Direct solution to the general reduced-order stochastic observation problem

The stochastic optimal state observation problem is considered for a general linear, continuous, time-invariant system with unmeasurable stationary inputs and measurement outputs that may be, at least in part, perfect. A general solution to the problem is obtained by processing the perfect measurements through a specific differentiation-transformation scheme in order to extract the maximum accurate information on the system states. Using this information the original system is transformed to a new reduced-order model whose measurements are corrupted by a white noise of non-singular intensity matrix. A minimum-order full-state estimator to the original system is then constructed by combining the outputs of any full-order observer to the reduced-order model and the perfect combinations of the system states that were derived by the differentiation-transformation scheme. A solution to the general singular Kalman filtering problem is then obtained by minimizing the variance of the estimation error of the observer to the reduced-order model.