Alpha Beta Filter - Filter Equations

Filter Equations

An alpha beta filter presumes that a system is adequately approximated by a model having two internal states, where the first state is obtained by integrating the value of the second state over time. Measured system output values correspond to observations of the first model state, plus disturbances. This very low order approximation is adequate for many simple systems, for example, mechanical systems where position is obtained as the time integral of velocity. 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.

Since velocity variable v is presumed constant, so its projected value at the next sampling time equals the current value.

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.

The output measurement is expected to deviate from the prediction because of noise and dynamic effects not included in the simplified dynamic model. This prediction error r is also called the residual or innovation, based on statistical or Kalman filtering interpretations

Suppose that residual r is positive. This could result because the previous x estimate was low, the previous v was low, or some combination of the two. The alpha beta filter takes selected alpha and beta constants (from which the filter gets its name), uses alpha times the deviation r to correct the position estimate, and uses beta times the deviation r to correct the velocity estimate. An extra ΔT factor conventionally serves to normalize magnitudes of the multipliers.

$\textbf{(4)} \quad \hat{\textbf{x}}_{k} \leftarrow \hat{\textbf{x}}_{k} + (\alpha)\ \textbf{r}_{k}$

$\textbf{(5)} \quad \hat{\textbf{v}}_{k} \leftarrow \hat{\textbf{v}}_{k} + ( \beta / )\ \textbf{r}_{k}$

The corrections can be considered small steps along an estimate of the gradient direction. As these adjustments accumulate, error in the state estimates is reduced. For convergence and stability, the values of the alpha and beta multipliers should be positive and small:

Noise is suppressed only if, otherwise the noise is amplified.

Values of alpha and beta typically are adjusted experimentally. 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. If a good balance between accurate tracking and noise reduction is found, and the algorithm is effective, filtered estimates are more accurate than the direct measurements. This motivates calling the alpha-beta process a filter.

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