Minimal order Wiener filter for a linear system with exact measurements

Formulas are derived for the minimal order Wiener filter for both a continuous and discrete detectable system when some measurements are noiseless and when such a filter exists. 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 for the continuous system. For the discrete time system, the estimator is a linear functional of the measurements and a finite number of time delays of the exact measurements. Our conditions are shown to be dual to the generalized Legendre-Clebsch conditions of the dual optimal singular regulator. It is shown that as all process and measurement noise vanishes the error covariance of our filter converges to a null matrix. A separation principle is derived.