Feedback stabilization, stability and chaotic dynamics

We examine the control of a (prototype) nonlinear pendulum under uncertainty in modelling. The uncertainty is sufficiently small and is a function of both state and time. We apply Liapunov direct method to prove the stability of an equilibrium point. Then, we apply the Melnikov method to introduce a criterion for the control to avoid deforming the region of stability of the equilibrium point. We demonstrate the cooperative relationship between the Liapunov and the Melnikov methods in assessing stability. Consequently, we propose preserving the size and shape of the region of stability, and preventing the creation of complicated dynamics on its boundary, as indicative of robust control. We also demonstrate the rise of chaotic dynamics in a prototype adaptive feedback system in the face of disturbances. In essence, robustness of controllers for nonlinear systems must be treated from (nonlocal) dynamical systems viewpoint. Simulation results are given which quantitatively support the analysis.