Robust Kalman filtering via Krein space estimation

A robust Kalman filter is proposed for the discrete-time system with norm-bounded parametric uncertainties. The uncertainties are described by the energy bound constraint, i.e. the sum quadratic constraint (SQC). It is shown that the SQC can be converted into an indefinite quadratic cost function to be minimised in the Krein space, and it is found that the Krein space Kalman filter is a solution of the minimisation problem. After introducing a Krein space state-space model, which includes the uncertainty, one can easily write a robust version of the Krein space Kalman filter by modifying the measurement matrix and the variance of measurement noises in the original Krein space Kalman filter. Since the resulting robust Kalman filter has the same recursive structure as a conventional Kalman filter, a robust filtering scheme can be readily designed using the proposed method. A numerical example demonstrates that the proposed filter achieves robustness against parameter variation and improvement in performance when compared with a conventional Kalman filter and an existing robust Kalman filter, respectively.