Enhance the stability and vibration control of the electrostatic micro-actuator with time-delayed fractional order PD controller

Abstract

This paper investigates the effect of a delayed fractional-order proportional-derivative (PD) controller on the stability and dynamic behavior of an electrostatic micro-actuator. After a brief description of the model, a semi-analytical method is applied to the linearized system to identify both delay-independent and delay-dependent stability regions, illustrating how time delay, control gains, and fractional order affect the system’s response. The results indicate that decreasing the fractional order reduces the delay-independent stability domain and leads to stability switches as the delay increases. Bifurcation diagrams confirm the presence of these switches, showing that specific control settings lead to more complex or chaotic motion, while others help reduce vibrations. Melnikov’s method is used to predict transitions to horseshoe chaos, and the numerical analysis of basins of attraction validates these predictions. Overall, the study demonstrates that the fractional-order delayed controller improves the stability and robustness of micro-actuators compared to conventional PD controllers.