In this paper, we present a unified optimal and exponentially stable filterfor linear discrete-time stochastic systems that simultaneously estimates thestates and unknown inputs in an unbiased minimum-variance sense, without makingany assumptions on the direct feedthrough matrix. We also derive input andstate observability/detectability conditions, and analyze their connection tothe convergence and stability of the estimator. We discuss two variations ofthe filter and their optimality and stability properties, and show that filtersin the literature, including the Kalman filter, are special cases of the filterderived in this paper. Finally, illustrative examples are given to demonstratethe performance of the unified unbiased minimum-variance filter.
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