Adaptive fractional-order backstepping sliding mode controller design for an electrostatically actuated size-dependent microplate

In this article, adaptive backstepping sliding mode controller and adaptive fractional backstepping sliding mode controller methods are proposed to control an electrostatic microplate with a piezoelectric layer. Based on the modified couple stress theory, a size-dependent mathematical model is proposed, in which the microplate is modeled using the Kirchhoff plate theory. To take into account the geometric nonlinearities, the von Kármán nonlinear strains are considered in the mathematical model. The Hamilton’s principle is used to obtain the nonlinear equation of motion of the system, which is then converted into a nonlinear ordinary differential equation via the Galerkin technique. The validity of the results obtained from the proposed reduced-order model is checked through direct numerical simulation of the partial differential equation using the finite element method. Finally, two Lyapunov-based control approaches that are adaptive backstepping sliding mode and adaptive fractional backstepping sliding mode are applied to the system. In the adaptive fractional backstepping sliding mode controller method, the adaptive control law is used to evaluate the upper bound of uncertainties and random disturbances. The fractional-order form of sliding surface is used to reduce the amount of chattering and also to improve the tracking error. In the results section, first, numerical studies are conducted to study the effect of system parameters, such as material length scale parameter, aspect ratio, fractional order, and robustness of the controller, on the performance of the system. In addition, the performances of the two control methods are compared, and the merits and demerits of each method are discussed.