The problem of joint input and state estimation is investigated for discrete-time stochastic systems with direct feedthrough from unknown inputs to outputs in this paper. A Kalman filter with unknown inputs (KF-UI) approach is derived with the weighted least-squares estimation method. The least-squares estimators for states and unknown inputs are proven inherently optimal in the minimum-variance and unbiased sense. In addition, the necessary and sufficient conditions for the least-squares estimation of unknown inputs and states are provided and it has been shown that no prior information of unknown inputs is required for the proposed KF-UI approach. Simulation results for a linear system are included to demonstrate the effectiveness and optimality of the proposed KF-UI approach.
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