Technical stability of stationary automatic control systems of variable structure with switched filters

Abstract

Sufficient conditions for the technical stability in measure of a nonstationary control system with variable structure are established. The controller of the system has feedback-switched filters functioning together with shaper and actuator. It is assumed that the nonstationary parameters of the system vary within given ranges, at a finite rate, with appropriate control laws, with adjustment against mismatch signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach for analysis of technical stability does not involve sliding mode conditions. Criteria of technical instability in measure for the control system under consideration are formulated using the properties of systems of comparison from below. The general criteria of technical stability and instability are applied to nonstationary filtered-control systems of variable structure of the third order. The comparison method based on normalized Lyapunov functions is used Keywords: technical stability and instability, nonstationary control systems with variable structure, switched filters, normalized Lyapunov functions, comparison method Introduction. Among various control systems that allow optimal control, of widespread use are nonstationary automatic-control systems of variable structure with a control filter subjected to internal feedback and functioning together with a shaper and an actuator [1, 3, 8, etc.]. Control actions are discontinuous functions of the phase coordinates and external actions [4,17,19]. Filters introduced into the control loop increase the amount of information needed to design controllers [2, 21, 22, etc.]. Such variable-structure systems with filters make it possible to gain the maximum effect, including optimality, from the use of automatics [9, 11, 12, etc.]. Under certain conditions, these systems may give rise to a specific motion, so-called sliding mode [8,9,19].The present study is concerned with the technical stability and instability in measure of nonstationary variable-structure automatic-control systems with filters in the control loop. The problem to be formulated below will be solved using the comparison and direct Lyapunov methods and is independent of the sliding-mode conditions on the switching hyperplane in the phase space. The present study is based on the results reported in [14,[21][22][23][24][25].1. Problem Formulation. Let us consider a nonstationary automatic-control system of variable structure that includes a feedback-switched filter to provide high-quality control of real processes [1, 9, etc.]. Let the following differential equation of the nth order describe the operation of the basic part of the system-the actuator and the control plant, which have time-dependent parameters [8,19]: