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)

Sätze und Probleme der kombinatorischen Geometrie

In the ninth chapter we derived upper estimates for the norm and the quotient norm of singular operators on spaces L p (Γ, ρ). Now we are going to show that these estimates are exact for the operator SΓ in the spaces L p (Γ, ρ) as well as for the operators aI + bSΓ in L2(Γ, ρ). These results will be used to obtain sufficient conditions for the Fredholmness of singular integral operators in L p (Γ, ρ).

Exact constants in theorems on the boundedness of singular operators

Second order parallel algorithms for Fredholm integral equations with piecewise smooth displacement kernels are derived. One is based on a difference scheme of Runge-Kutta type for an unusual partial differential equations for continuous functions of two variables. The other is based on the trapezoidal quadrature rule applied to a modified integral equations. It is found that the Runge-Kutta type algorithm exhibits certain advantages.

Second order parallel algorithms for piecewise smooth displacement kernels

In 1960 Kalman introduced the state space method as a systematic tool for the understanding of the structure of a linear dynamical system A model for a linear dynamical system is a system of vector differential equations 

$$\begin{gathered} \dot x = Ax + Bu,x\left( 0 \right) = {x_0} \hfill \\ y = Cx + Du \hfill \\ \end{gathered}$$

When one assumes that the state vector x(t) is subject to zero initial condition (x 0 = 0), then after Laplace transformation the relation between the input function u(s) and the output function y(s) is given by 

$$\hat y\left( s \right) = W\left( s \right)\hat u\left( s \right)$$

where 

$$W\left( s \right) = D + C{\left( {sI - A} \right)^{ - 1}}B$$

(1)

is the transfer function of the system The converse problem is to realize a given proper rational matrix function as the transfer function of a linear system, ie given W(s) find matrices A, B, C, D so that (1) holds Using the notions of controllability, observability and minimal realization, Kalman clarified the precise connection between an input-output or classical frequency domain description of a linear dynamical system on the one hand and a state space model on the other Kalman also introduced a primitive axiomatic framework for linear dynamical systems and pursued applications of the state space method to specific engineering problems, such as the linear quadratic regulator problem and the Kalman filter

The State Space Method in the Study of Interpolation by Rational Matrix Functions

In this communication Toeplitz matrices of the form ∥a j-k ∥ j,k=0 n , where a j (j=0,±1,...,±n are elements of some noncommutative algebra, and their continual analogues are considered. The theorems presented here are generalizations of theorems from [1] to the noncommutative case.

Israel Gohberg

Papers

Sätze und Probleme der kombinatorischen Geometrie

Exact constants in theorems on the boundedness of singular operators

Second order parallel algorithms for piecewise smooth displacement kernels

The State Space Method in the Study of Interpolation by Rational Matrix Functions

Inversion of Finite Toeplitz Matrices