Robust time-varying Kalman smoothers with Uncertain Noise Variances

This paper addresses the design of robust Kalman smoothers for the time-varying system with uncertain noise variances. According to the unbiased linear minimum variance (ULMV) optimal estimation rule, the robust time-varying Kalman smoothers are presented. Specially, two robust Kalman smoothing algorithms are presented by the augmented and non-augmented state approaches, respectively. They have the robustness in the sense that their actual estimation error variances are guaranteed to have a minimal upper bound for all admissible uncertainties of noise variances. Their robustness is proved by the Lyapunov equation approach, and their robust accuracy relations are proved. The corresponding steady-state robust Kalman smoothers are also presented for the time-invariant system, and the convergence in a realization between the time-varying and steady-state robust Kalman estimators is proved by the dynamic error system analysis (DESA) method and the dynamic variance error system analysis (DVESA) method. A simulation example is given to verify the robustness and robust accuracy relations.