Unbiased Minimum-variance Filtering for Input Reconstruction

In this paper, we introduce the concept of input and state observability, that is, conditions under which both the unknown input and state can be estimated from the output measurements. We discuss sufficient and necessary conditions for a discrete-time system to be input and state observable. Next, we derive an unbiased minimum-variance filter to estimate the unknown input and the state, when the state space matrices are known. We show that the Kalman filter and other filters in the literature are special cases of the filter derived in this paper. Finally, we present an illustrative example.