KF
How it works
Predict: x̂⁻ = F·x̂ + B·u; P⁻ = F·P·Fᵀ + Q. Update: K = P⁻·Hᵀ·(H·P⁻·Hᵀ + R)⁻¹; x̂ = x̂⁻ + K·(z - H·x̂⁻); P = (I - K·H)·P⁻. The residual (innovation) z - H·x̂⁻ serves as a signal for fault detection and isolation (FDI).
Problem solved
How to optimally estimate the state of a dynamic system from noisy measurements, without storing the full data history and under limited computational resources.
Components
Propagation of state and covariance through the dynamics model (matrices F, B, Q).
Update of the estimate using the new measurement z (matrices H, R), computation of Kalman gain K.
Difference between predicted and actual measurement: ν = z - H·x̂⁻. Foundation of FDI.
Implementation
Poorly tuned Q (process) and R (measurement) covariances lead to filter divergence or overly slow updates.
EKF loses optimality and may diverge for strongly nonlinear models.
Evolution
R. E. Kálmán publishes "A New Approach to Linear Filtering and Prediction Problems" in ASME Journal of Basic Engineering.
Extension to continuous time (Kálmán–Bucy Filter).
KF applied in the Apollo mission navigation computer — the first large-scale industrial implementation.
EKF becomes the standard in robotic navigation and SLAM after popularisation by Smith, Self & Cheeseman.
Julier & Uhlmann propose the UKF — a better nonlinearity approximation via sigma points than EKF linearisation.
Hardware requirements
KF runs on RT CPU in most embedded and robotic systems.
FPGAs used for very short cycles (e.g. IMU at 10 kHz) with deterministic latency.
Algorithm is hardware-agnostic — a deterministic scheduler is the key requirement.