Robust Kalman estimators for systems with multiplicative and uncertain-variance linearly correlated additive white noises

For linear discrete-time stochastic systems with mixed uncertainties including multiplicative and uncertain-variance linearly correlated additive white noises, this paper addresses the problem of designing robust Kalman estimators. By the fictitious noise-based Lyapunov equation approach, the system under consideration is converted into one with only uncertain noise variances. Based on the worst-case system with conservative upper bounds of actual noise variances, the minimax robust time-varying Kalman estimators (predictor, filter, and smoother) are presented in a unified framework. Their robustness is proved in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. The corresponding robust steady-state Kalman estimators are also presented. An innovation test rule and a search technique of selecting the less-conservative upper bounds of actual noise variances are presented, by which the two classes of guaranteed robust accuracy Kalman estimators are presented. One class is robust Kalman estimators with improved robust accuracy, the other class is ones with the prescribed robust accuracy index. Three modes of convergence in a realization among the time-varying and steady-state robust Kalman estimators for the time-varying and time-invariant systems are presented and proved by the dynamic error system analysis (DESA) method. Two simulation examples applied to autoregressive (AR) signal processing and uninterruptible power system (UPS) show the effectiveness of the proposed results.