Estimation and the Kalman Filter
Stub. Estimating a system's state from noisy observations: the general estimation problem (building on observability), and the Kalman filter as its recursive, optimal (minimum mean-square error) solution for linear systems with Gaussian noise.
Corresponds to lectures: Estimation; The Kalman filter. Observability (controllability-observability-and-canonical-forms) says the state is in principle recoverable from the output; estimation asks how to actually recover it in the presence of noise, in the best possible way. The Kalman filter is the answer for linear systems with Gaussian noise: a recursive algorithm that maintains a running state estimate and its uncertainty, updating both with each new noisy measurement — exactly the MMSE estimator of expectation-covariance-and-mmse-estimation specialized to a system that evolves over time, and (by duality) built from the same Riccati-equation machinery as LQR.
Outline (TODO — flesh out each)
- Estimation — the general problem, building on observability
- The Kalman filter — recursive state estimation
- Duality with LQR