Observation and Separation Principle for the Chattering-Free Control of Multi-Input Systems with Simplex Sliding Mode

In this paper a method to simultaneously achieving the two main objective of smoothing the control while maintaining ideally infinite frequency regimes is particularized to the simplex sliding mode method. The present paper proposes a simplex sliding mode control logic based on the time derivative of the control vector. This practice, beside the standard chattering reduction effect, has the important consequence of making possible the control of a rather wide class of uncertain systems nonlinear in the control. In the uncertain case the increment of the relative degree implicit in the proposed approach requires the introduction of second order sliding mode observer. A separation theorem is proven and sufficient conditions for the finite time convergence of both the estimation and tracking errors are found in terms of further requirements on the control amplitude.