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 $$\ddot{x}(t)+f_D(x(t),\dot{x}(t))+x(t)=1-\frac{e{\mathcal{V}}^2(t,x(t),x_d(t),\dot{x}(t),{\dot{x}}_d(t))}{x^2(t)},x(t)\in ]0,\infty [$$ where $x_d(t)=x(t-d)$ and ${\dot{x}}_d(t)=\dot{x}(t-d),$ denote position and velocity feedback respectively, and $$\mathcal{V}(t,x(t),x_d(t),\dot{x}(t),{\dot{x}}_d(t))=V(t)+g_1(x(t)-x_d(t))+g_2(\dot{x}(t)-{\dot{x}}_d(t)),$$ is the feedback voltage with positive input voltage $V(t)\in C(\mathbb{R}/T\mathbb{Z})$ for $e\in {\mathbb{R}}^{+},g_1,g_2\in \mathbb{R}$, $d\in [0,T[$. The damping force $f_D(x,\dot{x})$ can be linear, i.e., $f_D(x,\dot{x})=c\dot{x}$, $c\in {\mathbb{R}}^{+}$ or squeeze film type, i.e., $f_D(x,\dot{x})=\gamma \dot{x}/x^3$, $\gamma \in {\mathbb{R}}^{+}$. 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.

23 pages, 11 Figures