Optimality of the Adaptive Controllers

Results presented in [1] can be considered as the fundamental one in the adaptive control theory. Essentially, it is shown that if the system is exactly modelled, a self-tuning controller provides the same performance as the minimum-variance controller. It turns out that similar results are valid even in the presence of a modelling error. In this paper it is proved that in the presence of unmodelled dynamics, adaptive controller guarantees in a certain sense the same performance as the best non-adaptive controller. Despite the fact that the estimation algorithm has a vanishing gain sequence, uniform boundedness of all signals is established.