This paper proposes a novel adaptive sliding mode control (ASMC) for a class of polynomial systems comprising uncertain terms and input nonlinearities. In this approach, a new polynomial sliding surface is proposed and designed based on the sum-of-squares (SOS) decomposition. In the proposed method, an adaptive control law is derived such that the finite-time reachability of the state trajectories in the presence of input nonlinearity and uncertainties is guaranteed. To do this, it is assumed that the uncertain terms are bounded and the input nonlinearities belong to sectors with positive slope parameters. However, the bound of the uncertain terms is unknown and adaptation law is proposed to effectively estimate the uncertainty bounds. Furthermore, based on a novel polynomial Lyapunov function, the finite-time convergence of the sliding surface to a pre-chosen small neighborhood of the origin is guaranteed. To eliminate the time derivatives of the polynomial terms in the stability analysis conditions, the SOS variables of the Lyapunov matrix are optimally selected. In order to show the merits and the robust performance of the proposed controller, chaotic Chen system is provided. Numerical simulation results demonstrate chattering reduction in the proposed approach and the high accuracy in estimating the unknown parameters.

This paper considers the output tracking problem for micro-electro-mechanical systems (MEMS) under uncertainties and external disturbances. The robust non-linear controllers are designed by two methods. The first method consists of a backstepping strategy combined with a first-order sliding mode controller. Also, in order to reduce the chattering effect and to improve the robustness of the proposed scheme, a new variable universe fuzzy control action with an adaptive coefficient is used instead of the signum function in the switching control law. In the proposed fuzzy scheme, the centres of the output membership functions are optimized via three heuristic optimization algorithms including the artificial bee colony (ABC) algorithm, ant colony optimization (ACO) and particle swarm optimization (PSO). In the second method, a class of second-order sliding mode controller is combined with the backstepping strategy. The second controller includes the proposed optimal fuzzy controllers of the first method. The stability of the closed-loop systems in both approaches are proved via the Lyapunov stability criterion and the conditions of stabilization are provided by linear matrix inequalities (LMIs). Numerical simulations are carried out to verify the theoretical results and to demonstrate the robust performance of the proposed controller in output tracking of the time-varying reference signal.