Summary
We have derived all five Kalman Filter equations in matrix notation. Let us put them all together on a single page.
The Kalman Filter operates in a “predict-correct” loop, as shown in the diagram below.
Once initialized, the Kalman Filter predicts the system state at the next step. It also provides the uncertainty of the prediction.
Once the measurement is received, the Kalman Filter updates (or corrects) the prediction and the uncertainty of the current state. As well the Kalman Filter predicts the following states, and so on.
The following diagram provides a complete picture of the Kalman Filter operation.
The following table describes all Kalman Filter Equations.
| Equation | Equation Name | Alternative names | |
|---|---|---|---|
| Predict | \( \boldsymbol{\hat{x}}_{n+1,n} = \boldsymbol{F\hat{x}}_{n,n} + \boldsymbol{Gu}_{n} \) | State Extrapolation |
Predictor Equation Transition Equation Prediction Equation Dynamic Model State Space Model |
| \( \boldsymbol{P}_{n+1,n} = \boldsymbol{FP}_{n,n}\boldsymbol{F}^{T} + \boldsymbol{Q} \) | Covariance Extrapolation | Predictor Covariance Equation | |
| Update (correction) |
\( \boldsymbol{\hat{x}}_{n,n} = \boldsymbol{\hat{x}}_{n,n-1} + \boldsymbol{K}_{n} (\boldsymbol{z}_{n} - \boldsymbol{H\hat{x}}_{n,n-1} ) \) | State Update | Filtering Equation |
| \( \boldsymbol{P}_{n,n} = \left(\boldsymbol{I} - \boldsymbol{K}_{n}\boldsymbol{H} \right) \boldsymbol{P}_{n,n-1} \left(\boldsymbol{I} - \boldsymbol{K}_{n}\boldsymbol{H} \right)^{T} + \boldsymbol{K}_{n}\boldsymbol{R}_{n}\boldsymbol{K}_{n}^{T} \) | Covariance Update | Corrector Equation | |
| \( \boldsymbol{K}_{n} = \boldsymbol{P}_{n,n-1}\boldsymbol{H}^{T}\left(\boldsymbol{HP}_{n,n-1}\boldsymbol{H}^{T} + \boldsymbol{R}_{n} \right)^{-1} \) | Kalman Gain | Weight Equation | |
| Auxiliary | \( \boldsymbol{z}_{n} = \boldsymbol{Hx}_{n} \) | Measurement Equation | |
| \( \boldsymbol{R}_{n} = E\left( \boldsymbol{v}_{n}\boldsymbol{v}_{n}^{T} \right) \) | Measurement Uncertainty | Measurement Error | |
| \( \boldsymbol{Q}_{n} = E\left( \boldsymbol{w}_{n}\boldsymbol{w}_{n}^{T} \right) \) | Process Noise Uncertainty | Process Noise Error | |
| \( \boldsymbol{P}_{n,n} = E\left( \boldsymbol{e}_{n}\boldsymbol{e}_{n}^{T} \right) = E\left( \left( \boldsymbol{x}_{n} - \boldsymbol{\hat{x}}_{n,n} \right) \left( \boldsymbol{x}_{n} - \boldsymbol{\hat{x}}_{n,n} \right)^{T} \right) \) | Estimation Uncertainty | Estimation Error |
The following table summarizes notation (including differences found in the literature) and dimensions.
| Term | Name | Alternative term | Dimensions |
|---|---|---|---|
| \( \boldsymbol{x} \) | State Vector | \( n_{x} \times 1 \) | |
| \( \boldsymbol{z} \) | Output Vector | \( \boldsymbol{y} \) | \( n_{z} \times 1 \) |
| \( \boldsymbol{F} \) | State Transition Matrix | \( \boldsymbol{\Phi,A} \) | \( n_{x} \times n_{x} \) |
| \( \boldsymbol{u} \) | Input Variable | \( n_{u} \times 1 \) | |
| \( \boldsymbol{G} \) | Control Matrix | \( \boldsymbol{B} \) | \( n_{x} \times n_{u} \) |
| \( \boldsymbol{P} \) | Estimate Uncertainty | \( \boldsymbol{\Sigma} \) | \( n_{x} \times n_{x} \) |
| \( \boldsymbol{Q} \) | Process Noise Uncertainty | \( n_{x} \times n_{x} \) | |
| \( \boldsymbol{R} \) | Measurement Uncertainty | \( n_{z} \times n_{z} \) | |
| \( \boldsymbol{w} \) | Process Noise Vector | \( \boldsymbol{y} \) | \( n_{x} \times 1 \) |
| \( \boldsymbol{v} \) | Measurement Noise Vector | \( n_{z} \times 1 \) | |
| \( \boldsymbol{H} \) | Observation Matrix | \( \boldsymbol{C} \) | \( n_{z} \times n_{x} \) |
| \( \boldsymbol{K} \) | Kalman Gain | \( n_{x} \times n_{z} \) | |
| \( \boldsymbol{n} \) | Discrete-Time Index | \( \boldsymbol{k} \) |
Dimensions notation:
- \( n_{x} \) is a number of states in a state vector
- \( n_{z} \) is a number of measured states
- \( n_{u} \) is a number of elements of the input variable