Optimally Robust Kalman Filtering

We present some optimality results for robust Kalman filtering. To this end, we introduce the general setup of state space models which will not be limited to a Euclidean or time-discrete framework. We pose the problem of state reconstruction and repeat the classical existing algorithms in this context. We then extend the ideal-model setup allowing for outliers which in this context may be system-endogenous or -exogenous, inducing the somewhat conflicting goals of tracking and attenuation. In quite a general framework, we solve corresponding minimax MSE-problems for both types of outliers separately, resulting in saddle-points consisting of an optimally-robust procedure and a corresponding least favorable outlier situation. Still insisting on recursivity, we obtain an operational solution, the rLS filter and variants of it. Exactly robust-optimal filters would need knowledge of certain hard-to-compute conditional means in the ideal model; things would be much easier if these conditional means were linear. Hence, it is important to quantify the deviation of the exact conditional mean from linearity. We obtain a somewhat surprising characterization of linearity for the conditional expectation in this setting. Combining both optimal filter types (for system-endogenous and -exogenous situation) we come up with a delayed hybrid filter which is able to treat both types of outliers simultaneously. Keywords: robustness, Kalman Filter, innovation outlier, additive outlie