KalmanBucy

The Kalman-Bucy filter is the continuous-time counterpart of the Kalman filter, developed by Rudolf E. Kalman and Richard Bucy in 1961. It provides optimal, real-time estimates of the state of a linear dynamic system subject to Gaussian process and measurement noise. The standard formulation models the state as dx/dt = A x + B u + w, and the measurements as y = C x + v, where w and v are zero-mean white Gaussian noises with covariances Q and R, respectively.

The estimator evolves according to a differential equation for the state estimate x̂(t): dot{x̂} = A x̂ + B u + K (y − C x̂). The Kalman gain K is time-varying and given by K = P C^T R^−1, where P(t) is the error covariance matrix. The covariance P obeys a Riccati differential equation: dot{P} = A P + P A^T + Q − P C^T R^−1 C P. Under mild conditions, P converges to a steady state, yielding a constant gain K in the time-invariant case.

The Kalman-Bucy filter yields the posterior distribution of the state given all measurements up to time t, which, for linear Gaussian models, remains Gaussian with mean x̂ and covariance P. It is widely used in control, navigation, aerospace, and signal processing. While optimal for linear systems with Gaussian noise, its accuracy depends on correct model parameters; mismatches and nonlinearity require extensions such as linearization, adaptive methods, or nonlinear filters.