  In this paper, we present an optimal filter for linear time-varying
continuous-time stochastic systems that simultaneously estimates the states and
unknown inputs in an unbiased minimum-variance sense. We first show that the
unknown inputs cannot be estimated without additional assumptions. Then, we
discuss two complementary variants of the filter: (i) for the case when an
additional measurement containing information about the state derivative is
available, and (ii) for the case without the additional measurement but the
input signals are assumed to be sufficiently smooth and have bounded
derivatives. Conditions for uniform asymptotic stability and the existence of a
steady-state solution for the proposed filter, as well as the convergence rate
of the state and input estimate biases are given. Moreover, we show that a
principle of separation of estimation and control holds and that the unknown
inputs may be rejected. Two examples, including a nonlinear vehicle reentry
example, are given to illustrate that our filter is applicable even when some
strong assumptions do not hold.
