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. In such cases, optimal estimation of the state involves augmenting the system, thus a higher-order observer is required. 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