The Kalman Filter

The Kalman Filter developed in the early sixties by R.E. Kalman [7, 8] is a recursive state estimator for partially observed non-stationary stochastic processes. It gives an optimal estimate in the least squares sense of the actual value of a state vector from noisy observations. Consider a discrete-time stochastic process x k+1 = f (x k , u k , v k) (1) with system input u and unmodeled process dynamics plus noise v. The task at hand is to find an estimate of the state vector x. However, x is only accessible from noise distorted sensor measurements z k = h(x k , w k) (2) in which as with the process model, w represents observation model inaccuracies and sensor noise. Recursive state estimation consists on iteratively reconstructing the state vector from our knowledge of the process dynamics, the measurement model, and the sensed data.