Kalman Filtering with State Equality Constraints

For linear dynamic systems with white process and measurement noise, the Kalman filter is known to be an optimal estimator. In the application of Kalman filters there is often known model or signal information that is either ignored or dealt with heuristically [1]. This work presents a way to generalize the Kalman filter in such a way that known relations among the state variables (i.e., state constraints) are satisfied by the state estimate. Some researchers have treated state constraints by reducing the system model parameterization [2, 3], but there are a couple of disadvantages with this approach. First, it may be desirable to maintain the form and structure of the state equations due to the physical meaning associated with each state. The reduction of the state equations makes their interpretation less natural and more difficult. Second, the equality constraint solution presented here can be extended to inequality constraints by checking the inequality constraints at each time step of the filter [4]. If the inequality constraints are satisified at a given time step, then the inequality constrained problem is solved. If the inequality constraints are not satisifed, then the constrained solution presented here can be used to enforce the constraints. Some researchers treat state constraints as perfect measurements [5, 6]. This results in a singular covariance matrix but does not present any theoretical problems [7]. In fact, Kalman’s original paper [8] presents an example that uses perfect measurements (i.e., no measurement noise). However, there are a couple of considerations that indicate against the use of perfect measurements in a Kalman filter implementation. First of all, although the Kalman filter does not formally require a nonsingular covariance matrix, in practice a singular covariance increases the possibility of numerical problems [9, p. 249], [10, p. 365]. Secondly, the incorporation of state constraints as perfect measurements increases the dimension of the problem, which in turn increases the size of the matrix that needs to be inverted in the Kalman gain computation. For instance, if we have a Kalman filtering problem with m measurements, then we need to invert an m£m matrix in order to compute the Kalman gain. If in addition we have s state constraints that we treat as perfect measurements, then we need to instead invert an (m+ s)£ (m+ s) matrix. An alternative method, a constrained Kalman filter, which incorporates the state constraints into the state estimation framework is proposed. The proposed solution does not have any numerical problems, does not increase the dimension of the problem, and has known and proven statistical properties. At each time step the unconstrained Kalman filter solution is projected onto the constraint surface. This is similar to