Adaptive smooth control for nonlinear uncertain systems

This paper presents an adaptive smooth controller for a class of nonlinear dynamical systems in the presence of bounded uncertainties with unknown bounds. Motivated by the concept of sliding mode control, a continuous control law that drives the sliding variable to a user-specified small domain in a finite time is developed. Also, an adaptation law is obtained to ensure that the gain update is performed in real time and the error is always bounded within a desired range without a priori knowledge of the uncertainties. Only the magnitude of the control input is required for online gain adaptation, and hence, the gain update is readily implemented in real time, which greatly eases the application of the suggested algorithm to real-world systems. In addition, by adding nonlinear terms in the sliding manifold, the reaching phase vanishes and the transient performance of the controlled system is considerably improved. Numerical examples serve to demonstrate the effectiveness and the robustness of the control methodology proposed herein.