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 the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.

Spectrum estimation and harmonic analysis

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.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

Essentials of Robust Control

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

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

Asymptotic properties of Toeplitz matrix factorization

On the Separation of Certain Spectral Components of Selfadjoint Operator Matrices.- 1. Introduction.- 2. Conditions for the Separation of Spectral Components.- 3. Example.- References.- A Coisometric Realization for Triangular Integral Operators.- 1. Introduction.- 2. Preliminaries and Notations.- 3. Resolvent Operators and Resolvent Equations.- 4. The State SpacesHL(S) andHR(S).- 5. The Coisometric Realization.- References.- Some Remarks on the Inverse Monodromy Problem for 2 x 2 Canonical Differential Systems.- 1. Introduction.- 2. Monodromy Matrices with Zero Type in the Upper Half Plane.- 3. Monodromy Matrices with Zero Type in the Lower Half Plane.- 4. Monodromy Matrices with Nonzero Type in Both Halfplanes.- 5. Reparametrizations.- 6. Three Classes of J-inner mvf's and Some Examples.- 7. Another Parametrization.- References.- Interpolation and Commutant Lifting for Multipliers on Reproducing Kernel Hilbert Spaces.- 1. Introduction.- 2. The Multiplier Space forH(kd).- 3. Multipliers for Nevanlinna-Pick-type Kernels.- 4. Interpolation by Multipliers.- 5. The Commutant Lifting Theorem forMk(?,?*).- 6. Examples and Applications.- References.- Sums of Idempotents and Logarithmic Residues in Matrix Algebras.- 1. Introduction.- 2. Preliminaries.- 3. Matrix Algebras Generated by a Single Matrix.- 4 Rank, Trace and Decomposition of Matrices.- 5. Logarithmic Residues of Matrix and Fredholm Operator Valued Functions.- 6. The Algebra of Block Upper Triangular Matrices.- References.- Generalized Nevanlinna Functions with Polynomial Asymptotic Behaviour at Infinity and Regular Perturbations.- 1. Introduction.- 2. Preliminaries.- 3. Multiplicity of the Generalized Poles for the Sum of Nk -functions.- 4. Polynomial Behaviour at Infinity.- 5. The Subclasses Induced via Polynomial Asymptotics to Nk0.- 6. Spectral Characterizations of Regular Rank One Perturbations.- References.- Extensions of Matrix Valued Inner Products on Modules and the Inversion Formula for Block Toeplitz Matrices.- 0. Introduction.- 1. The Scalar Case.- 1.1 Extension of a Scalar Product.- 1.2 Connections with Prediction Theory and the Extension Theorem.- 1.3 Properties of the Extended Scalar Product.- 1.4 The Inversion Formula.- 2. The Matrix Case.- 2.1 Matrix Valued Inner Products on Modules.- 2.2 Extension of a Matrix-valued Inner Product.- 2.3 Properties of the Extended Matrix-valued Inner Product.- 2.4 The Gohberg-Heinig Formula.- References.- Linear Independence of Jordan Chains.- 1. Introduction.- 2. Linearly Independent Chains with Respect to a Sequence of Operators.- 3. Jordan Chains of Holomorphic Operator Functions.- 4. Right (spectral) Roots of a Regular Holomorphic Matrix Function.- References.- Weighted Nevanlinna-Pick Interpolation.- 0. Introduction.- 1. Preliminaries.- 2. Some State Space Existence Results.- 3. The Outer Spectral Factor Case.- 4. A State Space Solution.- 5. A State Space Computation for $$ \tilde B $$ and $$ {\tilde B_h} $$.- 6. The Case whenT QT*?Q.- References.- Effects of Small Delays on Stability and Control.- 1. Introduction.- 2. The Abstract Setting of the Problem.- 3. Difference Equations.- 4. Neutral Delay Differential Equations.- 5. Delayed Boundary Control in a Hyperbolic Equation.- References.- Generalized Bezoutian, Factorization of Rational Matrix Functions and Matrix Quadratic Equations.- 0. Introduction.- 1. Bezoutian of Rational Matrix Functions.- and Matrix Quadratic Equations.- 2. Generalized T-Bezoutian of Rational Matrix Functions.- 3. Discrete Quadratic Equation and Factorizations of Rational Matrix Functions.- References.- A Note on Factorization of Analytic Matrix Functions.- 1. Introduction.- 2. Preliminary Results.- 3. Spectral Divisors and the Casec(F) = 1.- 4. When ? is a Circle.- References.- Diagonalization of certain Block Operator Matrices and Applications to Dirac Operators.- 0. Introduction.- 1. Basic Propositions.- 2. Block Operator Matrices.- 3. The Self-adjoint Case.- 4. The Non-self-On the Separation of Certain Spectral Components of Selfadjoint Operator Matrices.- 1. Introduction.- 2. Conditions for the Separation of Spectral Components.- 3. Example.- References.- A Coisometric Realization for Triangular Integral Operators.- 1. Introduction.- 2. Preliminaries and Notations.- 3. Resolvent Operators and Resolvent Equations.- 4. The State SpacesHL(S) andHR(S).- 5. The Coisometric Realization.- References.- Some Remarks on the Inverse Monodromy Problem for 2 x 2 Canonical Differential Systems.- 1. Introduction.- 2. Monodromy Matrices with Zero Type in the Upper Half Plane.- 3. Monodromy Matrices with Zero Type in the Lower Half Plane.- 4. Monodromy Matrices with Nonzero Type in Both Halfplanes.- 5. Reparametrizations.- 6. Three Classes of J-inner mvf's and Some Examples.- 7. Another Parametrization.- References.- Interpolation and Commutant Lifting for Multipliers on Reproducing Kernel Hilbert Spaces.- 1. Introduction.- 2. The Multiplier Space forH(kd).- 3. Multipliers for Nevanlinna-Pick-type Kernels.- 4. Interpolation by Multipliers.- 5. The Commutant Lifting Theorem forMk(?,?*).- 6. Examples and Applications.- References.- Sums of Idempotents and Logarithmic Residues in Matrix Algebras.- 1. Introduction.- 2. Preliminaries.- 3. Matrix Algebras Generated by a Single Matrix.- 4 Rank, Trace and Decomposition of Matrices.- 5. Logarithmic Residues of Matrix and Fredholm Operator Valued Functions.- 6. The Algebra of Block Upper Triangular Matrices.- References.- Generalized Nevanlinna Functions with Polynomial Asymptotic Behaviour at Infinity and Regular Perturbations.- 1. Introduction.- 2. Preliminaries.- 3. Multiplicity of the Generalized Poles for the Sum of Nk -functions.- 4. Polynomial Behaviour at Infinity.- 5. The Subclasses Induced via Polynomial Asymptotics to Nk0.- 6. Spectral Characterizations of Regular Rank One Perturbations.- References.- Extensions of Matrix Valued Inner Products on Modules and the Inversion Formula for Block Toeplitz Matrices.- 0. Introduction.- 1. The Scalar Case.- 1.1 Extension of a Scalar Product.- 1.2 Connections with Prediction Theory and the Extension Theorem.- 1.3 Properties of the Extended Scalar Product.- 1.4 The Inversion Formula.- 2. The Matrix Case.- 2.1 Matrix Valued Inner Products on Modules.- 2.2 Extension of a Matrix-valued Inner Product.- 2.3 Properties of the Extended Matrix-valued Inner Product.- 2.4 The Gohberg-Heinig Formula.- References.- Linear Independence of Jordan Chains.- 1. Introduction.- 2. Linearly Independent Chains with Respect to a Sequence of Operators.- 3. Jordan Chains of Holomorphic Operator Functions.- 4. Right (spectral) Roots of a Regular Holomorphic Matrix Function.- References.- Weighted Nevanlinna-Pick Interpolation.- 0. Introduction.- 1. Preliminaries.- 2. Some State Space Existence Results.- 3. The Outer Spectral Factor Case.- 4. A State Space Solution.- 5. A State Space Computation for $$ \tilde B $$ and $$ {\tilde B_h} $$.- 6. The Case whenT QT*?Q.- References.- Effects of Small Delays on Stability and Control.- 1. Introduction.- 2. The Abstract Setting of the Problem.- 3. Difference Equations.- 4. Neutral Delay Differential Equations.- 5. Delayed Boundary Control in a Hyperbolic Equation.- References.- Generalized Bezoutian, Factorization of Rational Matrix Functions and Matrix Quadratic Equations.- 0. Introduction.- 1. Bezoutian of Rational Matrix Functions.- and Matrix Quadratic Equations.- 2. Generalized T-Bezoutian of Rational Matrix Functions.- 3. Discrete Quadratic Equation and Factorizations of Rational Matrix Functions.- References.- A Note on Factorization of Analytic Matrix Functions.- 1. Introduction.- 2. Preliminary Results.- 3. Spectral Divisors and the Casec(F) = 1.- 4. When ? is a Circle.- References.- Diagonalization of certain Block Operator Matrices and Applications to Dirac Operators.- 0. Introduction.- 1. Basic Propositions.- 2. Block Operator Matrices.- 3. The Self-adjoint Case.- 4. The Non-self-adjoint Case.- 5. Dirac Operators with Potential.- References.- Stability of Pseudospectral Factorizations.- 1. Introduction.- 2. Dissipative Matrices and their Invariant Subspaces.- 3. Functions of the Form Identity Plus a Contraction.- 4. Positive Real Functions.- References.- Factorization of Almost Periodic Matrix Functions of Several Variables and Toeplitz Operators.- 1. Introduction.- 2. Algebras of Almost Periodic Functions and Factorizations.- 3. Toeplitz Operators.- 4. Factorization of Sectorial Matrix Functions.- 5. Factorization of Hermitian Matrix Functions.- 6. One-sided Invertibility of Toeplitz Operators.- 7. Robustness and Continuity of Factorizations.- References.- Simultaneous Similarity of Pairs of Companions to their Transposes.- 1. Introduction.- 2. Companion and Bezoutian Matrices.- 3. Simultaneous Similarity.- References.- Conference Programm.- List of Participants.

Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume Workshop in Amsterdam, November 12-14, 1997

In this paper the following completion problem for square matrices is solved and analyzed. Given a square matrix with unidentified last row, except the diagonal entry. Describe the invariant polynomials of all possible completions of this matrix.

On a Completion Problem for Matrices

Direct approach to the band completion problem

A finite section method is developed for linear difference equations over an infinite time interval. A necessary and sufficient condition is given in order that the solutions of such equations may be obtained as limits of solutions of corresponding equations over a finite time interval. Both the time-variant and the time-invariant case are considered. For the time-invariant case the condition reduces to the requirement that two subspaces defined in terms of the equations should be complementary. The results obtained extend those derived earlier for linear ordinary differential equations.

Finite Section Method for Difference Equations

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