Realization of sliding motion in a nonlinear stabilization system based on the method of the Lyapunov function

Abstract

In this article, an approach is suggested for obtaining a sliding mode in nonlinear stabilization systems on the basis of the general stability theory. As an indicator of the stability, a time derivative is used of a decomposable quadratic form in phase variables. From the condition that the derivative is negative, two simultaneously operating kinds of control are obtained: an equivalent control that transforms the system into a linear one with a switching curve as its phase trajectory, and a high-frequency control that provides sliding motion along this curve. For the purpose of saving energy, a border layer is introduced. In this case, the sliding motion takes place at the border layer boundary, while inside the zero control is enabled. The results are modeled by means of the Simulink package, which is a part of the Matlab application.