A reduced-order stochastic observer approach to optimal state estimation with noise-free measurements

In a continuous-time Kalman filter, it is required that the measurement noise covariance be non-singular. If the measurements are noise-free, then this condition does not hold and, in practice, the measurement data are differentiated to define a derived measurement function to build what is known as Deyst filter. It is proposed here that a reduced-order observer be used in deriving the linear minimum-variance filter to construct state estimates based on the original measurement data with no need for differentiation. This filter is of dimension (n - p) where n and p are the state and measurement vector dimensions, respectively. In this work, we consider both the finite-time and infinite-time results. The set of all assignable estimation error covariances are characterized and the set of all estimator gains are parametrized in addition to the linear minimum variance optimal results. The conditions for the existence of the optimal steady-state filter are obtained in terms of the system theoretic properties of the original signal model. A simple example is included to illustrate the effectiveness of the proposed technique.