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)

I. Main problems and motivation.- I.1 Eigenvalue completion problems and first examples.- I.2 Reduction by similarity.- I.3 Blocks.- I.4 Block similarity.- I.5 Special cases of block similarity.- I.6 Eigenvalue completion and restriction problems.- Notes.- II. Elementary operations on blocks.- II.1 Block-invariant subspaces.- II.2 Direct sums of blocks and decomposable blocks.- II.3 Indecomposable blocks.- II.4 Duality of blocks.- Notes.- III. Full length blocks.- III.1 Structure theorems for full length blocks.- III.2 Finite dimensional operator pencils.- III.3 Similarity of non-everywhere defined linear operators.- III.4 Dual sequences.- Notes.- IV. The eigenvalue completion problem for full length blocks.- IV.1 Main theorems.- IV.2 Reduction to a problem on matrix polynomials.- IV.3 A one column completion problem for matrix polynomials.- IV.4 Proof of the first main theorem.- IV.5 Some applications of the restriction problem.- IV.6 A matrix equation.- Notes.- V. Full width blocks.- V.1 Structure theorems for full width blocks.- V.2 Finite dimensional operator pencils.- V.3 Similarity of operators modulo a subspace.- V.4 Duality.- V.5 The eigenvalue completion problem and related problems.- V.6 A matrix equation.- Notes.- VI. Principal blocks.- VI.1 Structure theorem for principal blocks.- VI.2 The eigenvalue completion problem for principal blocks.- VI.3 The eigenvalue restriction problem for principal blocks.- Notes.- VII. General blocks.- VII.1 Block similarity invariants, completion and restriction problems.- VII.2 Structure theorems and canonical form.- VII.3 Proof of Proposition 2.2.- VII.4 Proof of Theorems 1.1 and 2.1.- VII.5 Finite dimensional operator pencils.- VII.6 Non-everywhere defined operators modulo a subspace.- VII.7 Duality of operator blocks.- VII.8 The eigenvalue completion problem.- Notes.- VIII. Off-diagonal blocks.- VIII.1 Structure theorems for off-diagonal blocks.- VIII.2 The eigenvalue completion and restriction problems.- Notes.- IX. Connections with linear systems.- IX.1 Linear input/output systems and transfer functions.- IX.2 Blocks and controllability.- IX.3 Blocks and observability.- IX.4 Minimal systems.- IX.5 Feedback and block similarity.- IX.6 Eigenvalue assignment and eigenvalue completion.- IX.7 Assignment of controllability indices and eigenvalue restriction.- IX.8 (A, B)-invariant subspaces.- IX.9 Output stabilization by state feedback.- IX.10 Output injection.- Notes.- X. Applications to matrix polynomials.- X.1 Preliminaries.- X.2 Matrix polynomials with prescribed zero structure.- X.3 Wiener-Hopf factorization and indices.- Notes.- XI. Applications to rational matrix functions.- XI.1 Preliminaries on pole pairs and null pairs.- XI.2 The one sided homogeneous interpolation problem.- XI.3 Homogeneous two sided interpolation.- XI.4 An auxiliary result on block similarity.- XI.5 Factorization indices for rational matrix functions.- Notes.- XII. Infinite dimensional operator blocks.- XII.1 Preliminaries.- XII.2 Main theorems about (P, I)-blocks.- XII.3 Main theorems for (I, Q)-blocks.- XII.4 Operator blocks on a separable Hilbert space.- XII.5 Spectral completion and assignment problems.- Notes.- XIII. Factorization of operator polynomials.- XIII.1 Preliminaries on null pairs and spectral triples.- XIII.2 Wiener-Hopf equivalence.- XIII.3 Wiener-Hopf factorization.- XIII.4 Wiener-Hopf factorization and strict equivalence.- XIII.5 The Fredholm case.- Notes.- XIV. Factorization of analytic operator functions.- XIV.1 Preliminaries on spectral triples.- XIV.2 Wiener-Hopf equivalence.- XIV.3 Wiener-Hopf factorization.- Notes.- XV. Eigenvalue completion problems for triangular matrices.- XV.1 U-similar and decomposed U-specified matrices.- XV.2 Invariants for U-similarity.- XV.3 Invariants and U-canonical form in the generic case.- XV.4 The diagonal of U-similar matrices.- XV.5 An eigenvalue completion problem.- XV.6 Applications.- Notes.- Append.- A.1 Root functions of regular analytic matrix functions.- A.2 Right Jordan pairs of regular analytic matrix functions.- A.3 Left Jordan pairs.- A.4 Jordan pairs and Laurent principal parts.- A.5 Global spectral data for regular analytic matrix functions.- Notes.- List of notations.

Partially Specified Matrices and Operators: Classification, Completion, Applications

I: Canonical and Minimal Factorization.- Editorial introduction.- Left Versus Right Canonical Factorization.- 1. Introduction.- 2. Left and right canonical Wiener-Hopf factorization.- 3. Application to singular integral operators.- 4. Spectral and antispectral factorization on the unit circle.- 5. Symmetrized left and right canonical spectral factorization on the imaginary axis.- References.- Wiener-Hopf Equations With Symbols Analytic In A Strip.- 0. Introduction.- I. Realization.- 1. Preliminaries.- 2. Realization triples.- 3. The realization theorem.- 4. Construction of realization triples.- 5. Basic properties of realization triples.- II. Applications.- 1. Inverse Fourier transforms.- 2. Coupling.- 3. Inversion and Fredholm properties.- 4. Canonical Wiener-Hopf factorization.- 5. The Riemann-Hilbert boundary value problem.- References.- On Toeplitz and Wiener-Hopf Operators with Contour-Wise Rational Matrix and Operator Symbols.- 0. Introduction.- 1. Indicator.- 2. Toeplitz operators on compounded contours.- 3. Proof of the main theorems.- 4. The barrier problem.- 5. Canonical factorization.- 6. Unbounded domains.- 7. The pair equation.- 8. Wiener-Hopf equation with two kernels.- 9. The discrete case.- References.- Canonical Pseudo-Spectral Factorization and Wiener-Hopf Integral Equations.- 0. Introduction.- 1. Canonical pseudo-spectral factorizations.- 2. Pseudo-?-spectral subspaces.- 3. Description of all canonical pseudo-?-spectral factorizations.- 4. Non-negative rational matrix functions.- 5. Wiener-Hopf integral equations of non-normal type.- 6. Pairs of function spaces of unique solvability.- References.- Minimal Factorization of Integral operators and Cascade Decompositions of Systems.- 0. Introduction.- I. Main results.- 1. Minimal representation and degree.- 2. Minimal factorization (1).- 3. Minimal factorization of Volterra integral operators (1).- 4. Stationary causal operators and transfer functions.- 5. SB-minimal factorization (1).- 6. SB-minimal factorization in the class (USB)..- 7. Analytic semi-separable kernels.- 8. LU- and UL-factorizations (1).- II. Cascade decomposition of systems.- 1. Preliminaries about systems with boundary conditions.- 2. Cascade decompositions.- 3. Decomposing projections.- 4. Main decomposition theorems.- 5. Proof of Theorem II.4.1.- 6. Proof of Theorem II.4.2.- 7. Proof of Theorem II.4.3.- 8. Decomposing projections for inverse systems..- III. Proofs of the main theorems.- 1. A factorization lemma.- 2. Minimal factorization (2).- 3. SB-minimal factorization (2).- 4. Proof of Theorem I.6.1.- 5. Minimal factorization of Volterra integral operators (2).- 6. Proof of Theorem I.4.1.- 7. A remark about minimal factorization and inversion.- 8. LU- and UL-f actorizations (2).- 9. Causal/anticausal decompositions.- References.- II: Non-Canonical Wiener-Hopf Factorization.- Editorial introduction.- Explicit Wiener-Hopf Factorization and Realization.- 0. Introduction.- 1. Preliminaries.- 1. Peliminaries about transfer functions.- 2. Preliminaries about Wiener-Hopf factorization.- 3. Reduction of factorization to nodes with centralized singularities.- II. Incoming characteristics.- 1. Incoming bases.- 2. Feedback operators related to incoming bases.- 3. Factorization with non-negative indices.- III. Outgoing characteristics.- 1. Outgoing bases.- 2. Output injection operators related to outgoing bases.- 3. Factorization with non-positive indices.- IV. Main results.- 1. Intertwining relations for incoming and outgoing data.- 2. Dilation to a node with centralized singularities.- 3. Main theorem and corollaries.- References,.- Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions.- 1. Introduction and main result.- 2. Simple nodes with centralized singularities.- 3. Multiplication by plus and minus terms.- 4. Dilation.- 5. Spectral characteristics of transfer functions: outgoing spaces.- 6. Spectral characteristics of transfer functions: incoming spaces.- 7. Spectral characteristics and Wiener-Hopf equivalence.- References.- Multiplication by Diagonals and Reduction to Canonical Factorization.- 1. Introduction.- 2. Spectral pairs associated with products of nodes.- 3. Multiplication by diagonals.- References.- Symmetric Wiener-Hopf Factorization of Self-Adjoint Rational Matrix Functions and Realization.- 0. Introduction and summary.- 1. Introduction.- 2. Summary.- I. Wiener-Hopf factorization.- 1. Realizations with centralized singularities..- 2. Incoming data and related feedback operators.- 3. Outgoing data and related output injection operators.- 4. Dilation to realizations with centralized singularities.- 5. The final formulas.- II. Symmetric Wiener-Hopf factorization.- 1. Duality between incoming and outgoing operators.- 2. The basis in (C and duality between the feedback operators and the output injection operators.- 3. Proof of the main theorems.- References.

Constructive methods of Wiener-Hopf factorization

A linear ordinary differential operator with bounded coefficients satisfying certain homogeneous initial conditions is shown to be invertible onLn2(0, ∞) if and only if the underlying system of differential equations has a dichotomy. Moreover, in that case the operator is proved to be a direct sum of two infinitesimal generators ofC0-semigroups, one of which has support on the negative half-line and the other on the positive half-line. The effect of perturbations of the initial values on the dichotomy is also described.

Israel Gohberg

Papers

Partially Specified Matrices and Operators: Classification, Completion, Applications

Constructive methods of Wiener-Hopf factorization

Invertibility and dichotomy of differential operators on a half-line

Analytic matrix functions with prescribed local data

The Fast Generalized Parker-Traub Algorithm for Inversion of Vandermonde and Related Matrices