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

"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

/pdf/matrix-mathematics-theory-facts-and-formulas-with-8swhnd659u.pdf

Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

This document focuses on translating various information-theoretic measures of distinguishability for probability distributions into measures of distin- guishability for quantum states. These measures should have important appli- cations in quantum cryptography and quantum computation theory. The results reported include the following. An exact expression for the quantum fidelity between two mixed states is derived. The optimal measurement that gives rise to it is studied in detail. Several upper and lower bounds on the quantum mutual information are derived via similar techniques and compared to each other. Of note is a simple derivation of the important upper bound first proved by Holevo and an explicit expression for another (tighter) upper bound that appears implicitly in the same derivation. Several upper and lower bounds to the quan- tum Kullback relative information are derived. The measures developed are also applied to ferreting out the extent to which quantum systems must be disturbed by information gathering measurements. This is tackled in two ways. The first is in setting up a general formalism for describing the tradeoff between inference and disturbance. The main point of this is that it gives a way of expressing the problem so that it appears as algebraic as that of the problem of finding quantum distinguishability measures. The second result on this theme is a theorem that prohibits "broadcasting" an unknown (mixed) quantum state. That is to say, there is no way to replicate an unknown quantum state onto two separate quantum systems when each system is considered without regard to the other. This includes the possibility of correlation or quantum entanglement between the systems. This result is a significant extension and generalization of the standard "no-cloning" theorem for pure states.

https://arxiv.org/pdf/quant-ph/9601020.pdf

Distinguishability and Accessible Information in Quantum Theory

Introduction.- Aspects of stochastic control theory.- Optimal stabilization of linear stochastic systems.- Linear mappings on ordered vector spaces.- Newtons method.- Solution of the Riccati equation.

Rational Matrix Equations in Stochastic Control

Existence and comparison theorems for algebraic Riccati equations for continuous- and discrete-time systems

The structure of the set of hermitian solutions of the matrix quadratic equation $XDX - XA - A^ * X - C = 0$ is studied under the conditions that $C = C^ * $, D is positive semidefinite and $(A,D)$ is stabilizable. New features (e.g., nonexistence of the minimal solution) appear in contrast with the known case when $(A,D)$ is controllable.

On hermitian solutions of the symmetric algebraic Riccati equation

11. Factorization of Self-Adjoint Matrix Polynomials

The structure of the set of hermitian solutions of the matrix quadratic equationXDX XA A*X C 0 is studied underthe conditions thatC C*,D is positive semidefinite and (A, D) is stabilizable. New features (e.g., nonexistence of the minimal solution) appear in contrast with the known case when (A, D) is controllable.

P. Lancaster

Papers

On hermitian solutions of the symmetric algebraic Riccati equation

11. Factorization of Self-Adjoint Matrix Polynomials

On hermitian solutions of the symmetric

2. Representation of Monic Matrix Polynomials