Optimal bilinear observers for bilinear state-space models by interaction matrices

This paper formulates optimal bilinear observers for bilinear state-space models. Observers in bilinear form, as opposed to other nonlinear forms, are required to develop an extension of observer/Kalman filter identification for simultaneous identification of a bilinear state-space model and an associated bilinear observer from noisy input-output measurements. The paper establishes the relationship between the bilinear observer gains and the interaction matrices which are used to convert the original bilinear state-space model to a form that simplifies the identification of such a model. Techniques to find the interaction matrices are developed. In the absence of noises, these matrices produce the gains of the fastest converging observer. In the presence of noises, they minimise the state estimation error in the same manner as a standard steady-state Kalman filter. Numerical examples illustrate both the theoretical and computational aspects of the proposed algorithms.