Analysis of adaptive identifiers in the presence of unmodelled dynamics: Averaging and tuned parameters

Adaptive identifiers are designed with the assumption that the order of the plant is known. In this paper we analyze the behavior of a standard identifier when the plant contains additional dynamics, called unmodelled dynamics, which invalidate the known order assumption. The first result of our analysis is an input richness condition which does not depend on the order of the unmodelled dynamics to guarantee persistency of excitation of the regressor. Then we show that the PE condition leads to a BIBO stability property for the identifier. We use the method of averaging to formally define the notion of tuned parameters as the equilibrium of the identifier averaged system. It is shown that the tuned parameters always exist and that the actual parameters converge to some neighborhood of the tuned parameters. From the definition of the tuned parameters, we derive an explicit expression to calculate them and interpret them as the fixed parameter values which minimize the mean squared output error.