On the error behavior in linear minimum variance estimation problems

For linear systems the error covariance matrix for the unbiased, minimum variance estimate of the state does not depend upon any specific realization of the measurement sequence. Thus it can be examined to determine the expected behavior of the error in the estimate before actually using the filter in practice. In this paper, the general linear system that contains both plant and measurement noise is shown to exhibit a decomposition property that permits the derivation of upper and lower bounds upon the error covariance matrix. This decomposition allows systems containing either plant or measurement noise, but not both, to be considered separately. Some general characteristics of these simpler systems are discussed and conditions for the positive definiteness and vanishing of the error covariance matrix are established. It is seen that the presence of plant noise, in general, prevents the error from vanishing. Alternatively, the condition of
q
-stage observability is seen to be sufficient to insure that the error covariance matrix asymptotically approaches the zero matrix for systems with noise-free plants. These results are used to establish very specific lower bounds. Through the application of the duality principle, they can be applied directly to the analysis of the linear regulator problem.