A Proportional Derivative (PD) Controller for Suppression the Vibrations of a Contact-Mode AFM Model

The nonlinear dynamics control of a contact-mode atomic force microscopy (AFM) system with multi forces (harmonic and parametric excitation force) utilizing the time delay proportional derivative (PD) controller is investigated. The perturbation method is utilized to calculate the first-order approximate solutions for the AFM system. The stability of the AFM system is investigated at the worst resonance case by Lyapunov's first method. We also show the bifurcation diagrams of response curves using frequency response equations before and after control are performed. Furthermore, we focus on the effect of the time-delayed control and returns signals gain on the vibration amplitude and the stability analysis of the controlled system in primary, sub-harmonic resonance for different parameter variations. In addition, the numerical results are procured using MATLAB program. Eventually, validation curves are presented to estimate the nearness degree between the analytical predictions also numerical simulation. The obtained results exhibited that, the time-delayed PD control efficiency to put down the nonlinear oscillations of the system.