Bias Issues in Closed Loop Identification with Application to Adaptive Control∗

It is well known in system identification that, under output feedback control without persistent excitation, the resultant estimate of the process will be biased. In non-parametric identification the estimate is biased towards the negative inverse of the controller. A similar relationship is shown to exist for parametric identification. Indeed, we show the existence of a fundamental sensitivity to the noise model in the parametric case. However, in the parametric case, constraints are generally required, depending on the relative degree of the controller, to ensure a causal estimate. The implications of this bias to a simple adaptive control algorithm are also examined.