Input and State Estimation for Linear Systems: A Least Squares Estimation Approach

The problem of joint input and state estimation is addressed in this paper for linear discrete-time stochastic systems without direct feedthrough from unknown inputs to outputs. With the weighted least squares estimation for an extended state vector including unknown inputs and states, a recursive filter approach referred to as Kalman filter with unknown inputs without direct feedthrough (KF-UI-WDF) is derived. It is shown that the proposed KF-UI-WDF approach is uniquely optimal in sense of both least-squares (LS) and minimum-variances unbiased (MVU) over a category of MVU filters (e.g., [4], [5], [10]). The global optimality of the proposed KF-UI-WDF approach is also discussed. Due to the limited space, an illustrative example is omitted.