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

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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)

Table of Contents

Table of Contents

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



“The sensitivity of a given mathematical object (or of the corresponding computational problem) is among its most important properties. It shows how the solution of the problem varies under the perturbations of small changes in the data. This property is subject of the so called Perturbation Theory which is widely used in Science and Engineering. In the separate scientific disciplines various perturbation theories have been developed which differ in the problems solved and mathematical methods used, for instance in the Celestial Mechanics, Theory of Nonlinear Oscillations and Control Theory. All these theories are based on the idea to investigate a system whose behavior deviates slightly from the behavior of a simple ideal system for which the full solution of the problem under consideration is known. Perturbation Theory for Linear Operators, which is relevant to the given case, was created by the physicists Strutt and Lord Rayleigh [17] and Schrödinger [16] and the modern perturbation theory for linear operators is developed by Kato [7]…The authors present new, original results in perturbation linear algebra and control, based on the Method of Splitting Operators and Lyapunov Majorant Functions. Combined with the Schauder or Banach fixed point principles, this method allows to obtain rigorous non-local perturbation bounds for a set of important objects in matrix analysis and control theory. Thus, the perturbation problems in these important fields are investigated in a uniform way, which presents a significant contribution to perturbation theory. As a direction of further work, I would recommend to extend the results obtained to the case of component-wise perturbation analysis in order to find perturbation bounds for the individual super-diagonal elements of the Schur form and for the angles between the perturbed and unperturbed invariant subspaces of the matrix. This will allow to develop full perturbation theory in this important form and go deeper into the properties of the corresponding problems…READ MORE-Reviewed by Ivan Popchev for the Bulgarian Academy of Sciences, Cybernetics and Information Technologies, Volume 20, No 4, Sofia, 2020

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