Delayed recursive state and input reconstruction

The unknown inputs in a dynamical system may represent unknown external drivers, input uncertainty, state uncertainty, or instrument faults and thus unknown-input reconstruction has several wide-spread applications. In this paper we consider delayed recursive reconstruction of states and unknown inputs for both square and non-square systems. That is, we develop filters that use current measurements to estimate past states and reconstruct past inputs. We further derive necessary and sufficient conditions for convergence of filter estimates and show that these convergence properties are related to multivariable zeros of the system. With the help of illustrative examples we highlight the key contributions of this paper in relation with the existing literature. Finally, we also show that existing unbiased minimum-variance filters are special cases of the proposed filters and as a consequence the convergence results in this paper also apply to existing unbiased minimum-variance filters.