Stability and Convergence of Adaptive Control Algorithms: A Survey and Some New Results.

Abstract

Abstract : Although adaptive controllers have been designed for a number of years, the central question of global asymptotic stability of the overall feedback adaptive loop and its associated error equations remained open until recently. The two main approaches used to design controllers from a stability viewpoint - Lyapunov's Direct Method and Popov's Hyperstability Theory - only assure boundedness of the (augmented) state and parameter errors. The difficulties encountered in both approaches are analyzed and shown to be exactly analogous. Subsequently, a generic model for an adaptive controller is proposed from which already existing adaptive algorithms can be derived with minor modifications. The model unifies various algorithms, under the essential requirement for positive reality of the associated error transfer function, and its proof of stability contains others suggested to date as special cases. Finally some general comments are made regarding rates of convergence, performance of the algorithms in the presence of disturbances and unmodeled dynamics, ease of implementation, etc. (Author)