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Linear Matrix Inequalities in System and Control Theory

Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan

Studies in Applied Mathematics 15

In this book the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimization problems that involve linear matrix inequalities. These optimization problems can be solved using recently developed numerical algorithms that not only are polynomial-time but also work very well in practice; the reduction therefore can be considered a solution to the original problems. This book opens up an important new research area in which convex optimization is combined with system and control theory, resulting in the solution of a large number of previously unsolved problems.


This book is primarily intended for researchers in system and control theory; both the beginner and the advanced researcher will find the book useful.

Researchers in convex optimization will find this book a source of optimization problems for which algorithms need to be devised. A background in linear algebra, elementary analysis, and exposure to differential equations and system and control theory is recommended.


Preface; Chapter 1: Introduction. Overview; A Brief History of LMIs in Control Theory; Notes on the Style of the Book; Origin of the Book; Chapter 2: Some Standard Problems Involving LMIs. Linear Matrix Inequalities; Some Standard Problems; Ellipsoid Algorithm; Interior-Point Methods; Strict and Nonstrict LMIs; Miscellaneous Results on Matrix Inequalities; Some LMI Problems with Analytic Solutions; Chapter 3: Some Matrix Problems. Minimizing Condition Number by Scaling; Minimizing Condition Number of a Positive-Definite Matrix; Minimizing Norm by Scaling; Rescaling a Matrix Positive-Definite; Matrix Completion Problems; Quadratic Approximation of a Polytopic Norm; Ellipsoidal Approximation; Chapter 4: Linear Differential Inclusions. Differential Inclusions; Some Specific LDIs; Nonlinear System Analysis via LDIs; Chapter 5: Analysis of LDIs: State Properties. Quadratic Stability; Invariant Ellipsoids; Chapter 6: Analysis of LDIs: Input/Output Properties. Input-to-State Properties; State-to-Output Properties; Input-to-Output Properties; Chapter 7: State-Feedback Synthesis for LDIs. Static State-Feedback Controllers; State Properties; Input-to-State Properties; State-to-Output Properties; Input-to-Output Properties; Observer-Based Controllers for Nonlinear Systems; Chapter 8: Luré and Multiplier Methods. Analysis of Luré Systems; Integral Quadratic Constraints; Multipliers for Systems with Unknown Parameters; Chapter 9: Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise; State-Feedback Synthesis; Chapter 10: Miscellaneous Problems. Optimization over an Affine Family of Linear Systems; Analysis of Systems with LTI Perturbations; Positive Orthant Stabilizability; Linear Systems with Delays; Interpolation Problems; The Inverse Problem of Optimal Control; System Realization Problems; Multi-Criterion LQG; Nonconvex Multi-Criterion Quadratic Problems; Notation; List of Acronyms; Bibliography; Index.

Royalties from the sale of this book are contributed to the SIAM Student Travel Fund.

1994 / ix + 193 pages / Softcover / ISBN-13: 978-0-898714-85-2 / ISBN-10: 0-89871-485-0 /
List Price $71.50 / SIAM Member Price $50.05 / Order Code AM15
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