Convergence studies of adaptive control algorithms part I: Analysis

The so-called "adaptive control problem" has received attention by theoreticians and practitioners alike for the past twenty-five years. Elegant and useful theoretical advances have been made in the last decade, and especially in the past three years, that have unified diverse approaches. However, despite the fact that the central question of global asymptotic stability of the overall feedback adaptive loop has been resolved for a substantial class of adaptive controllers--at least in the deterministic case--the mechanism of adaptation and the convergence properties in general of already existing algorithms are still far from being well understood. As a consequence, there is a significant gap between the available (theoretical) methodologies and the potential applications of such adaptive schemes. This paper contains the first results of a research effort whose long range objective is in developing a methodology of design for adaptive control systems by attempting to unify promising concepts based primarily upon hyperstability theory and stochastic optimal control, respectively, with some common sense engineering techniques. In this context, special emphasis is given on the convergence properties--in terms of rates and transient behavior--of a large class of adaptive control algorithms.