KalmanFilterAlgorithmen

The Kalman filter is an optimal recursive data processing algorithm that estimates the state of a dynamic system from a series of incomplete and noisy measurements. In its standard form, it provides the best linear unbiased estimate for systems with Gaussian noise by combining a model of the system dynamics with incoming measurements.

In the discrete-time linear case, the system is described by x_k = F_k x_{k-1} + B_k u_k + w_k and z_k = H_k x_k + v_k, where x_k is the state vector, z_k the observation, u_k a control input, and w_k, v_k are process and measurement noise with zero-mean Gaussian distributions, covariance Q_k and R_k.

The algorithm proceeds in two steps: prediction and update. Prediction computes the a priori state estimate x_{k|k-1} = F_k x_{k-1|k-1} + B_k u_k and covariance P_{k|k-1} = F_k P_{k-1|k-1} F_k^T + Q_k. The update uses the measurement z_k to refine the estimate. Innovation y_k = z_k - H_k x_{k|k-1}, innovation covariance S_k = H_k P_{k|k-1} H_k^T + R_k, Kalman gain K_k = P_{k|k-1} H_k^T S_k^{-1}, and the corrected state x_{k|k} = x_{k|k-1} + K_k y_k with updated covariance P_{k|k} = (I - K_k H_k) P_{k|k-1}.

Extensions include the Extended Kalman Filter (EKF) for nonlinear systems, which linearizes around the current estimate, and the Unscented Kalman Filter (UKF) which uses a deterministic sampling method to better handle nonlinearity. Applications span navigation, tracking, robotics, economics, and signal processing. Limitations include sensitivity to model mismatch, the assumption of Gaussian noise, need for appropriate tuning of Q and R, and computational considerations for high-dimensional systems.