Nonlinear sliding surfaces; Computing and existence of solution

In this paper, we have concentrated on real systems consisting of structural uncertainties and affected by external disturbances. In this regard, Sliding Mode Control (S.M.C.) is utilized. To decrease energy consumption, arising from chattering phenomenon, a smooth switch has been used in design procedure. Consequently, sliding equation will play a dominant controlling role in its neighborhood. The converging property of sliding motion towards the origin is a challenging issue. In this article we present a new method to prove the stability of the sliding phase which means, state trajectories on the sliding surface move toward the origin. At the beginning, the equivalent control method is reestablished such that makes this purpose accessible. The modification bounds the sliding equation to a converging set. Then to improve main factors of closed loop system, such as, transient behavior, energy consumption and the domain of attraction, the optimal control theory is used to compute the optimized sliding surface in the stabilizing set. Generally, desired surface has nonlinear terms. Finally, we propose an elaborate algorithm for computing optimized nonlinear surfaces. The designed controller is applied to a flexible-link setup. Simulation results show the efficiency of the proposed approach.