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

1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.

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Spectral analysis of signals

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)

This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

Linear Robust Control

This paper contains a detailed treatment of the Nevanlinna–Pick interpolation problem for matrix-valued functions. A close relationship with the trigonometric moment problem is put into light. Matrix extensions of Pick’s solvability criterion and Nevanlinna’s iterative algorithm are presented. Necessary and sufficient conditions are obtained for uniqueness of the solution of the infinite interpolation problem. The approach is essentially based upon the theories of J-contractive transformations and of Weyl matrix circles.

The Nevanlinna–Pick Problem for Matrix-Valued Functions

The classical Levinson algorithm for computing the predictor polynomial relative to a real positive definite Toeplitz matrix is shown to be redundant in complexity. It can be broken down into two simpler algorithms, either of which needs only to be processed. This result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows: the symmetric and antisymmetric parts of the predictors relative to the sequence of Toeplitz matrices constitute two families of polynomials orthogonal on the interval [- 1,1] with respect to some even weight functions. It turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor. The resulting "split Levinson algorithm" requires roughly one-half the number of multiplications and the same number of additions as the classical Levinson algorithm. A simple derivation of Cybenko's method for computing the Pisarenko frequencies is obtained from the recurrence relations underlying the split Levinson algorithm.

The split Levinson algorithm

This paper proposes a natural matrix extension of the classical theory of orthogonal polynomials on the unit circle introduced by Szego. As a result, orthogonal polynomial matrices appear to be a unifying concept in various mathematical aspects of circuit and system theory.

Orthogonal polynomial matrices on the unit circle

Bounds are obtained for the cardinality of sets of lines having a prescribed number of angles, both in real and in complex Euclidean n-space. Extremal sets provide combinatorial configurations with a particular algebraic structure, such as association schemes and regular two-graphs. The bounds are derived by use of matrix techniques and the addition formula for Jacobi polynomials.

Bounds for systems of lines, and jacobi polynomials

The paper is concerned with applications of the standard Nevanlinna–Pick problem in various technical domains, namely the following: interpolation by reflectance functions, polynomial stability checking, cascade synthesis of passive one-ports, and model reduction with a Hankel norm criterion. Some fundamental results on the Nevanlinna–Pick problem are shown to be of a definite interest in each of these subjects.

P. Delsarte

Papers

The Nevanlinna–Pick Problem for Matrix-Valued Functions

The split Levinson algorithm

Orthogonal polynomial matrices on the unit circle

Bounds for systems of lines, and jacobi polynomials

On the role of the Nevanlinna–Pick problem in circuit and system theory†