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)

Introduction to the theory of linear nonselfadjoint operators

Preface to Volume II Table of contents of Volume II Introduction PART V: TRIANGULAR REPRESENTATIONS XX Additive lower-upper triangular decompositions of operators 1 Additive lower-upper triangular decompositions relative to finite chains 2 Preliminaries about chains 3 Diagonals 4 Chains on Hilbert space 5 Triangular algebras 6 Riemann-Stieltjes integration along chains 7 Additive lower-upper decomposition theorem 8 Additive lower-upper decomposition of a Hilbert-Schmidt operator 9 Multiplicative integration along chains 10 Basic properties of reproducing kernel Hilbert spaces and chains 11 Example of an additive LU-decomposition in a RKHS XXI Operators in triangular form 1 Triangular representation 2 Intermezzo about completely nonselfadjoint operators 3 Volterra operators with a one-dimensional imaginary part 4 Unicellular operators XXII Multiplicative lower-upper triangular decompositions of operators 1 LU-factorization with respect to a finite chain 2 The LU-factorization theorem 3 LU-factorizations of compact perturbations of the identity 4 LU-factorizatioris of Hilbert-Schmidt perturbations of the identity 5 LU-factorizations of integral operators 6 Triangular representations of operators close to unitary 7 LU-factorization of semi-separable integral operators 8 Generalised Wiener-Hopf equations 9 Generalised LU-factorization relative to discrete chains Comments on Part V Exercises to Part V PART VI: CLASSES OF TOEPLITZ OPERATORS XXIII Block Toeplitz operators 1 Preliminaries 2 Block Laurent operators 3 Block Toeplitz operators 4 Block Toeplitz operators defined by continuous functions 5 The Fredholm index of a block Toeplitz operator defined by a continuous function XXIV Toeplitz operators defined by rational matrix functions 1 Preliminaries 2 Invertibility and Fredholm index (scalar case) 3 Wiener-Hopf factorization 4 Invertibility and Fredholm index (matrix case) 5 Intermezzo about realisation 6 Inversion of a block Laurent operator 7 Explicit canonical factorization 8 Explicit inversion formulas 9 Explicit formulas for Fredholm characteristics 10 An example 11 Asymptotic formulas for determinants of block Toeplitz matrices XXV Toeplitz operators defined by piecewise continuous matrix functions 1 Piecewise continuous functions 2 Symbol and Fredholm index (scalar case) 3 Symbol and Fredholm index (matrix case) 4 Sums of products of Toeplitz operators defined by piecewise continuous functions 5 Sums of products of block Toeplitz operators defined by piecewise continuous functions Comments on Part VI Exercises to Part VI PART VII: CONTRACTIVE OPERATORS AND CHARACTERISTIC OPERATOR FUNCTIONS XXVI Block shift operators 1 Forward shifts and isometries 2 Parts of block shift operators 3 Invariant subspaces of forward shift operators XXVII Dilation theory 1 Preliminaries about contractions 2 Preliminaries about dilations 3 Isometric dilations 4 Unitary dilations

Classes of Linear Operators

Convolution Equations and Projection Methods for Their Solution

I Homogeneous Interpolation Problems with Standard Data.- 1. Null Structure for Analytic Matrix Functions.- 2. Null Structure and Interpolation Problems for Matrix Polynomials.- 3. Local Data for Meromorphic Matrix Functions.- 4. Rational Matrix Functions.- 5. Rational Matrix Functions with Null and Pole Structure at Infinity.- 6. Rational Matrix Functions with J-Unitary Values on the Imaginary Line.- 7. Rational Matrix Functions with J-Unitary Values on the Unit Circle.- II Homogeneous Interpolation Problems with Other Forms of Local Data.- 8. Interpolation Problems with Null and Pole Pairs.- 9. Interpolation Problems for Rational Matrix Functions Based on Divisibility.- 10. Polynomial Interpolation Problems Based on Divisibility.- 11. Coprime Representations and an Interpolation Problem.- III Subspace Interpolation Problems.- 12. Null-Pole Subspaces: Elementary Properties.- 13. Null-Pole Subspaces for Matrix Functions with J-Unitary Values on the Imaginary Axis or Unit Circle.- 14. Subspace Interpolation Problems.- 15. Subspace Interpolation with Data at Infinity.- IV Nonhomogeneous Interpolation Problems.- 16. Interpolation Problems for Matrix Polynomials and Rational Matrix Functions.- 17. Partial Realization as an Interpolation Problem.- V Nonhomogeneous Interpolation Problems with Metric Constraints.- 18. Matrix Nevanlinna-Pick Interpolation and Generalizations.- 19. Matrix Nevanlinna-Pick-Takagi Interpolation.- 20. Nehari Interpolation Problem.- 21. Boundary Nevanlinna-Pick Interpolation.- 22. Caratheodory-Toeplitz Interpolation.- VI Some Applications to Control and Systems Theory.- 23. Sensitivity Minimization.- 24. Model Reduction.- 25. Robust Stabilizations.- Appendix. Sylvester, Lyapunov and Stein Equations.- A.1 Sylvester equations.- A.2 Stein equations.- A.3 Lyapunov and symmetric Stein equations.- Notes.- References.- Notations and Conventions.

Israel Gohberg

Papers

Introduction to the theory of linear nonselfadjoint operators

Classes of Linear Operators

Convolution Equations and Projection Methods for Their Solution

Interpolation of Rational Matrix Functions

Minimal Factorization of Matrix and Operator Functions