Analysis of sliding-mode control systems with relative degree altering perturbations

Abstract

We consider sliding-mode control systems subject to unmatched perturbations. Classical first-order sliding-mode techniques are capable to compensate unmatched perturbations if differentiations of the output of sufficiently high order are included in the sliding variable. For such perturbations it is commonly assumed that they do not affect the relative degree of the system. In this contribution we consider perturbations that alter the relative degree of the process and study their impact on the closed-loop control system with a classical first-order sliding-mode design. In particular we consider systems with full (nominal) relative degree subject to a perturbation reducing the relative degree by one and analyse the resulting closed-loop system. It turns out that the sliding-manifold is not of reduced dimension and the uniqueness of the solution may be lost. Also attractivity of the sliding-manifold and global stability of the origin may be lost whereas the disturbance rejection properties of the sliding-mode control are not impaired. We present a necessary and sufficient condition for the existence of unique solutions for the closed-loop system. The second-order case is studied in great detail and allows to parametrically specify the conditions obtained before. We derive a necessary condition for the global asymptotic stability of the closed-loop system. Further we present a constructive condition for the global asymptotic stability of the closed-loop system using a piece-wise linear Lyapunov function. Each of the prominent results is illustrated by a numerical example.