The Minimal Bound on the Estimation Error Covariance Matrix in the Presence of Correlated Driving Noise

Abstract

The application of optimal linear filter theory to situations in which the state forcing function is correlated with the state in an unknown way can present serious problems. In many instances, the cross-correlation and forcing function must be modeled, for if they are ignored the filter gains tend to sink below their optimal levels and useful measurement information is discarded. This paper presents a conservative and minimal formula for bounding cross-correlation between a random forcing function and the state error when this correlation is unknown. The bound is conservative in the sense that its use always results in overestimating the estimation error covariance, and it is minimal in the sense that given any conservative cross-correlation estimate, a bound of the minimal form can always be found which is no more conservative than the given estimate. When this minimal bound is used to approximate the differential equation for the estimation error covariance matrix, there remains the problem of finding the free parameter associated with the minimal bound. This paper presents a noniterative expression for this parameter as the solution to an optimal control problem in which the cost function is a linear combination of the elements of the covariance matrix at a final time of interest. Simulation results are given for a satellite in orbit around a model earth.