Bifurcations and chaos in a periodically forced prototype adaptive control system

An adaptive control system with a first-order plant and the so-called cr-modification adaptation law is analyzed in the case of periodic disturbance or reference input. The local bifurcations of the low-period solutions are numerically detected by means of a continuation method, and the different modes of behavior are classified as well as the transitions among them. As predicted by the theory, the control system is robust in the sense that all trajectories are bounded regardless to the action of the disturbance. However, the periodicity of the input can give rise to chaotic behavior. The result of the analysis will aid the designer in selecting the controller parameters in order to achieve an acceptable behavior.