Initial-condition robustness of linear least squares filtering algorithms

We give some sufficient conditions under which the mean-square error of linear least squares (lls) estimates converges to its true steady-state value despite perturbations due to uncertainties in initial conditions, round-off errors in calculation, etc. For state-variable estimators, this property, called initial-condition robustness, is implied by the exponential asymptotic stability of the estimating filter, but this latter property though desirable is of course far from necessary for the more basic (since mean-square error is the ultimate criterion) property of robustness. We present a general sufficient condition for such robustness of lls predictors of stochastic processes. This condition is then specialized to lls estimators for processes described by state-variable models and by autoregressive-moving agerage difference equation models. It is shown that our conditions can establish robustness in cases where previous criteria either fail or are inconclusive.