Kalman filter
A Kalman filter is a method for estimating a changing quantity, such as the position of a car, from noisy measurements made over time. At each step, it predicts the quantity from a model of how the quantity changes, then corrects the prediction toward the new measurement, trusting whichever of the two is less uncertain more.
Consider a discrete-time linear system with state $x_k$, input $u_k$, and measured output $y_k$, given by
\[x_{k+1} = A x_k + B u_k + w_k, \qquad y_k = C x_k + v_k,\]where the process noise $w_k$ and the measurement noise $v_k$ are independent, zero-mean, Gaussian, and white, with covariances $Q$ and $R$, and the initial state $x_0$ has mean $\hat{x}0$ and covariance $P_0$. The _Kalman filter for this system is the filter that computes a state estimate $\hat{x}_k$ and its error covariance $P_k$ by repeating two steps. First, it predicts
\[\hat{x}_k^- = A \hat{x}_{k-1} + B u_{k-1}, \qquad P_k^- = A P_{k-1} A^\top + Q.\]Then, it corrects the prediction using the measurement $y_k$:
\[\hat{x}_k = \hat{x}_k^- + K_k (y_k - C \hat{x}_k^-), \qquad P_k = (I - K_k C) P_k^-,\]where $K_k = P_k^- C^\top (C P_k^- C^\top + R)^{-1}$ is the Kalman gain.
Properties
- For a linear system with Gaussian noise, the Kalman filter’s estimate $\hat{x}_k$ is the conditional mean of $x_k$ given the measurements $y_1, \dots, y_k$, so it has the smallest mean-square error of every estimator of $x_k$ from those measurements.
- The Kalman filter is an observer with the same structure as a Luenberger observer, but with a time-varying gain $K_k$ chosen from the noise covariances $Q$ and $R$.
- If $(A, C)$ is detectable and $(A, Q^{1/2})$ is stabilizable, then the Kalman gain $K_k$ converges to a constant gain as $k \to \infty$.
Notes
For nonlinear systems, variants such as the extended Kalman filter and the unscented Kalman filter apply the same predict-and-correct steps to approximations of the system. These variants are not optimal in general.
The continuous-time version is called the Kalman–Bucy filter.