KalmanVerfahren

KalmanVerfahren, often translated as the Kalman filter, is a powerful mathematical algorithm used for estimating the state of a dynamic system from a series of noisy measurements. Developed by Rudolf E. Kálmán in the late 1950s and early 1960s, it is a recursive Bayesian estimator that efficiently predicts the next state of a system and then updates this prediction based on incoming measurement data. The core idea is to combine a system's predicted state, based on a mathematical model of its behavior, with a measured state, taking into account the uncertainty in both.

The Kalman filter operates in two main stages: a prediction step and an update step. In the prediction step, the algorithm uses the system's mathematical model to forecast its state at the next time step, along with an estimate of the uncertainty in this prediction. In the update step, when a new measurement becomes available, the filter combines this measurement with the predicted state. It weighs the prediction and the measurement based on their respective uncertainties, producing a refined estimate of the system's state that is more accurate than either the prediction or the measurement alone.

The Kalman filter is optimal in the sense that it minimizes the mean squared error of the state estimate, provided that the system dynamics are linear and the noise is Gaussian. While the original Kalman filter is for linear systems, extensions such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) have been developed to handle nonlinear systems. The Kalman filter finds wide applications in various fields, including navigation (GPS), control systems, economics, and signal processing, wherever precise estimation in the presence of noise is critical.