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

Linear Robust Control

A control scheme called repetitive control is proposed, in which the controlled variables follow periodic reference commands. A high-accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period L into the closed-loop system. Sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small-gain theorem and the stability theorem for time-lag systems. Synthesis algorithms are presented by both the state-space approach and the factorization approach. In the former approach, the technique of the Kalman filter and perfect regulation is utilized, while coprime factorization over the matrix ring of proper stable rational functions and the solution of the Hankel norm approximation are used in the latter one. >

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Repetitive control system: a new type servo system for periodic exogenous signals

Introduction to Functional Analysis.

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

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.

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

The parameters occurring in Szego’s recurrence relations associated with a class C function are known to be the same as the Schur parameters of the corresponding class S function The present paper contains a matrix extension of this result Certain important questions about the matrix classes S and C, related to spectral factorization, are studied by means of their Schur–Szego parameters

Y. Genin

Papers

The Nevanlinna–Pick Problem for Matrix-Valued Functions

The split Levinson algorithm

Orthogonal polynomial matrices on the unit circle

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

Schur Parametrization of Positive Definite Block-Toeplitz Systems