Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough

This paper extends previous work on joint input and state estimation to systems with direct feedthrough of the unknown input to the output. Using linear minimum-variance unbiased estimation, a recursive filter is derived where the estimation of the state and the input are interconnected. The derivation is based on the assumption that no prior knowledge about the dynamical evolution of the unknown input is available. The resulting filter has the structure of the Kalman filter, except that the true value of the input is replaced by an optimal estimate.