In this paper we pose the state estimation problem for linear systems with Gaussian noise and disturbances and independently distributed measurements outliers as that of finding the joint a posteriori most probable (JAPMP) state and outlier sequence given the observations. We show that this problem can be reformulated as an optimal reference tracking problem for switched linear systems, which we call the dual problem. By using techniques from optimal and approximate control of switched linear systems we are able to solve this computationally challenging problem in an attractive manner. In particular, we can provide state estimators which guarantee to be within a constant likelihood factor from the optimal as well as state estimators which guarantee a better likelihood than that of other, suboptimal state estimators.
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