Robust minimax estimation applied to kalman filtering

Abstract

Kalman filtering is one of the most essential tools in estimating an unknown state of a dynamic system from measured data, where the measurements and the previous states have a known relation with the present state. It has generally two steps, prediction and update. This filtering method yields the minimum mean-square error when the noise in the system is Gaussian and the best linear estimate when the noise is arbitrary. But, Kalman filtering performance degrades significantly with the model uncertainty in the state dynamics or observations. In this thesis, we consider the problem of estimating an unknown vector x in a statespace model that may be subject to uncertainties. We assume that the model uncertainty has a known bound and we seek a robust linear estimator for x that minimizes the worst case mean-square error across all possible values of x and all possible values of the model matrix. Robust minimax estimation technique is derived and analyzed in this thesis, then applied to the state-space model and simulation results with different noise perturbation models are presented. Also, a radar tracking application assuming a linear state dynamics is also investigated. Modifications to the James-Stein estimator are made according to the scheme we develop in this thesis, so that some of its limitations are dealt with. In our scheme, James-Stein estimation can be applied even if the observation equation is perturbed and the number of observations are less than the number of states, still yielding robust estimations.