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Traditional statements of the Kalman filter focus on the estimation of states rather than outputs. An output of interest may contain feedthrough from both known inputs u and unknown inputs w, yn=Cxn+Dun+Hwn. It is usually assumed that the posterior state estimate and known inputs are enough to generate the minimum-variance output estimate, given by which is the same equation implemented in most popular...
If a system and its observation are both represented in state space with linear equations, the system noise and the measurement noise are white, Gaussian, and mutually uncorrelated, and the system and measurement noise statistics are known exactly; then, a Kalman filter (KF) [1] with the same order as the system provides optimal state estimates in a way that is simple and fast and uses little memory...
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