Kalman Filtering with Statistical State Constraints

For linear dynamic systems with white process and measurement noise, the Kalman filter is known to be the minimum variance linear state estimator. In the case that the random quantities are Gaussian, then the Kalman filter is the minimim variance state estimator. However, in the application of Kalman filters 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 fit easily into the structure of the optimal filter. Previous work by the authors demonstrated an analytic method of incorporating deterministic state equality constraints in the Kalman filter. This paper extends that work to develop the properties of Kalman filters 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 filter estimates to satisfy those constraints. This results in a family of constrained filters with each member parameterized by a weighting matrix. This paper derives several interesting properties of the constrained Kalman filters