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)

Topics in Matrix and Operator Theory

Gohberg, I. and M.A. Kaashoek, Asymptotic formulas of Szego-Kac-Achiezer type, Asymptotic Analysis 5 (1992) 187-220. A generalization of Szego's first limit theorem is deduced for a general class of block Toeplitz operators with Hilbert -Schmidt entries. Analogous formulas for Wiener-Hopf integral operators are derived, and by discretization a direct connection between the two types of asymptotics is established. For the continuous case the results are specified further for operators with a rational matrix symbol.

Asymptotic formulas of Szegö–Kac–Achiezer type

A (X) = F(X)B(X)E(X), \\eSl. The concept of equivalence is also of interest in the case that (X = Y and) A and B are linear functions of the form C XI. In that case it provides a language in which the spectral structure of a linear operator at a point may be classified. Two operators T and S are said to have the same spectral structure at a point X0 if the operator functions T-XI and S XI are equivalent on an open neighbourhood of X0. More generally, we say that two operator functions A and B belong to the same spectral class at a point X0 if there exists an open neighbourhood of X0 on which A and B are equivalent. Let D(X) = (X X0) P1 + . . . 4(X X0) \"Pn + P0, where kl9...9kn are positive integers, Px, . . . , Pn are mutually disjoint one dimensional projections and P0 = I (Px 4• • • 4Pn). An operator function A belongs to the spectral class generated by D at X0 if and only if ^(X) is bijective for X near X0, A(X0) is Fredholm and the partial multiplicities of A at X0 are given by the numbers kl9 . . . , kn (see [5]). Other examples of spectral classes can be found in [ l ] a n d [ 7 ] . The problem of finding the simplest representative of a spectral class containing a given operator function A has many variants. One possibility is to look for functions of the form T X/. It is clear that in this way the problem is not always solvable. However after a suitable extension of the operator function A and the underlying spaces the problem has a positive solution. If Z is a Banach space, the Z-extension of A is the operator function whose value at X is the operator A(X) 0 Iz in L(X 0 Z, Y 0 Z), i.e., the direct sum of

https://projecteuclid.org/download/pdf_1/euclid.bams/1183538134

Spectral classification of operators and operator functions

We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower triangular part of a special form. For such a specified part the unitary completion is a structured matrix, and we derive in this paper the formulas for its structure. Next we apply the unitary completion method to the solution of the eigenvalue problem for a class of structured matrices via structured QR iterations.

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Israel Gohberg

Papers

Topics in Matrix and Operator Theory

Asymptotic formulas of Szegö–Kac–Achiezer type

Spectral classification of operators and operator functions

The unitary completion and QR iterations for a class of structured matrices

Matrix and Operator Valued Functions