Adaptively Controlled Systems

Abstract

In the preceding chapters, we saw how to transform non-strict Lyapunov functions into explicit strict Lyapunov functions for cases where the system dynamics are completely known. However, there are important cases where the system parameters are unknown, and where the objectives are to simultaneously (a) design controllers that force the trajectories to track a prescribed reference trajectory and (b) estimate the unknown parameters. In this chapter, we present a generalization of strictification that can be used to meet these two objectives. It involves constructing global strict Lyapunov functions for an augmented system that includes the tracking error and the parameter estimation error. Our strict Lyapunov function approach makes it possible to quantify the effects of other types of uncertainty in the model as well, using the input-to-state stability framework. We illustrate our results using Rössler’s dynamics and Lorenz systems.