (α(ζ), β(ζ)). Since the pair (aN (ζ), ζbN (ζ)) is irreducible, the controller (α(ζ), β(ζ)) can be chosen in such a way that the polynomial ∆d (ζ) is equal to any given (stable) polynomial. In this connection, the roots of the second group will be called controllable

7.7 Modal Controllability and the Set of Stabilising Controllers

305

2. The standard sampled-data system with the plant (7.92) will be called modal controllable, if all roots of its characteristic polynomial are controllable, i.e., (ζ) = const. = 0. Under the strict conditions for non-pathological behavior, necessary and sufficient conditions for the system to be modal controllable are given by the following theorem. Theorem 7.17. Let the poles of the matrix

Or DL K(p) L(p) C1 −1 B1 B2 + (pIχ − A) (7.96) w(p) = = C2 On Onm M (p) N (p) satisfy Conditions (7.80) and (7.81). Then, a necessary and sufficient condition for the standard sampled-data system to be modal controllable is that the matrix N (p) dominates in the matrix w(p). Proof. Sufficiency: Without loss of generality, we take DL = Orm and assume that the standard representation is minimal. Let the matrix N (p) dominate in Matrix (7.96). Then due to Theorem 2.67, the realisation (A, B2 , C2 ) on the right-hand side of (7.96) is minimal. Construct the discrete model Dwµ (T, ζ, t) of the matrix w(p)µ(p). Obviously, we have

DKµ (T, ζ, t) DLµ (T, ζ, t) . Dwµ (T, ζ, t) = DM µ (T, ζ, t) DN µ (T, ζ, t) Using the second formula in (6.122) and (7.96), we obtain Dwµ (T, ζ, t) = D1 (ζ, t) + D2 (t) , where

C1 D1 (ζ, t) = − (ζIχ − e−AT )−1 eA(t−T ) µ(A) B1 B2 , C2 T

C1 D2 (t) = − eA(t−τ ) µ(τ ) dτ B1 B2 . C2 t

By virtue of Theorem 6.30, the right-hand side of the first equation defines a minimal standard representation of the matrix D1 (ζ, t). At the same time, the realisation eAT , eA(t−T ) µ(A)B2 , C2 is also minimal. Therefore, we can take A = A˜ν in (7.89). Hence r1 (s) = const. = 0 and (ζ) = const. = 0, and the sufficiency has been proven. The necessity of the conditions of the theorem is seen by reversing the above derivations. 3. Under the stabilisability condition, the set of stabilising controllers is completely determined by the properties of the matrix N (p), and is defined as the set of pairs (α(ζ), β(ζ)) satisfying (7.94) for all possible stable polynomials ∆d (ζ). The form of Equation (7.94) coincides with (5.163), where the matrix Ql (ζ, α, β) is given by (5.160). Therefore, to describe the set of stabilising controllers, all the results of Section 5.8 can be used.

306

4.


As a special case, the following propositions hold:

a) Let (α0 (ζ), β0 (ζ)) be a controller, such that

aN (ζ) −ζbN (ζ) det = const. = 0 . −β0 (ζ) α0 (ζ) Then the set of all stabilising controllers for the stabilisable standard sampled-data system is given by α(ζ) = Dl (ζ)α0 (ζ) − ζMl (ζ)bN (ζ) , β(ζ) = Dl (ζ)β0 (ζ) − Ml (ζ)aN (ζ) , where Dl (ζ) and Ml (ζ) are any polynomial matrices, but Dl (ζ) has to be stable. b) Together with the ILMFD (7.95), let us have an IRMFD wN (ζ) = ζC2 (I − ζeAT )−1 eAT µ(A)B2 = ζbr (ζ)a−1 r (ζ) . Then the set of stabilising controllers (α(ζ), β(ζ)) for the standard sampled-data system coincides with the set of solutions of the Diophantine equation α(ζ)ar (ζ) − ζβ(ζ)br (ζ) = Dl (ζ) , where Dl (ζ) is any stable polynomial matrix. 5. Any stabilising controller (α(ζ), β(ζ)) for the standard sampled-data system fulfills det α(0) = 0, i.e., the matrix α(ζ) is invertible. Therefore, any stabilising controller has a transfer matrix wd (ζ) = α−1 (ζ)β(ζ) . The following propositions hold: c) The set of transfer matrices of all stabilising controllers for the standard sampled-data system can be written in the form −1

wd (ζ) = [α0 (ζ) − ζφ(ζ)bN (ζ)]

[β0 (ζ) − φ(ζ)aN (ζ)] ,

where φ(ζ) is any stable rational matrix of compatible dimension. d) The rational matrix wd (ζ) is associated with a stabilising controller for a stabilisable standard system, if and only if there exists any of the following representations: wd (ζ) = F1−1 (ζ)F2 (ζ),

wd (ζ) = G2 (ζ)G−1 1 (ζ) ,

where the pairs of rational matrices (F1 (ζ), F2 (ζ)) and [G1 (ζ), G2 (ζ)] are stable and satisfy the equations F1 (ζ)ar (ζ) − ζF2 (ζ)br (ζ) = Im , al (ζ)G1 (ζ) − ζbl (ζ)G2 (ζ) = In .

8 Analysis and Synthesis of SD Systems Under Stochastic Excitation

8.1 Quasi-stationary Stochastic Processes in the Standard SD System 1. Let the input of the standard sampled-data system be acted upon by a vector signal x(t) that is modelled as a centered stochastic process with the autocorrelation matrix Kx (τ ) = E [x(t)x (t + τ )] , where E[·] denotes the operator of mathematical expectation. Assume that the integral ∞ Kx (τ )e−sτ dτ , Φx (s) = −∞

which will be called the spectral density of the input signal, converges absolutely in some stripe −α0 ≤ Re s ≤ α0 , where α0 is a positive number. 2. Let the block L(p) in the matrix w(p) (7.2) be at least proper and the remaining blocks be strictly proper. Let also the system (7.3)–(7.7) be internally stable. When the input of the standard sampled-data system is the above mentioned signal, after fading away of transient processes, the steady-state stochastic process z∞ (t) is characterised by the covariance matrix [143, 148] j∞ 1 Kz (t1 , t2 ) = w(−s, t1 )Φx (s)w (s, t2 )es(t2 −t1 ) ds , (8.1) 2πj −j∞ where w(s, t) = w(s, t + T ) is the PTM of the system. Hereinafter, the stochastic process z∞ (t) with the correlation matrix (8.1) will be called quasistationary. As follows from (8.1), the covariance matrix Kz (t1 , t2 ) depends separately on each of its arguments t1 and t2 rather than on their difference. Therefore, the quasi-stationary output z∞ (t) is a non-stationary stochastic process. Since

308


w(−s, t1 ) = w(−s, t1 + T ),

w (s, t2 ) = w (s, t2 + T ) ,

(8.2)

we have Kz (t1 , t2 ) = Kz (t1 + T, t2 + T ) . Stochastic processes satisfying this condition will be called periodically nonstationary, or shortly periodical. Using this term, we state that the steadystate (quasi-stationary) response of a stable standard sampled-data system to a stationary input signal is a periodically non-stationary stochastic process. 3.

The scalar function dz (t) = trace Kz (t, t)

(8.3)

will be called the variance of the quasi-stationary output. Here ‘trace’ denotes the trace of a matrix defined as the sum of its diagonal elements. From (8.1) for t1 = t2 = t and (8.3), we find j∞ 1 trace [w(−s, t)Φx (s)w (s, t)] ds . (8.4) dz (t) = 2πj −j∞ Then using (8.2), we obtain dz (t) = dz (t + T ) , i.e., the variance of the quasi-stationary output is a periodic function of its argument t. For matrices A, B of compatible dimensions, the relation trace(AB) = trace(BA)

(8.5)

is well known. Thus in addition to (8.4), the following equivalent relations hold: j∞ 1 dz (t) = trace [Φx (s)w (s, t)w(−s, t)] ds , (8.6) 2πj −j∞ j∞ 1 dz (t) = trace [w (s, t)w(−s, t)Φx (s)] ds . (8.7) 2πj −j∞ 4. Assume in particular that Φx (s) = I , i.e. the input signal is white noise with uncorrelated components. For this case, we denote

rz (t) = dz (t) . Then (8.6) and (8.4) yield 1 rz (t) = 2πj =

1 2πj

j∞

−j∞ j∞ −j∞

trace [w (s, t)w(−s, t)] ds , trace [w(−s, t)w (s, t)] ds .

8.1 Quasi-stationary Stochastic Processes in the Standard SD System

309

Substituting here the variable −s for s, we find also j∞ 1 rz (t) = trace [w (−s, t)w(s, t)] ds , 2πj −j∞ j∞ 1 = trace [w(s, t)w (−s, t)] ds . 2πj −j∞ 5. A practical calculation of the variance dz (t) using Formulae (8.4), (8.6) and (8.7) causes some technical difficulties, because the integrands of these formulae are transcendent functions of the argument s. To solve the problem, it is reasonable to transform these integrals to those with finite integration limits. The corresponding equations, which stem from (8.6) and (8.7) have the form jω/2 jω/2 T T trace U 1 (T, s, t) ds = trace U 2 (T, s, t) ds , dz (t) = 2πj −jω/2 2πj −jω/2 (8.8) where ω = 2π/T and

U 1 (T, s, t) =

U 2 (T, s, t) =

∞ 1 Φx (s + kjω)w (s + kjω, t)w(s + kjω, t) , T

1 T

k=−∞ ∞

w (s + kjω, t)w(s + kjω, t)Φx (s + kjω) .

(8.9)

(8.10)

k=−∞

In (8.9) and (8.10) for any function f (s), we denote

f (s) = f (−s) . Moreover, for any function (matrix) g(ζ), we use as before the notation

g (s) = g(ζ) | ζ=e−sT .

Obviously, the following reciprocal relations hold:

g (s) = g(ζ) | ζ=e−sT ,

g(ζ) = g (s) | e−sT =ζ

(8.11)

and per construction

g (s) = g (s + jω) ,

ω = 2π/T .

(8.12)

As follows from [148] for a rational matrix Φx (s), the matrices (8.9) and (8.10) are rational matrices of the argument ζ = e−sT . Therefore, to calculate the integrals (8.8), we could take profit from the technique described in [148]. There exists an alternative way to compute the integrals in (8.8). With this aim in view, we pass to the integration variable ζ in (8.8), such that

310


dz (t) =

1 2πj

# trace U1 (T, ζ, t) Γ

1 dζ = ζ 2πj

# trace U2 (T, ζ, t) Γ

dζ , ζ

(8.13)

where, according to the notation (8.11),

Ui (T, ζ, t) = U i (T, s, t) | e−sT =ζ ,

(i = 1, 2)

are rational matrices in ζ. The integration in (8.13) is performed along the unit circle Γ in positive direction (anti-clockwise). The integrals (8.13) can be easily computed using the residue theorem with account for the fact that all poles of the PTM of a stable standard sampled-data system lie in the open left half-plane. Other ways of calculating these integrals are described in [11] and [177]. 6. Example 8.1. Let us find the variance of the quasi-stationary output for the simple single-loop system shown in Fig. 8.1, where the forming element is a x

z

1 s

-g 6 v

C

•

-

Fig. 8.1. Single sampled-data control loop

zero-order hold with transfer function µ(s) = µ0 (s) =

1 − e−sT s

and Φx (s) = 1. Using Formulae (7.40)-(7.42), it can be shown that in this case the PTM w(s, t) can be written in the form w(s, t) = e−st [v0 (s) + tψ0 (s) + c(s, t)] ,

0≤t≤T

(8.14)

with

v0 (s) =

1 − e−sT α(s) , s ∆(s)

1 − e−sT β (s) ψ0 (s) = , s ∆(s) est − 1 , c(s, t) = s

(8.15)

8.1 Quasi-stationary Stochastic Processes in the Standard SD System

311

where

α(s) = α0 + α1 e−sT + . . . + αρ e−ρsT ,

β (s) = β0 + β1 e and

−sT

+ . . . + βρ e

(8.16)

−ρsT

∆(s) = 1 − e−sT α(s) − T e−sT β (s) .

(8.17)

Using (8.14)–(8.17) from (8.9) and (8.10) after fairly tedious calculations, it is found that for 0 ≤ t ≤ T U 1 (T, s, t)

= U 2 (T, s, t)

⎛

⎞

α(s) α(−s) α(s) β (−s) α(−s) β (s) ⎠ =T + tT ⎝ + ∆(s)∆(−s) ∆(s)∆(−s) ∆(s)∆(−s)

$

β (s) β (−s) α(s) α(−s) + t e−sT + esT + t2 T ∆(s)∆(−s) ∆(s) ∆(−s) ⎛ ⎞ β (s) β (−s) ⎠ + t2 ⎝e−sT + esT + t. ∆(s) ∆(−s)

% (8.18)

To derive Formula (8.18), we employed expressions for the sums of the following series: ∞ 1 α(s) α(−s) v0 (s + kjω)v 0 (s + kjω) = T , T ∆(s)∆(−s) k=−∞

∞ 1 β (s) β (−s) ψ0 (s + kjω)ψ 0 (s + kjω) = T , T ∆(s)∆(−s) k=−∞

∞ 1 α(s) β (−s) v0 (s + kjω)ψ 0 (s + kjω) = T , T ∆(s)∆(−s) k=−∞ ∞ 1 α(s) −sT v0 (s + kjω)c(s + kjω, t) = t, e T ∆(s) k=−∞

∞ 1 β (s) −sT ψ0 (s + kjω)c(s + kjω, t) = t, e T ∆(s) k=−∞ ∞ 1 c(s + kjω, t)c(s + kjω, t) = t T k=−∞

312


and ∞ (1 − e−sT )t + T e−sT 1 e(s+kjω)t = , 2 T (s + kjω) (1 − e−sT )2 k=−∞

∞ 1 e−(s+kjω)t −(1 − e−sT )e−sT t + T e−sT = , T (s + kjω)2 (1 − e−sT )2

0≤t≤T.

k=−∞

After substiting e−sT = ζ in (8.18), we find a rational function of the argument ζ, for which the integrals (8.13) can be calculated elementary.

8.2 Mean Variance and H2 -norm of the Standard SD System 1. Let dz (t) be the variance of the quasi-stationary output determined by anyone of Formulae (8.4), (8.6) or (8.7). Then the value 1 d¯z = T

T

dz (t) dt 0

will be called the mean variance of the quasi-stationary output. Using here (8.6) and (8.7), we obtain 1 d¯z = 2πj

j∞

−j∞

trace [Φx (s)w ˜1 (s)] ds =

1 2πj

j∞

−j∞

trace [w ˜1 (s)Φx (s)] ds , (8.19)

where

w ˜1 (s) = 2.

1 T

T

w (s, t)w(−s, t) dt .

(8.20)

0

When Φx (s) = I , for the mean variance, we will use the special notation 1 r¯z = T

T

rz (t) dt . 0

The value r¯z is determined by the properties of the standard sampled-data system and does not depend on the properties of the exogenous excitations. Formulae for calculating r¯z can be derived from (8.19) and (8.20) with Φ(s) = I . In particular, assuming

= w(s) ˜

1 T

T

0

from the formulae in (8.19), we find

w (−s, t)w(s, t) dt ,

(8.21)

8.2 Mean Variance and H2 -norm of the Standard SD System

r¯z =

1 2πj

The value

313

j∞

trace [w(s)] ˜ ds .

(8.22)

−j∞

√ S2 = + r¯z

(8.23)

henceforth, will be called the H2 -norm of the stable standard sampled-data system S. Hence j∞ 1 S22 = trace w(s) ˜ ds . (8.24) 2πj −j∞ For further transformations, we write the right-hand side of (8.24) in the form S22 =

T 2πj

jω/2

−jω/2

trace Dw˜ (T, s, 0) ds ,

where ω = 2π/T and

Dw˜ (T, s, 0) =

∞ 1 w(s ˜ + kjω) T

(8.25)

k=−∞

is a rational matrix in e−sT . Using the substitution e−sT = ζ, similarly to (8.13), we obtain # 1 dζ 2 . (8.26) trace Dw˜ (T, ζ, 0) S2 = 2πj Γ ζ Example 8.2. Under the conditions of Example 8.1, we get T 2 α(ζ)β(ζ −1 ) α(ζ −1 )β(ζ) α(ζ)α(ζ −1 ) Dw˜ (T, ζ, 0) = T + + ∆(ζ)∆(ζ −1 ) 2 ∆(ζ)∆(ζ −1 ) ∆(ζ)∆(ζ −1 ) T ζα(ζ) ζ −1 α(ζ −1 ) T 3 β(ζ)β(ζ −1 ) + + + 3 ∆(ζ)∆(ζ −1 ) 2 ζ∆(ζ) ∆(ζ −1 ) T 2 ζβ(ζ) ζ −1 β(ζ −1 ) T + + + 3 ∆(ζ) ∆(ζ −1 ) 2 and the integral (8.26) is calculated elementary.

Remark 8.3. The H2 -norm is defined directly by the PTM. This approach opens the possibility to define the H2 -norm for any system possessing a PTM. Interesting results have been already published by the authors for the class of linear periodically time-varying systems [98, 100, 101, 88, 89]. In contrast to other approaches like [200, 32, 203, 28, 204], the norm computation over the PTM yields closed formulae and needs to evaluate matrices of only finite dimensions.

314


3. Let us find a general expression for the H2 -norm of the standard sampleddata system (6.3)–(6.6). Using the assumptions of Section 8.1 and notation (8.11), the PTM of the system can be written in the form

w(s, t) = ϕLµ (T, s, t)RN (s)M (s) + K(s) .

(8.27)

Substituting −s for s after transposition, we receive

w (s, t) = w (−s, t) = M (s)RN (s)ϕL µ (T, −s, t) + K (s) . Multiplying the last two equations, we find

w (s, t)w(s, t) = M (s)R N (s)ϕL µ (T, −s, t)ϕLµ (T, s, t)RN (s)M (s)

+ M (s)R N (s)ϕL µ (T, −s, t)K(s)

+ K (s)ϕLµ (T, s, t)RN (s)M (s) + K (s)K(s) . Using this in (8.21) yields

w(s) ˜ = M (s)R N (s)DL (s)RN (s)M (s) + M (s)R N (s)Q L (s)K(s)

+ K (s)QL (s)RN (s)M (s) + K (s)K(s) ,

(8.28)

where

DL (s) = QL (s) = Q L (s) =

1 T 1 T 1 T

T

ϕL µ (T, −s, t)ϕLµ (T, s, t) dt , 0

T

ϕLµ (T, s, t) dt = 0

1 L(s)µ(s) , T

T

ϕL µ (T, −s, t) dt = 0

1 L (−s)µ(−s) . T

Using (8.28) in (8.24), we obtain j∞ 1 S22 = trace M (s)R N (s)DL (s)RN (s)M (s) 2πj −j∞

+ M (s)R N (s)Q L (s)K(s)

+ K (s)QL (s)RN (s)M (s) + K (s)K(s) ds .

(8.29)

All matrices in the integrand, except for the matrix RN (s), are determined by the transfer matrix w(s) of the continuous plant and are independent of the transfer matrix w d (s) of the controller. Moreover, each transfer matrix

8.3 Representing the PTM in Terms of the System Function

315

w d (s)

of a stabilising controller is associated with a nonnegative value S22 . Therefore, the right-hand side of (8.29) can be considered as a functional defined over the set of transfer functions of stabilising controllers w d (s). Hence the following optimisation problem arises naturally. H2 -problem. Let the matrix w(p) in (7.2) be given, where the matrix L(p) is at least proper and the remaining elements are strictly proper. Furthermore, the sampling period T and the impulse form m(t) are fixed. Find the transfer function of a stabilising controller w d (s), which minimises the functional (8.29).

8.3 Representing the PTM in Terms of the System Function 1. Equation (8.29) is not fairly convenient for solving the H2 -optimisation problem. As will be shown below, a representation of the H2 -norm in terms of the so-called system function is more suitable for this purpose. To construct such a representation, we must at first write the PTM in terms of the system function. This topic is considered in the present section. 2. To simplify the further reading, we summarise some relations obtained above. Heed that the notation slightly differs from that in the previous exposition. Using (8.11), we present the PTM of the standard sampled-data system (7.2)–(7.7) in the form (8.27), where the matrix ϕLµ (T, s, t) is determined by (7.29) −1 RN (s) = w d (s) In − DN µ (T, s, 0)w d (s) (8.30) with ∞ 1 DN µ (T, s, 0) = ϕN µ (T, s, 0) = N (s + kjω)µ(s + kjω) T

k=−∞

and

−1

w d (s) = α l (s) β l (s) ,

(8.31)

where

α l (s) = α0 + α1 e−sT + . . . + αρ e−ρsT ,

β l (s) = β0 + β1 e−sT + . . . + βρ e−ρsT

are polynomial matrices in the variable ζ = e−sT . Moreover, µ(s) is the transfer function of the forming element (6.38). Matrix (8.31) will be called the transfer function of the controller.

316

3.


The PTM (8.27) is associated with the rational matrix

m

K(p) L(p) w(p) = M (p) N (p)

κ n

.

Henceforth as above, we assume that the matrix L(p) is at least proper and the remaining elements are strictly proper. The above matrix can be associated with the state equations (7.10) dv = Av + B1 x + B2 u dt z = C1 v + DL u ,

y = C2 v ,

where A is a constant χ × χ matrix. Without loss of generality, we can assume that the pairs

C1 (A, B1 B2 ), [A, ] C2 are controllable and observable, respectively. 4. As follows from Theorems 7.3 and 7.4, the PTM (8.27) admits a representation of the form Pw (s, t) w(s, t) = , (8.32) ∆(s) where Pw (s, t) = Pw (s, t + T ) is a κ × matrix, whose elements are integer

functions in s for all t and the function ∆(s) is given by

∆(s) = det Q(s, α l , β l ) ,

(8.33)

where Q(s, α l , β l ) is a matrix of the form ⎡ ⎤ Iχ − e−sT eAT Oχn −e−sT eAT µ(A)B2 ⎢ ⎥ ⎥. −C2 In Onm Q(s, α l , β l ) = ⎢ ⎣ ⎦ Omχ − β l (s) α l (s) Assuming e−sT = ζ in (8.33), we find the characteristic polynomial

∆(ζ) = ∆(s) | e−sT =ζ = det Q(ζ, αl , βl ) , where

⎡ ⎢ Q(ζ, αl , βl ) = ⎣

Iχ − ζeAT

Oχn −ζeAT µ(A)B2

−C2

In

Onm

Omχ

−βl (ζ)

αl (ζ)

(8.34) ⎤ ⎥ ⎦


317

with αl (ζ) = α0 + α1 ζ + . . . + αρ ζ ρ , βl (ζ) = β0 + β1 ζ + . . . + βρ ζ ρ .

(8.35)

For brevity, we will refer to the matrices (8.35) as a controller and the matrix

wd (ζ) = w d (s) | e−T =ζ = αl−1 (ζ)βl (ζ)

as well as w d (s) are transfer functions of this controller. 5.

As was shown in Chapter 6, the following equations hold:

DN µ (T, ζ, 0) = DN µ (T, s, 0) | e−sT =ζ ∞ 1 = N (s + kjω)µ(s + kjω) | e−sT =ζ T k=−∞

= ζC2 (Iχ − ζeAT )−1 eAT µ(A)B2 = wN (ζ) . If this rational matrix is associated with an ILMFD wN (ζ) = DN µ (T, ζ, 0) = ζa−1 l (ζ)bl (ζ) , then the function (ζ) =

det(Iχ − ζeAT ) det al (ζ)

(8.36)

(8.37)

is a polynomial, which is independent of the choice of the controller (8.35). The characteristic polynomial (8.34) has the form ∆(ζ) ≈ (ζ)∆d (ζ) ,

(8.38)

where ∆d (ζ) is a polynomial determined by the choice of the controller (8.35). Moreover, the standard system is stabilisable, if and only if the polynomial (ζ) is stable. The polynomial ∆d (ζ) appearing in (8.38) satisfies the relation ∆d (ζ) ≈ det QN (ζ, αl , βl ) , where

al (ζ) −ζbl (ζ) QN (ζ, α, β) = −βl (ζ) αl (ζ)

(8.39)

is a polynomial matrix (7.76). If the stabilisability conditions hold, then the set of stabilising controllers for the standard sampled-data system coincide with the set of controllers (8.35) with stable matrices (8.39).

318

6.


Let (α0l (ζ), β0l (ζ)) be a basic controller such that ∆d (ζ) ≈ det QN (ζ, α0l , β0l ) = const. = 0 .

(8.40)

Then, as was proved before, the set of all causal stabilising controllers can be given by αl (ζ) = Dl (ζ)α0l (ζ) − ζMl (ζ)bl (ζ) ,

(8.41)

βl (ζ) = Dl (ζ)β0l (ζ) − Ml (ζ)al (ζ) ,

where Ml (ζ) and Dl (ζ) are polynomial matrices, the first of them can be chosen arbitrarily, while the second one must be stable. Then the transfer matrix of any stabilising controller for a given basic controller has a unique left representation of the form −1

wd (ζ) = αl−1 (ζ)βl (ζ) = [α0l (ζ) − ζθ(ζ)bl (ζ)] where

[β0l (ζ) − θ(ζ)al (ζ)] ,

θ(ζ) = Dl−1 (ζ)Ml (ζ)

is a stable rational matrix, which will hereinafter be called the system function of the standard sampled-data system. 7.

Together with an ILMFD (8.36), let us have an IRMFD wN (ζ) = DN µ (T, ζ, 0) = ζbr (ζ)a−1 r (ζ)

(8.42)

and let (α0l (ζ), β0l (ζ)) and [α0r (ζ), β0r (ζ)] be two dual basic controllers corresponding to the IMFDs (8.36) and (8.42). These controllers will be called initial controllers. Then the transfer matrix of a stabilising controller admits the right representation wd (ζ) = βr (ζ)αr−1 (ζ)

(8.43)

with αr (ζ) = α0r (ζ)Dr (ζ) − ζbr (ζ)Mr (ζ) ,

(8.44)

βr (ζ) = β0r (ζ)Dr (ζ) − ar (ζ)Mr (ζ) ,

where Dr (ζ) is a stable polynomial matrix and Mr (ζ) is an arbitrary polynomial matrix. Thus Mr (ζ)Dr−1 (ζ) = θ(ζ) and Equation (8.43) appears as −1

wd (ζ) = [β0r (ζ) − ar (ζ)θ(ζ)] [α0r (ζ) − ζbr (ζ)θ(ζ)]

.

(8.45)


319

Moreover, according to (5.182), (5.183) and (5.181), we obtain wd (ζ) = V2 (ζ)V1−1 (ζ) ,

(8.46)

where −1 V1 (ζ) = al (ζ) − ζbl (ζ)wd (ζ) = α0r (ζ) − ζbr (ζ)θ(ζ) , −1 = β0r (ζ) − ar (ζ)θ(ζ) . V2 (ζ) = wd (ζ) al (ζ) − ζbl (ζ)wd (ζ)

(8.47)

8. Using (8.47), we can write the matrix RN (s) in (8.30) in terms of the system function θ(ζ). Indeed, using (8.30), (8.36), and (8.47), we find

−1

RN (ζ) = RN (s) | e−sT =ζ = wd (ζ) [In − DN µ (T, ζ, 0)wd (ζ)]

−1 = wd (ζ) In − ζa−1 l (ζ)bl (ζ)wd (ζ) = wd (ζ) [al (ζ) − ζbl (ζ)wd (ζ)]

−1

(8.48)

al (ζ) = V2 (ζ)al (ζ) .

From (8.47) and (8.48), we obtain RN (ζ) = β0r (ζ)al (ζ) − ar (ζ)θ(ζ)al (ζ) .

(8.49)

Hence

RN (s) = RN (ζ) | ζ=e−sT = β 0r (s) a l (s) − a r (s) θ (s) a l (s) .

(8.50)

Substituting (8.50) into (8.27), we obtain

w(s, t) = ψ(s, t) θ (s)ξ(s) + η(s, t) ,

(8.51)

where

ψ(s, t) = −ϕLµ (T, s, t) a r (s) ,

ξ(s) = a l (s)M (s) ,

(8.52)

η(s, t) = ϕLµ (T, s, t) β 0r (s) a l (s)M (s) + K(s) . Equation (8.51) will hereinafter be called a representation of the PTM in terms of the system function. The matrices (8.52) will be called the coefficients of this representation. 9. Below, we will prove several propositions showing that the coefficients (matrices) (8.52) should be calculated with account for a number of important cancellations.

320


Theorem 8.4. The poles of the matrix η(s, t) belong to the set of roots of the function (s) = (e−sT ), where ρ(ζ) is the polynomial given by (8.37). Proof. Assume Dl (ζ) = I and Ml (ζ) = Omn in (8.41), i.e. we choose the

initial controller (α0l (ζ), β0l (ζ)). In this case, θ (s) = Omn and w(s, t) = η(s, t) . Since the controller (α0l (ζ), β0l (ζ)) is a basic controller, we have (8.40) and from (8.38), it follows ∆(ζ) ≈ (ζ) . Assuming this, from (8.32) we get w(s, t) = η(s, t) =

Pη (s, t)

(s)

,

(8.53)

where the matrix Pη (s, t) is an integral function of the argument s. The claim of the theorem follows from (8.53). Corollary 8.5. Let the standard sampled-data system be modal controllable, i.e. (s) = const. = 0. Then the matrix η(s, t) is an integral function in s. Theorem 8.6. For any polynomial matrix θ(ζ), the set of poles of the matrix

G(s, t) = ψ(s, t) θ (s)ξ(s)

(8.54)

belongs to the set of roots of the function (s).

Proof. Let θ (s) = θ(ζ) | ζ=e−sT with a polynomial matrix θ(ζ). Then for any ILMFD θ(ζ) = Dl−1 (ζ)Ml (ζ) , the matrix Dl (ζ) is unimodular. Therefore, due to Theorem 4.1, the controller (8.41) is a basic controller. Hence we have ∆d (ζ) = const = 0. In this case,

∆d (s) = const. = 0 and (8.38) yields ∆(ζ) ≈ (ζ) . From this relation and (8.32), we obtain w(s, t) =

P˜w (s, t)

(s)

,

where the matrix P˜w (s, t) is an integral function in s. Using (8.53) and the last equation, we obtain G(s, t) = w(s, t) − η(s, t) =

PG (s, t)

(s)

where the matrix PG (s, t) is an integral function in s.

,

(8.55)


321

Corollary 8.7. If the standard sampled-data system is modal controllable, then the matrix G(s, t) is an integral function of the argument s for any polynomial θ(ζ). 10. In principle for any θ(ζ), the right-hand side of (8.54) can be cancellee by a function 1 (s), where 1 (ζ) is a polynomial independent of t. In this case after cancellation, we obtain an expression similar to (8.55): G(s, t) =

PGm (s, t)

m (s)

,

(8.56)

where deg m (ζ) < deg (ζ). If deg m (ζ) has the minimal possible value independent of the choice of θ(ζ), the function (8.56) will be called globally irreducible. Using (8.52), we can represent (8.56) in the form

−ϕLµ (T, s, t) a r (s) θ (s) a l (s)M (s) =

PGm (s, t)

m (s)

= G(s, t) .

Multiplying this by est , we find

est G(s, t) = G1 (s, t) = −DLµ (T, s, t) a r (s) θ (s) a l (s)M (s) . Hence ∞ 1 G1 (s + kjω, t)e(s+kjω)t = −DLµ (T, s, t) a r (s) θ (s) a l (s)DM (T, s, t) T k=0 (8.57) N G1 (s, t) , = m (s)

where NG1 (ζ, t) is a polynomial matrix in ζ for any 0 < t < T . 11. The following propositions prove some further cancellations in the calculation of the matrices ψ(s, t) and ξ(s) appearing in (8.52). Theorem 8.8. For 0 < t < T , let us have the irreducible representations

DLµ (T, s, t) a r (s) =

N L (s, t) L (s)

,

(8.58)

a l (s)DM (T, s, t) =

N M (s, t) M (s)

,

(8.59)

where NL (ζ, t) and NM (ζ, t) are polynomial matrices in ζ, and M (ζ) and M (ζ) are scalar polynomials. Let also the fractions

322


NL (ζ, t) , M (ζ)

NM (ζ, t) L (ζ)

be irreducible and the fraction (8.57) be globally irreducible. Then the function

γ(ζ) =

(ζ) L (ζ)M (ζ)

is a polynomial. Moreover, L (ζ)M (ζ) ≈ m (ζ) . The proof of the theorem is preceded by two auxiliary claims. Lemma 8.9. Let A and B be constant matrices of dimensions n × m and × κ, respectively. Moreover, for any m × matrix Ω, the equation (8.60)

AΩB = Onκ should hold. Then at least one of the matrices A, B is a zero matrix.

Proof. Assume the converse, namely, let us have (8.60) for any Ω, where A and B are both nonzero. Let the elements aij and bpq of the matrices A and B be nonzero. Assume Ω = Ijp , where Ijp is an m × matrix having the single unit element at the cross of the j-th row and the p-th column, while all other elements are zero. It can be easily verified that the product AIjp B is nonzero. This contradiction proves the Lemma. Lemma 8.10. Let us have two irreducible rational n × m and × κ matrices in the standard form A(λ) =

NA (λ) , dA (λ)

B(λ) =

NA (λ) , dB (λ)

NB (λ) dA (λ)

NB (λ) . dB (λ)

(8.61)

Let also the fractions (8.62)

be irreducible. For any m × polynomial matrix Ω(λ), let us have NA (λ) NB (λ) N (λ) Ω(λ) = , dA (λ) dB (λ) d0 (λ)

deg d0 (λ) = δ ,

(8.63)

where d0 (λ) is a fixed polynomial and N (λ) is a polynomial matrix. Then the function d0 (λ) (8.64) γ˜ (λ) = dA (λ)dB (λ) is a polynomial. Moreover, if the right-hand side of (8.63) is globally irreducible, then dAB (λ) = dA (λ)dB (λ) ≈ d0 (λ) . (8.65)


323

Proof. Assume that the function (8.64) is not a polynomial. If p(λ) is a GCD of the the polynomials d0 (λ) and dAB (λ), then d0 (λ) = p(λ)d1 (λ),

dAB (λ) = p(λ)d2 (λ) ,

where the polynomials d1 (λ) and d2 (λ) are coprime and deg d2 (λ) > 0. Substituting these equations into (8.63), we obtain NA (λ)Ω(λ)NB (λ) = N (λ)

d2 (λ) . d1 (λ)

Let λ0 be a root of the polynomial d2 (λ). Then for λ = λ0 , the equality NA (λ0 )Ω(λ0 )NB (λ0 ) = Onκ can be written for any constant matrix Ω(λ0 ). Then with the help of Lemma 8.9, it follows that at least one of the following two equations holds: NA (λ0 ) = Onm or NB (λ0 ) = Oκ . In this case, at least one of the rational matrices (8.61) or (8.62) appears to be reducible. This contradicts the assumptions. Thus, deg d2 (λ) = 0 and γ˜ (λ) is a polynomial. Now, let the right-hand side of (8.63) be globally irreducible. We show that in this case deg d1 (λ) = 0 and we have (8.65). Indeed, if we assume the converse, we have deg d0 (λ) > deg dAB (λ). This contradicts the assumption that the right-hand side of (8.63) is globally irreducible. Proof (of Theorem 8.8). From (8.57)-(8.59) for e−sT = ζ, we obtain NL (ζ, t) NM (ζ, t) NG1 (ζ) θ(ζ) = . L (ζ) M (ζ) m (ζ) Since here the polynomial matrix θ(ζ) can be chosen arbitrarily, the claim of the theorem stems directly from Lemma 8.10. Corollary 8.11. When under the conditions of Theorem 8.8, the right-hand side of (8.55) is globally irreducible, then we have L (ζ)M (ζ) ≈ (ζ) .

(8.66)

Corollary 8.12. As follows from the above reasoning, the converse proposition is also valid: When under the conditions of Theorem 8.8, Equation (8.66) holds, then the representations (8.53) and (8.55) are globally irreducible.

324


Theorem 8.13. Let the conditions of Theorem 8.8 hold. Then we have the irreducible representations

ψ(s, t) = −ϕLµ (T, s, t) a r (s) =

ξ(s) = a l (s)M (s) =

Pξ (s) M (s)

Pψ (s, t)

L (s)

,

(8.67)

,

(8.68)

where the numerators are integral functions of the argument s. Proof. Multiplying the first equation in (8.58) by e−st , we obtain

ψ(s, t) = −ϕLµ (T, s, t) a r (s) =

e−st N L (s, t)

L (s)

.

The matrix e−st N L (s, t) is an integral function and the fraction on the righthand side is irreducible, because the function e−st has no zeros. Thus, (8.67) has been proven. Further, multiplying (8.59) by e−st and integrating by t, we find

ξ(s) = a l (s)M (s) = & T −st T e N M (s, t) dt −st . = a l (s)e DM (T, s, t) dt = 0 M (s) 0 The numerator of the latter expression is an integral function in s, i.e., we have (8.68). It remains to prove that representation (8.68) is irreducible. Assume the converse, i.e., let the representation (8.68) be reducible. Then

ξ(s) = a l (s)M (s) =

Pξ1 (s)

M 1 (s)

,

where deg M 1 (ζ) < deg M (ζ). With respect to (8.59), we thus obtain an expression of the form

∞ N M 1 (s, t) 1 , a l (s)DM (T, s, t) = ξ(s + kjω)e(s+kjω)t = T M 1 (s) k=−∞

(8.69)

where NM 1 (ζ, t) is a polynomial matrix in ζ for 0 ≤ t ≤ T . This contradicts the irreducibility assumption of the right-hand side of (8.59). Hence (8.68) is irreducible. Corollary 8.14. In case of modal controllability, Matrices (8.67), (8.68) and (8.69) are integral functions of the argument s for 0 ≤ t ≤ T .

8.4 Representing the H2 -norm in Terms of the System Function

325

Corollary 8.15. In a similar way, it can be proved that for an irreducible representation (8.58), the following irreducible representation holds:

L(s) a r (s)µ(s) =

PL (s) L (s)

,

where PL (s) is an integral function in s.

8.4 Representing the H2 -norm in Terms of the System Function 1. In this section on the basis of (8.21)–(8.25), we construct expressions for the value S22 for the standard sampled-data system, using the representation of the PTM w(s, t) in terms of the system function θ(ζ) defined by (8.51). From (8.51), we have

w (s, t) = w (−s, t) = ξ (−s) θ (−s)ψ (−s, t) + η (−s, t) . Multiplying this with the function (8.51), we receive

w (−s, t)w(s, t) = η (−s, t)η(s, t) + ξ (−s) θ (−s)ψ (−s, t)ψ(s, t) θ (s)ξ(s)

+ ξ (−s) θ (−s)ψ (−s, t)η(s, t) + η (−s, t)ψ(s, t) θ (s)ξ(s) . Substituting this into (8.21), we obtain 1 w(s) ˜ = T

T

w (−s, t)w(s, t) dt = g1 (s) − g2 (s) − g3 (s) + g4 (s) ,

0

where

g1 (s) = ξ (−s) θ (−s)

1 T

T

ψ (−s, t)ψ(s, t) dt θ (s)ξ(s) ,

0

1 T ψ (−s, t)η(s, t) dt , (8.70) g2 (s) = −ξ (−s) θ (−s) T 0 1 T η (−s, t)ψ(s, t) dt θ (s)ξ(s) , g3 (s) = g2 (−s) = − T 0

and 1 g4 (s) = T

0

T

η (−s, t)η(s, t) dt .

326


2. Next, we calculate Matrices (8.70). First of all with regard to (8.52), we find T 1 AL (s) = ψ (−s, t)ψ(s, t) dt T 0 (8.71) 1 T ϕL µ (T, −s, t)ϕLµ (T, s, t) dt a r (s) . = a r (−s) T 0 Since ϕLµ (T, s, t) =

∞ 1 L(s + kjω)µ(s + kjω)ekjωt , T k=−∞

∞ 1 L (−s + kjω)µ(−s + kjω)ekjωt , ϕL µ (T, −s, t) = T

(8.72)

k=−∞

after substituting (8.72) into (8.71) and integration, we receive 1 AL (s) = a r (−s) DL Lµµ (T, s, 0) a r (s) , T

(8.73)

where

DL Lµµ (T, s, 0) =

∞ 1 L (−s − kjω)L(s + kjω)µ(s + kjω)µ(−s − kjω) . T k=−∞

Using (8.73) in (8.70), we obtain

g1 (s) = ξ (−s) θ (−s)AL (s) θ (s)ξ(s) . 3.

To calculate the matrices g2 (s) and g3 (s), we denote

Q(s) = −

1 T

Then, 1 Q (−s) = − T

T

η (−s, t)ψ(s, t) dt .

(8.74)

0

T

ψ (−s, t)η(s, t) dt .

(8.75)

0

Using (8.52), we find

−ψ (−s, t)η(s, t) = a r (−s)ϕL µ (T, −s, t)ϕLµ (T, s, t) β 0r (s) a l (s)M (s)

+ a r (−s)ϕL µ (T, −s, t)K(s) . Substituting this into (8.75) and taking account of (8.72) after integration, we find


327

1 Q (−s) = a r (−s) DL Lµµ (T, s, 0) β 0r (s) a l (s)M (s) T 1 + a r (−s)L (−s)µ(−s)K(s) T

and 1 Q(s) = M (−s) a l (−s) β 0r (−s) DL Lµµ (T, s, 0) a r (s) T 1 + K (−s)L(s)µ(s) a r (s) T

considering the identity

DL Lµµ (T, −s, 0) = DL Lµµ (T, s, 0) . 4.

Using the above relations and (8.22)–(8.24), we obtain S22

1 = 2πj

j∞

−j∞

trace w(s) ˜ ds = J1 + J2 ,

(8.76)

where 1 J1 = 2πj

j∞

−j∞

trace ξ (−s) θ (−s)AL (s) θ (s)ξ(s) (8.77)

− ξ (−s) θ (−s)Q (−s) − Q(s) θ (s)ξ(s) ds ,

J2 =

1 2πj

j∞

−j∞

trace g4 (s) ds .

Under the given assumptions, these integrals converge absolutely, i.e. all the integrands as |s| → ∞ tend to zero as |s|−2 . 5. Since for a given initial controller, the value J2 is a constant, we have to consider only (8.77). With regard to (8.5) from (8.77), we obtain J1 =

1 2πj

j∞

−j∞

trace θ (−s)AL (s) θ (s)ξ(s)ξ (−s)

− θ (−s)Q (−s)ξ (−s) − ξ(s)Q(s) θ (s) ds ,

where the integral on the right-hand side converges absolutely. From (8.52), (8.74) and (8.75), we have

328


1 ξ(s)Q(s) = a l (s)M (s)M (−s) a l (−s) β 0r (−s) DL Lµµ (T, s, 0) a r (s) T 1 + a l (s)M (s)K (−s)L(s)µ(s) a r (s) , T 1 Q (−s)ξ (−s) = a r (−s) DL Lµµ (T, s, 0) β 0r (s) a l (s)M (s)M (−s) a l (−s) T 1 + a r (−s)L (−s)K(s)M (−s)µ(−s) a l (−s) . T

Substitute this into the equation before and pass to an integral with finite integration limits. Then using (8.12), we obtain the functional J1 = T 2πj

jω/2

−jω/2

(8.78)

trace θ (−s)AL (s) θ (s)AM (s) − θ (−s)C (−s) − C(s) θ (s) ds ,

where ω = 2π/T , the matrix AL (s) is given by (8.73), AM (s)

∞ 1 ξ(s + kjω)ξ (−s − kjω) T

=

k=−∞

=

a l (s)

∞ 1 M (s + kjω)M (−s − kjω) a l (−s) T

(8.79)

k=−∞

= a l (s)DM M (T, s, 0) a l (−s) and

C(s) =

∞ 1 ξ(s + kjω)Q(s + kjω) T k=−∞

1 = AM (s) β 0r (−s) DL Lµµ (T, s, 0) a r (s) T 1 + a l (s) DM K Lµ (T, s, 0) a r (s) , T 1 C (−s) = a r (−s) DL Lµµ (T, s, 0) β 0r (s)AM (s) T 1 + a r (−s) DL KM µ (T, s, 0) a l (−s) . T

6.

(8.80)

Let us note several useful properties of the matrices AL (s), AM (s) and

C(s) appearing in the functional (8.78). We shall assume that (8.66) holds, because this is true in almost all applied problems.


329

Theorem 8.16. The matrices AL (s) (8.73), AM (s) (8.79) and C(s) (8.80) are rational periodic and admit the representations

C(s) =

B C (s)

B C (s)

L (s) L (−s) M (s) M (−s)

=

AL (s)

=

(s) (−s)

, (8.81)

B L (s) L (s) L (−s)

AM (s) =

,

B M (s) M (s) M (−s)

,

where the numerators are finite sums of the forms

B L (s) =

α

lk e−ksT ,

lk = l−k ,

k=−α

B M (s) =

β

mk e−ksT ,

mk = m−k ,

(8.82)

k=−β

B C (s) =

δ

ck e−ksT .

k=−γ

Herein, α, β, γ and δ are non-negative integers and lk , mk and ck are constant real matrices.

Proof. Let us prove the claim for AL (s). First of all, the matrix AL (s) is rational periodical, as follows from the general properties given in Chapter 6. Due to Corollary 8.15, the following irreducible representations exist:

L(s)µ(s) a r (s) =

PL (s) L (s)

,

a r (−s)L (−s)µ(−s)

=

PL (−s) L (−s)

,

where the matrix PL (s) is an integral function of s. Using (8.73), we can write T AL (s)

∞ 1 = a r (−s − kjω)L (−s − kjω)µ(−s − kjω) T k=−∞ · L(s + kjω)µ(s + kjω) a r (s + kjω) .

(8.83)

Each summand on the right-hand side can have poles only at the roots of the product 1 (s) = L (s) L (−s). Under the given assumptions, the series (8.83) converges absolutely and uniformly in any restricted part of the complex plain containing no roots of the function 1 (s). Hence the sum of the series (8.83) can have poles only at the roots of the function 1 (s). Therefore, the

matrix B L (s) = T A(s) ρ 1 (s) has no poles. Moreover, since AL (−s) = AL (s),

we have B L (−s) = B L (s), and the first relation in (8.82) is proven. The remaining formulae in (8.82) are proved in a similar way using (8.66).

330


Corollary 8.17. When the original system is stabilisable, then the matrices (8.81) do not possess poles on the imaginary axis. Corollary 8.18. If the standard sampled-data system is modal controllable, then the matrices (8.81) are free of poles, i.e., they are integral functions of the argument s. Passing in the integral (8.78) to the variable ζ = e−sT , we obtain # dζ 1 ˆ ˆ ˆ (8.84) J1 = trace θ(ζ)A L (ζ)θ(ζ)AM (ζ) − θ(ζ)C(ζ) − C(ζ)θ(ζ) 2πj Γ ζ

7.

with the notation

Fˆ (ζ) = F (ζ −1 ) .

Perform integration along the unit circle Γ in positive direction (antiˆ clockwise). The matrices AL (ζ), AM (ζ), C(ζ) and C(ζ) appearing in (8.84) admit the representations AL (ζ) =

1 a ˆr (ζ)DL Lµµ (T, ζ, 0)ar (ζ) , T

(8.85)

al (ζ) AM (ζ) = al (ζ)DM M (T, ζ, 0)ˆ and 1 1 C(ζ) = AM (ζ)βˆ0r (ζ) DL Lµµ (T, ζ, 0)ar (ζ) + al (ζ)DM K Lµ (T, ζ, 0)ar (ζ) , T T (8.86) 1 1 ˆ ˆr (ζ)DL KM µ (T, ζ, 0)ˆ C(ζ) =a ˆr (ζ) DL Lµµ (T, ζ, 0)β0r (ζ)AM (ζ) + a al (ζ) . T T Per construction, AˆL (ζ) = AL (ζ),

AˆM (ζ) = AM (ζ) .

8. A rational matrix F (ζ) having no poles except for ζ = 0 will be called a quasi-polynomial matrix . Any quasi-polynomial matrix F (ζ) can be written in the form ν F (ζ) = Fk ζ k , k=−µ

where µ and ν are nonnegative integers and the Fk are constant matrices. Substituting e−sT = ζ in (8.81) and (8.82), we obtain AL (ζ) = C(ζ) =

BL (ζ) , L (ζ)L (ζ −1 )

AM (ζ) =

BM (ζ) , M (ζ)M (ζ −1 )

BC (ζ) , L (ζ)L (ζ −1 )M (ζ)M (ζ −1 )

(8.87)

8.5 Wiener-Hopf Method

331

where the numerators are the quasi-polynomial matrices BL (ζ) =

α

lk ζ k ,

lk = l−k

k=−α

(8.88) BM (ζ) =

β

mk ζ k ,

mk = m−k

k=−β

BC (ζ) =

δ

ck ζ k .

k=−γ

Here, α, β, γ, and δ are nonnegative integers and lk , mk , and ck are constant real matrices. Per construction, ˆL (ζ) = BL (ζ), B

ˆM (ζ) = BM (ζ) . B

(8.89)

Remark 8.19. If the standard sampled-data system is modal controllable, then the matrices (8.87) are quasi-polynomial matrices. Remark 8.20. Hereinafter, the matrix BM (ζ) will be called a quasi-polynomial matrix of type 1, and the matrix BL (ζ) a quasi-polynomial matrix of type 2.

8.5 Wiener-Hopf Method 1. In this section, we consider a method for solution of the H2 -optimisation problem for the standard sampled-data system based on minimisation of the integral in (8.84). Such an approach was previously applied for solving H2 problems for continuous-time and discrete-time LTI systems and there, it was called Wiener-Hopf method [196, 47, 5, 80]. 2. The following theorem provides a substantiation for applying the WienerHopf method to sampled-data systems. Theorem 8.21. Suppose the standard sampled-data system be stabilisable and let us have the IMFDs (8.36) and (8.42). Furthermore, let (α0r (ζ), β0r (ζ)) be any right initial controller, which has a dual left controller and the stable rational matrix θo (ζ) ensures the minimal value J1 min of the integral (8.84). Then the transfer function of the optimal controller wdo (ζ), for which the standard sampled-data system is stable and the functional (8.29) approaches the minimal value, has the form −1 wdo (ζ) = V2o (ζ)V1o (ζ) ,

where

(8.90)

332


V1o (ζ) = α0r (ζ) − ζbr (ζ)θo (ζ) ,

(8.91)

V2o (ζ) = β0r (ζ) − ar (ζ)θo (ζ) . If in addition, we have IMFDs θo (ζ) = Dl−1 (ζ)Ml (ζ) = Mr (ζ)Dr−1 (ζ) ,

then the characteristic polynomial ∆o (ζ) of the optimal system satisfies the relation ∆o (ζ) ≈ (ζ) det Dl (ζ) ≈ (ζ) det Dr (ζ) . Proof. Since the matrix θo (ζ) is stable, the rational matrices (8.91) are also stable. Then using the Bezout identity and the ILMFD (8.36), we have al (ζ)V1o (ζ) − ζbl (ζ)V2o (ζ) = al (ζ)α0r (ζ) − ζbl (ζ)β0r (ζ) − ζ [al (ζ)br (ζ) − bl (ζ)ar (ζ)] θo (ζ) = In . Hence due to Theorem 5.58, wdo (ζ) is the transfer function of a stabilising controller. Using the ILMFDs (8.36) and (8.48), we find Matrix (8.49) as o (ζ) = wdo (ζ) [In − DN µ (T, ζ, 0)wdo (ζ)] RN

−1

= β0r (ζ)al (ζ) − ar (ζ)θo (ζ)al (ζ) . For ζ = e−sT , we have −1 o o RN (s) = w d (s) In − DN µ (T, s, 0)w d (s)

o

o

= β 0r (s) a l (s) − a r (s) θ (s) a l (s) . After substituting this equation into (8.29) and some transformations, the integral (8.29) can be reduced to the form S22 = J1o + J2 , where J1o is given by (8.84) for θ(ζ) = θo (ζ) and the value J2 is constant. Per construction, the value S22 is minimal. Therefore, Formula (8.90) gives the transfer matrix of an optimal stabilising controller.

8.6 Algorithm for Realisation of Wiener-Hopf Method 1. According to the aforesaid, we shall consider the problem of minimising the functional (8.84) over the set of stable rational matrices, where the matrices AL (ζ), AM (ζ) and C(ζ) satisfy Conditions (8.87), (8.88) and (8.89). Moreover, if the stabilisability conditions hold for the system, then the matrices AL (ζ), AM (ζ), and C(ζ) do not possess poles on the integration path.

8.6 Algorithm for Realisation of Wiener-Hopf Method

333

2. The following proposition presents a theoretical basis for the application of the Wiener-Hopf method to the solution of the H2 -problem. Lemma 8.22. Let us have a functional of the form # dζ 1 ˆ ˆ Jw = (ζ)Γ (ζ) − Ψˆ (ζ)C(ζ) − C(ζ)Ψ (ζ) , trace Γˆ (ζ)Ψˆ (ζ)Π(ζ)Π(ζ)Ψ 2πj ζ (8.92) where the integration is performed along the unit circle in positive direction and Γ (ζ), Π(ζ) and C(ζ) are rational matrices having no poles on the integration path. Furthermore, let the matrices Π(ζ), Γ (ζ) be invertible and stable together with their inverses. Then, there exists a stable matrix Ψ o (ζ) that minimises the functional (8.92). The matrix Ψ o (ζ) can be constructed using the following algorithm: a) Construct the rational matrix ˆ −1 (ζ)C(ζ) ˆ Γˆ −1 (ζ) . R(ζ) = Π

(8.93)

b) Perform the separation R(ζ) = R+ (ζ) + R− (ζ) ,

(8.94)

where the rational matrix R− (ζ) is strictly proper and its poles incorporate all unstable poles of R(ζ). Such a separation will be called principal separation. c) The optimal matrix Ψ o (ζ) is determined by the formula Ψ o (ζ) = Π −1 (ζ)R+ (ζ)Γ −1 (ζ) .

(8.95)

Proof. Using (8.5), we have trace[C(ζ)Ψ (ζ)] = trace[C(ζ)Ψ (ζ)Γ (ζ)Γ −1 (ζ)] = trace[Γ −1 (ζ)C(ζ)Ψ (ζ)Γ (ζ)] . Therefore, the functional (8.92) can be represented in the form # 1 ˆ Jw = trace Γˆ (ζ)Ψˆ (ζ)Π(ζ)Π(ζ)Ψ (ζ)Γ (ζ) 2πj dζ ˆ , − Γˆ (ζ)Ψˆ (ζ)Φ(ζ) − Φ(ζ)Ψ (ζ)Γ (z) ζ where

Φ(ζ) = Γ −1 (ζ)C(ζ) .

The identity ˆ ˆ Γˆ (ζ)Ψˆ (ζ)Π(ζ)Π(ζ)Ψ (ζ)Γ (ζ) − Γˆ (ζ)Ψˆ (ζ)Φ(ζ) − Φ(ζ)Ψ (ζ)Γ (ζ) ˆ = [Π(ζ)Ψ (ζ)Γ (ζ) − R(ζ)]' [Π(ζ)Ψ (ζ)Γ (ζ) − R(ζ)] − R(ζ)R(ζ)

(8.96)

334


can easily be verified. Using the separation (8.94), the last relation can be transformed into ˆ ˆ Γˆ (ζ)Ψˆ (ζ)Π(ζ)Π(ζ)Ψ (ζ)Γ (ζ) − Γˆ (ζ)Ψˆ (ζ)Φ(ζ) − Φ(ζ)Ψ (ζ)Γ (ζ) = [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)]' [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)] ˆ − (ζ)R− (ζ) − R ˆ − (ζ) [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)] +R ˆ − [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)]'R− (ζ) − R(ζ)R(ζ) . Hence the functional (8.96) can be written in the form Jw = Jw1 + Jw2 + Jw3 + Jw4 , where Jw1 =

1 2πj

Jw2 = − Jw3 Jw4

#

1 2πj

trace [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)]' [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)] #

dζ , ζ

ˆ − (ζ) [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)] dζ , trace R ζ

# 1 dζ trace [Π(ζ)Ψ (ζ)Γ (ζ) − R+ (ζ)]'R− (ζ) , =− 2πj ζ # 1 dζ ˆ − (ζ)R− (ζ) − R(ζ)R(ζ) ˆ trace R . = 2πj ζ

(8.97)

The integral Jw4 is independent of Ψ (ζ). As for the scalar case [146], it can be shown that Jw2 = Jw3 = 0. The integral Jw1 is nonnegative, its minimal value Jw1 = 0 can be reached for (8.95). Corollary 8.23. The minimal value of the integral (8.92) is Jw min = Jw4 . 3. Using Lemma 8.22, we can formulate a proposition deriving a solution to the H2 -optimisation problem for the standard sampled-data system. Theorem 8.24. Let the quasi-polynomial matrices BL (ζ) and BM (ζ) in (8.88) admit the factorisations ˆ L (ζ)λL (ζ), BL (ζ) = λ

ˆ M (ζ) , BM (ζ) = λM (ζ)λ

(8.98)

where λL (ζ) and λM (ζ) are invertible real stable polynomial matrices. Let also Condition (8.66) hold. Then the optimal matrix θo (ζ) can be found using the following algorithm: a) Construct the matrix R(ζ) =

ˆ −1 (ζ) ˆ −1 (ζ) ˆ −1 (ζ)B ˆ −1 (ζ)B (ζ −1 )λ ˆC (ζ)λ λ λ C L M M = L . L (ζ)M (ζ) (ζ)

(8.99)

8.6 Algorithm for Realisation of Wiener-Hopf Method

335

b) Perform the principal separation R(ζ) = R+ (ζ) + R− (ζ) , where R+ (ζ) =

˜ + (ζ) ˜ + (ζ) R R = L (ζ)M (ζ) (ζ)

(8.100)

˜ + (ζ). with a polynomial matrix R c) The optimal system function θo (ζ) is given by the formula −1 ˜ θo (ζ) = λ−1 L (ζ)R+ (ζ)λM (ζ) .

(8.101)

Proof. Let the factorisations (8.98) hold. Since for the stabilisability of the system the polynomials L (ζ) and M (ζ) must be stable, the following factorisations hold: ˆ , AL (ζ) = Π(ζ)Π(ζ) where Π(ζ) =

λL (ζ) , L (ζ)

AM (ζ) = Γ (ζ)Γˆ (ζ) ,

Γ (ζ) =

λM (ζ) M (ζ)

(8.102)

(8.103)

are rational matrices, which are stable together with their inverses. From (8.103) we also have ˆ L (ζ) λ ˆ Π(ζ) = , L (ζ −1 )

ˆ M (ζ) λ Γˆ (ζ) = . M (ζ −1 )

(8.104)

Regarding (8.102)–(8.104), the integral (8.84) can be represented in the form (8.92) with BC (ζ) , (ζ)(ζ −1 )

C(ζ) =

BC (ζ) −1 L (ζ)L (ζ )M (ζ)M (ζ −1 )

ˆ C(ζ) =

ˆC (ζ) B BC (ζ −1 ) = . L (ζ)L (ζ −1 )M (ζ)M (ζ −1 ) (ζ)(ζ −1 )

=

Then the matrix R(ζ) in (8.93) appears to be equal to (8.99). ˆ −1 (ζ) can have only unstable poles, because the ˆ −1 (ζ) and λ The matrices λ L M (ζ −1 ) can polynomial matrices λL (ζ) and λM (ζ) are stable. The matrix BC have unstable pole only at ζ = 0. Therefore, the set of stable poles of Matrix (8.99) belongs to the set of roots of the polynomial (ζ) ≈ L (ζ)M (ζ). Hence the matrix R+ (ζ) in the principal separation (8.99) has the form (8.100), ˜ + (ζ) is a polynomial matrix. Equation (8.101) can be derived from where R (8.100), (8.103), and (8.95). Corollary 8.25. If the system is modal controllable, then R+ (ζ) is a polynomial matrix.

336


Corollary 8.26. The characteristic polynomial of a modal controllable closedloop system ∆d (ζ) ≈ det Dl (ζ) ≈ det Dr (ζ) is a divisor of the polynomial det λL (ζ) det λM (ζ). Hereby, if the right-hand side of (8.101) is an irreducible DMFD (this is often the case in applications), then ∆d (ζ) ≈ det λL (ζ) det λM (ζ) . 4.

From (8.76) and (8.97), it follows that the minimal value of S22 is

S22

1 = 2πj

#

j∞ dζ 1 ˆ ˆ trace R− (ζ)R− (ζ) − R(ζ)R(ζ) + trace g4 (s) ds . ζ 2πj −j∞

8.7 Modified Optimisation Algorithm 1. The method for solving the H2 -problems described in Section 8.5 requires for given IMFDs (8.36) and (8.42) of the plant, that the basic controller [α0r (ζ), β0r (ζ)] has to be previously found. This causes some numerical difficulties. In the present section, we describe a modified optimisation procedure, which does not need the basic controller. This method will be called the modified Wiener-Hopf method. 2. Let F (ζ) be a rational matrix and F+ (ζ), F− (ζ) be the results of the principal separation (8.94). Then for the matrix F− (ζ), we shall use the notation F− (ζ) = F (ζ)− . Obviously, F1 (ζ) + F2 (ζ)− = F1 (ζ)− + F2 (ζ)− . 3. Consider Matrix (8.93) in detail. Using the above relations as well as (8.86) and (8.102), we find ˆ Γˆ −1 (ζ) = R1 (ζ) + R2 (ζ) , ˆ −1 (ζ)C(ζ) R(ζ) = Π

(8.105)

where 1 T 1 R2 (ζ) = T

R1 (ζ) =

ˆ −1 (ζ)ˆ Π ar (ζ)DL Lµµ (T, ζ, 0)β0r (ζ)AM (ζ)Γˆ −1 (ζ) ,

(8.106)

ˆ −1 (ζ)ˆ Π ar (ζ)DL KM µ (T, ζ, 0)ˆ al (ζ)Γˆ −1 (ζ) .

(8.107)

Since

1 DL Lµµ (T, ζ, 0)ar (ζ) = AL (ζ) , T Matrix (8.106) can be written in the form a ˆr (ζ)

(8.108)

8.7 Modified Optimisation Algorithm

ˆ −1 (ζ)AL (ζ)a−1 ˆ −1 (ζ) . R1 (ζ) = Π r (ζ)β0r (ζ)AM (ζ)Γ

337

(8.109)

With respect to (8.102), we obtain R1 (ζ) = Π(ζ)a−1 r (ζ)β0r (ζ)Γ (ζ) .

(8.110)

On the basis of (8.105)–(8.110), the following lemma can be proved. Lemma 8.27. In the principal separation R(ζ) = R+ (ζ) + R(ζ)− ,

(8.111)

the matrix R(ζ)− is independent of the choice of the basic controller. ∗ Proof. If we choose another right initial controller with a matrix β0r (ζ), we ∗ obtain a new matrix R (ζ) of the form

R∗ (ζ) = R1∗ (ζ) + R2 (ζ) with

∗ R1∗ (ζ) = Π(ζ)a−1 r (ζ)β0r (ζ)Γ (ζ) ,

(8.112)

where the matrix R2 (ζ) is the same as in (8.105). Therefore, to prove the lemma, it is sufficient to show R1 (ζ)− = R1∗ (ζ)− .

(8.113)

But from (4.39), it follows that ∗ β0r (ζ) = β0r (ζ) − ar (ζ)Q(ζ)

(8.114)

with a polynomial matrix Q(ζ). Substituting this formula into (8.112), we find R1∗ (ζ) = R1 (ζ) − Π(ζ)Q(ζ)Γ (ζ) , where the second term on the right-hand side is a stable matrix, because the matrices Π(ζ) and Γ (ζ) are stable. Therefore, (8.113) holds.

4. Lemma 8.28. The transfer matrix of an optimal controllers wdo (ζ) is independent of the choice of the initial controller. Proof. Using (8.105)–(8.111), we have R+ (ζ) = R(ζ) − R(ζ)− = Π(ζ)a−1 r (ζ)β0r (ζ)Γ (ζ) + Λ(ζ) , where

(8.115)

338


Λ(ζ) =

1 ˆ −1 Π (ζ)ˆ ar (ζ)DL KM µ (T, ζ, 0)ˆ al (ζ)Γˆ −1 (ζ) − R(ζ)− . T

(8.116)

Using Lemma 8.27 and (8.116), we find that the matrix Λ(ζ) does not depend on the choice of initial controller. Hence only the first term on the right-hand side of (8.115) depends on the initial controller. From (8.115) and (8.95), we find the optimal system function −1 (ζ)Λ(ζ)Γ −1 (ζ) . θo (ζ) = a−1 r (ζ)β0r (ζ) + Π

Using (8.91), we obtain the optimal matrices V1o (ζ) and V2o (ζ): V2o (ζ) = β0r (ζ) − ar (ζ)θo (ζ) = −ar (ζ)Π −1 (ζ)Λ(ζ)Γ −1 (ζ)

(8.117)

and V1o (ζ) = α0r (ζ) − ζbr (ζ)θo (ζ) −1 = α0r (ζ) − ζbr (ζ)a−1 (ζ)Λ(ζ)Γ −1 (ζ) . r (ζ)β0r (ζ) − ζbr (ζ)Π

The Bezout identity guarantees −1 α0r (ζ) − ζbr (ζ)a−1 r (ζ)β0r (ζ) = α0r (ζ) − ζal (ζ)bl (ζ)β0r (ζ) −1 = a−1 l (ζ) [al (ζ)α0r (ζ) − ζbl (ζ)β0r (ζ)] = al (ζ) .

Together these relations yield −1 V1o (ζ) = a−1 (ζ)Λ(ζ)Γ −1 (ζ) . l (ζ) − ζbr (ζ)Π

(8.118)

The matrices (8.117) and (8.118) are independent of the matrix β0r (ζ). Then using (8.90), we can find an expression for the transfer matrix of the optimal controller that is independent of β0r (ζ). 5. As follows from Lemma 8.28, we can find an expression for the optimal matrix wdo (ζ), when we manage to find an expression for R(ζ)− that is independent of β0r . We will show a possibility for deriving such an expression. Assume that the matrix γl (ζ) = bl (ζ)bl (ζ −1 ) = bl (ζ)ˆbl (ζ)

is invertible and the poles of the matrix

−1 δl (ζ) = ζ −1 bl (ζ −1 ) bl (ζ)bl (ζ −1 )

(8.119)

coincide neither with eigenvalues of the matrix ar (ζ) nor with poles of the matrices Π(ζ) and Γ (ζ). Then, Equation (8.110) can be written in the form

−1 −1

−1 bl (ζ ) bl (ζ)bl (ζ −1 ) Γ (ζ) . (8.120) R1 (ζ) = Π(ζ)a−1 r (ζ)β0r (ζ)bl (ζ) ζ

8.7 Modified Optimisation Algorithm

339

Due to the inverse Bezout identity (4.32), we have −ζβ0r (ζ)bl (ζ) + ar (ζ)α0l (ζ) = Im . Thus,

−1 ζa−1 r (ζ)β0r (ζ)bl (ζ) = α0l (ζ) − ar (ζ) .

Hence Equation (8.120) yields R1 (ζ) = R11 (ζ) + R12 (ζ) , where

−1 −1 −1 R11 (ζ) = −Π(ζ)a−1 bl (ζ ) bl (ζ)bl (ζ −1 ) Γ (ζ) r (ζ)ζ

(8.121)

= −Π(ζ)a−1 r (ζ)δl (ζ)Γ (ζ) ,

−1 R12 (ζ) = Π(ζ)α0l (ζ)ζ −1 bl (ζ −1 ) bl (ζ)bl (ζ −1 ) Γ (ζ)

(8.122)

= Π(ζ)α0l (ζ)δl (ζ)Γ (ζ) . Consider the separation a

δ

+ R11 (ζ) = R11 (ζ) + R11 (ζ)− + R11 (ζ) ,

(8.123)

a

where R11 (ζ)− is a strictly proper function, whose poles are the unstable δ eigenvalues of the matrix ar (ζ); R11 is a strictly proper function, whose + poles are the poles of Matrix (8.119); and R11 (ζ) is a rational function, whose poles are the stable eigenvalues of the matrix ar (ζ) as well as the poles of the matrices Π(ζ) and Γ (ζ). Similarly to (8.123) for (8.122), we find δ

+ R12 (ζ) = R12 (ζ) + R12 (ζ) ,

(8.124)

δ

where R12 (ζ) is a strictly proper function, whose poles are the poles of + Matrix (8.119), and R12 (ζ) is a stable rational matrix. Summing up Equations (8.123) and (8.124), we obtain R1 (ζ) = R11 (ζ) + R12 (ζ) a

δ

δ

+ + (ζ) + R12 (ζ) + R11 (ζ)− + R11 (ζ) + R12 (ζ) . = R11

But δ

δ

δ

R11 (ζ) + R12 (ζ) = R11 (ζ) + R12 (ζ) = Omn , because from (8.110), it follows that under the given assumptions, the matrix R1 (ζ) has no poles that are simultaneously poles of the matrix δl (ζ). Then, a

R1 (ζ)− = R11 (ζ)− , where the matrix on the right-hand side is independent of the choice of the initial controller. Using the last relation and (8.105), we obtain

340

8 Analysis and Synthesis of SD Systems Under Stochastic Excitation a

R(ζ)− = R1 (ζ) + R2 (ζ)− = R11 (ζ)− + R2 (ζ)− . Per construction, this matrix is also independent of the choice of the initial controller. Substituting the last equation into (8.116) and using (8.117), (8.118) and (8.90), an expression can be derived for the optimal transfer matrix wdo (ζ) that is independent of the choice of the initial controller. 6. A similar approach to the modified optimisation method can be proposed for the case, when the matrix ˆbr (ζ)br (ζ) is invertible, where br (ζ) is the polynomial matrix appearing in the IRMFD (8.42). In this case, (8.110) can be written in the form −1 ζ −1ˆbr (ζ)ζbr (ζ)a−1 R1 (ζ) = Π(ζ) ˆbr (ζ)br (ζ) r (ζ)β0r (ζ)Γ (ζ) . Due to the inverse Bezout identity, we have −1 a−1 r (ζ)β0r (ζ) = β0l (ζ)al (ζ) ,

α0r (ζ)al (ζ) − ζbr (ζ)β0l (ζ) = In , −1 ζbr (ζ)β0l (ζ)a−1 l (ζ) = α0r (ζ) − al (ζ) .

Using these relations, we obtain −1 R1 (ζ) = Π(ζ) ˆbr (ζ)br (ζ) ζ −1ˆbr (ζ)α0r (ζ)Γ (ζ) − −1 − Π(ζ) ˆbr (ζ)br (ζ) ζ −1ˆbr (ζ)a−1 l (ζ)Γ (ζ) . Assuming that no pole of the matrix −1 ˆbr (ζ) δr (ζ) = ζ −1 ˆbr (ζ)br (ζ) coincides with an eigenvalue of the matrices al (ζ), Π(ζ) or Γ (ζ), we find that the further procedure of constructing the optimal controller is similar to that described above.

8.8 Transformation to Forward Model 1. To make the reading easier, we will use some additional terminology and notation. The optimisation algorithms described above make it possible to find the optimal system matrix θo (ζ). Using the ILMFD θo (ζ) = Dl−1 (ζ)Ml (ζ) = Mr (ζ)Dr−1 (ζ)

8.8 Transformation to Forward Model

341

and Formulae (8.41), (8.44) and (8.46), we are able to construct the transfer o (ζ), which will be called the backward matrix of the optimal controller wdb transfer matrix. Using the ILMFD o wdb (ζ) = αl−1 (ζ)βl (ζ) ,

the matrix ⎡ ⎢ Qb (ζ, αl , βl ) = ⎣

Iχ − ζeAT

Oχn −ζeAT µ(A)B2

−C2

In

Onm

Omχ

−βl (ζ)

αl (ζ)

⎤ ⎥ ⎦

(8.125)

can be constructed. This matrix is called the backward characteristic matrix. As shown above, the characteristic polynomial of the backward model ∆b (ζ) is determined by the relation ∆b (ζ) = det Qb (ζ, αl , βl ) = b (ζ)∆db (ζ) , where b (ζ) and ∆db (ζ) are polynomials. Especially, we know al (ζ) −ζbl (ζ) , ∆db (ζ) = det −βl (ζ) αl (ζ)

(8.126)

(8.127)

where the matrices al (ζ) and bl (ζ) are given by the ILMFD −1 AT wN (ζ) = ζC2 Iχ − ζeAT e µ(A)B2 = ζa−1 l (ζ)bl (ζ) .

(8.128)

Moreover, the polynomial b (ζ) appearing in (8.126) is determined by the relation det(Iχ − ζeAT ) b (ζ) = (8.129) det al (ζ) and is independent of the choice of the controller. Further, the value

δb = ord Qb (ζ, αl , βl ) = deg det Qb (ζ, αl , βl ) = deg b (ζ) + deg ∆db (ζ) will be called the order of the optimal backward model. As shown above, deg det Dl (ζ) = deg ∆db (ζ) . 2. For applied calculations and simulation, it is often convenient to use the forward system model instead of the backward model. In the present section we consider the realisation of such a transformation and investigate some general properties of the forward model. Hereinafter, the matrix o o (z) = wdb (z −1 ) wdf

342


will be called the forward transfer function of the optimal controller. Using the ILMFD o (z) = αf−1 (z)bf (z) , wdf a controllable forward model of the optimal discrete controller is found: αf (q)ψk = βf (q)yk . Together with (7.16), this equation determines a discrete forward model of the optimal system qvk = eAT vk + eAT µ(A)B2 ψk + gk yk = C2 vk αf (q)ψk = βf (q)yk . These difference equations are associated with the matrix ⎤ ⎡ zIχ − eAT Oχn −eAT µ(A)B2 ⎥ ⎢ In Onm Qf (z, αf , βf ) = ⎣ −C2 ⎦ Omχ

−βf (z)

(8.130)

αf (z)

that will be called the forward characteristic matrix of the optimal system. Below, we formulate some propositions determining some properties of the characteristic matrices (8.125) and (8.130). 3.

Similarly to (8.126)–(8.129), it can be shown that the polynomial

∆f (z) = det Qf (z, αf , βf ) , which is called the characteristic polynomial of the forward model, satisfies the relation ∆f (z) = f (z)∆df (z) , (8.131) where f (z) and ∆df (z) are polynomials. Moreover,

af (z) −bf (z) ∆df (z) = det , −βf (z) αf (z) where the matrices af (z) and bf (z) are determined by the ILMFD −1 AT e µ(A)B2 = a−1 wf (z) = C2 zIχ − eAT f (z)bf (z) .

(8.132)

(8.133)

The polynomial f (z) appearing in (8.131) satisfies the equation f (z) =

det(zIχ − eAT ) . det af (z)

(8.134)

The value

δf = ord Qf (z, αf , βf ) = deg det Qf (z, αf , βf ) = deg f (z) + deg ∆df (z) will be called the order of the optimal forward model.


343

4. A connection between the polynomials (8.129) and (8.134) is determined by the following lemma. Lemma 8.29. The following equation holds:

δ = deg b (ζ) = deg f (z) . Moreover,

f (z) ≈ z δ b (z −1 ) ,

b (ζ) ≈ ζ δ f (ζ −1 ) .

(8.135) (8.136)

Proof. Since Matrix (8.133) is strictly proper, there exists a minimal standard realisation in the form wf (z) = Cν (zIq − U )−1 Bν ,

q ≤ χ.

Then for the ILMFD on the right-hand side of (8.133), we have det af (z) ≈ det(zIq − U ) .

(8.137)

det(zIχ − eAT ) . det(zIq − U )

(8.138)

Hence f (z) ≈

From (8.138), it follows that the matrix U is nonsingular (this is a consequence of the non-singularity of the matrix eAT ). Comparing (8.128) with (8.133), we find wN (ζ) = ζCν (Iq − ζU )−1 Bν , where the PMD (Iq − ζU, ζBν , Cν ) is irreducible due to Lemma 5.35. Thus for the ILMFD (8.128), we obtain det al (ζ) ≈ det(Iq − ζU ) ,

(8.139)

where ord al (ζ) = deg det(Iq − ζU ) = q , because the matrix U is nonsingular. From (8.129) and (8.139), we obtain b (ζ) ≈

det(Iχ − ζeAT ) det(Iq − ζU )

and comparing this with (8.138) results in (8.135) and (8.136). 5.

The following propositions determine a property of the matrices

al (ζ) −ζbl (ζ) af (z) −bf (z) Q1b (ζ) = , Q1f (z) = . −βl (ζ) αl (ζ) −βf (z) αf (z)

Denote γf = deg det αf (z) ,

γb = deg det αl (ζ) .

344


Lemma 8.30. The following equation holds: ord Q1f (z) = q + γf . Proof. Applying Formula (4.12) to Q1f (z), we obtain

o det Q1f (z) = det af (z) det αf (z) det Im − wdf (z)wf (z) .

(8.140)

(8.141)

Since the optimal controller (αl (ζ), βl (ζ)) is stabilising, it is causal, i.e., the o (ζ) is analytical at the point ζ = 0. Hence backward transfer function wdb o (z) is at least proper. Therethe forward transfer matrix of the controller wdf o (z)wf (z) is a fore, the matrix wf (z) is strictly proper. Thus, the product wdf o strictly proper matrix. Moreover, the rational fraction det Im − wdf (z)wf (z) is proper, because of

o lim det Im − wdf (z)wf (z) = 1 . z→∞

Thus, it follows that there exists a representation of the form

z λ + b1 z λ−1 + . . . o det Im − wdf , (z)wf (z) = λ z + a1 z λ−1 + . . . where λ a nonnegative integer. Substituting this into (8.141), we find det Q1f (z) = det af (z) det αf (z) ·

z λ + b1 z λ−1 + . . . . z λ + a1 z λ−1 + . . .

Since the right-hand side is a polynomial, we obtain deg det Q1f (z) = deg det af (z) + deg det αf (z) . With regard to (8.137), this equation is equivalent to (8.140). Lemma 8.31. Let us have unimodular matrices m(z) and µ(z), such that the matrices ˜ f (z) = µ(z)αf (z) a ˜f (z) = m(z)af (z) , α are row reduced. Then the matrix ˜ 1f (z) = diag{m(z), µ(z)} Q1f (z) Q

(8.142)

is row reduced. Proof. Rewrite (8.142) in the form ˜ 1f (z) = Q

m(z)af (z) −m(z)bf (z) a ˜f (z) −˜bf (z) = , −µ(z)βf (z) µ(z)αf (z) −β˜f (z) α ˜ f (z)

(8.143)


345

where the pairs (˜ af (z), ˜bf (z)) and (˜ αf (z), β˜f (z)) are irreducible. Since the ˜ f (z) are row reduced, we receive a representation of the matrices a ˜f (z) and α form (1.21) a ˜f (z) = diag {z a1 , . . . , z an } A0 + a ˜1f (z) ,

a1 + . . . + an = q ,

α ˜ f (z) = diag {z

α1 + . . . + αn = γf ,

α1

,...,z

αn

} B0 + α ˜ 1f (z) ,

where det A0 = 0 and det B0 = 0. Hereby, the degree of the i-th row of the matrix a ˜1f (z) is less than ai , and the degree of the i-th row of the matrix α ˜ 1f (z) is less than αi . Moreover, −1 ˜ a ˜−1 f (z)bf (z) = af (z)bf (z) = wf (z) , o α ˜ f−1 (z)β˜f (z) = αf−1 (z)βf (z) = wdf (z) . o Since the matrix wf (z) is strictly proper and the matrix wdf (z) is at least ˜ proper, the degree of the i-th row of the matrix bf (z) is less than αi − 1, and the degree of the i-th row of β˜f (z) is not more than αi . Therefore, Matrix (8.143) can be represented in the form (1.21)

˜ 1f (z) = diag{z a1 , . . . , z an , z α1 , . . . , z αm } D0 + Q ˜ 2f (z) , Q

(8.144)

where D0 is the constant matrix D0 =

A0 Onm . C0 B0

Since det D0 = 0, Matrix (8.142) is row reduced. Lemma 8.32. The polynomials ∆db (ζ) and ∆df (z) given by (8.127) and (8.132) are connected by ∆db (ζ) ≈ ζ q+γf ∆df (ζ −1 ) ,

(8.145)

i.e., ∆db (ζ) is equivalent to the reciprocal polynomial for ∆df (z). Proof. Since Matrix (8.143) is row reduced and has the form (8.144), the matrix ˜ 1b (ζ) = diag{ζ a1 , . . . , ζ an , ζ α1 , . . . , ζ αm } Q ˜ 1f (ζ −1 ) Q

(8.146)

˜ 1f (z). Then defines a backward eigenoperator associated with the operator Q the following formula stems from Corollary 5.39: ˜ 1b (ζ) ≈ ζ q+γf det Q ˜ 1f (ζ −1 ) . det Q Per construction,

(8.147)

346

8 Analysis and Synthesis of SD Systems Under Stochastic Excitation ˜ 1f (z) ≈ det Q1f (z) = ∆df (z) , ∆˜df (z) = det Q

(8.148)

˜ 1f (z) are left-equivalent. Let us show the because the matrices Q1f (z) and Q relation ˜ 1b (ζ) ≈ det Q1b (ζ) = ∆db (ζ) . (8.149) ∆˜db (ζ) = det Q Notice that Matrix (8.146) can be represented in the form diag{ζ a1 , . . . , ζ an } a ˜f (ζ −1 ) − diag{ζ a1 , . . . , ζ an } ˜bf (ζ −1 ) ˜ 1b (ζ) = Q − diag{ζ α1 , . . . , ζ αm } β˜f (ζ −1 ) diag{ζ α1 , . . . , ζ αm } α ˜ f (ζ −1 )

(8.150) a1l (ζ) −ζb1l (ζ) , = −β1l (ζ) α1l (ζ) where the pairs (a1l (ζ), ζb1l (ζ)) and (α1l (ζ), β1l (ζ)) are irreducible due to Lemma 5.34. We have ζa−1 (ζ)b1l (ζ) = a ˜−1 (ζ −1 )˜bf (ζ −1 ) = wf (ζ −1 ) = wb (ζ) 1l

f

and the left-hand side is an ILMFD. On the other hand, the right-hand side of (8.128) is also an ILMFD and the following equations hold: a1l (ζ) = φ(ζ)al (ζ),

b1l (ζ) = φ(ζ)bl (ζ) ,

(8.151)

where φ(ζ) is a unimodular matrix. In a similar way, it can be shown that α1l (ζ) = ψ(ζ)αl (ζ),

β1l (ζ) = ψ(ζ)βl (ζ)

(8.152)

with a unimodular matrix ψ(ζ). Substituting (8.151) and (8.152) into (8.150), we find ˜ 1l (ζ) = diag{φ(ζ), ψ(ζ)} Q1b (ζ) , Q ˜ 1b (ζ) are left-equivalent. Therefore, Relations i.e., the matrices Q1b (ζ) and Q (8.149) hold. Then Relation (8.145) directly follows from (8.147)–(8.149). On the basis of Lemmata 8.29–8.32, the following theorem will be proved. Theorem 8.33. Let ∆f (z) and ∆b (ζ) be the forward and backward characteristic polynomials of the optimal system. Then, δf = deg ∆f (z) = δ + q + γf , δb = deg ∆b (ζ) = δ + deg det Dl (ζ) , where the number δ is determined by (8.135) and the number δ0 of zero roots of the polynomial ∆f (z) is δ0 = q + γf − deg det Dl (ζ) . In this case, the polynomials ∆f (z) and ∆b (ζ) are related by ∆b (ζ) = ζ δf ∆f (ζ −1 ) , i.e., ∆b (ζ) is equivalent to the reciprocal polynomial for ∆f (z). Proof. The proof is left as exercise for the reader.

9 H2 Optimisation of a Single-loop Multivariable SD System

9.1 Single-loop Multivariable SD System 1. The aforesaid approach for solving the H2 -problem is fairly general and can be applied to any sampled-data system that can be represented in the standard form. Nevertheless, as will be shown in this chapter, the specific structure of a system and algebraic properties of the transfer matrices of the continuous-time blocks can play an important role for solving the H2 -problem. In this case, we can find possibilities for some additional cancellations, extracting matrix divisors and so on. Moreover, this makes it possible to investigate important additional properties of the optimal solutions. 2. In this chapter, the above ideas are exemplarily illustrated by the singleloop system shown in Fig. 9.1, where F (s), Q(s) and G(s) are rational matrices h 6

x

κ y

T

-

C m×n

u

- G(s)

h1 6

×m

q

? F (s) - fκ×

q v-

Q(s)

n×κ

Fig. 9.1. Single-loop sampled-data system

of compatible dimensions and κ is a constant. The vector

h(t) κh1 (t) = z(t) = v(t) v(t)

(9.1)

348

9 H2 Optimisation of a Single-loop System

will be taken as the output of the system. If the system is internally stable and x(t) is a stationary centered vector, the covariance matrix of the quasistationary output is given by Kz (t1 , t2 ) = E [z(t1 )z (t2 )] .

Since

z(t1 )z (t2 ) =

κ 2 h1 (t1 )h1 (t2 ) κh1 (t1 )v (t2 ) κv(t1 )h1 (t2 )

v(t1 )v (t2 )

,

we have trace[z(t1 )z (t2 )] = trace[v(t1 )v (t2 )] + κ 2 trace[h1 (t1 )h1 (t2 )] . For Φx (s) = I , we obtain the square of the H2 -norm of the system S in Fig. 9.1 as 1 T trace [Kz (t, t)] dt = κ 2 d¯h1 + d¯v , (9.2) S22 = r¯z = T 0 where d¯h1 and d¯v are the mean variances of the corresponding output vectors. To solve the H2 -problem, it is required to find a stabilising discrete controller, such that the right-hand side of (9.2) reaches the minimum.

9.2 General Properties 1. Using the general methods described in Section 7.3, we construct the PTM w(s, t) of the single-loop system from the input x to the output (9.1). Let us show that such a construction can easily be done directly on the basis of the block-diagram shown in Fig. 9.1 without transformation to the standard form. In the given case, we realise

whx (s, t) w(s, t) = wvx (s, t)

(9.3)

κ

,

where whx (s, t) and wvx (s, t) are the PTMs of the system from the input x to the outputs h and v, respectively. 2. To find the PTM wvx (s, t), we assume according to the previously exposed approach x(t) = est I ,

v(t) = wvx (s, t)est ,

wvx (s, t) = wvx (s, t + T )

and y(t) = wyx (s, t)est ,

wyx (s, t) = wyx (s, t + T ) ,

(9.4)

9.2 General Properties

349

x wyx (s, 0)est

-

u C

- G(s)

?-f F (s)

v

- Q(s)

y

-

Fig. 9.2. Open-loop sampled-data system

where wyx (s, t) is the PTM from the input x to the output y. The matrix wyx (s, t) is assumed to be continuous in t. For our purpose, it suffices to assume that the matrix Q(s)F (s)G(s) is strictly proper. Consider the open-loop system shown in Fig. 9.2. The exp.per. output y(t) is expressed by

y(t) = ϕQF Gµ (T, s, t)w d (s)wyx (s, 0)est + Q(s)F (s)est . Comparing the formulae for y(t) here and in (9.4), we obtain

wyx (s, t) = ϕQF Gµ (T, s, t)w d (s)wyx (s, 0) + Q(s)F (s) . Hence for t = 0, we have −1 wyx (s, 0) = In − DQF Gµ (T, s, 0)w d (s) Q(s)F (s) . Returning to Fig. 9.1 and using the last equation, we immediately get

wvx (s, t) = ϕF Gµ (T, s, t)RQF G (s)Q(s)F (s) , +F (s) where

(9.5)

−1 RQF G (s) = w d (s) In − DQF Gµ (T, s, 0)w d (s) .

Comparing (9.5) and (7.30), we find that in the given case K(p) = F (p), M (p) = Q(p)F (p), i.e., Matrix (7.2) has the form wv (p) =

F (p)

L(p) = F (p)G(p) , N (p) = Q(p)F (p)G(p) ,

F (p)G(p)

Q(p)F (p) Q(p)F (p)G(p)

(9.6)

.

The matrix F (p)G(p) is assumed to be at least proper and the remaining blocks should be strictly proper. In a similar way, it can be shown that

whx (s, t) = κϕGµ (T, s, t)RQF G (s)Q(s)F (s) and the corresponding matrix wh (p) (7.2) is equal to

(9.7)


350

wh (p) =

κG(p)

O

Q(p)F (p) Q(p)F (p)G(p)

,

where the matrix G(p) is assumed to be at least proper. Combining (9.5) and (9.7), we find that the PTM (9.3) has the form

w(s, t) = ϕLµ (T, s, t)RN (s)M (s) + K(s)

(9.8)

with

O K(s) = F (s)

κ

,

m

κG(s) L(s) = F (s)G(s)

κ

, (9.9)

m

M (s) = Q(s)F (s) n ,

N (s) = Q(s)F (s)G(s)

n

.

Under the given assumptions, the matrix L(s) is at least proper and the remaining matrices in (9.9) are strictly proper. The matrix w(p) associated with the PTM (9.8) has the form m ⎤ .. O . κG(p) ⎥ ⎢ ⎢ ⎥ . K(p) L(p) ⎢ F (p) .. F (p)G(p) ⎥ w(p) = =⎢ ⎥ .. ⎢ ⎥ M (p) N (p) ⎣ ......... ................⎦ . Q(p)F (p) .. Q(p)F (p)G(p)

⎡

3. as

(9.10)

κ

n

.

Hereinafter the standard form (2.21) of a rational matrix R(s) is written R(s) =

NR (s) . dR (s)

(9.11)

The further exposition is based on the following three assumptions I–III, which usually hold in applications. I The matrices Q(s) =

NQ (s) , dQ (s)

F (s) =

NF (s) , dF (s)

NG (s) dG (s)

(9.12)

NQ (s)NF (s)NG (s) dQ (s)dF (s)dG (s)

(9.13)

G(s) =

are normal. II The fraction N (s) = Q(s)F (s)G(s) = is irreducible.

9.2 General Properties

351

III The poles of the matrix N (s) should satisfy the strict conditions for nonpathological behavior (6.124) and (6.125). Moreover, it is assumed that the number of inputs and outputs of any continuous-time block does not exceed the McMillan-degree of its transfer function. These assumptions are introduced for the sake of simplicity of the solution. They are satisfied for the vast majority of applied problems. 4. Let us formulate a number of propositions following from the above assumptions. Lemma 9.1. The following subordination relations hold: F (s)G(s) ≺ N (s),

Q(s)F (s) ≺ N (s),

r

l

G(s) ≺ F (s)G(s) . r

(9.14)

Proof. The proof follows immediately from Theorem 3.14. Lemma 9.2. All matrices (9.6) are normal. Proof. The claim follows immediately from Theorem 3.8. Lemma 9.3. All matrices (9.9) are normal. Proof. Obviously, it suffices to prove the claim for L(s). But L(s) = L1 (s)G(s) ,

where L1 (s) =

κI = F (s)

κ dF (s)I NF (s) dF (s)

(9.15)

=

NL1 (s) . dF (s)

Let us show that this matrix is normal. Indeed, since the matrix F (s) is normal, all second-order minors of the matrix NF (s) are divisible by dF (s). Obviously, the same is true for all second-order minors of the matrix NL1 (s). Thus, both factors on the right-hand side of (9.15) are normal matrices and its product is irreducible, because the product F (s)G(s) is irreducible. Therefore, the matrix L(s) is normal. Corollary 9.4. The matrix F (s)G(s) dominates in the matrix L(s). Lemma 9.5. Matrix (9.10) is normal. Moreover, the matrix N (s) dominates in w(s). Proof. Matrix (9.10) can be written in the form ⎤ ⎡ O κI w(p) = diag {I , Iκ , Q(p)} ⎣ F (p) F (p) ⎦ diag {I , G(p)} . F (p) F (p)

(9.16)


352

Each factor on the right-hand side of (9.16) is a normal matrix. This statement is proved similarly to the proof of Lemma 9.3. Moreover, Matrix (9.13) is irreducible, such that the product on the right-hand side of (9.16) is irreducible. Hence Matrix (9.16) is normal. It remains to prove that the matrix N (s) = Q(s)F (s)G(s) dominates in Matrix (9.10). Denote

δQ = deg dQ (s),

δF = deg dF (s),

δG = deg dG (s) .

Then by virtue of Theorem 3.13,

Mdeg N (p) = δQ + δF + δG = χ . On the other hand, using similar considerations, we find for Matrix (9.16) Mdeg w(p) = δQ + δF + δG = χ . Hence Mdeg w(p) = Mdeg N (p) . This equation means that the matrix N (p) dominates in Matrix (9.10). 5.

Let a minimal standard realisation of the matrix N (p) have the form N (p) = C2 (pIχ − A)−1 B2 ,

(9.17)

where the matrix A is cyclic. Then, as follows from Theorem 2.67, the minimal standard realisation of the matrix w(p) can be written in the form C1 (pIχ − A)−1 B1 C1 (pIχ − A)−1 B2 + DL w(p) = , (9.18) C2 (pIχ − A)−1 B1 C2 (pIχ − A)−1 B2 or equivalently, w(p) =

O+κ, DL C1 (pIχ − A)−1 B1 B2 + . C2 On Onm

Equation (9.19) is associated with the state equations dv = Av + B1 x + B2 u dt z = C1 v + DL u ,

y = C2 v .

(9.19)

9.3 Stabilisation

353

9.3 Stabilisation 1. Theorem 9.6. Let Assumptions I–III on page 350 hold. Then the single-loop system shown in Fig. 9.1 is modal controllable (hence, is stabilisable). Proof. Under the given assumptions, the matrix −1 AT DN µ (T, ζ, 0) = DQF Gµ (T, s, 0) | e−sT =ζ = ζC2 Iχ − ζeAT e µ(A)B2

is normal. Thus, in the IMFDs −1 DN µ (T, ζ, 0) = ζa−1 l (ζ)bl (ζ) = ζbr (ζ)ar (ζ) ,

the matrices al (ζ), ar (ζ) are simple and we have det al (ζ) ≈ det ar (ζ) ≈ det Iχ − ζeAT ≈ ∆Q (ζ)∆F (ζ)∆G (ζ) .

(9.20)

(9.21)

In (9.21) and below, ∆R (ζ) denotes the discretisation of the polynomial dR (s) given by (9.11). Thus, in the given case, we have (ζ) =

det(Iχ − ζeAT ) = const. = 0 , det al (ζ)

(9.22)

whence the claim follows. 2. Remark 9.7. In the general case, the assumption on irreducibility of the righthand side of (9.13) is essential. If the right-hand side of (9.13) is reducible by an unstable factor, then the system in Fig. 9.1 is not stabilisable despite the fact that all other assumptions of Theorem 9.6 hold. Example 9.8. Let us have instead of (9.12), the irreducible representations Q(p) =

NQ (p) , dQ (p)

F (p) =

N1F (p) , p d1F (p)

G(p) =

p N1G (p) , dG (p)

(9.23)

where d1F (0) = 0, dQ (0) = 0, dG (0) = 0 and the matrix ˜ (p) = NQ (p)N1F (p)N1G (p) N dQ (p)d1F (p)dG (p) is irreducible. Then, Matrix (9.10) takes the form

(9.24)

354


⎡

⎤ .. . (p) p N 1G ⎢ ⎥ .. O ⎢ ⎥ dG (p) . ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ˜ ˜ ⎢ ⎥ N1F (p)N1G (p) N1F (p) K(p) L(p) ⎥= .. . w(p) ˜ =⎢ ⎢ ⎥ ˜ (p) N ˜ (p) p d1F (p) d1F (p)dG (p) . M ⎢ ⎥ . ⎢ · · · · · · · · · · · · · · · .. · · · · · · · · · · · · · · · · · · · · · ⎥ ⎢ ⎥ . ⎢ ⎥ ⎣ NQ (p)N1F (p) .. NQ (p)N1F (p)N1G (p) ⎦ . p dQ (p)d1F (p) .. dQ (p)d1F (p)dG (p) ˜ (p) is not dominant in the matrix w(p), In this case, the matrix N ˜ because it is analytical for p = 0, while some elements of the matrix w(p) ˜ have poles at p = 0. Then the matrix A in the minimal standard realisation (9.19) will have the eigenvalue zero. At the same time, the matrix A˜ in the minimal representation ˜ (p) = C˜2 (pIχ − A) ˜2 ˜ −1 B N has no eigenvalue zero, because the right-hand side of (9.24) is analytical for p = 0. Therefore, in the given case, (9.22) is not a stable polynomial. Hence the single-loop system with (9.23) is not stabilisable.

9.4 Wiener-Hopf Method 1. Using the results of Chapter 8, we find that in this case, the H2 -problem reduces to the minimisation of a functional of the form (8.84) # dζ 1 ˆ ˆ ˆ J1 = trace θ(ζ)A L (ζ)θ(ζ)AM (ζ) − θ(ζ)C(ζ) − C(ζ)θ(ζ) 2πj ζ over the set of stable rational matrices. The matrices AL (ζ), AM (ζ), C(ζ) and ˆ C(ζ) can be calculated using Formulae (8.85) and (8.86). Then referring to (9.9) and (8.86), we find 1 a ˆr (ζ)DL Lµµ (T, ζ, 0)ar (ζ) T 1 ˆr (ζ) κ 2 DG Gµµ (T, ζ, 0) + DG F F Gµµ (T, ζ, 0) ar (ζ) , = a T

AL (ζ) =

al (ζ) = al (ζ)DQF F Q (T, ζ, 0)ˆ al (ζ) . AM (ζ) = al (ζ)DM M (T, ζ, 0)ˆ Applying (9.9) and (8.78), we obtain

(9.25)

9.5 Factorisation of Quasi-polynomials of Type 1

355

1 1 DL Lµµ (T, ζ, 0)ar (ζ) + al (ζ)DM K Lµ (T, ζ, 0)ar (ζ) T T 1 2 = AM (ζ)βˆ0r (ζ) κ DG Gµµ (T, ζ, 0) + DG F F Gµµ (T, ζ, 0) ar (ζ) + T 1 + al (ζ)DQF F F Gµ (T, ζ, 0)ar (ζ) , T (9.26) 1 2 ˆ C(ζ) = a ˆr (ζ) κ DG Gµµ (T, ζ, 0) + DG F F Gµµ (T, ζ, 0) β0r (ζ)AM (ζ) T 1 ˆr (ζ)DG F F F Q µ (T, ζ, 0)ˆ + a al (ζ) , T

C(ζ) = AM (ζ)βˆ0r (ζ)

where the matrices al (ζ) and ar (ζ) are determined by the IMFDs (9.20). 2. Since under the given assumptions, the single-loop system is modal controllable, all matrices in (9.25) and (9.26) are quasi-polynomials and can have poles only at ζ = 0. Assume the following factorisations ˆ AL (ζ) = Π(ζ)Π(ζ),

AM (ζ) = Γˆ (ζ)Γ (ζ) ,

(9.27)

where the polynomial matrices Π(ζ) and Γ (ζ) are stable. Then, there exists an optimal controller, which can be found using the algorithm described in Chapter 8: a) Calculate the matrix ˆ −1 (ζ)C(ζ) ˆ Γˆ −1 (ζ) . R(ζ) = Π b) Perform the principal separation R(ζ) = R+ (ζ) + R(ζ)− ,

(9.28)

where R+ (ζ) is a polynomial matrix and the matrix R(ζ)− is strictly proper. c) The optimal system function is given by the formula θo (ζ) = Π −1 (ζ)R+ (ζ)Γ −1 (ζ) . d) The transfer matrix of the optimal controller wdo (ζ) is given by (8.90) and (8.91).

9.5 Factorisation of Quasi-polynomials of Type 1 1. One of the fundamental steps in the Wiener-Hopf-method requires to factorise the quasi-polynomial (9.25) according to (9.27). In the present section, we investigate special features of the factorisation of a quasi-polynomial AM (ζ) of type 1 determined by the given assumptions.


356

2.

Let us formulate some auxiliary propositions. Suppose M (s) = Q(s)F (s) =

NM (s) dM (s)

with dM (s) = dQ (s)dF (s) = (s − m1 )µ1 · · · (s − mρ )µρ ,

(9.29)

deg dM (s) = µ1 + . . . + µρ = deg dQ (s) + deg dF (s) = δQ + δF = γ . Let us have a corresponding minimal standard realisation similar to (9.17) M (s) = CM (sIγ − AM )−1 BM , where Iγ is the identity matrix of compatible dimension. As follows from the subordination relations (9.14) and (9.18), M (s) = C2 (sIχ − A)−1 B1 ,

(9.30)

where B1 and C2 are constant matrices. For 0 < t < T , let us have as before −1 AM t DM (T, ζ, t) = CM Iγ − ζeAM T e BM .

(9.31)

Matrix (9.31) is normal for all t. Let us have an ILMFD −1 CM Iγ − ζeAM T = a−1 M (ζ)bM (ζ) .

(9.32)

Then under the given assumptions, the formulae DM (T, ζ, t) = a−1 M (ζ)bM (ζ, t),

bM (ζ, t) = bM (ζ)eAM t BM

(9.33)

determine an ILMFD of Matrix (9.31). Moreover, the matrix aM (ζ) is simple and (9.34) det aM (ζ) ≈ ∆Q (ζ)∆F (ζ) . On the other hand using (9.30), we have −1 At DM (T, ζ, t) = C2 Iχ − ζeAT e B1 .

(9.35)

Consider the ILMFD −1 ˜ C2 Iχ − ζeAT = a−1 l (ζ)bl (ζ) .

(9.36)

The set of matrices al (ζ) satisfying (9.36) coincides with the set of matrices al (ζ) for the ILMFD (9.20), which satisfy Condition (9.21). This fact follows from the minimality of the PMD Iχ − ζeAT , ζeAt B2 , C2 . From (9.35) and (9.36), we obtain an LMFD for the matrix DM (T, ζ, t)


357

˜bl (ζ)eAt B1 = a−1 (ζ)˜bM (ζ, t) . DM (T, ζ, t) = a−1 l l Since Equation (9.33) is an ILMFD, we obtain al (ζ) = a1 (ζ)aM (ζ) ,

(9.37)

where a1 (ζ) is a polynomial matrix. Here the matrix a1 (ζ) is simple and with respect to (9.34) and (9.35), we find det a1 (ζ) = 3.

det al (ζ) ≈ ∆G (ζ) . det aM (ζ)

Consider the sum of the series ∞ 1 DM M (T, ζ, 0) = M (s + kjω)M (−s − kjω) T k=−∞

(9.38)

| e−sT =ζ

and the matrix aM (ζ) . PM (ζ) = aM (ζ)DM M (T, ζ, 0)ˆ

(9.39)

Lemma 9.9. Matrix (9.39) is a symmetric quasi-polynomial matrix of the form (8.88). Proof. Since

DM (T, s, t) =

∞ 1 M (s + kjω)e(s+kjω)t , T k=−∞

∞ 1 DM (T, −s, t) = M (−s + kjω)e(−s+kjω)t , T

(9.40)

k=−∞

we have the equality

T

DM (T, s, t)D M (T, −s, t) dt .

DM M (T, s, 0) =

(9.41)

0

This result can be proved by substituting (9.40) into (9.41) and integrating term-wise. Substituting ζ for e−sT in (9.41), we find DM M (T, ζ, 0) =

T

DM (T, ζ, t)DM (T, ζ −1 , t) dt .

0

As follows from (9.33) for 0 < t < T ,

−1 (T, ζ −1 , t) = bM (ζ −1 , t) aM (ζ −1 ) DM (T, ζ −1 , t) = DM = ˆbM (ζ, t)ˆ a−1 M (ζ) .

(9.42)

358


Then substituting this and (9.33) into (9.42), we receive DM M (T, ζ, 0) = a−1 M (ζ)

T

bM (ζ, t)ˆbM (ζ, t) dt a ˆ−1 M (ζ) .

(9.43)

0

Hence with account for (9.39),

T

bM (ζ, t)ˆbM (ζ, t) dt .

PM (ζ) =

(9.44)

0

Obviously, the right-hand side of (9.44) is a quasi-polynomial matrix and PˆM (ζ) = PM (ζ −1 ) = PM (ζ) ,

i.e., the quasi-polynomial (9.39) is symmetric. 4. The symmetric quasi-polynomial P (ζ) = Pˆ (ζ) of dimension n × n will be called nonnegative (positive) on the unit circle, if for any vector x ∈ C1n and |ζ| = 1, we have x > 0) , xP (ζ)¯ x ≥ 0, (xP (ζ)¯ where the overbar denotes the complex conjugate value [133]. Lemma 9.10. The quasi-polynomial (9.44) is nonnegative on the unit circle. Proof. Since we have ζ −1 = ζ¯ on the unit circle, Equation (9.44) yields

xPM (ζ)¯ x =

T

0

T 2

|xbM (ζ, t)| dt ≥ 0 ,

[xbM (ζ, t)] [xbM (ζ, t) ] dt = 0

where | · | denotes the absolute value of the complex row vector. Corollary 9.11. Since under the given assumptions, the matrix bM (ζ, t) is continuous with respect to t, the quasi-polynomial PM (ζ) is nonnegative on the unit circle, if and only if there exists a constant nonzero row x0 such that x0 bM (ζ, t) = O1n . If such row does not exist, then the quasi-polynomial matrix PM (ζ) is positive on the unit circle. Remark 9.12. In applied problems, the quasi-polynomial matrix PM (ζ) is usually positive on the unit circle.


359

5. Lemma 9.13. Let under the given assumptions the matrix aM (ζ) in the ILMFD (9.32) be row reduced. Then, λ

PM (ζ) =

mk ζ k ,

mk = m−k

(9.45)

k=−λ

with where

0 ≤ λ ≤ ρM − 1 ,

(9.46)

ρM = deg aM (ζ) ≤ deg det aM (ζ) = γ .

(9.47)

Proof. Since the matrix DM (T, ζ, t) is strictly proper for 0 < t < T , due to Corollary 2.23 for the ILMFD (9.33), we have deg bM (ζ, t) < deg aM (ζ),

(0 < t < T ) .

At the same time, since the matrix aM (ζ) is row reduced, Equation (9.47) follows from (9.29). Then using (9.44), we obtain (9.45) and (9.46). 6.

Denote

qM (ζ) = det PM (ζ) . Lemma 9.14. Let the matrix M (s) be normal and the product (−s) NM (s)NM ˜ (s) = M M (s)M (−s) = dM (s)dM (−s)

(9.48)

be irreducible. Let also the roots of the polynomial

gM (s) = dM (s)dM (−s) satisfy the conditions for non-pathological behavior (6.106). Then, qM (ζ) =

ξ

qk ζ k ,

qk = q−k ,

(9.49)

k=−ξ

where the qk are real constants and 0 ≤ ξ ≤ γ − n, where n is the dimension of the vector y in Fig. 9.1.

(9.50)

360


Proof. a) From (9.31) after transposition and substitution of ζ −1 for ζ, we obtain −1 DM (T, ζ −1 , t) = BM eAM t Iγ − ζ −1 eAM T CM . Substituting this and (9.31) into (9.42), we find −1 −1 DM M (T, ζ, 0) = CM Iγ − ζeAM T J Iγ − ζ −1 eAM T CM , (9.51) where

J=

T

eAM t BM BM eAM t dt .

0

From (9.51), it follows that the matrix DM M (T, ζ, 0) is strictly proper and can be written in the form DM M (T, ζ, 0) =

ζK(ζ) , ∆M (ζ)∆M (ζ)

(9.52)

where µ1 µρ ∆M (ζ) = ζ − e−m1 T · · · ζ − e−mρ T , µ1 µρ ∆M (ζ) = ζ − em1 T · · · ζ − emρ T ,

(9.53)

and K(ζ) is a polynomial matrix. b) Let us show that Matrix (9.52) is normal. Indeed, using (9.48), we can write DM M (T, ζ, 0) = DM˜ (T, ζ, 0) . Since the matrix M (s) is normal, the matrix M (s) = M (−s) is normal. ˜ (s) is also normal as a product of irreducible normal matrices. Hence M ˜ (s) satisfy Condition (6.106), the normality Moreover, since the poles of M of Matrix (9.52) follows from Corollary 6.25. c) Since Matrix (9.52) is strictly proper and normal, using (2.92) and (2.93), we have ζ n uM (ζ) , (9.54) fM (ζ) = det DM M (T, ζ, 0) = ∆M (ζ)∆M (ζ) where uM (ζ) is a polynomial, such that deg uM (ζ) ≤ 2γ −2n. From (9.53), we find (ρ (9.55) ∆M (ζ) = (−1)γ eT i=1 mi µi ∆M (ζ −1 )ζ γ . Substituting (9.55) into (9.54), we obtain fM (ζ) =

ψM (ζ) , ∆M (ζ)∆M (ζ −1 )

(9.56)

where ψM (ζ) is a quasi-polynomial of the form ψM (ζ) = (−1)γ e−T

(ρ i=1

mi µi (n−γ)

ζ

uM (ζ) .

(9.57)


361

Substituting ζ −1 for ζ in (9.56) and using the fact that fM (ζ) = fM (ζ −1 ), we obtain (9.58) ψM (ζ) = ψM (ζ −1 ) , i.e., the quasi-polynomial (9.57) is symmetric. Moreover, the product ζ γ−n ψ(ζ) is a polynomial due to (9.57). Calculating the determinants of both sides on (9.43), we find fM (ζ) =

qM (ζ) . det aM (ζ) det aM (ζ −1 )

Since det aM (ζ) = λ∆M (ζ) ,

det aM (ζ −1 ) = λ∆M (ζ −1 )

with λ = const. = 0, a comparison of (9.56) with (9.58) yields ψ(ζ) = νqM (ζ) with ν = const. = 0. Therefore, the product ζ γ−n qM (ζ) is a polynomial as (9.49) and (9.50) claim.

7. Next, we provide some important properties of the quasi-polynomial matrix AM (ζ) in (9.25). Theorem 9.15. Let Assumptions I-III on page 350 hold and let us have the ILMFD (9.36) −1 ˜ = a−1 (9.59) C2 Iχ − ζeAT l (ζ)bl (ζ) . Then, the matrices al (ζ) in (9.59) and aM (ζ) in the ILMFD (9.32) can be chosen in such a way that the following equality holds: al (ζ) = AM (ζ) = al (ζ)DM M (T, ζ, 0)ˆ

η

ak ζ k ,

ak = a−k ,

(9.60)

k=−η

where 0≤η ≤χ−1

(9.61)

and χ = deg dN (s) = deg dQ (s) + deg dF (s) + deg dG (s) . Proof. As was proved above, the sets of matrices al (ζ) from (9.59) and (9.20) coincide. Taking into account (9.37) and (9.43), we rewrite the matrix AM (ζ) in the form a1 (ζ) , (9.62) AM (ζ) = a1 (ζ)PM (ζ)ˆ where PM (ζ) is the quasi-polynomial (9.39). Using (9.44) from (9.62), we obtain

362


T

AM (ζ) =

[a1 (ζ)bM (ζ, t)] [a1 (ζ)bM (ζ, t)]' dt .

(9.63)

0

Let the matrix aM (ζ) be row reduced. Then we have deg bM (ζ, t) < deg aM (ζ) ≤ deg det aM (ζ) = γ .

(9.64)

Moreover, if we have the ILMFD (9.59), any pair (ξ(ζ)al (ζ), ξ(ζ)˜bl (ζ)) with any unimodular matrix ξ(ζ) is also an ILMFD for Matrix (9.59). As a special case, the matrix ξ(ζ) can be chosen in such a way, that the matrix a1 (ζ) in (9.37) is row reduced. Then with respect to (9.38), we obtain deg a1 (ζ) ≤ deg det a1 (ζ) = deg ∆G (ζ) = δG . If this and (9.64) hold, we have deg [a1 (ζ)bM (ζ, t)] ≤ δM + δG − 1 = χ − 1 . From this and (9.63), the validity of (9.60) and (9.61) follows. Theorem 9.16. Denote

rM (ζ) = det AM (ζ) . Under Assumptions I-III on page 350, we have rM (ζ) =

σ

rk ζ k ,

rk = r−k

(9.65)

k=−σ

with 0 ≤ σ ≤ χ − n.

(9.66)

Proof. From (9.62), we have rM (ζ) = det a1 (ζ) det a1 (ζ −1 ) det PM (ζ) = det a1 (ζ) det a1 (ζ −1 )qM (ζ) . (9.67) Then with regard to (9.49) and (9.50), rM (ζ) = det a1 (ζ) det a1 (ζ −1 )

ξ

qk ζ k ,

k=−ξ

which is equivalent to (9.65) and (9.66), because deg det a1 (ζ) = δG , 0 ≤ ξ ≤ γ − n and δG + γ = χ. Corollary 9.17. As follows from (9.67), the set of zeros of the function rM (ζ) includes the set of roots of the polynomial ∆G (ζ) as well as those of the quasipolynomial ∆G (ζ −1 ).


363

8. Using the above auxiliary relations, we can formulate an important proposition about the factorisation of quasi-polynomials of type 1. Theorem 9.18. Let Assumptions I-III on page 350 and the propositions of Lemma 9.14 hold. Let also the quasi-polynomial AM (ζ) be positive on the unit circle. Then, there exists a factorisation AM (ζ) = Γ (ζ)Γˆ (ζ) = Γ (ζ)Γ (ζ −1 ) ,

(9.68)

where Γ (ζ) is a stable real n × n polynomial matrix. Under these conditions, there exists a factorisation + + rM (ζ) = det AM (ζ) = rM (ζ)rM (ζ −1 ) ,

(9.69)

+ + where rM (ζ) is a real stable polynomial with deg rM (ζ) ≤ χ − n. Moreover, + det Γ (ζ) ≈ rM (ζ)

and the matrices al (ζ), aM (ζ) in the ILMFDs (9.32), (9.59) can be chosen in such a way that deg Γ (ζ) ≤ χ − 1 . Proof. With respect to the above results, the proof is a direct corollary of the general theorem about factorisation given in [133]. Remark 9.19. As follows from Corollary 9.17, if the polynomial dG (s) has roots on the imaginary axis, then the quasi-polynomial rM (ζ) has roots on the unit circle. In this case, the factorisations (9.68) and (9.69) are impossible. Remark 9.20. Let the polynomial dG (s) = (s − g1 )λ1 · · · (s − gσ )λσ be free of roots on the imaginary axis. Let also be Re gi < 0,

(i = 1, . . . , κ); Re gi > 0,

(i = κ + 1, . . . , σ) .

+ Then the polynomial rM (ζ) can be represented in the form + + rM (ζ) = ∆+ G (ζ)r1M (ζ) , + where ∆+ G (ζ), r1M (ζ) are stable polynomials and

λκ λκ+1 λσ −g1 T λ1 ∆+ ζ − egκ+1 T · · · ζ − e−gκ T · · · ζ − eg σ T , G (ζ) = ζ − e i.e., the numbers e−gi T , (i = 1, . . . , κ) and egi T , (i = κ + 1, . . . , σ) are found + among the roots of the polynomial rM (ζ).


364

9.6 Factorisation of Quasi-polynomials of Type 2 Let the matrix L(s) be at least proper and have the standard form

1.

L(s) =

NL (s) , dL (s)

Mdeg L(s) = β ,

where dL (s) = (s − λ1 )η1 · · · (s − λm )ηm ,

η1 + . . . + ηm = β

with the minimal standard representation L(s) = CL (sIβ − AL )−1 BL + DL . Let also

T

e−sτ m(τ ) dτ

µ(s) = 0

be the transfer function of the forming element. Then for 0 < t < T from (6.90), (6.100) and (6.103), we have

DLµ (T, s, t) =

∞ 1 L(s + kjω)µ(s + kjω)e(s+kjω)t T k=−∞

=

CL h∗µ (AL , t)BL

+ CL µ(AL )e

where h∗µ (AL , t) = −

AL t

Iβ − e

−sT AL T −1 e

(9.70) BL + DL m(t) ,

T

eAL (t−τ ) m(τ ) dτ . t

Replacing e−sT = ζ in (9.70), we find the rational matrix

DLµ (T, ζ, t) = DLµ (T, s, t) | e−sT =ζ = CL µ(AL )eAL t wL (ζ) + DL (t) , where and

(9.71)

−1 wL (ζ) = Iβ − ζeAL T BL ,

(9.72)

DL (t) = CL h∗µ (AL , t)BL + DL m(t)

(9.73)

is a matrix independent of ζ. Let us have an IRMFD −1 wL (ζ) = Iβ − ζeAL T BL = bL (ζ)a−1 L (ζ) .

(9.74)

Then using (9.71), we find an RMFD DLµ (T, ζ, t) = bL (ζ, t)a−1 L (ζ) ,

(9.75)


365

where the matrix bL (ζ, t) = CL eAL t µ(AL )bL (ζ) + DL (t)aL (ζ) is a polynomial in ζ for all t. When Assumptions I-III on page 350 hold, then the matrix aL (ζ) is simple and we have det aL (ζ) ≈ ∆F (ζ)∆G (ζ) . 2.

Consider the sum of the series ∞ 1 L (−s − kjω)L(s + kjω)µ(s + kjω)µ(−s − kjω) T

DL Lµµ (T, s, 0) =

k=−∞

and the rational matrices

and

DL Lµµ (T, ζ, 0) = DL Lµµ (T, s, 0) | e−sT =ζ

(9.76)

PL (ζ) = aL (ζ −1 )DL Lµµ (T, ζ, 0)aL (ζ) .

(9.77)

Let us formulate a number of propositions determining some properties of the matrices (9.76) and (9.77) required below. Lemma 9.21. Matrix (9.77) is a symmetric quasi-polynomial. Proof. Since

DL µ (T, −s, t) =

∞ 1 L (−s + kjω)µ(−s + kjω)e(−s+kjω)t T k=−∞

regarding (9.70) after integration, we find

T

DL µ (T, −s, t)D Lµ (T, s, t) dt

0

∞ 1 L (−s − kjω)L(s + kjω)µ(−s − kjω)µ(s + kjω) = T k=−∞

= DL Lµµ (T, s, 0) . Substituting ζ for e−sT , we find

T

DL Lµµ (T, ζ, 0) = 0

Nevertheless from (9.75), we have

DL µ (T, ζ −1 , t)DLµ (T, ζ, t) dt .

(9.78)

366

9 H2 Optimisation of a Single-loop System DL µ (T, ζ −1 , t) = DLµ (T, ζ −1 , t)

−1 −1 −1 ˆ = aL (ζ ) bL (ζ , t) = a ˆ−1 L (ζ)bL (ζ, t) .

(9.79)

Using (9.75) and (9.79) in (9.78), we find DL Lµµ (T, ζ, 0) = a ˆ−1 L (ζ)

T

ˆbL (ζ, t)bL (ζ, t) dt a−1 (ζ) . L

0

Hence

T

ˆbL (ζ, t)bL (ζ, t) dt

PL (ζ) =

(9.80)

0

is a symmetric quasi-polynomial. Lemma 9.22. The quasi-polynomial PL (ζ) is nonnegative on the unit circle. Proof. The proof is similar to that given for Lemma 9.10. Lemma 9.23. Let Assumptions I-III on page 350 hold and the matrix aL (ζ) from the IRMFD (9.74) be column reduced. Then, PL (ζ) =

ν

k ζ k ,

k = −k

k=−ν

with 0 ≤ ν ≤ ρL , where

ρL = deg aL (ζ) ≤ deg det aL (ζ) = β . Proof. Under the given assumptions, the matrix wL (ζ) is normal. Hence deg det aL (ζ) = deg ∆L (ζ) = β = δF + δG and since the matrix aL (ζ) is column reduced, we have deg aL (ζ) ≤ β . Moreover, since Matrix (9.71) is at least proper, due to Corollary 2.23, we obtain deg bL (ζ, t) ≤ deg aL (ζ) ≤ β . The claim of the lemma follows from (9.80) and the last relations.


3.

367

Introduce the following additional notations

qL (ζ) = det PL (ζ) and

µ ˜(s) = µ(s)µ(−s) . Lemma 9.24. Let Assumptions I-III on page 350 hold and the product N (s)NL (s) ˜ L(s) = L (−s)L(s) = L dL (s)dL (−s)

(9.81)

˜1, . . . , λ ˜ ρ of the polynomial be irreducible. Let also the roots λ d˜L (s) = dL (s)dL (−s)

satisfy the conditions for non-pathological behavior (6.106) and moreover, ˜ i ) = 0, µ ˜(λ

(i = 1, . . . , ρ) .

(9.82)

Then, qL (ζ) =

ν ˜

q˜k ζ k ,

k=−˜ ν

where the q˜k =

q˜ −k

are real constants and 0 ≤ ν˜ ≤ β .

Proof. a) First of all, we show that the rational matrix DL Lµµ (T, ζ, 0) is at least proper. With this aim in view, recall that (9.71) yields

DL µ (T, ζ −1 , t) = wL (ζ −1 )µ (AL )eAL t CL + DL (t) .

Using (9.72) and (9.73), it can be easily established that this matrix is at least proper. Then the product DL µ (T, ζ −1 , t)DLµ (T, ζ, t) is also at least proper. Further from (9.78), it follows that the matrix DL Lµµ (T, ζ, 0) is at least proper. Then, if dL (s) = (s − λ1 )η1 · · · (s − λm )ηm , then DL Lµµ (T, ζ, 0) = with

η1 + . . . + ηM = β ,

L(ζ) ∆L (ζ)∆L (ζ)

η1 ηm ∆L (ζ) = ζ − e−λ1 T · · · ζ − e−λm T , η1 ηm · · · ζ − eλm T , ∆L (ζ) = ζ − eλ1 T

where L(ζ) is a polynomial matrix, such that deg L(ζ) ≤ 2β.

(9.83)

(9.84)


368

b) Let us show that Matrix (9.83) is normal. For this purpose using (9.81), we write

DL Lµµ (T, s, 0) = DLµµ (T, s, 0) ˜ ∞ 1 ˜ L(s + kjω)µ(s + kjω)µ(−s − kjω) . = T

(9.85)

k=−∞

Using the fact that

T

esτ m(τ ) dτ

µ(−s) = 0

from (9.85), we can derive

T ∞ 1 ˜ L(s + kjω)µ(s + kjω) e(s+kjω)τ m(τ ) dτ T 0 k=−∞ ∞ 1 ˜ (s+kjω)τ L(s + kjω)µ(s + kjω)e m(τ ) dτ (9.86) T

DL Lµµ (T, s, 0) = = 0

=

T

k=−∞

T

DLµ ˜ (T, s, τ )m(τ ) dτ . 0

Under the given assumptions, the matrix L (−s) is normal. Hence Matrix (9.81) is also normal as a product of irreducible normal matrices. ˜ Therefore, the minimal standard representation of the matrix L(s) can be written in the form −1 ˜L + D ˜ ˜L . B L(s) = C˜L sI2β − A˜L Then similarly to (9.70)–(9.73) for 0 < t < T , we obtain −1 ˜L + D ˜ A˜L t µ(A˜L ) sI2β − e−sT eA˜L T ˜ L (t) , (9.87) B DLµ ˜ (T, s, t) = CL e where

˜ L (t) = C˜2 h∗ (A˜L , t)B ˜L + D ˜ L m(t) . D µ

Using (9.87) in (9.86), after integration and substitution e−sT = ζ, we obtain −1 ˜ ˜L B DL Lµµ (T, ζ, 0) = C˜L µ(−A˜L )µ(A˜L ) I2β − ζeAL T T ˜ L (t)m(t) dt , D + 0

where


T

µ(−A˜L ) =

369

˜

eAL t m(t) dt . 0

˜ Under the given assumptions, the matrices A˜L and eAL T are cyclic, the ˜ ˜ ˜L ) is controllable and the pair [eAL T , C˜L ] is observable. Morepair (eAL T , B over, the matrix µ ˜(A˜L ) = µ(−A˜L )µ(A˜L ) ˜

is commutative with the matrix eAL and due to (9.82), it is nonsingular. Therefore, Matrix (9.83) is normal. c) With respect to the normality of Matrix (9.83), calculating the determinants on both sides of (9.83), we find

fL (ζ) = det DL Lµµ (T, ζ, 0) =

uL (ζ) , ∆L (ζ)∆L (ζ)

(9.88)

where ∆L (ζ), ∆L (ζ) are the polynomials (9.84), and uL (ζ) is a polynomial with deg uL (ζ) ≤ 2β. Per construction, we have fL (ζ) = fL (ζ −1 ). Therefore, similarly to (9.54) from (9.88), we obtain fL (ζ) = where

ψL (ζ) , ∆L (ζ)∆L (ζ −1 )

ψL (ζ) = (−1)β e−T

(n

i=1

λi ηi

(9.89)

ζ −β uL (ζ)

is a symmetric quasi-polynomial. Moreover, since the product ψL (ζ)ζ β is a polynomial, we receive ψL (ζ) =

ν ˜

ψk ζ k ,

ψk = ψ−k ,

k=−˜ ν

where 0 ≤ ν˜ ≤ β . Using (9.89), (9.77) and the relations det aL (ζ) ≈ ∆L (ζ),

det aL (ζ −1 ) = det aL (ζ −1 ) ,

we obtain the equality qL (ζ) = det aL (ζ) det aL (ζ −1 )fL (ζ) = kL ψL (ζ), This completes the proof.

kL = const.


370

4. Using the above auxiliary results under Assumptions I-III on page 350, we consider some properties of the quasi-polynomial AL (ζ). Using a minimal standard representation (9.18), introduce the matrix −1 B2 (9.90) w ˜L (ζ) = Iχ − ζeAT and an arbitrary IRMFD w ˜L (ζ) = ˜bL (ζ)a−1 r (ζ) .

(9.91)

Since Matrix (9.90) is normal, the matrix ar (ζ) is simple and det ar (ζ) ≈ det Iχ − ζeAT ≈ ∆Q (ζ)∆F (ζ)∆G (ζ) . From (9.18), it follows that the representation L(s) = C1 (sIχ − A)−1 B2 + DL exists. Hence together with (9.71), we have −1 B2 + m(t)DL . DLµ (T, ζ, t) = C1 h∗µ (A, t)B2 + C1 µ(A)eAt Iχ − ζeAT Thus with account for (9.91), we obtain the RMFD DLµ (T, ζ, t) = ˜bL (ζ, t)a−1 r (ζ) ,

(9.92)

where ˜bL (ζ, t) = C1 h∗ (A, t)B2 ar (ζ) + C1 µ(A)eAt˜bL (ζ) + m(t)DL ar (ζ) . µ Simultaneously with the RMFD (9.92), we have the IRMFD (9.75), therefore ar (ζ) = aL (ζ)a2 (ζ) .

(9.93)

Moreover, the polynomial matrix a2 (ζ) is simple and det a2 (ζ) =

det ar (ζ) ≈ ∆Q (ζ) . det aL (ζ)

(9.94)

Theorem 9.25. The set of matrices ar (ζ) in the IRMFD (9.91) coincides with the set of matrices ar (ζ) in the IRMFD (9.20). Moreover, the matrices ar (ζ) in (9.91) and aL (ζ) in the IRMFD (9.74) can be chosen in such a way, that the following representation holds: AL (ζ) =

ν 1 a ˆr (ζ)DL Lµµ (T, ζ, 0)ar (ζ) = ak ζ k , T

ak = a−k ,

(9.95)

k=−ν

where 0 ≤ ν ≤ χ.

(9.96)


371

Proof. The coincidence of the sets of matrices ar (ζ) in (9.80) and (9.91) stems from the minimality of the PMD Iχ − ζeAT , eAt µ(A)B2 , C2 . Using (9.79), (9.80) and (9.93), the matrix AL (ζ) can be written in the form AL (ζ) = a ˆ2 (ζ)PL (ζ)a2 (ζ) ,

(9.97)

where PL (ζ) is the quasi-polynomial matrix (9.77). Using (9.80) from (9.97), we find 1 T AL (ζ) = t)a2 (ζ)] [bL (ζ, t)a2 (ζ)] dt . (9.98) [bL (ζ, T 0 Let the matrix aL (ζ) be column reduced. Then as before, we have deg bL (ζ, t) ≤ deg aL (ζ) ≤ deg det al (ζ) = β . Moreover, if we have the IRMFD (9.91), then any pair [ar (ζ)φ(ζ), ˜bL (ζ)φ(ζ)], where φ(ζ) is any unimodular matrix, determines an IRMFD for Matrix (9.91). In particular, the matrix φ(ζ) can be chosen in such a way that the matrix a2 (ζ) in (9.93) becomes column reduced. In this case, we receive deg a2 (ζ) ≤ deg det a2 (ζ) = deg ∆Q (ζ) = δq . The last two estimates yield deg [bL (ζ, t)a2 (ζ)] ≤ β + δq = χ . Equations (9.95) and (9.96) follow from (9.98) and the last estimate. Theorem 9.26. Denote

rL (ζ) = det AL (ζ) . Then under Assumptions I-III on page 350 and the conditions of Lemma 9.23, we have κ rL (ζ) = r˜k ζ k , r˜k = r˜ −k , k=−κ

where 0 ≤ κ ≤ χ. Proof. From (9.97), we have rL (ζ) = δ det a2 (ζ) det a2 (ζ −1 )qL (ζ),

δ = const. = 0

(9.99)

that is equivalent to the claim, because deg det a2 (ζ) = δQ and δQ + β = χ. Corollary 9.27. From (9.99), it follows that the set of roots of the function rL (ζ) includes the set of roots of the polynomial ∆Q (ζ) and the set of roots of the quasi-polynomial ∆Q (ζ −1 ).

372


5. Using the above results, we prove a theorem about factorisation of quasipolynomials of type 2. Theorem 9.28. Let Assumptions I-III on page 350 and the conditions of Lemmata 9.23 and 9.24 hold. Let also the quasi-polynomial AL (ζ) be positive on the unit circle. Then, there exists a factorisation ˆ AL (ζ) = Π(ζ)Π(ζ) = Π (ζ −1 )Π(ζ) ,

(9.100)

where Π(ζ) is a stable polynomial matrix. Under the same conditions, the following factorisation is possible: + + −1 rL (ζ) = det AL (ζ) = rL (ζ)rL (ζ ) ,

(9.101)

+ + where rL (ζ) is a real stable polynomial with deg rL (ζ) ≤ χ. Moreover, + det Π(ζ) ≈ rL (ζ)

and the matrices ar (ζ), aL (ζ) in the IRMFDs (9.91), (9.74) can be chosen, such that deg Π(ζ) ≤ χ . Proof. As for Theorem 9.16, the proof is a direct corollary of the theorem about factorisation from [133] with account for our auxiliary results. Remark 9.29. From Corollary 9.27, it follows that, when the polynomial dQ (s) has roots on the imaginary axis, then the quasi-polynomial (9.99) has roots on the unit circle and the factorisations (9.100) and (9.101) are impossible. Remark 9.30. Let the polynomial dQ (s) = (s − q1 )δ1 · · · (s − qλ )δλ ,

δ1 + . . . , δλ = δQ

be free of roots on the imaginary axis. Let also be Re qi < 0 , (i = 1, . . . , m);

Re qi > 0 , (i = m + 1, . . . , λ) .

+ Then the polynomial rL (ζ) can be represented in the form + + rL (ζ) = d+ Q (ζ)r1L (ζ) , + where d+ Q (ζ) and r1L (ζ) are stable polynomials and

δm δm+1 δλ −q1 T δ1 d+ ζ − eqm+1 T · · · ζ − e−qm T · · · ζ − eqλ T , Q (ζ) = ζ − e i.e., the numbers e−qi T , (i = 1, . . . , m) and eqi T , (i = m + 1, . . . , λ) are found + (ζ). among the roots of the polynomial rL

9.7 Characteristic Properties of Solution for Single-loop System

373

9.7 Characteristic Properties of Solution for Single-loop System 1. Using the above results, we can formulate some characteristic properties of the solution to the H2 -problem for the single-loop system shown in Fig. 9.1. We assume that Assumptions I-III on page 350 and the conditions of Lemmata 9.14, 9.24 hold. 2. Let the quasi-polynomials AM (ζ) and AL (ζ) be positive on the unit circle. Then, there exist factorisations (9.68) and (9.100). Moreover, the optimal system matrix θo (ζ) has the form θo (ζ) = Π −1 (ζ)R+ (ζ)Γ −1 (ζ) , where R+ (ζ) is a polynomial matrix and the relations deg det Π(ζ) ≤ χ and deg det Γ (ζ) ≤ χ − n hold. Due to Lemma 2.8, there exists an LMFD R+ (ζ)Γ −1 (ζ) = Γ1−1 (ζ)R1+ (ζ) with deg det Γ1 (ζ) = deg det Γ (ζ) ≤ χ − n. From the last two equations, we obtain the LMFD −1 θo (ζ) = [Γ1 (ζ)Π(ζ)] R1+ (ζ) , where deg det[Γ1 (ζ)Π(ζ)] ≤ 2χ − n. On the other hand, let us have an ILMFD θo (ζ) = Dl−1 (ζ)Ml (ζ) . Then the function det Γ1 (ζ) det Π(ζ) det Γ (ζ) det Π(ζ) = det Dl (ζ) det Dl (ζ) is a polynomial. Since the system under consideration is modal controllable, due to the properties of the system function, the polynomial det Dl (ζ) is equivalent to the characteristic polynomial of the optimal system ∆o (ζ). Then we obtain deg ∆o (ζ) = deg det Dl (ζ) ≤ 2χ − n . 3. Let g1 , . . . , gκ be the stable and gκ+1 , . . . , gσ the unstable poles of the matrix G(s); and q1 , . . . , qm ; qm+1 , . . . , qλ be the corresponding sequences of poles of the matrix Q(s). Then the characteristic polynomial has in the general case its roots at the points ζ1 = e−g1 T , . . . , ζκ = e−gκ T ; ζκ+1 = egκ+1 T , . . . , ζσ = egσ T ; and ζ˜1 = e−q1 T , . . . , ζ˜m = e−qm T ; ζ˜m+1 = eqm+1 T , . . . , ζ˜λ = eqλ T .

374


4. The single-loop system shown in Fig. 9.1 will be called critical, if at least one of the matrices Q(s), F (s) or G(s) has poles on the imaginary axis. These poles will also be called critical. The following important conclusions stem from the above reasoning. a) The presence of critical poles of the matrix F (s) does not change the H2 -optimisation procedure. b) If any of the matrices Q(s) or G(s) has a critical pole, then the corresponding factorisations (9.100) or (9.68) appear to be impossible, because at least one of the polynomials det a2 (ζ) or det a1 (ζ) has roots on the unit circle. In this case, formal following the Wiener-Hopf procedure leads to a controller that does not stabilise. c) As follows from the aforesaid, for solving the H2 -optimisation problems for sampled-data systems with critical continuous-time elements, it is necessary to take into account some special features of the system structure, as well as the placement of the critical elements with respect to the system input and output.

9.8 Simplified Method for Elementary System 1. In principle for the H2 -optimisation of the single-loop structure, we can use the modified Wiener-Hopf-method described in Section 8.6. However in some special cases, a simplified optimisation procedure can be used that does not need the inversion of the matrices bl (ζ)bl (ζ −1 ) or br (ζ −1 )br (ζ). In this section, such a possibility is illustrated by the example shown in Fig. 9.3, where F (s) ∈ Rnm (s). Hereinafter, such a system will be called elementary. x

u

y

- g - F (s) 6

κ

u1 •

C

•

-

Fig. 9.3. Simplified sampled-data system

The elementary system is a special case of the single-loop system in Fig. 9.1, when Q(s) = In and G(s) = Im . In this case from (9.9), we have

κIm Omm , L(s) = , K(s) = F (s) F (s) (9.102) M (s) = F (s), N (s) = F (s) . It is assumed that the matrix

9.8 Simplified Method for Elementary System

F (s) =

375

NF (s) dF (s)

with dF (s) = (s − f1 )ν1 · · · (s − fψ )νψ ,

ν1 + . . . + νψ = χ

(9.103)

is strictly proper and normal. Moreover, the fractions F (s)F (−s) =

NF (s)NF (−s) , dF (s)dF (−s)

F (−s)F (s) =

NF (−s)NF (s) dF (s)dF (−s)

are assumed to be irreducible. If

g(s) = dF (s)dF (−s) = (s − g1 )κ1 · · · (s − gρ )κρ , we shall assume that µ ˜(gi) = µ(gi )µ(−gi ) = 0,

(i = 1, . . . , ρ)

and the set of numbers gi satisfy Conditions (6.106). 2. To solve the H2 -optimisation problem, we apply the general relations of Chapter 8. Hereby, Equation (9.102) leads to a number of serious simplifications. Using (9.102), we have L (−s)L(s) = κ 2 Im + F (−s)F (s) . Then,

DL Lµµ (T, s, 0) = κ 2

∞ 1 µ(s + kjω)µ(−s − kjω) + DF F µµ (T, s, 0) . T k=−∞

This series can be easily summarised. Indeed, using (6.36) and (6.39), we obtain T ∞ ∞ 1 1 µ(s + kjω)µ(−s − kjω) = µ(s + kjω) e(s+kjω)t m(t) dt T T 0 k=−∞ k=−∞ T T ∞ 1 (s+kjω)t = µ(s + kjω)e m2 (t) dt = m2 . m(t) dt = T 0 0 k=−∞

Hence

DL Lµµ (T, s, 0) = κ 2 m2 + DF F µµ (T, s, 0) . Let us have a minimal standard realisation


376

F (s) = C(sIχ − A)−1 B and IMFDs −1 C Iχ − ζeAT = a−1 l (ζ)bl (ζ) , −1 Iχ − ζeAT B = br (ζ)a−1 r (ζ) , where ν1 νψ det al (ζ) ≈ det ar (ζ) ≈ ζ − e−f1 T · · · ζ − e−fψ T = ∆F (ζ) . In general, the determinant of the matrix 1 AL (ζ) = a ˆr (ζ) DL Lµµ (T, ζ, 0)ar (ζ) T =

1 2 1 κ m2 a ˆr (ζ)ar (ζ) + a ˆr (ζ) DF F µµ (T, ζ, 0)ar (ζ) T T

(9.104)

ˆr (ζ), because these will not vanish at the roots of the function det ar (ζ) det a roots are cancelled in the second summand. Similarly, the determinant of the quasi-polynomial al (ζ) AM (ζ) = al (ζ)DF F (T, ζ, 0)ˆ

(9.105)

is not zero at these points due to cancellations on the right-hand side. 3. Theorem 9.31. Let the above formulated assumptions hold in this section. Let the quasi-polynomials (9.104) and (9.105) be positive on the unit circle, so that there exist factorisations (9.68) and (9.100). Let also the set of eigenvalues of the matrices Π(ζ) and Γ (ζ) does not include the numbers ζi± = e±fi T , where fi are the roots of the polynomial (9.103). Then the following propositions hold: a) The matrix R2 (ζ) =

1 ˆ −1 Π (ζ)ˆ ar (ζ)DF F F µ (T, ζ, 0)ˆ al (ζ)Γˆ −1 (ζ) T

(9.106)

admits a unique separation R2 (ζ) = R21 (ζ) + R22 (ζ) ,

(9.107)

where R22 (ζ) is a strictly proper rational matrix having only unstable poles; it is analytical at the points ζi− = e−fi T . Moreover, R21 (ζ) is a rational matrix having its poles at the points ζi− .

9.8 Simplified Method for Elementary System

377

b) The transfer function of the optimal controller wdo (ζ) is given by −1 wdo (ζ) = V2o (ζ)V1o (ζ) ,

(9.108)

where −1 V1o (ζ) = a−1 (ζ)R21 (ζ)Γ −1 (ζ) , l (ζ) − br (ζ)Π

V2o (ζ) = −ar (ζ)Π −1 (ζ)R21 (ζ)Γ −1 (ζ) .

(9.109)

c) The matrices (9.109) are stable and analytical at the points ζi− , and the set of their poles is included in the set of poles of the matrices Π −1 (ζ) and Γ −1 (ζ). d) The characteristic polynomial of the optimal system ∆o (ζ) is a divisor of the polynomial det Π(ζ) det Γ (ζ). Proof. Applying (8.105)–(8.110) to the case under consideration, we have

where

ˆ −1 (ζ)C(ζ) ˆ Γˆ −1 (ζ) = R1 (ζ) + R2 (ζ) , R(ζ) = Π

(9.110)

R1 (ζ) = Π(ζ)a−1 r (ζ)β0r (ζ)Γ (ζ)

(9.111)

and the matrix R2 (ζ) is given by (9.106). Under the given assumptions owing ˆ to Remark 8.19, the matrix C(ζ) is a quasi-polynomial. Therefore, the matrix R(ζ) can have unstable poles only at the point ζ = 0 and at the poles of ˆ −1 (ζ) and Γˆ −1 (ζ). Hence under the given assumptions, Matrix the matrices Π (9.110) is analytical at the points ζi− = e−fi T . Simultaneously, all nonzero poles of the matrix al (ζ) a ˆr (ζ)DF F F µ (T, ζ, 0)ˆ belong to the set of the numbers ζi− , because the remaining poles are cancelled ˆl (ζ). Then it follows immediately that Matrix against the factors a ˆr (ζ) and a (9.106) admits a unique separation (9.107). Using (9.111) and (9.107) from (9.110), we obtain

R(ζ) = Π(ζ)a−1 (9.112) r (ζ)β0η (ζ)Γ (ζ) + R21 (ζ) + R22 (ζ) . Per construction, R22 (ζ) is a strictly proper rational matrix, whose poles include all poles of the matrix R(ζ), which are all unstable. Also per construction, the expression in the square brackets can have poles at the points ζi− . But under the given assumptions, the matrix R(ζ) is analytical at these points. Hence the matrix in the square brackets in (9.112) is a polynomial. Then the right-hand side of (9.112) coincides with the principal separation (9.28) and from (8.115), we obtain R+ (ζ) = Π(ζ)a−1 r (ζ)β0r (ζ)Γ (ζ) + R21 (ζ) , R− (ζ) = R(ζ)− = R22 (ζ) ,

R21 (ζ) = Λ(ζ) .


378

Using (8.95), we find the optimal system matrix −1 (ζ)R21 (ζ)Γ −1 (ζ) , θo (ζ) = a−1 r (ζ)β0r (ζ) + Π

which is stable and analytical at the points ζi− . Therefore, the matrices (9.109) calculated by (8.117)–(8.118) are stable and analytical at the points ζi− . The remaining claims of the theorem follow from the constructions of Section 8.7. Example 9.32. Consider the simple SISO system shown in Fig. 9.4 with x

u

κ

y

K -g s−a 6 u1 C

•

•

-

Fig. 9.4. Example of elementary sampled-data system

K , s−a

F (s) =

where K and a are constants. Moreover, assume that x(t) is unit white noise. It is required to find the transfer function of a discrete controller wdo (ζ), which stabilises the closed-loop system and minimises the value S22 = κ 2 d¯u1 + d¯y . In the given case from (6.72) and (6.86), it follows that

DF (T, s, t) =

Keat , 1 − eaT e−sT

0 0 and ν are constants, such that |ν| < 1. As follows from (9.113) and (9.78), a ˆl (ζ) = a ˆr (ζ) = 1 − ζ −1 eaT =

ζ − eaT . ζ

(9.117)

Then using (8.85), (9.115) and (9.116), we find AM (ζ) = γ,

AL (ζ) = K1 (1 − ζν)(1 − ζ −1 ν) ,

where K1 > 0 is a constant. Thus, we obtain that in the factorisations (9.68), (9.100), we can take Γ (ζ) = η1 ,

Π(ζ) = η2 (1 − ζν) ,

(9.118)

where η1 and η2 are real constants. For further calculations, we notice that in the given case, Formulae (6.92) and (6.93) yield DF F F µ (T, ζ, 0) =

ζ(2 + 1 ζ + 0 ζ 2 ) (1 − ζeaT )2 (1 − ζe−aT )

(9.119)

with constants 0 , 1 , and 2 . Since DF F F µ (T, ζ, 0) = DF F F µ (T, ζ −1 , 0) , from (9.119), we find DF F F µ (T, ζ, 0) =

0 + 1 ζ + 2 ζ 2 . (ζ − eaT )2 (ζ − e−aT )

Hence using (9.117), we obtain 1 eaT 0 + 1 ζ + 2 ζ 2 a ˆr (ζ)DF F F µ (T, ζ, 0)ˆ . al (ζ) = − T T ζ 2 (1 − ζeaT ) Owing to

(9.120)

380


m2 (ζ − ν) ˆ Π(ζ) = m2 (1 − ζ −1 ν) = , ζ

Γˆ (ζ) = m1

and using (9.120), we find the function (9.106) in the form R2 (ζ) =

n0 + n1 ζ + n2 ζ 2 , ζ(ζ − ν)(1 − ζeaT )

where n0 , n1 , n2 are known constants. Performing the separation (9.107) with |ν| < 1, we obtain λ R21 (ζ) = , 1 − ζeaT where λ is a known constant. From this form and (9.118), we find Π −1 (ζ)R21 (ζ)Γ −1 (ζ) =

λ1 (1 − ζeaT )(1 − ζν)

(9.121)

with a known constant λ1 . Taking into account (9.114), we obtain the function V1o (ζ) in (9.109): V1o (ζ) =

(1 −

ζλ2 aT ζe )(1

− ζν)

+

1 1 − ζeaT

with a known constant λ2 . Due to Theorem 9.31, the function V1o (ζ) is analytical at the point ζ = e−aT . Hence 1 + e−aT λ2 − e−aT ν = 0 . From the last two equations, we receive V1o (ζ) =

1 . 1 − ζν

Furthermore using (9.121) and (9.109), we obtain V2o (ζ) =

λ3 , 1 − ζν

λ3 = const.

Therefore, Formula (9.108) yields wdo (ζ) = λ3 = const. and the characteristic polynomial of the closed-loop appears as ∆o (ζ) ≈ 1 − ζν .

10 L2 -Design of SD Systems for 0 < t < ∞

10.1 Problem Statement 1. Let the input of the standard sampled-data system for t ≥ 0 be acted upon by a vector input signal x(t) of dimension × 1, and let z(t) be the r × 1 output vector under zero initial energy. Then the system performance can be evaluated by the value J˜ =

∞

z (t)z(t) dt = 0

r i=1

∞

zi2 (t) dt ,

(10.1)

0

where zi (t), (i = 1, . . . , r) are the components of the output vector z(t). It is assumed that the conditions for the convergence of the integral (10.1) hold. It is known [206] that the value ) zL = + J˜ 2

determines the L2 -norm of the output signal z(t). Thus, the following optimisation problem is formulated. L2 -problem. Given the matrix w(p) in (7.2), the input vector x(t), the sampling period T and the form of the control impulse m(t). Find a stabilising controller (8.35) that ensures the internal stability of the standard sampled-data system and the minimal value of z(t)L2 . 2. It should be noted that the general problem formulated above include, for different choice of the vector z(t), many important applied problems, also including the tracking problem. Indeed, let us consider the block-diagram shown in Fig. 10.1, where the dotted box denotes the initial standard system that will be called nominal. Moreover in Fig. 10.1, Q(p) denotes the transfer matrix of an ideal transition. To evaluate the tracking performance, it is natural to use the value

382

10 L2 -Design of SD Systems x

-

•

z w(p) y

u

C

-

? h −6

e

-

Q(p) z˜

Fig. 10.1. Tracking control loop

J˜e = 0

∞

e (t)e(t) dt =

∞

[z(t) − z˜(t)] [z(t) − z˜(t)] dt .

(10.2)

0

If the PTM of the nominal system w(s, t) has the form (7.30)

w(s, t) = ϕL (T, s, t)RN (s)M (s) + K(s) ,

(10.3)

then the tracking error e(t) can be considered as a transformed result of the input signal x(t) by a new standard sampled-data system with the PTM

we (s, t) = w(s, t) − Q(s) = ϕL (T, s, t)RN (s)M (s) + K(s) − Q(s) .

(10.4)

This system is fenced by a dashed line in Fig. 10.1. The standard sampleddata system with the PTM (10.4) is associated with a continuous-time LTI plant having the transfer matrix

K(p) − Q(p) L(p) . we (p) = M (p) N (p) Then, the integral (10.2) coincides with (10.1) for the new standard sampleddata system. 3. Under some restrictions formulated below and using Parseval’s formula [181], the integral (10.1) can be transformed into j∞ 1 J˜ = Z (−s)Z(s) ds , (10.5) 2πj −j∞ where Z(s) is the Laplace transform of the output z(t). Thus, the L2 -problem formulated above can be considered as a problem of choosing a stabilising controller which minimises the integral (10.5). This problem will be considered in the present chapter.

10.2 Pseudo-rational Laplace Transforms

383

10.2 Pseudo-rational Laplace Transforms 1. According to the above statement of the problem, to consider the integral (10.5), we have to find the Laplace transform of the output z(t) for the standard sampled-data system under zero initial energy and investigate its properties as a function of s. In this section, we describe some properties of a class of transforms used below. 2. Henceforth, we denote by Λγ the set of functions (matrices) f (t) that are zero for t < 0, have bounded variation for t ≥ 0 and satisfy the estimation |f (t)| < deγt ,

t > 0,

where d > 0 and γ are constants. It is known [39] that for any function f (t) ∈ Λγ for Re s > γ, there exists the Laplace transform ∞ F (s) = f (t)e−st dt (10.6) 0

and for any t ≥ 0 the following inversion formula holds: c+ja 1 ˜ f (t) = lim F (s)est ds , c > γ , 2πj a→∞ c−ja where

f (t − 0) + f (t + 0) f˜(t) = . 2 As follows from the general properties of the Laplace transformation [39], under the given assumptions in any half-plane Re s ≥ γ1 > γ, we have lim |F (s)| = 0

s→∞

for s increasing to infinity along any contour. Then for f (t) ∈ Λγ and s = x + jy, x ≥ γ1 > γ, the following estimation holds [22] : |F (x + kjy)| ≤

c , |k|

c = const.

(10.7)

Hereinafter, the elements f (t) of the set Λγ will be called originals and denoted by small letters, while the corresponding Laplace transforms (10.6) will be called images and denoted by capital letters. 3. For any original f (t) ∈ Λγ for Re s ≥ γ1 > γ, the following series converges: ϕf (T, s, t) =

∞ k=−∞

f (t + kT )e−s(t+kT ) ,

−∞ < t < ∞ .

(10.8)

384

10 L2 -Design of SD Systems

According to [148], ϕf (T, s, t) is the displaced pulse frequency response (DPFR). The function ϕf (T, s, t) is periodic in t, therefore, it can be associated with a Fourier series of the form (6.15) ϕF (T, s, t) =

∞ 1 F (s + kjω)ekjωt , T

ω=

k=−∞

2π . T

(10.9)

Hereinafter, we shall assume that the function ϕf (T, s, t) is of bounded variation over the interval 0 ≤ t ≤ T . This holds as a rule in applications. Then, we have the equality ϕf (T, s, t) = ϕF (T, s, t) , that should be understood in the sense that ϕf (T, s, t0 ) = ϕF (T, s, t0 ) for any t0 , where the function ϕf (T, s, t) is continuous and ϕF (T, s, t0 ) =

ϕf (T, s, t0 − 0) + ϕf (T, s, t0 + 0) , 2

if ϕf (T, s, t) at t = t0 has a break of the first kind (finite break). 4.

Together with the DPFR (10.8), we consider the discrete Laplace trans

form (DLT) Df (T, s, t) of the function f (t):

Df (T, s, t) =

∞

f (t + kT )e−ksT = ϕf (T, s, t)est

(10.10)

k=−∞

and the associated series (6.67) ∞ 1 DF (T, s, t) = F (s + kjω)e(s+kjω)t = ϕF (T, s, t)est , T

(10.11)

k=−∞

which will be called the discrete Laplace transform of the image F (s). Let us have a strictly proper rational matrix F (s) =

NF (s) , dF (s)

where NF (s) is a polynomial matrix and dF (s) = (s − f1 )ν1 · · · (s − fψ )νψ ,

ν1 + . . . + νψ = χ .

Then as follows from (6.70)–(6.72), the function (matrix)

DF (T, ζ, t) = DF (T, s, t) | e−sT =ζ

10.2 Pseudo-rational Laplace Transforms

385

is rational in ζ for all t, and for 0 < t < T it can be represented in the form (m dk (t)ζ k , (10.12) DF (T, ζ, t) = k=0 ∆F (ζ) where

ν1 νψ ∆F (ζ) = ζ − e−f1 T · · · ζ − e−fψ T

is the discretisation of the polynomial dF (s), and dk (t) are functions of bounded variation on the interval 0 < t < T . 5. Below it will be shown that some images F (s), which are not rational functions, may possess a DLT of the form (10.12). Henceforth, the image F (s) will be called pseudo-rational, if its DLT for 0 < t < T can be represented in the form (10.12). The set of all pseudorational images are determined by the following Lemma. Lemma 10.1. A necessary and sufficient condition for an image F (s) to be pseudo-rational is, that it can be represented as (m −ksT & T e dk (t)e−st dt 0 , (10.13) F (s) = k=0 ∆F (s) where ν1 νψ ∆F (s) = ∆F (ζ) | ζ=e−sT = e−sT − e−f1 T · · · e−sT − e−fψ T .

Proof. Necessity: Let the matrix DF (T, ζ, t) have the form (10.12). Then we have (m −ksT e dk (t) DF (T, s, t) = D(T, ζ, t) | ζ=e−sT = k=0 . (10.14) ∆F (s) Moreover for the image F (s), we have [148] T F (s) = DF (T, s, t)e−st dt .

(10.15)

0

With respect to (10.14), this yields (10.13). Sufficiency: Denote T dk (t)e−st dt . Dk (s) = 0

Then, (6.36)–(6.39) yield ∞ 1 Dk (s + njω)e(s+njω)t = dk (t) , T n=−∞

0

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