Stabilization of electrostatic MEMS resonators using a stochastic optimal control

Noise-induced motions are a significant source of uncertainty in the response of electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and dramatic loss of stability. In this manuscript, we present an analysis for the stabilization of an electrostatically actuated MEMS resonator in the presence of noise processes using a stochastic optimal control scheme. A stochastic lumped-mass model that accounts for the uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude of the resonator is presented. The Ito^ equations, describing the averaged modulations of the resonator's total energy and phase difference, are obtained via the stochastic averaging of energy envelope. The stochastic optimal control law aiming at minimization the pull-in probability of the resonator is determined via the stochastic dynamic programming equations associated with the Ito^ equations. We show that the resulting stochastic optimal feedback control, with a careful selection of its control gain, can be used to suppress the noise-activated pull-in instability of the disturbed resonator. The impact of the time delay inherent in the feedback control input on the control effectiveness is investigated. In addition, the control scheme also shows superior performance in counteracting the deterministic pull-in instability of the resonator operating in the dynamic pull-in frequency band. Good agreement between the theoretical results and numerical simulation is demonstrated.