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 chapter is introduced a new form of local and global null-pole data. The null-pole data are now presented as a subspace of rational vector functions, called the null-pole subspace. Here we study how to express the null pole subspace via an admissible data set, via a null-pole triple. and in terms of null and pole chains and Laurent coefficients. The latter two are done in the local version only. The development of this chapter is preparatory for the interpolation problems to be solved in the succeeding chapters.

Null-Pole Subspaces: Elementary Properties

In Part I we were considering interpolation problems with standard combinations of the data. There we considered interpolation problems with given right null pair and left pole pair; the analogous interpolation problem where a left null pair and right pole pair are given can be reduced to the previous one by taking transpose. In this chapter we consider the interpolation problems for the remaining nonstandard combinations, namely the case where a null pair and pole pair from the same side are given. This problem differs considerably from the case of standard local data, both the procedure to find a solution and also in the properties of the solution. We also consider in this chapter the problem of interpolation when all four pairs (right and left null pair, right and left pole pair) are given. This problem can be regarded as an overdetermined version of the problem with standard data.

Interpolation Problems with Null and Pole Pairs

This paper describes the main properties of analytic common monic left multiples of a system of monic operator polynomials whose coefficients depend analytically on e.

Common multiples of operator polynomials with analytic coefficients

In this chapter we develop further the results from Chapter 1 for meromorphic matrix valued functions. In addition to null structure, pole structure is introduced and studied. In contrast to the scalar case, a new feature appears in the matrix case, namely, a meromorphic matrix function can have a zero and a pole at the same point. This chapter contains the analysis of null and pole chains, null and pole pairs, null and pole triples and singular parts, as well as connections among them.

Local Data for Meromorphic Matrix Functions

One can associate to a rational function which is moreover strictly positive on the unit circle two sequences of numbers in the open unit disk, called the Szego sequence and the Nehari sequence. In the scalar case, they coincide up to multiplication by −1. We study the corresponding result in the matrix-valued case.

Israel Gohberg

Papers

Null-Pole Subspaces: Elementary Properties

Interpolation Problems with Null and Pole Pairs

Common multiples of operator polynomials with analytic coefficients

Local Data for Meromorphic Matrix Functions

A Connection between Szegö and Nehari Sequences in the Matrix-valued Case