Kalman Filtering with Statistical State Constraints

For linear dynamic systems with white process and measurement noise, the Kalman ̄lter is known to be the minimum variance linear state estimator. In the case that the random quantities are Gaussian, then the Kalman ̄lter is the minimim variance state estimator. However, in the application of Kalman ̄lters known signal information is often either ignored or dealt with heuristically. For instance, state variable constraints (which may be based on physical considerations) are often neglected because they do not ̄t easily into the structure of the optimal ̄lter. Previous work by the authors demonstrated an analytic method of incorporating deterministic state equality constraints in the Kalman ̄lter. This paper extends that work to develop the properties of Kalman ̄lters in the presence of statistical state constraints. That is, given a linear system such that the expected values of the state variables satisfy some linear equality, we can constrain the Kalman ̄lter estimates to sat-