On differential inclusions arising from some discontinuous systems

TL;DR

This paper develops a method for analyzing systems of differential equations with discontinuous controls causing sliding modes, focusing on differential inclusions and nonsmooth functions to understand system trajectories.

Contribution

It introduces a novel approach for solving differential inclusions in discontinuous systems, especially those with nonsmooth controls, and extends methods to systems with new types of controls.

Findings

A method for finding trajectories in sliding mode systems is proposed.

Differential properties of nonsmooth controls are analyzed.

New control types with useful differential properties are introduced.

Abstract

The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known definitions of the solution of discontinuous systems, then the motion of an object while being on some surface can be described in terms of differential inclusions. With the help of the previously developed apparatus for solving differential inclusions, a method is constructed for finding the trajectories of a system moving in a such a mode. Since some of frequently used discontinuous controls contain nonsmooth functions of phase variables, the paper pays special attention to study the differential properties of such systems. At the end of the paper controls of a slightly different, in contrast to the classical, type are considered which have useful differential properties, and a method is constructed for solving systems with such controls considered both before hitting the required surface and moving in its vicinity.