Stability in sliding mode of nonstationary automatic-control systems of variable structure with switched filters

Abstract

Stability conditions for a nonstationary automatic-control system of variable structure in sliding mode are established. The controller of the system has feedback-switched filters functioning together with the shaper and actuator. The nonstationary parameters of the system vary within given ranges, at a finite rate, under appropriate control laws, with adjustment for the error signal, its derivatives of finite order, and all variable parameters of the filter. The parameters of the switching hyperplane remain constant. This approach to stability analysis is based on the existence conditions for the sliding mode at the switching boundary in the phase space. The general stability and instability criteria are applied to nonstationary automatic filtered-control systems of variable structure of the third order Keywords: nonstationary automatic-control systems of variable structure, switched filters, sliding mode, stability and instability, error gain, nonstationary filtered-control system of variable structure of the third order Introduction. Filtered-control systems of variable structure are important, in particular, for solving practical problems in flight automation and automatic control of surface and subsurface ships, in various nonstationary filtration of program automatic control signals to reject artificial or natural interference, and in dynamic elastic processes and objects that should be controlled taking the time-dependence of their parameters into account [5, 10, 11, etc.]. The nonstationary parameters of the variable-structure system under consideration vary within given ranges, at a finite rate, under appropriate control laws, and with adjustment for the error signal, its derivatives of finite order, and all independent variable parameters of the filter [5,9,13], the parameters of the switching hyperplane remaining constant.The present paper addresses the stability of dynamic states of nonstationary automatic-control systems with variable structure in sliding modes. The controller in such systems includes control filters functionally commuted with the shaper and actuator. The approach proposed here for the stability analysis of such variable-structure systems is based on the existence conditions for the sliding mode at the switching boundary in the phase space [1][2][3][4]. Error gain is used for stabilization purposes [5,17]. The established stability criteria are applied to a nonstationary filtered-control system of variable structure of the third order. The study is based on the results from [1-20].1. Problem Formulation. Consider a nonstationary automatic-control system of variable structure that functionally includes filters of control signals [1,13]. The dynamic states of the system are described by the following equations [5,[18][19][20]: