Sliding Controller-Sliding Observer Design for Non-linear Systems

We study the combination of sliding observers and sliding controllers for affine non-linear systems. First, for systems in controllable canonical form we derive conditions for convergence of the observer and the combined sliding controller-sliding observer scheme. The main result is that under certain conditions uniform ultimately boundedness of the tracking error can be achieved. The analysis is extended to affine feedback linearisable systems. Affine systems with a relative degree lower than the system order are also studied, showing the necessity of some 'matching' conditions on the uncertainties to allow a theoretically perfect performance of the sliding observer. If those restrictive matching conditions are not imposed, the observer-controller scheme is still adequate if some bounds on the uncertainties are fulfilled, provided the sliding variable depends on the 'certain' part of the state. The application to robot manipulators is developed in some detail. The performance of the observer-controller scheme is tested by simulations on a two-degree of freedom rigid robot and a one-link flexible robot, showing the effects of finite switching frequency and measurement noise.