Constraints on state variables are commonly encountered in dynamic state estimation. These constraints can come in the form of algebraic equality and/or inequality constraints. For instance, mole fractions must always be greater than or equal to zero and the sum of mole fractions must be equal to one. Until recently, methods of state estimation for both linear and nonlinear dynamic systems either ignored the constraints or incorporated them in an ad hoc manner. For linear systems, the Kalman filter is the optimal state estimator. However, the structure of the filter does not allow for the imposition of the state constraints readily. For weakly nonlinear systems, the Extended Kalman filter (EKF) has found numerous uses as a suboptimal state estimator, which is once again not constrained in any way. For strongly nonlinear systems, the EKF is widely known to diverge. The arrival of the optimization based state estimation methods changed the scope of the problem.

The lack of constraints in Kalman filter and the stability problems in the EKF have been used numerous times by many researchers as motivation for the popular Moving Horizon Estimation methods for constrained state estimation. In fact the MHE class of estimators have become the natural choice whenever there are state equality and/or inequality constraints and nonlinear system dynamics. The MHE pose a least squares optimization problem in a moving horizon subject to constraints. Numerous researchers have presented the failure of EKF to obey constraints to justify other suboptimal estimators. However, we find that work on actually imposing the constraints in the existing EKF framework is scarce.

The main complicating issue in MHE is how the past information is summarized in the so called arrival cost inside the window. Several schemes are proposed such as the EKF filtered cost and smoothing updated cost to approximately calculate this cost. If the arrival cost is obtained exactly in a recursive form the MHE with window size one is the correct formulation. Larger window sizes are unnecessary. For the unconstrained case the Kalman filter is derived this way with a window size of one.

In this paper we present analytical solutions to the state constrained Kalman filter; both linear algebraic equality and inequality constraints are tackled as well as constraints with uncertainties. To the best of our knowledge, such a comprehensive solution has not been presented before. In addition, we show that the constrained filter can be implemented without any matrix inversions for an efficient implementation. We further prove that the constrained filter inherits all the stability properties of a standard Kalman filter.

Similar to the derivation of the EKF based on the Kalman filter, we show the development of a constrained Extended Kalman filter (cEKF) for equality and inequality constraints. The importance of this result is appreciated by comparing it with the moving horizon formulation. The cEKF is equivalent to an MHE with window size one and incorporates the correct recursive arrival cost term. Because the cEKF is an analytical solution that can be efficiently programmed without matrix inversion, the MHE formulation containing expensive optimization routines is completely avoided. The utility of the MHE as a general suboptimal strategy is not questioned here. For example to impose constraints on noise processes and inputs the MHE is necessary. However, for imposing state constraints, the proposed cEKF is sufficient.

The performance of cEKF is illustrated with a simulation study of a nonlinear batch reactor, on which earlier researchers have argued for the merits of the MHE. It is shown that the cEKF provides more accurate estimation with constraints at a fraction of the computation cost of the MHE.