  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.
