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)

Extensions of kernels of Fredholm operators

Discrete Analogs of Canonical Systems with Pseudo-exponential Potential. Definitions and Formulas for the Spectral Matrix Functions.- Matrix-J-unitary Non-commutative Rational Formal Power Series.- State/Signal Linear Time-Invariant Systems Theory, Part I: Discrete Time Systems.- Conservative Structured Noncommutative Multidimensional Linear Systems.- The Bezout Integral Operator: Main Property and Underlying Abstract Scheme.

The state space method : generalizations and applications

Explicit formulas for Wiener-Hopf factorization of rational matrix and analytic operator functions relative to a closed contour are constructed. The formulas are given in terms of a realization of the functions. Also formulas for the factorization indices are given.

Explicit Wiener-Hopf Factorization and Realization

Description de tous les diviseurs minimaux en un point donne d'une fonction matricielle meromorphe en termes de paires spectrales locales et de matrice de couplage locale

Minimal factorization of meromorphic matrix functions in terms of local data

A general class of strictly contractive block matrix valued functions g(ζ) = [gij (ζ)], i,j = 1,…,k, whose block entries gij are completely specified for j < i+τ (i.e., for gij below the τ’th block diagonal of g) and partially specified for j = i + τ (i.e., for gij on the τ’th block diagonal of g) is analyzed. Necessary and sufficient conditions for this class of “interpolants” to be nonempty are deduced. A linear fractional description of this class is then obtained and used to establish a maximum entropy principle. In particular it is shown that when this class is nonempty, then it contains exactly one interpolant which achieves the maximum entropy. This maximum entropy interpolant is also characterized as the “band” interpolant.

Israel Gohberg

Papers

Extensions of kernels of Fredholm operators

The state space method : generalizations and applications

Explicit Wiener-Hopf Factorization and Realization

Minimal factorization of meromorphic matrix functions in terms of local data

A New Class of Contractive Interpolants and Maximum Entropy Principles