Sliding mode control of uncertain systems: a singular perturbation approach

In this paper we consider a nonlinear control system affected by deterministic uncertainty and described by a system of ordinary differential equations. The uncertainty is modelled by a multivalued map whose t-measurable and x-Lipschitz selections represent the possible system dynamics of the uncertain system. We propose a dynamical feedback control design, based on the singular perturbation theory, which allows all the possible system trajectories corresponding to the system dynamics to have the same prescribed behaviour. Specifically, given a manifold K of the state space, defined as the zeros of a smooth map, the proposed control steers and then holds, during finite or infinite time intervals, any possible system trajectory to any prescribed neighbourhood of K. A result ensuring the exact attainability of K is also provided. Some examples illustrating the obtained results are presented.