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

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 yet general bursting scenario to derive an analytical lower bound on the worst-case peak steady-state error for a wide class of parameter estimation and system identification algorithms. Our results show that in the absence of any input constraints, arbitrarily small perturbations impose a serious performance limitation, in the sense that the worst-case performance deteriorates proportionally with the size of the parametric uncertainty set.