Communications and Control Engineering Ai-Guo Wu Ying Zhang Complex Conjugate Matrix Equations for Systems and Control Communications and Control Engineering Series editors Alberto Isidori, Roma, Italy Jan H van Schuppen, Amsterdam, The Netherlands Eduardo D Sontag, Piscataway, USA Miroslav Krstic, La Jolla, USA More information about this series at http://www.springer.com/series/61 Ai-Guo Wu Ying Zhang • Complex Conjugate Matrix Equations for Systems and Control 123 Ai-Guo Wu Harbin Institute of Technology, Shenzhen University Town of Shenzhen Shenzhen China Ying Zhang Harbin Institute of Technology, Shenzhen University Town of Shenzhen Shenzhen China ISSN 0178-5354 ISSN 2197-7119 (electronic) Communications and Control Engineering ISBN 978-981-10-0635-7 ISBN 978-981-10-0637-1 (eBook) DOI 10.1007/978-981-10-0637-1 Library of Congress Control Number: 2016942040 Mathematics Subject Classification (2010): 15A06, 11Cxx © Springer Science+Business Media Singapore 2017 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd To our supervisor, Prof Guang-Ren Duan To Hong-Mei, and Yi-Tian To Rui, and Qi-Yu (Ai-Guo Wu) (Ying Zhang) Preface Theory of matrix equations is an important branch of mathematics, and has broad applications in many engineering fields, such as control theory, information theory, and signal processing Specifically, algebraic Lyapunov matrix equations play vital roles in stability analysis for linear systems, and coupled Lyapunov matrix equations appear in the analysis for Markovian jump linear systems; algebraic Riccati equations are encountered in optimal control Due to these reasons, matrix equations are extensively investigated by many scholars from various fields, and the content on matrix equations has been very rich Matrix equations are often covered in some books on linear algebra, matrix analysis, and numerical analysis We list several books here, for example, Topics in Matrix Analysis by R.A Horn and C.R Johnson [143], The Theory of Matrices by P Lancaster and M Tismenetsky [172], and Matrix Analysis and Applied Linear Algebra by C.D Meyer [187] In addition, there are some books on special matrix equations, for example, Lyapunov Matrix Equations in System Stability and Control by Z Gajic [128], Matrix Riccati Equations in Control and Systems Theory by H Abou-Kandil [2], and Generalized Sylvester Equations: Unified Parametric Solutions by Guang-Ren Duan [90] It should be pointed out that all the matrix equations investigated in the aforementioned books are in real domain By now, it seems that there is no book on complex matrix equations with the conjugate of unknown matrices For convenience, this class of equations is called complex conjugate matrix equations The first author of this book and his collaborators began to consider complex matrix equations with the conjugate of unknown matrices in 2005 inspired by the work [155] of Jiang published in Linear Algebra and Applications Since then, he and his collaborators have published many papers on complex conjugate matrix equations Recently, the second author of this book joined this field, and has obtained some interesting results In addition, some complex conjugate matrix equations have found applications in the analysis and design of antilinear systems This book aims to provide a relatively systematic introduction to complex conjugate matrix equations and its applications in discrete-time antilinear systems vii viii Preface The book has 12 chapters In Chap 1, first a survey is given on linear matrix equations, and then recent development on complex conjugate matrix equations is summarized Some mathematical preliminaries to be used in this book are collected in Chap Besides these two chapters, the rest of this book is partitioned into three parts The first part contains Chaps 3–5, and focuses on the iterative solutions for several types of complex conjugate matrix equations The second part consists of Chaps 6–10, and focuses on explicit closed-form solutions for some complex conjugate matrix equations In the third part, including Chaps 11 and 12, several applications of complex conjugate matrix equations are considered In Chap 11, stability analysis of discrete-time antilinear systems is investigated, and some stability criteria are given in terms of anti-Lyapunov matrix equations, which are special complex conjugate matrix equations In Chap 12, some feedback design problems are solved for discrete-time antilinear systems by using several types of complex conjugate matrix equations Except part of Chap and Subsection 6.1.1, the other materials of this book are based on our own research work, including some unpublished results The intended audience of this monograph includes students and researchers in areas of control theory, linear algebra, communication, numerical analysis, and so on An appropriate background for this monograph would be the first course on linear algebra and linear systems theory Since 1980s, many researchers have devoted much effort in complex conjugate matrix equations, and much contribution has been made to this area Owing to space limitation and the organization of the book, many of their published results are not included or even not cited We extend our apologies to these researchers It is under the supervision of our Ph.D advisor, Prof Guang-Ren Duan at Harbin Institute of Technology (HIT), that we entered the field of matrix equations with their applications in control systems design Moreover, Prof Duan has also made much contribution to the investigation of complex conjugate matrix equations, and has coauthored many papers with the first author Some results in these papers have been included in this book Therefore, at the beginning of preparing the manuscript, we intended to get Prof Duan as the first author of this book due to his contribution on complex conjugate matrix equations However, he thought that he did not make contribution to the writing of this book, and thus should not be an author of this book Here, we wish to express our sincere gratitude and appreciation to Prof Duan for his magnanimity and selflessness We also would like to express our profound gratitude to Prof Duan for his careful guidance, wholehearted support, insightful comments, and great contribution We also would like to give appreciation to our colleague, Prof Bin Zhou of HIT for his help The first author has coauthored some papers included in this book with Prof Gang Feng when he visited City University of Hong Kong as a Research Fellow The first author would like to express his sincere gratitude to Prof Feng for his help and contribution Dr Yan-Ming Fu, Dr Ming-Zhe Hou, Mr Yang-Yang Qian, and Dr Ling-Ling Lv have also coauthored with the first author a few papers included in this book The first author would extend his great thanks to all of them for their contribution Preface ix Great thanks also go to Mr Yang-Yang Qian and Mr Ming-Fang Chang, Ph.D students of the first author, who have helped us in typing a few sections of the manuscripts In addition, Mr Fang-Zhou Fu, Miss Dan Guo, Miss Xiao-Yan He, Mr Zhen-Peng Zeng, and Mr Tian-Long Qin, Master students of the first author, and Mr Yang-Yang Qian and Mr Ming-Fang Chang have provided tremendous help in finding errors and typos in the manuscripts Their help has significantly improved the quality of the manuscripts, and is much appreciated The first and second authors would like to thank his wife Ms Hong-Mei Wang and her husband Dr Rui Zhang, respectively, for their constant support in every aspect Part of the book was written when the first author visited the University of Western Australia (UWA) from July 2013 to July 2014 The first author would like to thank Prof Victor Sreeram at UWA for his help and invaluable suggestions We would like to gratefully acknowledge the financial support kindly provided by the National Natural Science Foundation of China under Grant Nos 60974044 and 61273094, by Program for New Century Excellent Talents in University under Grant No NCET-11-0808, by Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No 201342, by Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos 20132302110053 and 20122302120069, by the Foundation for Creative Research Groups of the National Natural Science Foundation of China under Grant Nos 61021002 and 61333003, by the National Program on Key Basic Research Project (973 Program) under Grant No 2012CB821205, by the Project for Distinguished Young Scholars of the Basic Research Plan in Shenzhen City under Contract No JCJ201110001, and by Key Laboratory of Electronics Engineering, College of Heilongjiang Province (Heilongjiang University) Lastly, we thank in advance all the readers for choosing to read this book It is much appreciated if readers could possibly provide, via email: agwu@163.com, feedback about any problems found July 2015 Ai-Guo Wu Ying Zhang Contents Introduction 1.1 Linear Equations 1.2 Univariate Linear Matrix Equations 1.2.1 Lyapunov Matrix Equations 1.2.2 Kalman-Yakubovich and Normal Sylvester Matrix Equations 1.2.3 Other Matrix Equations 1.3 Multivariate Linear Matrix Equations 1.3.1 Roth Matrix Equations 1.3.2 First-Order Generalized Sylvester Matrix Equations 1.3.3 Second-Order Generalized Sylvester Matrix Equations 1.3.4 High-Order Generalized Sylvester Matrix Equations 1.3.5 Linear Matrix Equations with More Than Two Unknowns 1.4 Coupled Linear Matrix Equations 1.5 Complex Conjugate Matrix Equations 1.6 Overview of This Monograph 5 13 16 16 18 24 25 26 27 30 33 Mathematical Preliminaries 2.1 Kronecker Products 2.2 Leverrier Algorithms 2.3 Generalized Leverrier Algorithms 2.4 Singular Value Decompositions 2.5 Vector Norms and Operator Norms 2.5.1 Vector Norms 2.5.2 Operator Norms 2.6 A Real Representation of a Complex 2.6.1 Basic Properties 2.6.2 Proof of Theorem 2.7 35 35 42 46 49 52 52 56 63 64 68 Matrix xi 472 References 19 Betser, A., Cohen, N., Zeheb, E.: On solving the Lyapunov and Stein equations for a companion matrix Syst Control Lett 25(3), 211–218 (1995) 20 Bevis, J.H., Hall, F.J., Hartwing, R.E.: The matrix equation AX¯ − XB = C and its special cases SIAM J Matrix Anal Appl 9(3), 348–359 (1988) 21 Bevis, J.H., Hall, F.J., Hartwing, R.E.: Consimilarity and the matrix equation AX¯ − XB = C In: Current Trends in Matrix Theory, Auburn, Ala., 1986, pp 51–64 North-Holland, New York (1987) 22 Bischof, C.H., Datta, B.N., Purkyastha, A.: A parallel algorithm for the Sylvester observer equation SIAM J Sci Comput 17(3), 686–698 (1996) 23 Bitmead, R.R.: Explicit solutions of the discrete-time Lyapunov matrix equation and KalmanYakubovich equations IEEE Trans Autom Control, AC 26(6), 1291–1294 (1981) 24 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Differ Equ Appl 17(4), 603–611 (2011) Index A Absolute homogeneity, 52 Adjoint matrix, 233, 244, 261, 270 Algebraic anti-Riccati matrix equation, 464 Alternating power left alternating power, 99 right alternating power, 99 Annihilating polynomial, 10 Anti-Lyapunov matrix equation, 410 Anti-Riccati matrix equation, 459, 462, 467 Antilinear mapping, 405 B Bartels-Stewart algorithm, 10 Bezout identity, 10 C Cayley-Hamilton identity, 44 Cayley-Hamilton theorem, 10 Characteristic polynomial, 6, 233, 244, 250, 261, 300 Column sum norm, 57 Common left divisor, 362 Common right divisor, 362 Companion form, Companion matrix, 11, 42 Con-controllability matrix, 339, 342 Con-Kalman-Yakubovich matrix equation, 31, 241, 294, 344 Con-observability matrix, 339, 342 Con-Sylvester mapping, 81 Con-Sylvester matrix equation, 250, 254, 336, 441, 446 Con-Sylvester-polynomial matrix, 394 Con-Sylvester sum, 389, 392, 398 Con-Yakubovich matrix equation, 31, 164, 259, 263, 265, 343 Condiagonalization, 74 Coneigenvalue, 73 Coneigenvector, 73 Conequivalence, 373 Conjugate gradient, 88 Conjugate gradient method, Conjugate product, 356, 368, 392 Conjugate symmetry, 83 Consimilarity, 30, 73, 227, 440, 441 Continuous-time Lyapunov matrix equation, Controllability matrix, 7, 10, 12, 225, 276, 280 Controllable canonical form, 6, Convergence linear convergence, 107 quadratic convergence, 107 superlinear convergence, 107 Coordinate, 80 Coprimeness left coprimeness, 365 right coprimeness, 365 Coupled anti-Lyapunov matrix equation, 416, 417, 423 Coupled con-Sylvester matrix equation, 135 Coupled Lyapunov matrix equation, 27 Cramer’s rule, © Springer Science+Business Media Singapore 2017 A.-G Wu and Y Zhang, Complex Conjugate Matrix Equations for Systems and Control, Communications and Control Engineering, DOI 10.1007/978-981-10-0637-1 485 486 D Degrees of freedom, 262, 277, 294, 298, 323, 338, 345 Diophantine polynomial matrix equation, 339 Discrete minimum principle, 451 Discrete-time antilinear system, 440 Discrete-time Lyapunov matrix equation, 5, Division with reminder, 362 Divisor left divisor, 359 right divisor, 359 Dynamic programming principle, 454 E Eigenstructure, 440 Elementary column transformation, 373 Elementary row transformation, 371 Euclidean space, 53 Explicit solution, 10, 233, 244 Extended con-Sylvester matrix equation, 121, 179, 267, 307, 447 F Feedback stabilizing gain, 444 Feedforward compensation gain, 444 Frobenius norm, 55, 66, 163 G Gauss-Seidel iteration, 3, Generalized con-Sylvester matrix equation, 163, 272, 321, 328, 330, 332 Generalized eigenstructure assignment, 440 Generalized inverse, 2, 15 Generalized Leverrier algorithm, 49, 244, 248, 261, 265, 299, 311, 315, 332 Generalized Sylvester matrix equation, 18, 174, 270 Gradient, 89 Greatest common left divisor, 362, 363 Greatest common right divisor, 362 H Hankel matrix, 12 Hessian matrix, 89 Hierarchical principle, 121, 150 H˝older inequality, 54 Homogeneous con-Sylvester matrix equation, 276 Index I Image, 81 Inner product, 30, 83 Inverse, 371 J Jacobi iteration, 3, Jordan form, 21 K Kalman-Yakubovich matrix equation, 9, 12, 13, 114 Kernel, 81 Kronecker product, 12, 36, 41, 125 left Kronecker product, 36 L Laurent expansion, 15 Left coprimeness, 393 Leverrier algorithm, 42, 233, 238, 250, 256, 279 Linear quadratic regulation, 450 Lyapunov matrix equation, M Markovian jump antilinear system, 410 Matrix exponential function, Maximal singular value norm, 58 Minkowski inequality, 55 Mixed-type Lyapunov matrix equation, 16 Model reference tracking control, 443 Multiple left multiple, 359 right multiple, 359 N Nonhomogeneous con-Sylvester matrix equation, 284, 289 Nonhomogeneous con-Sylvesterpolynomial matrix equation, 397 Nonhomogeneous con-Yakubovich matrix equation, 296, 299 Norm, 52 1-norm, 57 2-norm, 58, 66, 142 Normal con-Sylvester mapping, 230 Normal con-Sylvester matrix equation, 30, 164, 172, 226, 230, 238, 277, 293 Index Normal Sylvester matrix equation, 9, 11, 226, 229, 233, 281, 282 Normed space, 52 O Observability matrix, 10, 12, 225, 276, 280 Operator norm, 56 P Permutation matrix, 41 Polynomial matrix, Q Quadratic performance index, 450 R Real basis, 80, 202 Real dimension, 80, 86, 172, 191, 202 Real inner product, 84, 172, 191 Real inner product space, 84, 172, 191, 202 Real linear combination, 77 Real linear mapping, 81, 227, 228 Real linear space, 76 Real linear subspace, 77 Real representation, 31, 63, 123, 139, 230, 231, 242, 259, 262, 300, 335, 426 Relaxation factor, Roth matrix equation, 17 Roth’s removal rule, Roth’s theorem, 15 Routh table, Row sum norm, 59 S Schur-Cohn matrix, Schur decomposition, 487 Second-order generalized Sylvester matrix equation, 24 Singular value, 49 Singular value decomposition (SVD), 49, 50, 52 Singular vector left-singular vector, 49 right-singular vector, 49 Smith accelerative iteration, 13 Smith iteration, 8, 103 Smith normal form, 15, 275, 285, 295, 299, 324, 376 Solvability, 229, 241 Squared Smith iteration, 108 Stochastic Lyapunov function, 411, 417 Stochastic stability, 411, 417 Successive over-relaxation, Sylvester matrix equation, 250 Sylvester-observer matrix equation, 18 Symmetric operator matrix, 225 Symmetry, 84 T Toeplitz matrix, 12 Transformation approach, 3, 28 Triangle inequality, 52 U Unimodular matrix, 373, 392, 398 Unitary matrix, 38, 50, 62 Upper Hankle matrix, 11 V Vectorization, 39 Y Young’s inequality, 54 ... special matrix equations, for example, Lyapunov Matrix Equations in System Stability and Control by Z Gajic [128], Matrix Riccati Equations in Control and Systems Theory by H Abou-Kandil [2], and. .. book For two integers m ≤ n, the notation © Springer Science+Business Media Singapore 2017 A.-G Wu and Y Zhang, Complex Conjugate Matrix Equations for Systems and Control, Communications and Control. .. laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this 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