KalmanFilterungen

KalmanFilterungen refer to a family of estimation methods designed to infer the hidden state of a dynamic system from noisy measurements. Originating from the work of Rudolf Kalman, these methods are used in engineering, navigation, robotics, and signal processing to fuse information from multiple sensors and track evolving quantities over time.

The classic Kalman filter assumes a linear, Gaussian state-space model. The hidden state x_k evolves as x_k = F_k x_{k-1} + B_k u_k + w_k, with process noise w_k ~ N(0, Q_k), and measurements z_k = H_k x_k + v_k, with measurement noise v_k ~ N(0, R_k). The algorithm alternates prediction and update steps, computing a state estimate x_hat and its covariance P, then updating with the new measurement using the Kalman gain K. This yields optimal estimates in the sense of minimum mean squared error for linear Gaussian systems.

Beyond the linear case, extensions address nonlinearity: the extended Kalman filter linearizes around the current estimate, while the unscented Kalman filter uses deterministic sampling to better capture nonlinear transformations. Other variants include square-root, reduced-rank, and ensemble Kalman filters, which trade off numerical stability or scalability. Kalman smoothers extend real-time estimates to retrospectively refine past states.

Applications include navigation (inertial/GPS fusion), aerospace tracking, robotics, finance, and environmental monitoring. Practical implementation requires initial estimates of the state and covariance, as well as careful tuning of the process and measurement noise covariances. The KalmanFilterungen assume Gaussian noise and approximate linearity; violations can cause bias or divergence, in which case alternative filters or smoothing techniques may be preferred.