Perturbation Methods in Matrix Analysis and Control

Mihail M. Konstantinov
Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, Sofia, Bulgaria

Petko H. Petkov
Department of Systems and Control Technical University of Sofia, Sofia, Bulgaria

Series: Computational Mathematics and Analysis
BISAC: MAT034000



Volume 10

Issue 1

Volume 2

Volume 3

Special issue: Resilience in breaking the cycle of children’s environmental health disparities
Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick


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The book presents a unified approach to the perturbation analysis in Matrix Analysis and Control, based on the method of splitting operators and Lyapunov majorant functions. Combined with the Schauder or Banach fixed point principles, this approach allows to obtain rigorous non-local perturbation bounds for a set of important objects in Linear Algebra and Control Theory. Among them are the Schur system of a matrix, the QR decomposition of a matrix, the orthogonal canonical forms of time-invariant linear systems, the state and output feedback gains in pole assignment design, the generalized Schur system of a pair of matrices, the Hamiltonian-Schur and block Hamiltonian-Schur forms of Hamiltonian matrices, and others. In this way, the approach proposed can be used as a unified tool in deriving asymptotic and nonlocal perturbation bounds in matrix analysis and control theory.

An important technique of the method considered is the construction of an operator equation, which is equivalent to the perturbed problem. It is based on the splitting of a certain linear matrix operator and its argument into strictly lower, diagonal and strictly upper parts, respectively. This allows to unify the perturbation analysis of matrix problems, involving unitary matrices, in which the resulting matrix is upper triangular.

Some other issues such as perturbation analysis of problems with non-unique solution and construction of improved asymptotic perturbation bounds are also considered.

The book is intended as a reference in the area of matrix computations and control theory. It will be of interest to researchers in the area of matrix analysis, linear control theory and applied mathematics. The book may also be useful for graduate students in the area of applied mathematics.
(Imprint: Nova)

Chapter 1. Introduction

Chapter 2. Notation and Preliminaries

Chapter 3. Perturbation Problems

Chapter 4. Splitting Operators and Lyapunov Majorants

Chapter 5. Schur Decomposition

Chapter 6. Hamiltonian Matrices: Basic Relations

Chapter 7. Hamiltonian Matrices: Asymptotic Analysis

Chapter 8. Hamiltonian Matrices: Non-Local Analysis

Chapter 9. Orthogonal Canonical Forms

Chapter 10. Feedback Synthesis Problem


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