Kalmanfilteri

The Kalman filter is an algorithm for estimating the hidden state of a dynamic system from noisy measurements. It is optimal for linear systems with Gaussian process and measurement noise, providing a best linear unbiased estimate of the state by combining a predictive model with new observations.

In discrete-time form, the standard model uses a state vector x_k, a measurement z_k, a state transition F_k, a control input B_k u_k, and an observation model H_k. The process noise w_k and measurement noise v_k are assumed Gaussian with zero means and covariances Q_k and R_k. The prediction equations are x_k|k-1 = F_k x_{k-1|k-1} + B_k u_k and P_k|k-1 = F_k P_{k-1|k-1} F_k^T + Q_k. The update equations compute the Kalman gain K_k = P_k|k-1 H_k^T (H_k P_k|k-1 H_k^T + R_k)^{-1}, then x_k|k = x_k|k-1 + K_k (z_k - H_k x_k|k-1) and P_k|k = (I - K_k H_k) P_k|k-1.

Variants and 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 approach to handle nonlinearity. There are also continuous-time formulations and information-form versions.

Historically, the filter was developed by Rudolf Kalman in 1960 and has since found wide use in navigation, tracking, robotics, control systems, economics, and signal processing. Limitations include the Gaussian noise assumption, sensitivity to model mismatch, and potential divergence if the model or noise covariances are poorly chosen. Proper initialization and tuning of Q and R are important for reliable performance.