H∞ bounds for least-squares estimators

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