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

In this technical note, we consider the problem of optimal filtering for linear time-varying continuous-time stochastic systems with unknown inputs. We first show that the unknown inputs cannot be estimated without additional assumptions. Then, we discuss some conditions under which meaningful estimation is possible and propose an optimal filter that simultaneously estimates the states and unknown inputs in an unbiased minimum-variance sense. Conditions for uniform asymptotic stability, and the existence of a steady-state solution, 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. A nonlinear vehicle reentry example is given to illustrate that our filter is applicable even when some strong assumptions do not hold.