Adaptive control in nonlinear dynamics

Abstract

We extend an adaptive control algorithm recently suggested by Huberman and Lumer to multi-parameter and higher-dimensional nonlinear systems. This control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour. We find that in all cases, the recovery time is linearly proportional to the inverse of control stiffness (for small stiffness). In higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery. The control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated. We further suggest methods by which periodic motion such as limit cycles can be adaptively controlled, and demonstrate the robustness of the procedure in the presence of (additive) background noise.