Simultaneous Input and State Estimation for Linear Time-Varying Continuous-Time Stochastic Systems

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.