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Kalman and UFIR state estimation with coloured measurement noise using backward Euler method

Under coloured noise, known modifications of the Kalman filter (KF) exist only for discrete-time state-space models produced by the forward Euler (FE) method, which fits with feedback control. In this study, the authors modify the KF and unbiased finite impulse response (UFIR) filter using the backward Euler (BE) method for models with coloured measurement noise (CMN), which better fits systems without feedback. The FE- and BE-based models differ by time indexes in the system input and noise that is essential for time-varying and Markov jump systems. Employing measurement differencing, two KF algorithms and a unique UFIR algorithm are derived for time-correlated and de-correlated noise. An equivalence of the KF algorithms is proved analytically and confirmed by simulations. Numerical examples are given for target tracking and experimental verification is provided for visual object tracking. The high efficiency of the designed algorithms in removing CMN is demonstrated experimentally.