H∞ bounds for least-squares estimators

In this paper we obtain upper and lower bounds for the H 1 norm of the Kalman lter and RLS algorithm, with respect to prediction and ltered errors. These bounds can be used to study the robustness properties of such estimators. One main conclusion is that, unlike H 1-optimal estimators which do not allow for any ampliication of the disturbances, the least-squares estimators do allow for such ampliication. This fact can be especially pronounced in the prediction error case, whereas in the ltered error case the energy ampliication is at most four. Moreover, it is shown that the H 1 norm for RLS is data-dependent, whereas for LMS and normalized LMS the H 1 norm is simply unity.