Revisiting State Estimation for Linear Systems with State and/or Filter Equality Constraints*

Abstract

This article revisits optimal state estimation, in the mean squared error matrix sense, for linear systems with state and/or filter subject to linear equality constraints (LECs). First, it is shown that the conventional Wiener filter (WF) form incorporates any LECs on the state, thus yielding a filter subject to the same LECs. Conversely, an optimal linear filter subject to LECs (or linear equality gain constraints) in general does not exist. Therefore, adding LECs on the WF or WF gain matrix either leaves unchanged or degrades the constrained WF performance w.r.t. the unconstrained WF. Since the Kalman filter (KF) and Kalman predictor (KP) are recursive WF forms for linear discrete state-space (LDSS) systems, the same results hold for both estimators, which is in contradiction with several existing results in the literature. Actually, even if these existing results are mathematically correct, however, they have been derived for unsuitable assumed LDSS models where the state is surprisingly not compliant with the assumed LECs. Indeed, it is shown that for suitable assumed state models, both standard KF and KP forms satisfy the assumed LECs, making any additional projection step superfluous.