Performance limitations of adaptive parameter estimation and system identification algorithms in the absence of excitation

In this paper we address the problem of fundamental performance limitations in adaptive parameter estimation and system identification, occurring in environments where perturbations are present but there is lack of sufficient excitation. We construct a simple but general bursting scenario to derive an analytical lower bound on the worst-case lim sup performance of a wide class of parameter estimation and system identification algorithms. Our results show that in the absence of any excitation or other input constraints, arbitrarily small perturbations impose a serious limitation on the lim sup performance of these adaptive algorithms, in the sense that the worst-case performance deteriorates proportionally with the size of the parametric uncertainty set. Furthermore, this performance limitation is a consequence of some fairly general properties of adaptive algorithms and independent of their precise form.