Self excited closed loop parametric estimation in the presence of noise undermodelling

It is well known in non-parametric estimation that without a persistently exciting signal the estimate of the process will be biased towards the negative inverse of the controller. However it has recently been shown that with parametric estimation the maximum likelihood estimate will yield the true process dynamics provided one has a consummate noise model. These two results appear contradictory. The purpose of this paper is a rapproachement of the two cases. Indeed we show that there exists a fundamental sensitivity to the noise model in the parametric case. Specifically, with undermodelling in the noise model we show that the parametric estimates provided by maximum likelihood are biased towards the negative inverse of the controller.