Minimal-order Wiener filter for a system with exact measurements

Necessary and sufficient conditions for the existence of an optimal steady-state state estimator are derived under the assumption that this estimator is a linear functional of the measurements and a finite number of derivatives of the exact measurements. Our conditions are shown to be dual to the generalized Legendre-Clebsch conditions of the dual optimal singular regulator. A separation principle is derived and it is shown that as all process and measurement noise vanishes, the error covariance of our filter converges to a null matrix.