Notch Filter Feedback Control for k-Period Motion in a Chaotic System∗

Chaotic motion can sometimes be desirable or undesirable, and hence control over such a phenomenon has become a topic of considerable interest. Currently available methods involve making systematic time-varying small perturbations in the system parameters. A new method is presented here to achieve control over chaotic motion using notch filter output feedback control. The notch filter controller uses an active negative feedback with fixed controller parameters without affecting the original system parameters. The motivation for using a notch filter in the feedback is to disturb the balance of power at the lower end of the participating frequencies in the power spectrum. This results in a truncation of the period-doubling route to chaos. For low-period motions the harmonic balance method is used to show that a single participating frequency can indeed be eliminated. To deal with relatively complex nonlinear plants, and higher-period motions, a numerical optimal parameter selection scheme is presented to choose the notch filter parameters. The procedures are tested on Duffing’s oscillator with a notch filter feedback to achieve desired k-period motion.