Extension of the robust two-stage Kalman filtering for systems with unknown inputs

This paper considers the optimal unbiased minimum-variance estimation for systems with unknown inputs that affect both the system model and the measurements. By making use of the recently developed optimal unbiased minimum-variance filter (OUMVF) and a new proposed constrained relationship, an extension of the previous proposed robust two-stage Kalman filter (RTSKF), which is named as the ERTSKF, is presented. The proposed ERTSKF is the optimal solution of two-stage Kalman filters for the considered problem in the sense that it is equivalent to the OUMVF. This is formally established in a theorem. It is also shown that the ERTSKF is computationally more attractive than the OUMVF. Moreover, an alternative to the ERTSKF is also presented to further reduce the computational complexity. Simulation results confirm the effectiveness of the proposed results.