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)

In this paper, we prove continuous analogues of results concerning the distribution of zeros of orthogonal matrix functions related to the Nehari problem.

Distribution of Zeros of Orthogonal Functions Related to the Nehari Problem

In this chapter we study a variant of the Nevanlinna-Pick interpolation problem where one seeks a function analytic and bounded by 1 on a domain which meets a set of interpolation conditions, some of which may now be prescribed on the boundary of the domain. When the interpolation conditions are of a suitable form, it develops that the set of all solutions is parametrized by a linear fractional map induced by a J-inner function having poles on the boundary, a phenomenon originally observed by Nevanlinna. Matrix extensions of this result provide an area of application for results in Chapters 6 and 7 concerning the structure of such J-inner functions, and form a natural extension of the results of Chapters 18 and 19 to the boundary case. This boundary interpolation is also sometimes called Loewner interpolation.

Boundary Nevanlinna-Pick Interpolation

Editorial Introduction D. Alpay, P. Loubaton The tangential trigonometric moment problem on an interval and related topics 1. Introduction 2. Some lemmas on matrix-valued rational functions 3. The main result 4. The Nevanlinna Pick problem References. M. Bakonyi, V.G. Kaftal, G. Weiss, H.J. Woerdeman Maximum entropy and joint norm bounds for operator extensions 1. Introduction 2. A sharp bound in the 2 x 2 case 3. The maximum entropy method 4. An application to integral operators References. J.A. Ball, 1. Gohberg, M.A. Kaashoek Bitangential interpolation for input-output operators of time varying systems: the discrete time case 0. Introduction 1. Residue calculus and generalized point evaluation 2. Pairs of diagonal operators and homogeneous one-sided interpolation 3. Bitangential interpolation data set 4. Bitangential interpolation in geometric terms 5. Intermezzo about admissible Sylvester data sets 6. Construction of a particular solution 7. Parametrization of all solutions (without norm constraints) 8. Input-output operators of time-varying systems 9. Parametrization of all contractive input-output operators satisfying the bitangential interpolation conditions References. J.A.Ball, I.Gohberg, L. Rodman Two-sided tangential interpolation of real rational matrix functions 1. Introduction 2. Minimal realizations 3. Local data 4. Two-sided tangential interpolation: existence of real interpolants 5. Two-sided tangential interpolation with real-valued data: Description of interpolants 6. Degrees of interpolants 7. Generalized Nevanlinna-Pick interpolation for real rational matrix functions References. H. Du, C. Gu On the spectra of operator completion problems 1. Introduction 2. Case of finite dimensional spaces 3. Case of infinite dimensional spaces References. C. Foias, A.E. Frazho, W.S. Li The exact H2 estimate for the central H interpolant Introduction 1. An improved Kaftal-Larson-Weiss estimate 2. Some formulas for DB 3. The role of DA2II0 4. The four block problem 5. Optimal solutions References. A.E. Frazho, S.M. Kherat On mixed H2 - H tangential interpolation 1. Introduction 2. Formulas for the central solution 3. A state space approach 4. Applications of the H2-H tangential interpolation problem References. I. Gohberg, C. Gu On a completion problem for matrices 1. Introduction 2. Main theorems in the finite dimensional case 3. The full range case 4. The proof of the main theorems in the finite dimensional case 5. Infinite dimensional case References.

New aspects in interpolation and completion theories

In this chapter we deal with integral operators of the following type: 


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Wiener-Hopf Integral Operators

For a class of entire matrix-functions a continuous analogue of the classical Bezout matrix for scalar polynomials is introduced and studied. This analogue is an integral operator with a matrix-valued kernel. The null space of this operator is explicitly expressed in terms of the common eigenvectors and common Jordan chains of the two underlying entire matrix functions. Also a refinement of the abstract scheme from [17] for defining Bezout operators is presented and analyzed. The approach of the paper is based, to a large extent, on the state space method from mathematical system theory. In particular, an important role is played by the fact that the functions involved can be represented as transfer functions of certain infinite-dimensional input output systems.

Israel Gohberg

Papers

Distribution of Zeros of Orthogonal Functions Related to the Nehari Problem

Boundary Nevanlinna-Pick Interpolation

New aspects in interpolation and completion theories

Wiener-Hopf Integral Operators

The Bezout integral operator: main property and underlying abstract scheme. The state space method generalizations and applications