H-infinity bounds for least-squares estimators

In this note, we obtain upper and lower bounds for the H-infinity norm of t he Kalman filter and the recursive-least-squares (RLS) algorithm, with resp ect to prediction and filtered errors. These bounds can be used to study th e robustness properties of such estimators. One main conclusion is that, un like H-infinity-optimal estimators which do not allow for any amplification of the disturbances, the least-squares estimators do allow for such amplif ication. This fact can be especially pronounced in the prediction error cas e, whereas in the filtered error case the energy amplification is at most f our. Moreover, it is shown that the Nm norm for RLS is data dependent, wher eas for least-mean-squares (LMS) algorithms and normalized LMS, the H-infin ity norm is simply unity.