Optimum Continuous Linear Estimator for Systems with Correlated Measurement Noise

Abstract

The optimal continuous-time filtering problem for the case of linear dynamics, linear measurements, and Gaussian white disturbances and colored measurement noises generated by a limited class of shaping filters excited by white noise input has been solved. However, the colored noise cannot be assumed to take an arbitrary form, in order that the problem become solvable. In a scalar measurement, for example, the noise with exponential type autocorrelation function is acceptable but the noise with exponential-cosine type autocorrelation function, which is also important, is not acceptable. Therefore, it merits a further study to determine the optimal estimator for a colored noise generated by a more general class of shaping filters.
As is often done to solve such a problem, an augmented system is constructed combining the process and the shaping filter. Formulating the problem in this approach, it is known that the original problem is included in the optimal filtering problem for the case of Gaussian white disturbance and the measurements not corrupted by noise.
In this paper, it is shown that this problem is essentially solved by finding a set of bases of the greatest observability subspace with unknown inputs. First, the systematic algorithm is developed to find the bases and the form to which the standard Kalman filter is applied formally, and obtain the formal estimator. And then, it is proved that this estimator is in fact the optimal estimator.