Feedback control algorithms through Lyapunov optimizing control and trajectory following optimization

This research is concerned with the development of feedback control laws specified through the combination of Lyapunov optimizing control techniques with trajectory following optimization algorithms. Initially, this research extends the application of Lyapunov optimizing control to explicitly consider periods of singular control when a state-space switching surface is encountered. A detailed investigation of minimum-time optimal control problems, where the state equations are linear and decoupled from the bounded, scalar control, is presented. A primary outcome is the specification of control design steps for the selection of an appropriate descent function. This analysis explicitly considers how a quadratic approximation to the optimal return function affects stability, chatter, and the existence of singular control. An additional concern is the presence of two time scales; whether part of the original state-space system or induced by control gains. Such time scales complicate the analysis make efficient numerical implementation of any algorithm more difficult. These time scales, or, singular perturbations are examined and interesting results are obtained. The second major focus of this research is the combination of trajectory following optimization techniques with Lyapunov optimizing control methods to produce new control algorithms for generating feedback controls "on-line". During this analysis we are concerned with the presence of time scales that appear in the augmented set of differential equations. The resulting Lyapunov optimizing control algorithms via trajectory following optimization allow the analyst to specify efficient feedback control algorithms that are especially well suited for on-line applications.