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)

Classes of Linear Operators, Vol.I, Operator Theory

In this paper we continue the study of a class of structured matrices which admit a linear complexity inversion algorithm. This class contains in particular diagonal plus semiseparable matrices of order one and tridiagonal matrices. For this class explicit inversion formulas are obtained and linear complexity inversion algorithms are developed. The implementation of algorithms is illustrated by numerical experiments. The new class which is studied here appears naturally as the class of matrices of input output operators for descriptor linear systems.

Linear complexity inversion algorithms for a class of structured matrices

The paper contains characterizations of all rational matrix functions W(λ) which are contractions for every real λ and strict contractions at infinity. The results are obtained in terms of minimal realizations. A description of all minimal, unitary on the line completions of such functions is given. This description is used in the study of linear fractional decompositions of proper contractions W(λ).

Proper Contractions and their Unitary Minimal Completions

Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.

Convolution equations on finite intervals and factorization of matrix functions

Let W be a rational n×n matrix function, and let λ0∈θ. In a deleted neighbourhood of λ0 we have the following expansion 

$$ [tex]W(\lambda ) = {\sum\limits_{j=-q}^\infty {(\lambda - {\lambda _0})} ^j}{W_j}[/tex]$$

(4.1)

.

Israel Gohberg

Papers

Classes of Linear Operators, Vol.I, Operator Theory

Linear complexity inversion algorithms for a class of structured matrices

Proper Contractions and their Unitary Minimal Completions

Convolution equations on finite intervals and factorization of matrix functions

Minimal Factorizations of Rational Matrix Functions