Effects of fractional viscoelasticity material of electrostatic micro-resonators on performances of delayed proportional-derivative control

A brief modelling of electrostatically actuated micro-actuator based on the Kelvin Voigt viscoelastic model with fractional order derivative is derived. Then, the effect of the viscoelastic material with fractional order derivative on the performance of delayed proportional-derivative control is investigated. Thanks to the D-subdivision method, the stability of the linearized system is depicted in the space of control parameter gains. The bifurcation diagram is used to confirm the existence of Hopf Bifurcation and the viscoelastic parameters effect. It is shown that the decrease of fractional order and the increase of the time relaxation contribute to increase the stable area. Moreover, the threshold conditions for the appearance of homoclinic and heteroclinic bifurcations are investigated using the Melnikov's criterion and the basin of attraction of initial conditions is used to validate the Melnikov's predictions. It is found that the consideration of the viscoelastic material contributes to increase the regular motions domain for any values of delayed proportional derivative controller parameters.