Alpha beta filter

It is closely related to Kalman filters and to linear state observers used in control theory.

Measured system output values correspond to observations of the first model state, plus disturbances.

Based on a mechanical system analogy, the two states can be called position x and velocity v. Assuming that velocity remains approximately constant over the small time interval ΔT between measurements, the position state is projected forward to predict its value at the next sampling time using equation 1.

If additional information is known about how a driving function will change the v state during each time interval, equation 2 can be modified to include it.

An extra ΔT factor conventionally serves to normalize magnitudes of the multipliers.

For convergence and stability, the values of the alpha and beta multipliers should be positive and small:[2] Noise is suppressed only if

In general, larger alpha and beta gains tend to produce faster response for tracking transient changes, while smaller alpha and beta gains reduce the level of noise in the state estimates.

Repeat for each time step ΔT: Alpha Beta filter can be implemented in C[3] as follows: The following images depict the outcome of the above program in graphical format.

A low value of beta is effective in controlling sudden surges in velocity.

In the case of alpha beta filters, this gain matrix reduces to two terms.

The linear Luenberger observer equations reduce to the alpha beta filter by applying the following specializations and simplifications.

A Kalman filter designed to track a moving object using a constant-velocity target dynamics (process) model (i.e., constant velocity between measurement updates) with process noise covariance and measurement covariance held constant will converge to the same structure as an alpha-beta filter.

[1] Similar extensions to additional higher orders are possible, but most systems of higher order tend to have significant interactions among the multiple states, [citation needed] so approximating the system dynamics as a simple integrator chain is less likely to prove useful.