A Kalman Filter is a tool for incorporating noisy measurements over time to estimate the true state of a dynamic system. It is widely used in control systems, robotics, and signal processing.
An Underlying Dynamical System Model
A great way to understand the Kalman Filter is to think of it as a hidden Markov model where the hidden states evolve over time according to a linear dynamical system, and the observations are noisy measurements of these states.
Model underlying the Kalman filter. Squares represent matrices. Ellipses represent multivariate normal distributions (with the mean and covariance matrix enclosed). Unenclosed values are vectors. For the simple case, the various matrices are constant with time, and thus the subscripts are not used, but Kalman filtering allows any of them to change each time step.
Where:
- is the state transition model
- is the observation model
- is the Covariance of the process noise
- is the Covariance of the observation noise