Analysis of adaptive identifiers in the presence of unmodeled dynamics: averaging and tuned parameters

The authors analyze the behavior of a standard identifier when the plant contains additional dynamics, called unmodeled dynamics, which invalidate the known order assumption. The first result of the analysis is an input richness condition which does not depend on the order of the unmodeled dynamics to guarantee persistency of excitation of the regressor. Then it is shown that the persistently exciting (PE) condition leads to a BIBO (bounded-input bounded-output) stability property for the identifier. The method of averaging is used 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, an explicit expression to calculate and interpret them as the fixed parameter values that minimize the mean-squared output error is derived