Feedback stabilization of the nonlinear pendulum under uncertainty: A robustness issue
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's direct method to prove the stability of an equilibrium point. Then, we apply Melnikov's method to introduce a criterion for the control to avoid deforming the actual region of stability of the equilibrium point. We demonstrate the cooperative relationship between the Liapunov and 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. In essence, robustness of controllers for nonlinear systems must be treated from (nonlocal) dynamical systems viewpoint. A simulation result is given which quantitatively supports the analysis.
scientific article; zbMATH DE number 3946012
scientific article