Sliding mode control and a variable structure system observer as a dual problem for systems with non-linear uncertainties

Sliding mode control is described for continuous-time systems with uncertain nonlinear elements, in which the euclidean norm of the unknown element is bounded by a known value. For a case of complete state information, we first consider a sliding mode controller consisting of three parts: a linear state feedback control, an equivalent input and a min-max control. It is then shown that the present attractiveness condition is simpler than that for a case without using the concept of equivalent input. We next derive a variable structure system (VSS) observer as a completely dual form to the sliding mode controller. The attractiveness condition is also simpler than that for a case an equivalent term is not used, but a more severe assumption is required to solve such a VSS observer. Finally, we discuss a case of incomplete state information by applying the VSS observer.