Optimal observers for systems with colored noises

The problem of optimal full order observers for continuous-time linear systems with colored process and measurement noises is considered. Optimal estimation of the state in such cases involves augmentation of the system, thus a higher order observer is required. In this paper the structure of a full-order observer is assumed and necessary conditions for the optimal observer are derived. The conditions are given for the general case where the intensity of the white noise component of the measurement noise may be singular. The solution consists of a modified Riccati equation and a Lyapunov equation coupled by two projection matrices in the singular case and one projection matrix in the nonsingular case. The problem under consideration is distinct from that of the optimal reduced-order estimator since, in general, the latter is not an observer.