A Kalman-filtering derivation of input and state estimation for linear discrete-time systems with direct feedthrough

This paper is devoted to investigating the problem of simultaneous input and state estimation for linear discrete-time systems with direct feedthrough from the perspective of a limiting case of the Kalman filtering problem. First, when the unknown input of the underlying system is described as a white Gaussian noise with finite mean and finite variance, a Kalman filter is derived. Next, under the case that the variance of the unknown input tends to infinity, the Kalman filter and the existing simultaneous input and state estimator are unified. Finally, for linear discrete-time systems without direct feedthrough, the relationship between the Kalman filter and the simultaneous input and state estimator is established in a simple manner. The result of this study will pave the way for designing an unbiased estimator for linear systems with unknown inputs and packet drops.