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)

Publications of Bernd Silbermann

In this chapter we consider matrix polynomials, i.e. matrix functions L(z) of the form \(L(z) = \sum\nolimits_{i = 0}^e {{A_i}{z^i},} \) where A 0 A l,..., A l are n × n matrices. We will assume throughout also that L(z) is regular, i.e. det L(z) does not vanish identically. One of our aims is to specify for matrix polynomials in a detailed way the theory developed in the first chapter for analytic matrix functions. We also solve here the first interpolation problem in this book, namely, how to construct a matrix polynomial with given null structure. Explicit formulas for the solution are obtained.

Null Structure and Interpolation Problems for Matrix Polynomials

In this paper it is proved that the band extension on the real line (viewed as a convolution operator) may be obtained as a limit in the operator norm of block Laurent operators of which the symbols are band extensions of appropriate discrete approximations of the given data.

The Band Extension on the Real Line as a Limit of Discrete Band Extensions, I. The Main Limit Theorem

An Overview of Matrix Factorization Theory and Operator Applications.- 0 Introduction.- 1 General theorems.- 2 Factorization in decomposing Banach algebras.- 3 Generalized Lp factorization.- 4 State space method.- References.- Matrix Riemann-Hilbert Problems Related to Branched Coverings of ??1.- 1 Introduction.- 2 Riemann-Hilbert problem with quasi-permutation monodromies and algebraic curves.- 3 Riemann surfaces. Rauch variational formulas.- 4 Solution of Riemann-Hilbert problems with quasi-permutation monodromies and Szegoe kernel.- 5 Isomonodromic tau-function and Cauchy-Riemann determinants.- Acknowledgements.- References.- Quantum Groups and Integrable Models.- 1 Introduction.- 2 Algebraic Bethe Ansatz and QISM.- 3 Yang-Baxter equation.- 4 Thermodynamic limits.- 5 Conclusion.- Acknowledgements.- References.- Integrable Systems and Factorization Problems.- 1 Introduction.- 2 A few preliminaries: Poisson brackets, coadjoint orbits, etc..- 3 Classical r-matrices and Lax equations.- 4 Classical Yang-Baxter identity.- 5 A finite-dimensional example.- 6 Loop algebras and the Riemann problem.- 7 More examples.- 8 Zero curvature equations.- 9 Difference equations and Poisson-Lie groups.- Acknowledgements.- References.- Programme of the Summer School on Factorization and Integrable Systems.

Factorization and integrable systems : summer school in Faro, Portugal, September 2000

Using the theory developed in Chapter IV, we now present some fundamental theorems concerning the spectral theory of compact self adjoint integral operators In general, the spectral series representations of these operators converge in the L2-norm which is not strong enough for many applications Therefore we prove the Hilbert-Schmidt theorem and Mercer’s theorem since each of these theorems gives conditions for a uniform convergence of the spectral decomposition of the integral operators As a corollary of Mercer’s theorem we obtain the trace formula for positive integral operators with continuous kernel function

Israel Gohberg

Papers

Publications of Bernd Silbermann

Null Structure and Interpolation Problems for Matrix Polynomials

The Band Extension on the Real Line as a Limit of Discrete Band Extensions, I. The Main Limit Theorem

Factorization and integrable systems : summer school in Faro, Portugal, September 2000

Spectral Theory of Integral Operators