Alpha Beta Filter¶

• Also known as the g-h filter

• $g$ is the scaling used for measurement (also known as $\alpha$)
• $h$ is the scaling for the change in measurement over time (also known as $\beta$)

• Basis for a huge number of filters including the Kalman filter
• Each type of filter has a different way of assigning values to $g$ and $h$
• Kalman filter varies $g$ and $h$ dynamically at each time step

Formal Terminology¶

• System/plant: the object we want to estimate
• State: actual hidden value of system
• Measurement: measured, observable value of system
• (State) estimate: filter's estimate of state
• Process/system model: mathematical model of the system
• Process/system error: error in mathematical model of the system
• System propogation/evolution: uses process model to form new state estimate; imperfect because of process error

Algorithm¶

Initialization

1. Initialize the state of the filter
2. Initialize our belief in the state

Predict
1. Use system behavior to predict state at next time step
2. adjust belief to account for uncertainty in prediction

Update
1. Get measurement and associated belief about its accuracy
2. Compute residual between estimated state and measurement
3. New estimate is somewhere on residual line


Notation¶

• $z$: measurement (some literature uses $y$)
• $k$: time step
• $z_{k}$ is the measurement for time step $k$

• In general, we have n sensors and n measurements
• x denotes the state; represents both initial weight and initial weight gain rate
• $\dot{x}$ is the derivative of x with respect to time $$\mathbf{x} = \begin{bmatrix} x \ \dot{x} \end{bmatrix}$$
• e.g weight of 62 kg with a gain of 0.3 kg/day $$\mathbf{x} = \begin{bmatrix} 62\ 0.3 \end{bmatrix}$$
• The state is initialized with $x_{0}$, the initial estimate.
• We then enter a loop, predicting the state for time or step $k$ from the values from time or step $k-1$.
• We get the measurement $z_{k}$ and choose intermediate point between measurements and prediction, creating the estimate $x_{k}$