On partially augmented observers for systems with colored noises

The problem of optimal fixed-order observers for continuous-time linear systems with deterministic input and colored process and measurement noises is considered. Optimal estimation of the state involves augmentation of the system, and as a result the observer has the augmented dimension. In this paper, the order of the observer is prefixed to a value which ranges between the dimension of the original system and that of the augmented system. The presence of a deterministic input imposes the structure of an observer on the estimator. The optional observer-estimator, under these dimensional and structural constraints, is then derived. A solution is given for the general case where the intensity of the white noise component of the measurement noise may be singular. Necessary conditions fully characterizing the optimal observer-estimator are given. The solution consists of a modified Riccati and three modified Lyapunov equations coupled by three projection matrices. The problem is shown to be the most general reduced-order optimal estimation problem