Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On Analytic Solution of the Lyapunov Equation in the Problem of Analysis of a Linear Discrete Dynamic System

Author(s):

Nikolay Evgenievich Zubov

Doctor of Technical Sciences, Professor, Professor of Department of Automatic Control Systems,
Dean of Rocket and Space Techniques Faculty at Bauman Moscow State Technical University (Bauman MSTU),
Professor of Postgraduate Studies at S.P. Korolev Rocket and Space Corporation “Energia”
(S.P. Korolev RSC “Energia”)

Nik.Zubov@gmail.com

Vladimir Nikolaevich Ryabchenko

Doctor of Technical Sciences, Associate Professor, Professor of Department of Automatic Control Systems
at Bauman Moscow State Technical University (Bauman MSTU)

RyabchenkoVN@yandex.ru

Alexey Vladimirovich Lapin

Candidate of Technical Sciences, Associate Professor of Department of Automatic Control Systems
at Bauman Moscow State Technical University (Bauman MSTU), 2nd category engineer
at State Research Institute of Aviation Systems (GosNIIAS)

AlexeyPoeme@yandex.ru

Abstract:

The analysis of linear discrete dynamic systems using the algebraic discrete Lyapunov equation is considered. An approach to the formation of an analytic solution to the algebraic discrete Lyapunov equation based on a number of assumptions is presented. The first assumption concerns the requirement of asymptotic stability of the matrix describing the control object, that is, finding the spectrum of the matrix inside the unit circle with the exception of the origin of the complex plane. The second assumption is related to the simplicity of the matrix and, consequently, the presence of decomposition matrices of a certain type for it. In addition, a set of (right) eigenvectors of a given matrix is introduced into consideration. With their help, a number of equivalent transformations of the algebraic discrete Lyapunov equation were carried out. The transformation condition is the reversibility of the control object matrix. As a result of these transformations, an analytic formula for the solution of the algebraic discrete Lyapunov equation in block-matrix form is obtained. The use of the analytic formula is illustrated by an example of the analysis of the observability of a linear discrete dynamic system. For the example under consideration, the algebraic discrete Lyapunov equation has a definite form, and its solution is called the asymptotic controllability gramian of the system and serves as an important characteristic of the system. On its basis, in particular, control measures are determined.

Keywords

• analytic solution
• discrete system
• Lyapunov equation
• stability of a system

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