Terminal sliding mode control with non-symmetric input saturation for vibration suppression of electrostatically actuated nanobeams in the presence of Casimir force

Abstract

• The nonlinear ordinary differential equation is derived for an electrostatically actuated functionally graded nanobeam. • A robust sliding mode controller is designed for the uncertain nanobeam. • A disturbance-observer-based terminal sliding mode control is presented to suppress the nonlinear vibration of the nanobeam. • Stability of the closed-loop system is proved in the presence of unknown disturbance and non-symmetric input saturation. In the present article, a robust sliding mode controller and a disturbance-observer-based terminal sliding mode tracking controller with finite time convergence are proposed for stabilization of a functionally graded and homogeneous nanobeam. The considered nanobeam is modeled based on the nonlocal strain gradient theory and Euler–Bernoulli beam theory and subjected to Casimir force. Electrostatic actuation is considered in the governing equations of the system as the control input modeling. The Galerkin approach is employed to reduce the governing nonlinear partial differential equation of motion to a nonlinear ordinary differential equation with quadratic and cubic nonlinearities. The robust sliding mode controller is designed for stabilization of the nanobeam with the uncertain parameters which eliminates chattering. Furthermore, a disturbance-observer-based terminal sliding mode controller is used for stabilization of the nanobeam in the presence of an external load disturbance and unknown non-symmetric input saturation which is useful for when a constraint exists in the system control input due to the electrostatic actuation. Simulation results are presented and comparisons of the suggested control techniques are accomplished for the mentioned system and also for an atomic force microscope as the second case study in order to show the performance of the proposed controllers. Finally, experimental verification has been done by using the suggested terminal sliding mode controller to prove the high precision positioning.