Matrix methods for the analysis and synthesis of dynamical systems


The book outlines constructive methods for the analysis and synthesis of dynamical systems based on the application of matrix equations and inequalities. The generalizations of the Lyapunov equation are presented in the framework of stability and spectrum localization theory of linear systems. The properties of solutions of such equations are formulated as generalized Lyapunov and Ostrowsky-Schneider theorems describing the location of matrix eigenvalues with respect to wide classes of analytic curves. Systems of spectrum splitting and solutions of generalized block spectral problems for matrix polynomials and functions are determined and used. Classical methods for motion stability analysis, modern methods for robust stabilization and optimization of dynamical systems, as well as new approaches to solving generalized $H_\infty$-control problems for continuous and discrete-time systems are presented. Sufficient stability conditions for the zero state are formulated with the common quadratic Lyapunov function for a family of control systems with uncertain coefficient matrices. The solution of robust stabilization problem and evaluation of the quadratic performance criterion for nonlinear control systems are proposed. Methods for construction of control laws providing a robust stability and specified evaluation of the weighted damping level of input signals and initial perturbations are proposed for a class of linear systems with controllable and observable outputs. The application of the main results reduces to solving the systems of linear and quadratic matrix inequalities. Algorithms for estimating and minimizing the weighted damping level of bounded perturbations in standard and descriptor control systems are proposed. The stability theory of positive and monotone dynamical systems is developed. The main results of the research are the criteria of asymptotic stability of linear positive systems, stated in terms of positive operators, methods for the stability investigation of nonlinear monotone systems, as well as the development of methods of comparison of systems in a partially ordered space. The supplementary chapter also presents some new results of matrix analysis, which can be used in various problems of analysis and synthesis of dynamical systems.
The book is intended for scientists, engineers, graduate students and students of the corresponding specialties.

Project Scientific Book

Kyiv, Naukova Dumka, 2023

ISBN 978-966-00-1892-1

UDK 517.93; 519.71



Editor O. A. Mykytenko


A. A. Martynyuk, Head of department, Academician of National Academy of Sciences of Ukraine, Professor, Doctor of Physical and Mathematical Sciences,

O. A. Boichuk, Head of department, Corresponding member of National Academy of Sciences of Ukraine, Professor, Doctor of Physical and Mathematical Sciences

Copyright 2007 Institute of Mathematics