On Partial Stability and Detectability of Functional Differential Systems with Aftereffect

We consider a general class of the nonlinear non-stationary system of functional differential equations with aftereffect that admits a "partial" (with respect to a part of the variables) zero equilibrium position. We obtain conditions under which stability (asymptotic stability) with respect to a part of the variables of the "partial" equilibrium position implies its stability (asymptotic stability) in all variables. We analyze these conditions from the standpoint of the problem of partial detectability of the system in question and introduce the concept of its partial zero dynamics. We also study an application to the problem of partial stabilization of controllable systems

On stability with respect to a part of the variables for nonlinear discrete-time systems with a random disturbances [Об устойчивости по части переменных нелинейных дискретных систем со случайными параметрами]

Nonlinear discrete (finite-difference) system of equations subject to the influence of a random disturbances of the "white" noise type, which is a difference analog of systems of stochastic differential equations in the Ito form, is considered. The increased interest in such systems is associated with the use of digital control systems, financial mathematics, as well as with the numerical solution of systems of stochastic differential equations. Stability problems are among the main problems of qualitative analysis and synthesis of the systems under consideration. In this case, we mainly study the general problem of stability of the zero equilibrium position, within the framework of which stability is analyzed with respect to all variables that determine the state of the system. To solve it, a discrete-stochastic version of the method of Lyapunov functions has been developed. The central point here is the introduction by N. N. Krasovskii, the concept of the averaged finite difference of a Lyapunov function, for the calculation of which it is sufficient to know only the right-hand sides of the system and the probabilistic characteristics of a random process. In this paper, for the class of systems under consideration, a statement of a more general problem of stability of the zero equilibrium position is given: not for all, but for a given part of the variables defining it. For the case of deterministic systems of ordinary differential equations, the formulation of this problem goes back to the classical works of A. M. Lyapunov and V. V. Rumyantsev. To solve the problem posed, a discrete-stochastic version of the method of Lyapunov functions is used with a corresponding specification of the requirements for Lyapunov functions. In order to expand the capabilities of the method used, along with the main Lyapunov function, an additional (vector, generally speaking) auxiliary function is considered for correcting the region in which the main Lyapunov function is constructed. © 2021 New Technologies Publishing House. All Rights Reserved

On problem of partial detectability for nonlinear discrete-time systems [О детектируемости по части переменных нелинейных дискретных систем]

Discrete (finite-difference) systems are widely used in modern nonlinear control theory. One of the main problems of a qualitative study of such systems is the problem of stability of the zero equilibrium position, which has great generality. In most works, such a stability problem is analyzed with respect to all variables that determine the state of the system. However, for many cases important in applications, it becomes necessary to analyze a more general problem of partial stability: the stability of the zero equilibrium position not for all, but only with respect to some given part of the variables. Such a problem is often also considered as auxiliary problem in the study of stability with respect to all variables. In this way, the corresponding concepts and problems of detectability of the studied system arise, which play an important role in the process of analysis of nonlinear controlled systems. Then, more general problems of partial detectability were posed, within the framework of which the situation was studied when stability from a part of variables implies stability not with respect to all, but with respect to more part of the variables. This article studies a nonlinear discrete (finite-difference) system of a general form that admits a zero equilibrium position. Easily interpreted conditions are found on the structural form of the system under consideration that determine its partial detectability, for which stability over a given part of the variables of the zero equilibrium position means its stability with respect to the other, more part of the variables. In this case, the stability with respect to the remaining part of the variables is uncertain and can be investigated additionally. In the process of analyzing this problem of partial detectability, the concept of partial null-dynamics of the system under study is introduced. An application of the obtained results to the stabilization problem with respect to part of the variables of nonlinear discrete controlled systems is given. © 2020 New Technologies Publishing House. All right reserved

Approach to the Stability Analysis of Partial Equilibrium States of Nonlinear Discrete Systems

Abstract: A nonlinear system of finite-difference equations of a general form, which admits a partial (in part of variables) zero equilibrium position, is considered. An approach to studying the stability of this equilibrium position is described, based on a preliminary study of stability in a part of the variables determining it based on the Lyapunov function method, followed by an analysis of the structural form of the system. To expand the possibilities of this approach, it is proposed to correct the area in which the Lyapunov function is constructed; this is achieved by introducing a second (vector, generally speaking) auxiliary function. Examples are given that show the features of this approach. © 2022, Pleiades Publishing, Ltd

О детектируемости по части переменных нелинейных дискретных систем

Discrete (finite-difference) systems are widely used in modern nonlinear control theory. One of the main problems of a qualitative study of such systems is the problem of stability of the zero equilibrium position, which has great generality. In most works, such a stability problem is analyzed with respect to all variables that determine the state of the system. However, for many cases important in applications, it becomes necessary to analyze a more general problem of partial stability: the stability of the zero equilibrium position not for all, but only with respect to some given part of the variables. Such a problem is often also considered as auxiliary problem in the study of stability with respect to all variables. In this way, the corresponding concepts and problems of detectability of the studied system arise, which play an important role in the process of analysis of nonlinear controlled systems. Then, more general problems of partial detectability were posed, within the framework of which the situation was studied when stability from a part of variables implies stability not with respect to all, but with respect to more part of the variables. This article studies a nonlinear discrete (finite-difference) system of a general form that admits a zero equilibrium position. Easily interpreted conditions are found on the structural form of the system under consideration that determine its partial detectability, for which stability over a given part of the variables of the zero equilibrium position means its stability with respect to the other, more part of the variables. In this case, the stability with respect to the remaining part of the variables is uncertain and can be investigated additionally. In the process of analyzing this problem of partial detect-ability, the concept of partial null-dynamics of the system under study is introduced. An application of the obtained results to the stabilization problem with respect to part of the variables of nonlinear discrete controlled systems is given.Дискретные (конечно-разностные) системы широко используются в современной нелинейной теории управления. Одной из основных задач качественного исследования таких систем является обладающая большой общностью задача устойчивости нулевого положения равновесия. В большинстве работ такая задача устойчивости анализируется по отношению ко всем переменным, определяющим состояние системы. Однако для многих важных в приложениях случаев возникает необходимость анализа более общей задачи: об устойчивости нулевого положения равновесия не по всем переменным, а только по некоторой заданной части переменных. Такая задача часто рассматривается также как вспомогательная при исследовании устойчивости по всем переменным. На этом пути возникают соответствующие понятия и задачи детектируемости изучаемой системы, играющие важную роль в процессе анализа нелинейных управляемых систем. Затем были поставлены более общие задачи частичной детектируемости, в рамках которых изучается ситуация, когда из устойчивости по части переменных следует устойчивость не по всем, а по большей части переменных. В данной статье рассматривается нелинейная дискретная (конечно-разностная) система общего вида, допускающая нулевое положение равновесия. Находятся условия на структурную форму рассматриваемой системы, определяющие ее частичную детектируемость. При выполнении этих условий устойчивость по заданной части переменных нулевого положения равновесия системы означает его фактическую устойчивость по другой - большей части переменных. При этом устойчивость по оставшимся переменным является неопределенной и может исследоваться дополнительно. В процессе анализа указанной проблемы частичной детектируемости вводится понятие частичной нуль-динамики системы. Дается приложение полученных результатов к задаче стабилизации к части переменных нелинейных дискретных управляемых систем