Non linear control of variable structure systems

Abstract

Sliding modes in the control of systems described by ordinary differential equations have been investigated both from a theoretical point of view and from practical considerations, when the control variables enter linearly in the system. In this paper we extend the sliding mode control technique to fully non linear systems. We introduce a definition of equi valent control and show, under suitable conditions, the equivalence between the relevant Filippov solutions and the solutions obtained by using the equivalent control law. We discuss the physical meaning of such an equivalent control taking into account the behaviour of the corrisponding trajectories as far as approximability properties (physically relevant) are considered. We introduce a definition of approximability for nonlinear control systems with sliding surfaces, which shows the inherent limitations of the concept of equivalent control for general nonlinear dynamics. This validates some conjectures of [1] and extends basic results thereof to certain classes of fully nonlinear control systems. We show that the equivalent control may be defined and the approximability property holds for control systems of the following form:

$$z^{(n)} = g(t,z,z',...z^{(n - 1)} ,u);$$

(A))

$$\dot x = A(t,x) + B(t,x)h(u)$$

(B))

, under suitable explicit conditions about g, A, B, h and the sliding manifold.