Linear Matrix Inequalties in System and Control Theory: (Studies in Applied and Numerical Mathematics)

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. Special Features: The book identifies a handful of standard optimization problems that are general (a wide variety of problems from system and control theory can be reduced to them) as well as specific (specialized numerical algorithms can be devised for them). Catalogues a diverse list of problems in system and control theory that can be reduced to the standard optimization problems. Problems considered are analysis and state feedback design for uncertain systems, matrix analysis problems, and many others. Most of the the book is accessible to anyone with a basic mathematics background, e.g., linear algebra and differential equations.

Table of Contents:
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: Lur'e and Multiplier Methods. Analysis of Lur'e 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.