Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 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 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 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 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 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 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.

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Linear Matrix Inequalities in System and Control Theory

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

Semidefinite programming

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

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Essentials of Robust Control

First published in 1995, Wavelets and Subband Coding offered a unified view of the exciting field of wavelets and their discrete-time cousins, filter banks, or subband coding. The book developed the theory in both continuous and discrete time, and presented important applications. During the past decade, it filled a useful need in explaining a new view of signal processing based on flexible time-frequency analysis and its applications. Since 2007, the authors now retain the copyright and allow open access to the book.

Wavelets and Subband Coding

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Fast Algorithms with Preprocessing for Matrix-Vector Multiplication Problems

in die Theorie der eindimensionalen singularen Integraloperatoren von 1. Gohberg und N. Krupnik In Deutscher Sprache herausgegeben von Prof. Dr. E. Meister Deutsche Ubersetzungvon Dr. B. Schiippel Springer Basel AG 1979 H.:0;. rox6epr H H. R. KpynHHK BBe eHHe B TeOpHIO O HOMepHbIX CHHryIDIPHbIX HlITerpaJILHbIX OnepaTOpOB erschien im Verlag "Stiinca" Kisinev 1973 CIP-Kurztitelaufnahme der Deutschen Bibliothek Gobberg, Izrail' c.: Einflihrung in die Theorie der eindimensionalen singulliren Integraloperationenlvon I. Gohberg u. N. Krupnik. In dt. Sprache hrsg. von E. Mei ster. Dt. Obers. von B. Schiippel. - Basel, Boston, Stuttgart: Birkhliuser, 1979. (Lehrbiicher und Monographien aus dem Gebiete der exakten Wissenschaften: Math. Reihe; Bd. 63) Einheitssacht.: Vvedenie v teoriju odnomernych singuljarnych integral'nych operatorov NE: Krupnik, Naum: Nachdruck verboten, Aile Rechte, insbesondere das der Obersetzung in fremde Sprachen und der Reproduktion aufphotostatischem Wege oder durch Mikrofilm, vorbehalten. (c) Springer Basel AG 1979 Urspriinglich erschienen bei Birkhliuser Verlag Basel 1979. Softcover reprint of the hardcover 1st edition 1979 ISBN 978-3-0348-5556-3 ISBN 978-3-0348-5555-6 (eBook) DOI 10. 1007/978-3-0348-5555-6 Inhaltsverzeichnis 5 Vorwort zur deutschen Ausgabe 11 Vorwort zur russischen Ausgabe 12 Einleitung. . . . . . . . 13 Kapitel I Der Operator der singuHiren Integration 17 1 Bezeichnungen, Definitionen und Hilfssatze 17 1. Der Operator der singularen Integration 17 2. Der Raum 4(f, p). . . . . 19 3. Interpolationstheoreme 20 2 Die Beschranktheit des Operators Sr im Raum 4([), wobei f eine einfache Kurve ist 20 3 Nichteinfache Kurven . . . . . 26 4 Integraloperatoren im Raum 4 mit Gewicht 29 5 Unbeschrankte Kurve . . . . . . . ."

Einführung in die Theorie der eindimensionalen singulären Integraloperatoren

Spectral analysis of selfadjoint matrix polynomials

This paper is concerned with minimal factorizations of rational matrix functions. The treatment is based on a new geometrical principle. In fact, it is shown that there is a one-to-one correspondence between minimal factorizations on the one hand and certain projections on the other. Considerable attention is given to the problem of stability of a minimal factorization. Also the numerical aspects are discussed. Along the way, a stability theorem for solutions of the matrix Riccati equation is obtained.

Factorizations of Transfer Functions

These lecture notes present an extensive review of the factorization theory of matrix functions relative to a curve with emphasis on the developments of the last 20–25 years. The classes of functions considered range from rational and continuous matrix functions to matrix functions with almost periodic or even semi almost periodic entries. Also included are recent results about explicit factorization based on the state space method from systems theory, with examples from linear transport theory. Related applications to Riemann-Hilbert boundary value problems and the Fredholm theory of various classes of singular integral operators are described too. The applications also concern inversion of singular integral operators of different types, including Wiener-Hopf and Toeplitz operators.

Israel Gohberg

Papers

Fast Algorithms with Preprocessing for Matrix-Vector Multiplication Problems

Einführung in die Theorie der eindimensionalen singulären Integraloperatoren

Spectral analysis of selfadjoint matrix polynomials

Factorizations of Transfer Functions

An Overview of Matrix Factorization Theory and Operator Applications