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.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

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)

A general theory of tracestr

 D
 
A and determinantsdet

 D
 
(I+A) in normed algebrasD of operators acting in Banach spacesB is proposed. In this approach trace and determinant are defined as continuous extensions of the corresponding functionals from finite dimensional operators. We characterize the algebras for which such extensions exist and describe sets of possible values of traces and determinants for the same operator in different algebras. In spite of the fact that the extended traces and determinants may differ in different algebrasD, operatorI+A (withA ∈D) is invertible inB if and only ifdet

 D
 
(I+A) does not vanish. Cramer's rule and formulas for the resolvent are obtained and they are expressed in different algebras by the same formulas viadet

 D
 
(I+A) andtr

 D
 
(A). A large set of examples and illustrations are also presented.

http://www.gbv.de/dms/ilmenau/toc/31041914X.PDF

Traces and determinants of linear operators

1 The operator of singular integration.- 1.1 Notations, definitions and auxiliary statements.- 1.1.1 The operator of singular integration.- 1.1.2 The space Lp(?,?).- 1.1.3 Interpolation theorems.- 1.2 The boundedness of the operator S? in the space Lp(?) with ? being a simple curve.- 1.3 Nonsimple curves.- 1.4 Integral operators in weighted Lp spaces.- 1.5 Unbounded curves.- 1.6 The operator of singular integration in spaces of Holder continuous functions.- 1.7 The operator S?*.- 1.8 Exercises.- Comments and references.- 2 One-sided invertible operators.- 2.1 Direct sum of subspaces.- 2.2 The direct complement.- 2.3 Linear operators. Notations and simplest classes.- 2.4 Projectors connected with the operator of singular integration.- 2.5 One-sided invertible operators.- 2.6 Singular integral operators and related operators.- 2.7 Examples of one-sided invertible singular integral operators.- 2.8 Two lemmas on the spectrum of an element in a subalgebra of a Banach algebra.- 2.9 Subalgebras of a Banach algebra generated by one element.- 2.10 Exercises.- Comments and references.- 3 Singular integral operators with continuous coefficients.- 3.1 The index of a continuous function.- 3.2 Singular integral operators with rational coefficients.- 3.3 Factorization of functions.- 3.4 The canonical factorization in a commutative Banach algebra.- 3.5 Proof of the factorization theorem.- 3.6 The local factorization principle.- 3.7 Operators with continuous coefficients.- 3.8 Approximate solutions of singular integral equations.- 3.9 Generalized factorizations of continuous functions.- 3.10 Operators with continuous coefficients (continuation).- 3.11 Additional facts and generalizations.- 3.12 Operators with degenerating coefficients.- 3.13 A generalization of singular integral operators with continuous coefficients.- 3.14 Solution of Wiener-Hopf equations.- 3.15 Some applications.- 3.16 Exercises.- Comments and references.- 4 Fredholm operators.- 4.1 Normally solvable operators.- 4.2 The restriction of normally solvable operators.- 4.3 Perturbation of normally solvable operators.- 4.4 The normal solvability of the adjoint operator.- 4.5 Generalized invertible operators.- 4.6 Fredholm operators.- 4.7 Regularization of operators. Applications to singular integral operators.- 4.8 Index and trace.- 4.9 Functions of Fredholm operators and their index.- 4.10 The structure of the set of Fredholm operators.- 4.11 The Dependence of kerX and imX on the operator X.- 4.12 The continuity of the function kx.- 4.13 The case of a Hilbert space.- 4.14 The normal solvability of multiplication by a matrix function.- 4.15 ?+--operators.- 4.16 One-sided regularization of operators.- 4.17 Projections of invertible operators.- 4.18 Exercises.- Comments and references.- 5 Local Principles and their first applications.- 5.1 Localizing classes.- 5.2 Multipliers on $$ \mathop l\limits^ \sim _p $$.- 5.3 paired equations with continuous coefficients on $$ \mathop l\limits^ \sim _p $$.- 5.4 Operators of local type.- 5.5 Exercises.- Comments and references.- References.

One-dimensional linear singular integral equations

Classes of Linear Operators Vol. I

Matrices and indefinite scalar products

Preface.- 1. Introduction and Outline.- 2. Indefinite Inner Products.- 3. Orthogonalization and Orthogonal Polynomials.- 4. Classes of Linear Transformations.- 5. Canonical Forms.- 6. Real H-Selfadjoint Matrices.- 7. Functions of H-Selfadjoint Matrices.- 8. H-Normal Matrices.- 9. General Perturbations. Stability of Diagonalizable Matrices.- 10. Definite Invariant Subspaces.- 11. Differential Equations of First Order.- 12. Matrix Polynomials.- 13. Differential and Difference Equations of Higher Order.- 14. Algebraic Riccati Equations.- Appendix: Topics from Linear Algebra.- Bibliography.- Index

Israel Gohberg

Papers

Traces and determinants of linear operators

One-dimensional linear singular integral equations

Classes of Linear Operators Vol. I

Matrices and indefinite scalar products

Indefinite Linear Algebra and Applications