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 Bitmead, R.R., Weiss, H.: On the solution of the discrete-time Lyapunov matrix equation in controllable canonical form IEEE Trans Autom Control, AC 24(3), 481–482 (1979) 25 Borno, I.: Parallel computation of the solutions of coupled algebraic Lyapunov equations Automatica 31(9), 1345–1347 (1995) 26 Borno, I., Gajic, Z.: Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems Comput Math Appl 30(7), 1–4 (1995) 27 Boukas, E.K., Yang, H.: Stability of discrete-time linear systems with markovian jumping parameters Math Control Sig 8(4), 390–402 (1995) 28 Brockett, R.W.: Introduction to Matrix Analysis IEEE Transactions on Education, Baltimore (1970) 29 Budinich, P., Trautman, A.: The Spinorial Chessboard Springer, Berlin (1988) 30 Calvetti, D., Lewis, B., Reichel, L.: On the solution of large Sylvester-observer equations Numer Linear Algebra Appl 8, 435–451 (2001) 31 Carvalho, J., Datta, K., Hong, Y.: A new block algorithm for full-rank solution of the Sylvesterobserver equation IEEE Trans Autom Control 48(12), 2223–2228 (2003) 32 Carvalho, J.B., Datta, B.N.: An algorithm for generalized Sylvester-observer equation in state estimation of descriptor systems In: Proceedings of the 41th Conference on Decision and Control, pp 3021–3026 (2002) 33 Carvalho, J.B., Datta, B.N.: A new block algorithm for generalized Sylvester-observer equation and application to state estimation of vibrating systems In: Proceedings of the 43th Conference on Decision and Control, pp 3613–3618 (2004) 34 Carvalho, J.B., Datta, B.N.: A new algorithm for generalized Sylvester-observer equation and its application to state and velocity estimations in vibrating systems Numer Linear Algebra Appl 18(4), 719–732 (2011) 35 Chang, M.F., Wu, A.G., Zhang, Y.: On solutions of the con-Yakubovich matrix equation X − AXF = BY In: Proceedings of the 33rd Chinese Control Conference, pp 6148–6152 (2014) 36 Chen, B.S., Zhang, W.: Stochastic H2 /H∞ control with state-dependent noise IEEE Trans Autom Control 49(1), 45–57 (2004) 37 Chen, Y., Xiao, H.: The explicit solution of the matrix equation AX − XB = C-to the memory of Prof Guo Zhongheng Appl Math Mech 16(12), 1133–1141 (1995) 38 Chien, C.J., Fu, L.C.: A new approach to model reference control for a class of arbitrarily fast time-varying unknown plants Automatica 28(2), 437–440 (1992) 39 Choi, J.W.: Left eigenstructure assignment via Sylvester equation KSME Int J 12(6), 1034– 1040 (1998) 40 Chu, K.E.: Singular value and generalized singular value decompositions and the solution of linear matrix equations Linear Algebra Appl 88–89, 83–98 (1987) 41 Chu, K.E.: The solution of the matrix equations AXB − CXD = E and (YA−DZ, YC −BZ) = (E, F) Linear Algebra Appl 93, 93–105 (1987) 42 Costa, O., Fragoso, M.D., Marques, R.P.: Discrete-time Markovian Jump Linear Systems Springer, Berlin (2005) 43 Costa, O.L.V., Fragoso, M.D.: Stability results for discrete-time linear systems with markovian jumping parameters J Math Anal Appl 179(1), 154–178 (1993) References 473 44 Datta, B.N., Hetti, C.: Generalized Arnoldi methods for the Sylvester-observer equation and the multi-input pole placement problem In: Proceedings of the 36th Conference on Decision and Control, pp 4379–4383 (1997) 45 Datta, B.N., Saad, Y.: Arnoldi methods for large Sylvester-like observer matrix equations, and an associated algorithm for partial spectrum assignment Linear Algebra Appl 154–156, 225–244 (1991) 46 Datta, K.: The matrix equation XA − BX = R and its applications Linear Algebra Appl 109, 91–105 (1988) 47 de Souza, E., Bhattacharyya, S.P.: Controllability, observability and the solution of AX −XB = C Linear Algebra Appl 39, 167–188 (1981) 48 de Teran, F., Dopico, F.M.: Consistency and efficient solution of the Sylvester equation for *-congruence Electron J Linear Algebra 22, 849–863 (2011) 49 Dehghan, M., Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation Appl Math Comput 202(2), 571–588 (2008) 50 Dehghan, M., Hajarian, M.: Efficient iterative method for solving the second-order Sylvester matrix equation EV F − AV F − CV = BW IET Control Theor Appl 3(10), 1401–1408 (2009) 51 Dehghan, M., Hajarian, M.: An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices Appl Math Model 34(3), 639–654 (2010) 52 Ding, F., Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations IEEE Trans Autom Control 50(8), 1216–1221 (2005) 53 Ding, F., Chen, T.: Hierarchical gradient-based identification of multivariable discrete-time systems Automatica 41(2), 315–325 (2005) 54 Ding, F., Chen, T.: Hierarchical least squares identification methods for multivariable systems IEEE Trans Autom Control 50(3), 397–402 (2005) 55 Ding, F., Chen, T.: Iterative least squares solutions of coupled Sylvester matrix equations Syst Control Lett 54(2), 95–107 (2005) 56 Ding, F., Chen, T.: On iterative solutions of general coupled matrix equations SIAM J Control Optim 44(6), 2269–2284 (2006) 57 Ding, F., Ding, J.: Least squares parameter estimation for systems with irregularly missing data Int J Adapt Control Sig Proces 24(7), 540–553 (2010) 58 Ding, F., Liu, P.X., Ding, J.: Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle Appl Math Comput 197(1), 41–50 (2008) 59 Djaferis, T.E., Mitter, S.K.: Algebraic methods for the study of some linear matrix equations Linear Algebra Appl 44, 125–142 (1982) 60 Dooren, P.V.: Reduced order observers: a new algorithm and proof Syst Control Lett 4(5), 243–251 (1984) 61 Dorato, P., Levis, A.: Optimal linear regulators: the discrete-time case IEEE Trans Autom Control 16(6), 613–620 (1971) 62 Drakakis, K., Pearlmutter, B.A.: On the calculation of the l2 → l1 induced matrix norm Int J Algebra 3(5), 231–240 (2009) 63 Duan, G.R.: Simple algorithm for robust pole assignment in linear output feedback IEE Proc.-Control Theor Appl 139(5), 465–469 (1992) 64 Duan, G.R.: Solution to matrix equation AV + BW = EV F and eigenstructure assignment for descriptor linear systems Automatica 28(3), 639–643 (1992) 65 Duan, G.R.: Robust eigenstructure assignment via dynamical compensators Automatica 29(2), 469–474 (1993) 66 Duan, G.R.: Solutions of the equation AV +BW = V F and their application to eigenstructure assignment in linear systems IEEE Trans Autom Control 38(2), 276–280 (1993) 67 Duan, G.R.: Eigenstructure assignment by decentralized output feedback-a complete parametric approach IEEE Trans Autom Control 39(5), 1009–1014 (1994) 474 References 68 Duan, G.R.: Parametric approach for eigenstructure assignment in descriptor systems via output feedback IEE Proc.-Control Theor Appl 142(6), 611–616 (1995) 69 Duan, G.R.: On the solution to the sylvester matrix equation AV + BW = EV F IEEE Trans Autom Control 41(4), 612–614 (1996) 70 Duan, G.R.: Eigenstructure assignment and response analysis in descriptor linear systems with state feedback control Int J Control 69(5), 663–694 (1998) 71 Duan, G.R.: Right coprime factorisations using system upper hessenberg forms the multi-input system case IEE Proc.-Control Theor Appl 148(6), 433–441 (2001) 72 Duan, G.R.: Right coprime factorizations for single-input descriptor linear systems: a simple numerically stable algorithm Asian J Control 4(2), 146–158 (2002) 73 Duan, G.R.: Parametric eigenstructure assignment via output feedback based on singular value decompositions IET Control Theor Appl 150(1), 93–100 (2003) 74 Duan, G.R.: Two parametric approaches for eigenstructure assignment in second-order linear systems J Control Theor Appl 1(1), 59–64 (2003) 75 Duan, G.R.: Linear Systems Theory, 2nd edn (In Chinese) Harbin Institute of Technology Press, Harbin (2004) 76 Duan, G.R.: A note on combined generalised sylvester matrix equations J Control Theor Appl 2(4), 397–400 (2004) 77 Duan, G.R.: Parametric eigenstructure assignment in second-order descriptor linear systems IEEE Trans Autom Control 49(10), 1789–1795 (2004) 78 Duan, G.R.: The solution to the matrix equation AV + BW = EV J + R Appl Math Lett 17(10), 1197–1202 (2004) 79 Duan, G.R.: Parametric approaches for eigenstructure assignment in high-order linear systems Int J Control, Autom Syst 3(3), 419–429 (2005) 80 Duan, G.R.: Solution to high-order generalized Sylvester matrix equations In: Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference 2005, pp 7247–7252 Seville, Spain (2005) 81 Duan, G.R.: Analysis and Design of Descritor Linear Systems Springer, New York (2010) 82 Duan, G.R.: On a type of generalized Sylvester equations In: Proceedings of the 25th Chinese Control and Decision Conference, pp 1264–1269 (2013) 83 Duan, G.R.: On a type of high-order generalized Sylvester equations In: Proceedings of the 32th Chinese Control Conference, Xi’an, China, pp 328–333 (2013) 84 Duan, G.R On a type of second-order generalized Sylvester equations In: Proceedings of the 9th Asian Control Conference (2013) 85 Duan, G.R.: Solution to a type of high-order nonhomogeneous generalized Sylvester equations In: Proceedings of the 32th Chinese Control Conference, Xi’an, China, pp 322–327 (2013) 86 Duan, G.R.: Solution to a type of nonhomogeneous generalized Sylvester equations In: Proceedings of the 25th Chinese Control and Decision Conference, pp 163–168 (2013) 87 Duan, G.R.: Solution to second-order nonhomogeneous generalized Sylvester equations In Proceedings of the 9th Asian Control Conference (2013) 88 Duan, G.R.: Parametric solutions to rectangular high-order Sylvester equations–case of F arbitrary In: SICE Annual Conference 2014, Sapporo, Japan, pp 1827–1832 (2014) 89 Duan, G.R.: Parametric solutions to rectangular high-order Sylvester equations–case of F jordan In: SICE Annual Conference 2014, Sapporo, Japan, pp 1820–1826 (2014) 90 Duan, G.R.: Generalized Sylvester Equations: Unified Parametric Solutions CRC Press, Taylor and Francis Group, Boca Roton (2015) 91 Duan, G.R., Howe, D.: Robust magnetic bearing control via eigenstructure assignment dynamical compensation IEEE Trans Control Syst Technol 11(2), 204–215 (2003) 92 Duan, G.R., Hu, W.Y.: On matrix equation AX − XC = Y (In Chinese) Control Decis 7(2), 143–147 (1992) 93 Duan, G.R., Li, J., Zhou, L.: Design of robust Luenberger observer (In Chinese) Acta Automatica Sinica 18(6), 742–747 (1992) References 475 94 Duan, G.R., Liu, G.P., Thompson, S.: Disturbance decoupling in descriptor systems via output feedback-a parametric eigenstructure assignment approach In: Proceedings of the 39th IEEE Conference on Decision and Control, pp 3660–3665 (2000) 95 Duan, G.R., Liu, G.P., Thompson, S.: Eigenstructure assignment design for proportionalintegral observers: continuous-time case IEE Proc.-Control Theor Appl 148(3), 263–267 (2001) 96 Duan, G.R., Liu, W.Q., Liu, G.P.: Robust model reference control for multivariable linear systems subject to parameter uncertainties Proc Inst Mech Eng Part I: J Syst Control Eng 215(6), 599–610 (2001) 97 Duan, G.R., Patton, R.J.: Eigenstructure assignment in descriptor systems via proportional plus derivative state feedback Int J Control 68(5), 1147–1162 (1997) 98 Duan, G.R., Patton, R.J.: Robust fault detection using Luenberger-type unknown input observers–a parametric approach Int J Syst Sci 32(4), 533–540 (2001) 99 Duan, G.R., Qiang, W.: Design of Luenberger observers with loop transfer recovery(B) (In Chinese) Acta Aeronautica ET Astronautica Sinica 14(7), A433–A436 (1993) 100 Duan, G.R., Qiang, W., Feng, W., Sun, L.: A complete parametric approach for model reference control system design (In Chinese) J Astronaut 15(2), 7–13 (1994) 101 Duan, G.R., Wu, A.G.: Robust fault detection in linear systems based on PI observers Int J Syst Sci 37(12), 809–816 (2006) 102 Duan, G.R., Wu, G.Y.: Robustness analysis of pole assignment controllers for continuous systems (In Chinese) J Harbin Inst Technol 2, 62–69 (1989) 103 Duan, G.R., Wu, G.Y., Huang, W.H.: Analysis of the robust stabiltiy of discrete pole assignment control system (In Chinese) J Harbin Inst Technol 4, 57–64 (1988) 104 Duan, G.R., Wu, G.Y., Huang, W.H.: Robust stability analysis of pole assignment controllers for linear systems (In Chinese) Control Theor Appl 5(4), 103–108 (1988) 105 Duan, G.R., Wu, G.Y., Huang, W.H.: The eigenstructure assignment problem of linear timevarying systems (In Chinese) Sci China: Series A 20(7), 769–776 (1990) 106 Duan, G.R., Xu, S.J., Huang, W.H.: Generalized positive definite matrix and its application in stability analysis (In Chinese) Acta Mechanica Sinica 21(6), 754–757 (1989) 107 Duan, G.R., Yu, H.H.: Observer design in high-order descriptor linear systems In: Proceedings of SICE-ICASE International Joint Conference 2006, Busan, Korea, pp 870–875 (2006) 108 Duan, G.R., Yu, H.H.: Parametric approaches for eigenstructure assignment in high-order descriptor linear systems In: Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, USA, 2006, pp 1399–1404 (2006) 109 Duan, G.R., Yu, H.H.: Robust pole assignment in high-order descriptor linear systems via proportional plus derivative state feedback IET Control Theor Appl 2(4), 277–287 (2008) 110 Duan, G.R., Yu, H.H., Tan, F.: Parametric control systems design with applications in missile control Sci China Series F: Inform Sci 52(11), 2190–2200 (2009) 111 Duan, G.R., Zhang, B.: Robust model-reference control for descriptor linear systems subject to parameter uncertainties J Control Theor Appl 5(3), 213–220 (2007) 112 Duan, G.R., Zhang, X.: Dynamical order assignment in linear descriptor systems via state derivative feedback In: Proceedings of the 41st IEEE Conference on Decision and Control, pp 4533–4538 (2002) 113 Duan, G.R., Zhou, B.: Solution to the second-order sylvester matrix equation MV F +DV F + K V = BW IEEE Trans Autom Control 51(5), 805–809 (2006) 114 Duan, X., Li, C., Liao, A., Wang, R.: On two classes of mixed-type Lyapunov equations Appl Math Comput 219, 8486–8495 (2013) 115 Fahmy, M., O’Reilly, J.: Eigenstructure assignment in linear multivariable systems-a parametric solution IEEE Trans Autom Control 28(10), 990–994 (1983) 116 Fahmy, M.M., Hanafy, A.A.R.: A note on the extremal properties of the discrete-time Lyapunov matrix equation Inform Control 40, 285–290 (1979) 117 Fairman, F.W.: Linear Control Theory: The State Space Approach Wiely, West Sussex, England (1998) 476 References 118 Fang, C.: A simple approach to solving the Diophantine equation IEEE Trans Autom Control 37(1), 152–155 (1992) 119 Fang, C., Chang, F.: A novel approach for solving Diophantine equations IEEE Trans Circ Syst 37(11), 1455–1457 (1990) 120 Fang, Y.: A new general sufficient condition for almost sure stability of jump linear systems IEEE Trans Autom Control 42(3), 378–382 (1997) 121 Fang, Y., Loparo, K.A.: Stochastic stability of jump linear systems IEEE Trans Autom Control 47(7), 1204–1208 (2002) 122 Fang, Y., Loparo, K.A., Feng, X.: Almost sure and δ-moment stability of jump linear systems Int J Control 59(5), 1281–1307 (1994) 123 Fang, Y., Loparo, K.A., Feng, X.: Stability of discrete-time jump linear systems J Math Syst Estimation Control 5(3), 275–321 (1995) 124 Fang, Y., Loparo, K.A., Feng, X.: New estimates for solutions of Lyapunov equations IEEE Trans Autom Control 42(3), 408–411 (1997) 125 Ferrante, A., Pavon, M., Ramponi, F.: Hellinger versus kullback-leibler multivariable spectrum approximation IEEE Trans Autom Control 53(4), 954–967 (2008) 126 Flanders, H., Wimmer, H.K.: On the matrix equation AX − XB = C and AX − YB = C SIAM J Appl Math 32(4), 707–710 (1977) 127 Frank, P.M.: Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy-a survey and some new results Automatica 26(3), 459–474 (1990) 128 Gajic, Z.: Lyapunov Matrix Equations in System Stability and Control Academic Press, New York (1995) 129 Gardiner, J.D., Laub, A.J., Amato, J.J., Moler, C.B.: Solution of the Sylvester matrix equation AXBT + CXDT = E ACM Trans Math Softw 18(2), 223–231 (1992) 130 Gavin, K.R., Bhattacharyya, S.P.: Robust and well-conditioned eigenstructure assignment via sylvester’s equation Optimal Control Appl Meth 4, 205–212 (1983) 131 Ghavimi, A.R., Laub, A.J.: Backward error, sensitivity, and refinement of computed solutions of algebraic Riccati equations Numer Linear Algebra Appl 2(1), 29–49 (1995) 132 Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn Johns Hopkins, Baltimore (1996) 133 Golub, G.H., Nash, S., Loan, C.V.: A Hessenberg-Schur method for the problem AX+XB = C IEEE Trans Autom Control, AC 24(6), 909–913 (1979) 134 Gugercin, S., Sorensen, D.C., Antoulas, A.C.: A modified low-rank Smith method for largescale Lyapunov equations Numer Algorithms 32, 27–55 (2003) 135 Hajarian, M.: Matrix form of the Bi-CGSTAB method for solving the coupled Sylvester matrix equations IET Control Theor Appl 7(14), 1828–1833 (2013) 136 Hammarling, S.J.: Numerical solution of the stable, nonnegative definite Lyapunov equation IMA J Numer Anal 2(3), 303–323 (1982) 137 Hanzon, B., Peeters, R.L.M.: A Faddeev sequence method for solving Lyapunov and Sylvester equations Linear Algebra Appl 241–243, 401–430 (1996) 138 Hernandez, V., Gasso, M.: Explicit solution of the matrix equation AXB − CXD = E Linear Algebra Appl 121, 333–344 (1989) 139 Heyouni, M.: Extended arnoldi methods for large low-rank Sylvester matrix equations Appl Numer Math 60, 1171–1182 (2010) 140 Higham, N.J.: Perturbation theory and backward error for AX − XB = C BIT 33(1), 124–136 (1993) 141 Hong, Y.P., Horn, R.A.: A canonical form for matrices under consimilarity Linear Algebra Appl 102, 143–168 (1988) 142 Horn, R.A., Johnson, C.R.: Matrix Analysis Cambridge University Press, Cambridge (1990) 143 Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis Cambridge University Press, Cambridge (1991) 144 Hou, S.: A simple proof of the Leverrier-Faddeev characteristic polynomial algorithm SIAM REV 40(3), 706–709 (1998) 145 Hu, G.D., Hu, G.D.: A relation between the weighted logarithmic norm of a matrix and the Lyapunov equation BIT 40, 606–610 (2000) References 477 146 Huang, L., Liu, J.: The extension of Roth’s theorem for matrix equations over a ring Linear Algebra Appl 259, 229–235 (1997) 147 Huang, L.: The explicit solutions and solvability of linear matrix equations Linear Algebra Appl 311, 195–199 (2000) 148 Huang, L.: Consimilarity of quaternion matrices and complex matrices Linear Algebra Appl 331, 21–30 (2001) 149 Huang, L., Zeng, Q.: The matrix equation AXB + CYD = E over a simple artinian ring Linear Multilinear Algebra 38, 225–232 (1995) 150 Jacobson, N.: Pseudo-linear transformation Ann Math 38, 484–507 (1937) 151 Jadav, R.A., Patel, S.S.: Applications of singular value decomposition in image processing Indian J Sci Technol 3(2), 148–150 (2010) 152 Jameson, A.: Solutions of the equation AX − XB = C by inversion of an M × M or N × N matrix SIAM J Appl Math 16, 1020–1023 (1968) 153 Ji, Y., Chizeck, H.J., Feng, X., Loparo, K.: Stability and control of discrete-time jump linear systems Control-Theor Adv Tech 7(2), 247–270 (1991) 154 Jiang, T., Cheng, X., Chen, L.: An algebraic relation between consimilarity and similarity of complex matrices and its applications J Phys A: Math Gen 39, 9215–9222 (2006) ¯ = C 155 Jiang, T., Wei, M.: On solutions of the matrix equations X − AXB = C and X − AXB Linear Algebra Appl 367, 225–233 (2003) 156 Young Jr., D.M.: Iterative Solution of Large Linear Systems Academic Press (1971) 157 Jones Jr., J., Lew, C.: Solutions of the Lyapunov matrix equation BX − XA = C IEEE Trans Autom Control, AC 27(2), 464–466 (1982) 158 Kagstrom, B., Dooren, P.V.: A generalized state-space approach for the additive decomposition of a transfer matrix Int J Numer Linear Algebra Appl 1(2), 165–181 (1992) 159 Kagstrom, B., Westin, L.: Generalized Schur methods with condition estimators for solving the generalized sylvester equation IEEE Trans Autom Control 34(7), 745–751 (1989) 160 Kailath, T.: Linear Systems Prentice Hall, New Jersey (1980) 161 Kalman, R.: On the general theory of control systems IRE Trans Autom Control 4(3), 481–492 (1959) 162 Kalman, R., Koepcke, R.: Optimal synthesis of linear sampling control systems using generalized performance indices Trans ASME 80, 1820–1826 (1958) 163 Kantorovich, L.V., Akilov, G.P.: Functional Analysis in Normed Spaces Macmillan, New York (1964) 164 Kautsky, J.,Nichols, N.K., Van Dooren, P.: Robust pole assignment in linear state feedback Int J Control 41(5), 1129–1155 (1985) 165 Kim, H.C., Choi, C.H.: Closed-form solution of the continuous-time Lyapunov matrix equation IEE Proc.-Control Theor Appl 141(5), 350–356 (1994) 166 Kim, Y., Kim, H.S., Junkins, J.L.: Eigenstructure assignment algorithm for mechanical second-order systems J Guidance, Control Dyn 22(5), 729–731 (1999) 167 Kleinman, D.L., Rao, P.K.: Extensions to the Bartels-Stewart algorithm for linear matrix equations IEEE Trans Autom Control, AC 23(1), 85–87 (1978) 168 Kressner, D., Schroder, C., Watkins, D.S.: Implicit QR algorithms for palindromic and even eigenvalue problems Numer Algorithms 51, 209–238 (2009) 169 Kwon, B., Youn, M.: Eigenvalue-generalized eigenvector assignment by output feedback IEEE Trans Autom Control 32(5), 417–421 (1987) 170 Lai, Y.S.: An algorithm for solving the matrix polynomial equation A(s)X(s) + B(s)Y (s) = C(s) IEEE Trans Circ Syst 36(8), 1087–1089 (1989) 171 Lancaste, P.: Explicit solutions of linear matrix equations SIAM Rev 12(4), 544–566 (1970) 172 Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn Academic Press, London, United Kingdom (1998) 173 Lee, C.H., Kung, F.C.: Upper and lower matrix bounds of the solutions for the continuous and discrete Lyapunov equations J Franklin Inst 334B(4), 539–546 (1997) 174 Lewis, F.L.: Further remarks on the Cayley-Hamiltion theorem and Leverrier’s method for matrix pencil (sE − A) IEEE Trans Autom Control 31(9), 869–870 (1986) 478 References 175 Li, C.J., Ma, G.F.: Optimal Control (In Chinese) Science Press, Beijing (2011) 176 Li, H., Gao, H., Shi, P., Zhao, X.: Fault-tolerant control of markovian jump stochastic systems via the augmented sliding mode observer approach Automatica 50(7), 1825–1834 (2014) 177 Liao, A.P., Lei, Y.: Least-squares solution with the minimum-norm for the matrix equation (AXB, GXH) = (C, D) Comput Math Appl 50(3–4), 539–549 (2005) 178 Liu, Y., Wang, D., Ding, F.: Least-squares based iterative algorithms for identifying BoxJenkins models with finite measurement data Digit Sig Process 20(5), 1458–1467 (2010) 179 Liu, Y., Xiao, Y., Zhao, X.: Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model Appl Math Comput 215(4), 1477–1483 (2009) 180 Liu, Y.H.: Ranks of solutions of the linear matrix equation AX + YB = C Comput Math Appl 52(6–7), 861–872 (2006) 181 Liu, Y.J., Yu, L., Ding, F.: Multi-innovation extended stochastic gradient algorithm and its performance analysis Circ Syst Sig Process 29(4), 649–667 (2010) 182 Lu, T.: Solution of the matrix equation AX − XB = C Computing 37, 351–355 (1986) 183 Ma, E.C.: A finite series solution of the matrix equation AX − XB = C SIAM J Appl Math 14(3), 490–495 (1966) 184 Maligranda, L.: A simple proof of the Holder and the minkowski inequality Am Math Mon 102(3), 256–259 (1995) 185 Mariton, M.: Jump Linear Systems in Automatic Control Marcel Dekker, New York (1990) 186 Mertzios, B.G.: Leverrier’s algorithm for singular systems IEEE Trans Autom Control 29(7), 652–653 (1984) 187 Meyer, C.D.: Matrix Analysis and Applied Linear Algebra SIAM (2000) 188 Miller, D.F.: The iterative solution of the matrix equation XA + BX + C = Linear Algebra Appl 105, 131–137 (1988) 189 Misra, P., Quintana, E.S., Van Dooren, P.M.: Numerically reliable computation of characteristic polynomials In: Proceedings of the 1995 American Control Conference, USA, 1995, pp 4025–4029 (1995) 190 Mitra, S.K.: The matrix equation AXB + CXD = E SIAM J Appl Math 32(4), 823–825 (1977) 191 Mom, T., Fukuma, N., Kuwahara, M.: Eigenvalue bounds for the discrete Lyapunov matrix equation IEEE Trans Autom Control, AC 30(9), 925–926 (1985) 192 Moore, B.C.: On the flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment IEEE Trans Autom Control, AC 21(5), 689–692 (1976) 193 Mori, T., Fukuma, N., Kuwahara, M.: Explicit solution, eigenvalue bounds in the Lyapunov matrix equation IEEE Trans Autom Control, AC 31(7), 656–658 (1986) 194 Navarro, E., Company, R., Jodar, L.: Bessel matrix differential equations: explicit solutions of initial and two-point boundary value problems Applicationes Mathematicae 22(1), 11–23 (1993) 195 Nijmeijer, H., Savaresi, S.M.: On approximate model-reference control of SISO discrete-time nonlinear systems Automatica 34(10), 1261–1266 (1998) 196 Ozguler, A.B.: The equations AXB +CYD = E over a principal ideal domain SIAM J Matrix Anal Appl 12(3), 581–591 (1991) 197 Paraskevopoulos, P., Christodoulou, M., Boglu, A.: An algorithm for the computation of the transfer function matrix for singular systems Automatica 20(2), 259–260 (1984) 198 Pavon, M., Ferrante, A.: On the georgiou-lindquist approach to constrained kullback-leibler approximation of spectral densities IEEE Trans Autom Control 51(4), 639–644 (2006) 199 Peng, Z., Peng, Y.: An efficient iterative method for solving the matrix equation AXB+CYD = E Numer Linear Algebra Appl 13(6), 473–485 (2006) 200 Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations SIAM J Sci Comput 21(4), 1401–1418 (2000) 201 Qian, Y.Y., Pang, W.J.: An implicit sequential algorithm for solving coupled Lyapunov equations of continuous-time Markovian jump systems Automatica 60(8), 245–250 (2015) References 479 202 Qiao, Y.P., Qi, H.S., Cheng, D.Z.: Parameterized solution to a class of Sylvester matrix equations Int J Autom Comput 7(4), 479–483 (2010) 203 Qu, Z., Dorsey, J.F., Dawson, D.M.: Model reference robust control of a class of siso systems IEEE Trans Autom Control 39(11), 2219–2234 (1994) 204 Ramadan, M.A., El-Danaf, T.S., Bayoumi, A.M.E.: Finite iterative algorithm for solving a class of general complex matrix equation with two unknowns of general form Appl Comput Math 3(5), 273–284 (2014) 205 Ramadan, M.A., El-Shazly, N.M., Selim, B.I.: A Hessenberg method for the numerical solutions to types of block sylvester matrix equations Math Comput Model 52(9–10), 1716– 1727 (2010) 206 Rao, C.R.: A note on a generalized inverse of a matrix with applications to problems in mathematical statistics J Roy Statist Soc Ser B 24(1), 152–158 (1962) 207 Regalia, P.A., Mitra, S.K.: Kronecker products, unitary matrices and signal processing applications SIAM Rev 31(4), 586–613 (1989) 208 Rohde, C.A.: Generalized inverses of partitioned matrices J Soc Indust Appl Math 13(4), 1033–1035 (1965) 209 Rohn, J.: Computing the norm A ∞,1 is NP-hard Linear Multilinear Algebra 47(3), 195–204 (2000) 210 Roth, W.E.: The equations AX − YB = C and AX − XB = C in matrices Proc Am Math Soc 3(3), 392–396 (1952) 211 Saberi, A., Stoorvogel, A.A., Sannuti, P.: Control of Linear Systems with Regulation and Input Constraints Springer, Berlin (1999) 212 Sadkane, M.: Estimates from the discrete-time Lyapunov equation Appl Math Lett 16(3), 313–316 (2003) 213 Sakthivel, R., Raja, R., Anthoni, S.M.: Linear matrix inequality approach to stochastic stability of uncertain delayed bam neural networks IMA J Appl Math 78(6), 1156–1178 (2013) 214 Shafai, B., Bhattacharyya, S.P.: An algorithm for pole assignment in high order multivariable systems IEEE Trans Autom Control 33(9), 870–876 (1988) 215 Shi, Y., Ding, F., Chen, T.: 2-norm based recursive design of transmultiplexers with designable filter length Circ Syst Sig Process 25(4), 447–462 (2006) 216 Shi, Y., Ding, F., Chen, T.: Multirate crosstalk identification in xDSL systems IEEE Trans Commun 54(10), 1878–1886 (2006) 217 Smith, P.G.: Numerical solution of the matrix equation AX + XAT + B = IEEE Trans Autom Control, AC 16(3), 278–279 (1971) 218 Smith, R.A.: Matrix equation XA + BX = C SIAM J Appl Math 16(1), 198–201 (1968) 219 Song, C., Chen, G.: An efficient algorithm for solving extended Sylvester-conjugate transpose matrix equations Arab J Math Sci 17(2), 115–134 (2011) 220 Sorensen D C., Zhou, Y.: Direct methods for matrix Sylvester and Lyapunov equations J Appl Math 6, 277–303 (2003) 221 Sreeram, V., Agathoklis, P.: Solution of Lyapunov equation with system matrix in companion form IEE Proc.-D: Control Theor Appl 138(6), 529–534 (1991) 222 Sreeram, V., Agathoklis, P., Mansour, M.: The generation of discrete-time q-Markov covers via inverse solution of the Lyapunov equation IEEE Trans Autom Control 39(2), 381–385 (1994) 223 Sun, J., Olbrot, A.W., Polis, M.P.: Robust stabilisation and robust performance using model reference control and modelling error compensation IEEE Trans Autom Control 39(3), 630–635 (1994) 224 Syrmos, V.L., Lewis, F.L.: Output feedback eigenstructure assignment using two Sylvester equations IEEE Trans Autom Control 38(3), 495–499 (1993) 225 Teran, F.D., Dopico, F.M., Guillery, N., Montealegre, D., Reyes, N.: The solution of the equations AX + X ∗ B = Linear Algebra Appl 438, 2817–2860 (2013) 226 Tian, Y.: Ranks of solutions of the matrix equation AXB = C Linear Multilinear Algebra 51(2), 111–125 (2003) 480 References 227 Tippett, M.K., Marchesin, D.: Upper bounds for the solution of the discrete algebraic Lyapunov equation Automatica 35(8), 1485–1489 (1999) 228 Tismenetsky, M.: Factorizations of hermitian block hankel matrices Linear Algebra Appl 166, 45–63 (1992) 229 Tong, L., Wu, A.G., Duan, G.R.: Finite iterative algorithm for solving coupled Lyapunov equations appearing in discrete-time Markov jump linear systems IET Control Theor Appl 4(10), 2223–2231 (2010) 230 Tsiligkaridis T., Hero A O.: Covariance estimation in high dimensions via Kronecker product expansions IEEE Trans Sig Process 61(21), 5347–5360 (2013) 231 Tsui, C.C.: A complete analytical solution to the equation TA − FT = LC and its applications IEEE Trans Autom Control 32(8), 742–744 (1987) 232 VanLoan, C.F.: The ubiquitous Kronecker product J Comput Appl Math 123(1–2), 85–100 (2000) 233 Wan, K.C., Sreeram, V.: Solution of the bilinear matrix equation using Astrom-Jury-Agniel algorithm IEE Proc.-Control Theor Appl 142(6), 603–610 (1995) 234 Wang, G., Lin, Y.: A new extension of Leverrier’s algorithm Linear Algebra Appl 180, 227–238 (1993) 235 Wang, L., Xie, L., Wang, X.: The residual based interactive stochastic gradient algorithms for controlled moving average models Appl Math Comput 211(2), 442–449 (2009) 236 Wang Q.W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity Linear Algebra Appl 384, 43–54 (2004) 237 Wang, Q., Lam, J., Wei, Y., Chen, T.: Iterative solutions of coupled discrete Markovian jump Lyapunov equations Comput Math Appl 55(4), 843–850 (2008) 238 Wimmer, H.K.: The matrix equation X −AXB = C and an analogue of Roth’s theorem Linear Algebra Appl 109, 145–147 (1988) 239 Wimmer, H.K.: Explicit solutions of the matrix equation Ai XDi = C SIAM J Matrix Anal Appl 13(4), 1123–1130 (1992) 240 Wimmer, H.K.: Consistency of a pair of generalized Sylvester equations IEEE Trans Autom Control 39(5), 1014–1016 (1994) 241 Wimmer, H.K.: Roth’s theorems for matrix equations with symmetry constraints Linear Algebra Appl 199, 357–362 (1994) 242 Wu, A.G.: Explicit solutions to the matrix equation EXF − AX = C IET Control Theor Appl 7(12), 1589–1598 (2013) 243 Wu, A.G., Chang, M.F., Zhang, Y.: Model reference control for discrete-time antilinear systems In: Proceedings of the 33rd Chinese Control Conference, Nanjing, 2014, pp 6153–6157 (2014) 244 Wu, A.G., Duan, G.R.: Design of generalized PI observers in descriptor linear systems IEEE Trans Circ Syst I: Regular Papers 53(12), 2828–2837 (2006) 245 Wu, A.G., Duan, G.R.: Design of PI observers for continuous-time descriptor linear systems IEEE Trans Syst Man Cybern.-Part B: Cybern 36(6), 1423–1431 (2006) 246 Wu, A.G., Duan, G.R.: IP observer design for descriptor linear systems IEEE Trans Circ Syst.-II: Express Briefs 54(9), 815–819 (2007) 247 Wu, A.G., Duan, G.R.: On solution to the generalized sylvester matrix equation AV + BW = EV F IET Control Theor Appl 1(1), 402–408 (2007) 248 Wu, A.G., Duan, G.R.: New iterative algorithms for solving coupled Markovian jump Lyapunov equations IEEE Trans Autom Control 60(1), 289–294 (2015) 249 Wu, A.G., Duan, G.R, Dong, J., Fu, Y.M.: Design of proportional-integral observers for discrete-time descriptor linear systems IET Control Theor Appl 3(1), 79–87 (2009) 250 Wu, A.G., Duan, G.R., Feng, G., Liu, W.: On conjugate product of complex polynomials Appl Math Lett 24(5), 735–741 (2011) 251 Wu, A.G., Duan, G.R., Fu, Y.M.: Generalized PID observer design for descriptor linear systems IEEE Trans Syst Man Cybern.-Part B: Cybern 37(5), 1390–1395 (2007) 252 Wu, A.G., Duan, G.R., Fu, Y.M., Wu, W.J.: Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation AX + BY = EXF + S Computing 89, 147–170 (2010) References 481 253 Wu, A.G., Duan, G.R., Hou, M.Z.: Parametric design approach for proportional multipleintegral derivative observers in descriptor linear systems Asian J Control 14(6), 1683–1689 (2012) 254 Wu, A.G., Duan, G.R., Liu, W.: Proportional multiple-integral observer design for continuoustime descriptor linear systems Asian J Control 14(2), 476–488 (2012) 255 Wu, A.G., Duan, G.R., Liu, W.: Implicit iterative algorithms for continuous Markovian jump Lyapunov matrix equations IEEE Trans Autom Control, page to appear (2016) 256 Wu, A.G., Duan, G.R., Liu, W., Sreeram, V.: Controllability and stability of discrete-time antilinear systems In: Proceedings of 3rd Australian Control Conference, Perth, 2013, pp 403–408 (2013) 257 Wu, A.G., Duan, G.R., Xue, Y.: Kronecker maps and Sylvester-polynomial matrix equations IEEE Trans Autom Control 52(5), 905–910 (2007) 258 Wu, A.G., Duan, G.R., Yu, H.H.: On solutions of the matrix equations XF − AX = C and XF − AX¯ = C Appl Math Comput 183(2), 932–941 (2006) 259 Wu, A.G., Duan, G.R., Yu, H.H.: Impulsive-mode controllablizability revisited for descriptor linear systems Asian J Control 11(3), 358–365 (2009) 260 Wu, A.G., Duan, G.R., Zhao, S.M.: Impulsive-mode controllablisability in descriptor linear systems IET Control Theor Appl 1(3), 558–563 (2007) 261 Wu, A.G., Duan, G.R., Zhao, Y., Yu, H.H.: A revisit to I-controllablisability for descriptor linear systems IET Control Theor Appl 1(4), 975–978 (2007) 262 Wu, A.G., Duan, G.R., Zhou, B.: Solution to generalized Sylvester matrix equations IEEE Trans Autom Control 53(3), 811–815 (2008) 263 Wu, A.G., Feng, G., Duan, G.R.: Proportional multiple-integral observer design for discretetime descriptor linear systems Int J Syst Sci 43(8), 1492–1503 (2012) 264 Wu, A.G., Feng, G., Duan, G.R., Liu, W.: Iterative solutions to the Kalman-Yakubovichconjugate matrix equation Appl Math Comput 217(9), 4427–4438 (2011) 265 Wu, A.G., Feng, G., Duan, G.R., Wu, W.J.: Iterative solutions to coupled Sylvester-conjugate matrix equations Comput Math Appl 60(1), 54–66 (2010) 266 Wu, A.G., Feng, G., Hu, J., Duan, G.R.: Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation Appl Math Comput 214(2), 442–450 (2009) 267 Wu, A.G., Feng, G., Liu, W., Duan, G.R.: The complete solution to the Sylvester-polynomialconjugate matrix equations Math Comput Modell 53(9–10), 2044–2056 (2011) 268 Wu, A.G., Feng, G., Wu, W.J., Duan, G.R.: Closed-form solutions to Sylvester-conjugate matrix equations Comput Math Appl 60(1), 95–111 (2010) 269 Wu, A.G., Fu, Y.M., Duan, G.R.: On solutions of matrix equations V − AV F = BW and V − AV¯ F = BW Math Comput Modell 47(11–12), 1181–1197 (2008) 270 Wu, A.G., Li, B., Zhang, Y., Duan, G.R.: Finite iterative solutions to coupled Sylvesterconjugate matrix equations Appl Math Modell 35(3), 1065–1080 (2011) 271 Wu, A.G., Liu, W., Duan, G.R.: On the conjugate product of complex polynomial matrices Math Comput Modell 53(9–10), 2031–2043 (2011) 272 Wu, A.G., Lv, L., Duan, G.R.: Iterative algorithms for solving a class of complex conjugate and transpose matrix equations Appl Math Comput 217, 8343–8353 (2011) 273 Wu, A.G., Lv, L., Duan, G.R., Liu, W.: Parametric solutions to Sylvester-conjugate matrix equations Comput Math Appl 62(9), 3317–3325 (2011) 274 Wu, A.G., Lv, L., Hou, M.Z.: Finite iterative algorithms for a common solution to a group of complex matrix equations Appl Math Comput 218(4), 1191–1202 (2011) 275 Wu, A.G., Lv, L., Hou, M.Z.: Finite iterative algorithms for extended Sylvester-conjugate matrix equations Math Comput Modell 54(9–10), 2363–2384 (2011) 276 Wu, A.G., Qian, Y.Y., Liu, W.: Stochastic stability for discrete-time antilinear systems with Markovian jumping parameters IET Control Theor Appl 9(9), 1399–1410 (2015) 277 Wu, A.G., Qian, Y.Y., Liu, W., Sreeram, V.: Linear quadratic regulation for discrete-time antilinear systems: an anti-Riccati matrix equation approach J Franklin Inst 353(5), 1041– 1060 (2016) 482 References 278 Wu, A.G., Sun, Y., Feng, G.: Closed-form solution to the non-homogeneous generalised Sylvester matrix equation IET control Theory Appl 4(10), 1914–1921 (2010) 279 Wu, A.G., Tong, L., Duan, G.R.: Finite iterative algorithm for solving coupled Lyapunov equations appearing in continuous-time Markov jump linear systems Int J Syst Sci 44(11), 2082–2093 (2013) ¯ = C 280 Wu, A.G., Wang, H.Q., Duan, G.R.: On matrix equations X − AXF = C and X − AXF J Comput Appl Math 230, 690–698 (2009) 281 Wu, A.G., Zeng, X., Duan, G.R., Wu, W.J.: Iterative solutions to the extended Sylvesterconjugate matrix equations Appl Math Comput 217(1), 130–142 (2010) 282 Wu, A.G., Zhang, E., Liu, F.: On closed-form solutions to the generalized Sylvester-conjugate matrix equation Appl Math Comput 218, 9730–9741 (2012) 283 Wu, Y., Duan, G.R.: Design of dynamical compensators for matrix second-order linear systems: A parametric approach In: Proceedings of the 44th IEEE Conference on Decision and Control, SPAIN, 2005, pp 7253–7257 (2005) 284 Xiao, C.S., Feng, Z.M., Shan, X.M.: On the solution of the continuous-time Lyapunov matrix equation in two canonical forms IEE Proc.-D, Control Theor Appl 139(3), 286–290 (1992) 285 Xie, L., Ding, J., Ding, F.: Gradient based iterative solutions for general linear matrix equations Comput Math Appl 58(7), 1441–1448 (2009) 286 Xing, W., Zhang, Q., Zhang, J., Wang, Q.: Some geometric properties of Lyapunov equation and LTI system Automatica 37(2), 313–316 (2001) 287 Xiong, Z., Qin, Y., Yuan, S.: The maximal and minimal ranks of matrix expression with applications J Inequal Appl 54 accepted (2012) 288 Xu, G., Wei, M., Zheng, D.: On solutions of matrix equation AXB+CYD = F Linear Algebra Appl 279(1–3), 93–109 (1998) 289 Xu, S., Cheng, M.: On the solvability for the mixed-type Lyapunov equation IMA J Appl Math 71(2), 287–294 (2006) 290 Yamada, M., Zun, P.C., Funahashi, Y.: On solving Diophantine equations by real matrix manipulation IEEE Trans Autom Control 40(1), 118–122 (1995) 291 Yang, C., Liu, J., Liu, Y.: Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment Asian J Control 14(6), 1669–1675 (2012) 292 Yu, H.H., Duan, G.R.: ESA in high-order linear systems via output feedback Asian J Control 11(3), 336–343 (2009) 293 Yu, H.H., Duan, G.R.: ESA in high-order descriptor linear systems via output feedback Int J Control, Autom Syst 8(2), 408–417 (2010) 294 Yu, H.H., Duan, G.R., Huang, L.: Asymptotically tracking in high-order descriptor linear systems In: Proceedings of the 27th Chinese Control Conference, Kunming, China, 2008, pp 762–766 (2008) 295 Zhang, H., Duan, G.R., Xie, L.: Linear quadratic regulation for linear time-varying systems with multiple input delays Automatica 42, 1465–1476 (2006) 296 Zhang, L., Zhu, A.F., Wu, A.G., Lv, L.L.: Parametric solutions to the regulator-conjugate matrix equation J Ind Manage Optim (to appear) (2015) 297 Zhang, X.: Matrix Analysis and Applications Tsinghua University Press, Beijing (2004) 298 Zhang, X., Thompson, S., Duan, G.R.: Full-column rank solutions of the matrix equation AV = EV J Appl Math Comput 151(3), 815–826 (2004) 299 Zhao, H., Zhang, H., Wang, H., Zhang, C.: Linear quadratic regulation for discrete-time systems with input delay: spectral factorization approach J Syst Sci Complex 21(1), 46–59 (2008) 300 Zhou, B., Duan, G.R.: Parametric solutions to the generalized sylvester matrix equation AX − XF = BY and the regulator equation AX − XF = BY + R Asian J Control 9(4), 475–483 (2007) 301 Zhou, B., Duan, G.R.: An explicit solution to the matrix equation AX − XF = BY Linear Algebra Appl 402, 345–366 (2005) 302 Zhou, B., Duan, G.R.: An explicit solution to polynomial matrix right coprime factorization with application in eigenstructure assignment J Control Theor Appl 4(2), 147–154 (2006) References 483 303 Zhou, B., Duan, G.R.: A new solution to the generalized Sylvester matrix equation AV − EV F = BW Syst Control Lett 55(3), 193–198 (2006) 304 Zhou, B., Duan, G.R.: Parametric approach for the normal Luenberger function observer design in second-order linear systems In: Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA 2006, pp 1423–1428 (2006) 305 Gershon, Eli, Shaked, Uri: Applications In: Gershon, Eli, Shaked, Uri (eds.) Advanced Topics in Control and Estimation of State-multiplicative Noisy Systems LNCIS, vol 439, pp 201– 216 Springer, Heidelberg (2013) 306 Zhou, B., Duan, G.R., Li, Z.Y.: Gradient based iterative algorithm for solving coupled matrix equations Syst Control Lett 58(5), 327–333 (2009) 307 Zhou, B., Duan, G.R., Lin, Z.: A parametric Lyapunov equation approach to the design of low gain feedback IEEE Trans Autom Control 53(6), 1548–1554 (2008) 308 Zhou, B., Lam, J., Duan, G.R.: Toward solution of matrix equation X = Af (X)B + C Linear Algebra Appl 435(6), 1370–1398 (2011) 309 Zhou, B., Lam, J., Duan, G.R.: Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations Comput Math Appl 56(12), 3070–3078 (2008) 310 Zhou, B., Lam, J., Duan, G.R.: On Smith-type iterative algorithms for the Stein matrix equation Appl Math Lett 22(7), 1038–1044 (2009) 311 Zhou, B., Li, Z.Y., Duan, G., Wang, Y.: Weighted least squares solutions to general coupled sylvester matrix equations J Comput Appl Math 224(2), 759–776 (2009) 312 Zhou, B., Li, Z.Y., Duan, G.R., Wang, Y.: Solutions to a family of matrix equations by using the kronecker matrix polynomials Appl Math Comput 212, 327–336 (2009) 313 Zhou, B., Lin, Z.: Parametric Lyapunov equation approach to stabilization of discrete-time systems with input delay and saturation IEEE Trans Circ Syst.-I: Regular Papers 58(11), 2741–2754 (2011) 314 Zhou, B., Lin, Z., Duan, G.: A parametric Lyapunov equation approach to low gain feedback design for discrete-time systems Automatica 45(1), 238–244 (2009) 315 Zhou, B., Lin, Z., Duan, G.: Properties of the parametric Lyapunov equation-based low-gain design with applications in stabilization of time-delay systems IEEE Trans Autom Control 54(7), 1698–1704 (2009) 316 Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control Prentice Hall, New Jersey (1996) 317 Zhu, Q., Hu, G.D., Zeng, L.: Estimating the spectral radius of a real matrix by discrete Lyapunov equation J 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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