# Variables| Symbol | Name | Description ||--------|-------------------------------|--------------------------------------------------|| x | State Vector | Represents the system’s current state || z | Observation (Measurement) | Measured values from sensors || P | State Covariance | Uncertainty of the state estimate || F | State Transition Matrix | Predicts next state from current state || H | Observation Matrix | Maps state space to observation space || Q | Process Noise Covariance | Models uncertainty in the system dynamics || R | Measurement Noise Covariance | Models uncertainty in sensor measurements |
Algorithm
Predict (Time Update)
| Equation | Description ||----------|-------------|| x̂ₖ⁻ = F x̂ₖ₋₁ | Predict state forward || Pₖ⁻ = F Pₖ₋₁ Fᵀ + Q | Predict state covariance |
Update (Observation Update)
| Equation | Description ||----------|-------------|| yₖ = zₖ − H x̂ₖ⁻ | Innovation (residual) || Sₖ = H Pₖ⁻ Hᵀ + R | Innovation covariance || Kₖ = Pₖ⁻ Hᵀ Sₖ⁻¹ | Kalman gain || x̂ₖ = x̂ₖ⁻ + Kₖ yₖ | Updated state estimate || Pₖ = (I − Kₖ H) Pₖ⁻ | Updated covariance |
