The Response of a Parametrically Excited van der Pol Oscillator to a Time Delay State Feedback

We investigate the parametric resonance of a van der Pol oscillator under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow-flow equations on the amplitude and phase ofthe oscillator. Their fixed points correspond to a periodic motion forthe starting system and we show parametric excitation-response andfrequency-response curves. We analyze the effect of time delay andfeedback gains from the viewpoint of vibration control and use energyconsiderations to study the existence and characteristics of limit cycles of the slow-flow equations. A limit cycle corresponds to a two-periodmodulated motion for the van der Pol oscillator. Analytical results areverified with numerical simulations. In order to exclude the possibilityof quasi-periodic motion and to reduce the amplitude peak of theparametric resonance, we find the appropriate choices for the feedbackgains and the time delay.