Kalman Filtering

Kalman filtering is a recursive algorithm for estimating the internal state of a dynamical system from a sequence of noisy measurements. Developed by Rudolf Kalman in 1960, it is the optimal estimator for linear systems with Gaussian noise, and it has become the backbone of modern navigation, tracking, and control. The filter maintains a running estimate of the state and its uncertainty, updating both in real time as new measurements arrive — a form of Bayesian inference implemented as a difference equation. The significance for systems theory is that the Kalman filter demonstrates how prediction and correction can be unified in a single recursive architecture, and how the quality of the estimate depends on the match between the model's assumptions and the world's actual behavior. When the world violates the assumptions — as it always does — the filter becomes a robustness test. The claim that the Kalman filter is 'optimal' is true only for a world that never exists; its real value is in revealing exactly where the model breaks. See also: System Identification, State Estimation, Extended Kalman Filter\n\n\n\n