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Linear Matrix Inequalities in
System and Control Theory
SIAM Studies in Applied Mathematics
This series of monographs focuses on mathematics and its applications to problems
of current concern to industry, government, and society. These monographs will be of
interest to applied mathematicians, numerical analysts, statisticians, engineers, and
scientists who have an active need to learn useful methodology.
Series List
Vol. 1 Lie-B¨acklund Transformations in Applications
Robert L. Anderson and Nail H. Ibragimov
Vol. 2 Methods and Applications of Interval Analysis
Ramon E. Moore
Vol. 3 Ill-Posed Problems for Integrodifferential Equations in Mechanics and
Electromagnetic Theory
Frederick Bloom
Vol. 4 Solitons and the Inverse Scattering Transform
Mark J. Ablowitz and Harvey Segur
Vol. 5 Fourier Analysis of Numerical Approximations of Hyperbolic Equations
Robert Vichnevetsky and John B. Bowles
Vol. 6 Numerical Solution of Elliptic Problems
Garrett Birkhoff and Robert E. Lynch
Vol. 7 Analytical and Numerical Methods for Volterra Equations
Peter Linz
Vol. 8 Contact Problems in Elasticity: A Study of Variational Inequalities and
Finite Element Methods
N. Kikuchi and J. T. Oden
Vol. 9 Augmented Lagrangian and Operator-Splitting Methods in Nonlinear
Mechanics
Roland Glowinski and P. Le Tallec
Vol. 10 Boundary Stabilization of Thin Plate Splines
John E. Lagnese
Vol. 11 Electro-Diffusion of Ions
Isaak Rubinstein
Vol. 12 Mathematical Problems in Linear Viscoelasticity
Mauro Fabrizio and Angelo Morro
Vol. 13 Interior-Point Polynomial Algorithms in Convex Programming
Yurii Nesterov and Arkadii Nemirovskii
Vol. 14 The Boundary Function Method for Singular Perturbation Problems
Adelaida B. Vasil’eva, Valentin F. Butuzov, and Leonid V. Kalachev
Vol. 15 Linear Matrix Inequalities in System and Control Theory
Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan
Balakrishnan
Stephen Boyd, Laurent El Ghaoui,
Eric Feron, and Venkataramanan Balakrishnan
Linear Matrix Inequalities in
System and Control Theory
Society for Industrial and Applied Mathematics
®
Philadelphia
Copyright
c
° 1994 by the Society for Industrial and Applied Mathematics.
All rights reserved. No part of this book may be reproduced, stored, or transmitted in any
manner without the written permission of the Publisher. For information, write to the Society
for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia,
Pennsylvania 19104-2688.
The royalties from the sales of this book are being placed in a fund to help students
attend SIAM meetings and other SIAM related activities. This fund is administered by
SIAM and qualified individuals are encouraged to write directly to SIAM for guidelines.
Library of Congress Cataloging-in-Publication Data
Linear matrix inequalities in system and control theory / Stephen Boyd
. . . [et al.].
p. cm. -- (SIAM studies in applied mathematics ; vol. 15)
Includes bibliographical references and index.
ISBN 0-89871-334-X
1. Control theory. 2. Matrix inequalities. 3. Mathematical
optimization. I. Boyd, Stephen P. II. Series: SIAM studies in
applied mathematics : 15.
QA402.3.L489 1994
515’.64--dc20 94-10477
Contents
Preface vii
Acknowledgments ix
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Brief History of LMIs in Control Theory . . . . . . . . . . . . . . . 2
1.3 Notes on the Style of the Book . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Origin of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Some Standard Problems Involving LMIs 7
2.1 Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Some Standard Problems . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Ellipsoid Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Strict and Nonstrict LMIs . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Miscellaneous Results on Matrix Inequalities . . . . . . . . . . . . . . 22
2.7 Some LMI Problems with Analytic Solutions . . . . . . . . . . . . . . 24
Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Some Matrix Problems 37
3.1 Minimizing Condition Number by Scaling . . . . . . . . . . . . . . . . 37
3.2 Minimizing Condition Number of a Positive-Definite Matrix . . . . . . 38
3.3 Minimizing Norm by Scaling . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Rescaling a Matrix Positive-Definite . . . . . . . . . . . . . . . . . . . 39
3.5 Matrix Completion Problems . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Quadratic Approximation of a Polytopic Norm . . . . . . . . . . . . . 41
3.7 Ellipsoidal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 42
Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Linear Differential Inclusions 51
4.1 Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Some Specific LDIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Nonlinear System Analysis via LDIs . . . . . . . . . . . . . . . . . . . 54
Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Analysis of LDIs: State Properties 61
5.1 Quadratic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Invariant Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
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