A remark on worst-case output estimation and identification

The problem of estimator/identifier design is considered where the objective is to estimate the uncorrupted system output given its noisy measurements. Here it is shown that if the system has at least one unstable mode then any algorithm that assumes there is no measurement noise and gives an estimate consistent with the measurements is optimal for all l_p induced norms from noises to the output prediction error. Moreover, for such an estimation problem, worst-case performance cannot be improved by incorporating the future information, i.e., smoother estimates are no better than filter estimates. A special case of this class of problems is the standard identification problem with a worst-case performance criterion. From these observations one concludes that good worst-case performance for output estimation by an identification algorithm, by itself, does not guarantee good performance of the identification scheme