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2009年,作者Chesi, Graziano ,基于SOS的鲁棒时变不确定控制。Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems,
Graziano chesi. Andrea garulli. alberto tesi AntonioⅤ Icing Homogeneous polynomial Forms for robustness Analysis of Uncertain Systems Springer Series Advisory board P. Fleming, P. Kokotovic A.B. Kurzhanskih. Kwakernaak A. Rantzer J N. Tsitsiklis Authors Graziano chesi PhD Alberto tesi. PhD University of Hong Kong Universita di firenze Department of Electrical and Facolta di Ingegneria Electronic Engineering Dipartimento di Sistemi e Informatica Chow Yei ching building Via di santa marta 3 Pokfulam road 50139 Firenze Hong kong Ital P.R. China E-mail: atesi@ dsi, unifi.it E-mail: chesi@eee hku hl Andrea garulli. phD Antonio vicino. PhD Universita di siena Universita di siena Dipartimento di Ingegneria Dipartimento di ingegneria dell'informazione dell'informazione Via roma 56 Via roma 56 53100 Siena 53100 Siena Ital. Ita E-mail: garulli@ing. unisiit E-mail: vicino @ing. unisiit ISBN978-1-84882-780-6 e-ISBN978-1-84882-781-3 DOI10.10071978-1-84882-781-3 Lecture Notes in Control and Information Sciences ISsN 0170-8643 orary of Congress Control Number: Applied for C2009 Springer-Verlag berlin heidelberg MATLAB and Simulink are registered trademarks of The Math Works, Inc., 3 Apple Hill Drive, Natick,Mao1760-2098,U.s.a.http://www.mathworks.com This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typeset Cover Design: Scientific Publishing Services Pvt Ltd, Chennai, India. Printed in acid-free paper 543210 springer. com To Shing Chee and Isabella(GC) To Maria, Giovanni, and Giacomo(AG) To Nicoletta, Andrea, and Bianca(at) To Maria Teresa and ludovico(Av) Preface It is well known that a large number of problems relevant to the control field can be formulated as optimization problems. For long time, the classical approach has been to look for a closed form solution to the specific optimization problems at hand. The last decade has seen a noticeable shift in the meaning of "closed form?' solution The formidable increase of computational power has dramatically changed the feel- ing of theoreticians as well as of practitioners about what is meant by tractable and untractable problems. A main issue regards convexity. From a theoretical viewpoint, there has been a large amount of work in the directions of"convexifying "noncon vex problems and studying structural features of convex problems. On the other hand, extremely powerful algorithms for the solution of convex problems have been devised in the last two decades. Clearly, the fact that a wide variety of engineer ing problems can be formulated as convex problems has strongly motivated efforts in this direction. The control field is not an exception in this sense: many prob- lems in robust control, identification and nonlinear control have been recognized as convex problems. Moreover, convex relaxations of nonconvex problems have been intensively investigated, as they provide an effective tool for bounding the optimal solution of the original problem As far as robust control is concerned, it is known since long time that several classes of problems can be reduced to testing positivity of suitable polynomials. Re markable examples are: the construction of Lyapunov functions and the evaluation of the stability margin for systems affected by structured uncertainty; the estima tion of the domain of attraction of nonlinear systems; the synthesis of fixed-order foo optimal controllers; the robust disturbance rejection for nonlinear systems, and many others. In recent years, it has been recognized that positivity of polynom- als can be tackled effectively in the framework of linear matrix inequality (Lmi problems, which are a special class of convex optimizations problems enjoying a number of appealing properties, like solvability in polynomial time. A fundamental relaxation consists of replacing the condition for a polynomial to be positive with the condition that it is a sum of squares of polynomials (soS). The interest for this relaxation is motivated by the facts that testing whether a polynomial is sos boils VIl Preface down to an LMi problem, and that the conservatism of this relaxation can be reduced by suitably increasing the degree of the polynomial This book describes some techniques developed recently by the authors and a number of researchers in the field, in order to address stability and performance analysis of uncertain systems affected by structured parametric uncertainties. Con vex relaxations for different robustness problems are constructed by employing ho mogeneous forms(hereafter, simply forms), i.e. polynomials whose terms have the same degree. Such forms are used to parametrize various classes of lyapunov func tions. The proposed solutions are characterized in terms of LMI problems and show a number of nice theoretical and practical features, which are illustrated throughout the book Organization of the book Chapter introduces the square matricial representation(SMR), which is a powerful tool for the representation of forms as it allows one to establish whether a form is SOS via an LMi feasibility test. This tool is then extended to address the representa tion of matrix forms and the characterization of sos matrix forms. It is shown that for a given form, an SOs index can be computed by solving an eigenvalue problem (EVP), which is the minimization of a linear function subject to LMI constraints also known as semidefinite program. Moreover, it is shown how the positivity of a polynomial on an ellipsoid and the positivity of a matrix polynomial on the simplex can be cast in terms of positivity of suitable forms. The chapter also provides con ditions under which vectors related to a given homogeneous polynomial structure lie in assigned linear spaces. This is useful in order to study the conservatism of SOS-based relaxations Chapter 2 investigates the relationship between convex relaxations for positiv ity of forms and Hilberts 17th problem, which concerns the existence of positive forms which are not sos forms(PNS). The concepts of maximal Smr matrices and SMR-tight forms are introduced, which allow one to derive a posteriori tight ness conditions for LMI optimizations arising in SOS relaxations. Also, this chapter provides results on Hilberts 17th problem based on the Smr, showing that each PNS form is the vertex of a cone of pns forms, and providing a parametrization of the set of pns forms Chapter 3 addresses robust stability and robust performance analysis of time varying polytopic systems, i.e. uncertain systems affected by linear dependent time arying uncertainties constrained in a polytope. It is shown how robustness prop erties can be investigated by using homogeneous polynomial Lyapunov functions (HPLFS), a non-conservative class of Lyapunov functions whose construction can be tackled by solving LMI problems such as LMi feasibility tests or generalized eigenvalue problemS(GEVPs), being the latter a class of quasi-convex optimizations with LMi constraints. Moreover, a priori conditions for tightness of the considered relaxations are provided. The extension to the case of uncertain systems with ra tional dependence on the uncertain parameters is derived through linear fractional representations (LFRS Preface Chapter !! investigates robustness analysis of time-invariant polytopic systems by adopting homogeneous parameter-dependent quadratic Lyapunov functions(HPD QLFS), again a non-conservative class of Lyapunov functions. The chapter pro vides a posteriori tests for establishing non-conservatism of the bounds obtained for robust stability margin or robust performance. alternative conditions for assess ing robust stability and instability of time-invariant polytopic systems are provided through LMi optimizations resulting from classical stability criteria. Moreover, it is shown how such results can be extended to the case of uncertain systems with rational dependence on the uncertain parameter, and to the case of discrete-time systems Chapter 5 considers the case of polytopic systems with time-varying uncertainties and finite bounds on their variation rate and illustrates how robustness analysis can be addressed by using homogeneous parameter-dependent homogeneous Lyapunov functions(HPD-HLFs) This class of functions include all possible forms in the state variables and uncertain parameters and therefore recovers the classes of hplfs and HPD-QLFs as special cases. The chapter shows how the construction of HPD-QLFS can be formulated in terms of LMI problems, and highlights the role played by the degrees of the lyapunov function in the state variables and uncertain parameters Lastly, Chapter treats quadratic distance problems(QDPs), i. e the computation of the minimum weighted euclidean distance from a point to a surface defined by a polynomial equation. This special class of nonconvex optimization problems finds numerous applications in control systems. It is shown that a lower bound to the solution of a QDp can be obtained through a sequence of lMi feasibility tests. a priori and a posteriori necessary and sufficient conditions for tightness of the lower bound are derived. The proposed technique is applied to the computation of the parametric stability margin of time-invariant polytopic systems Acknowledgements We thank various colleagues for the fruitful discussions in last years that contributed to the composition of this book, in particular F. blanchini, P.-A. Bliman, S. Boyd P Colaneri, Y Ebihara, Y. Fujisaki, M. Fujita, R. Genesio, D. Henrion, Y.S. Hung J -B Lasserre, J Lofberg, M. Khammash, L. Ljung, A Masi, S Miani, Y Oishi, C. Papageorgiou, S Prajna, M. Sato, C.W. Scherer and A. Seuret This book was typeset by the authors using ATEX. All computations were done using matlab and SeDuMi as much as we wish the book to be free of errors we know this will not be the case. Therefore, we will greatly appreciate to receive reports of errors, which can be Prefa sent to the following address: chesideee. hku. hk. An up-to-date errata list will beavailableatthefollowinghomepagehttp://www.eee.hku.hk/chesi Hong Kong Graziano Chesi Siena Andrea garulli Firenze Alberto tesi lena Antonio vicino May 2009 Contents Notation ..ⅩV Abbreviations XVII 1 Positive forms 1.1 Forms and polynomials 1.2 Representation via Power Vectors 1. 3 Representation via SMr 1.3.1 Equivalent sMr matrices 1.3.2 Complete SMR 1. 4 SOS Forms 单···· 244799 1. 4.1 SOS Tests based on the smr 1. 4.2 SOS Index 1. 5 Matrix Forms 13 1.5.1 SMR of matrix Forms 14 1. 5.2 SOS Matrix Forms 16 1. 6 Positive forms 1.6. 1 Positivity Index 1.6.2 Sufficient Condition for Positivity 88o 1. 6. 3 Positive matrix Forms 20 1.7 Positive Polynomials on Ellipsoids 22 1.7.1 Solution via positivity test on a Form 22 1.7.2 Even polynomials 24 1.8 Positive Matrix Polynomials on the simplex.......... 25 1.9 Extracting Power Vectors from Linear Spaces 27 1. 9.1 Basic Procedure 28 1.9.2 Extended Proceed 32 1.10 Notes and references 36

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