Periodic oscillations in electrostatic actuators under time delayed feedback controller

Abstract

In this paper, we prove the existence of two positive 𝑇 -periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation ̈𝑥(𝑡) + 𝑓𝐷(𝑥(𝑡), ̇𝑥(𝑡)) + 𝑥(𝑡) = 1 − 𝑒 2 (𝑡, 𝑥(𝑡), 𝑥𝑑 (𝑡), ̇𝑥(𝑡), ̇𝑥𝑑 (𝑡)) 𝑥 2 (𝑡) , 𝑥(𝑡) ∈ ]0, ∞[ where 𝑥𝑑 (𝑡) = 𝑥(𝑡 − 𝑑) and ̇𝑥𝑑 (𝑡) = ̇𝑥(𝑡 − 𝑑), denote position and velocity feedback respectively, and (𝑡, 𝑥(𝑡), 𝑥𝑑 (𝑡), ̇𝑥(𝑡), ̇𝑥𝑑 (𝑡)) = 𝑉 (𝑡) + 𝑔1 (𝑥(𝑡) − 𝑥𝑑 (𝑡)) + 𝑔2 ( ̇𝑥(𝑡) − ̇𝑥𝑑 (𝑡)), is the feedback voltage with positive input voltage 𝑉 (𝑡) ∈ 𝐶(R∕𝑇Z) for 𝑒 ∈ R+, 𝑔1 , 𝑔2 ∈ R, 𝑑 ∈ 0, 𝑇 . The damping force 𝑓𝐷(𝑥, ̇𝑥) can be linear, i.e., 𝑓𝐷(𝑥, ̇𝑥) = 𝑐 ̇𝑥, 𝑐 ∈ R+ or squeeze film type, i.e., 𝑓𝐷(𝑥, ̇𝑥) = 𝛾 ̇𝑥∕𝑥 3 , 𝛾 ∈ R+. The fundamental tool to prove our result is a local continuation method of periodic solutions from the non-delayed case (𝑑 = 0). Our approach provides new insights into the delay phenomenon on microelectromechanical systems and can be used to study the dynamics of a large class of delayed Liénard equations that govern the motion of several actuators, including the comb-drive finger actuator and the torsional actuator. Some numerical examples are provided to illustrate our results.