On partially augmented observers for systems with coloured noises

Abstract

The problem of optimal fixed order observers for continuous-time linear systems with deterministic input and coloured 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 optimal observer-estimator, under these dimensional and structural constraints, is then derived. 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-estimate are given. The solution consists of a modified Riccati and three modified Lyapunov equations coupled by three projection matrices. The three projections correspond to the order reduction, the singularity of the measurement, and the structural constraint. The problem is shown to be the most general reduced order optimal estimation problem and previously solved cases are all specializations of our results.