Alpha-Beta Filter

The alpha-beta filter can be viewed as a simplified version of the Kalman filter in which the state is represented by only two variables, one of which is the integral of the other. By analogy with physical systems, these variables can be called position $\mathbf {x}$ and velocity $\mathbf{v}$. If the velocity is assumed to remain constant over the short time interval $\Delta T$, the prior estimate (prediction) of the position at time $k$ is

$$\hat{\mathbf{x}}^{-}_k = \hat{\mathbf{x}}_{k-1} + \Delta T \mathbf{v}_{k-1}$$ (3.52)

whereas the velocity is always assumed to remain constant:

$$\hat{\mathbf{v}}^{-}_k = \hat{\mathbf{v}_{k-1}}$$ (3.53)

The output is nevertheless affected by noise, and the observed value $\mathbf{x}_k$ differs from the predicted value $\hat{\mathbf{x}}^{-}_k$. This prediction error $\mathbf{r}$ is called the residual (the posterior error estimate):

$$\mathbf{r}_k = \mathbf{x}_k - \hat{\mathbf{x}}^{-}_k$$ (3.54)

We define two parameters $\alpha$ and $\beta$ to obtain the posterior estimate as

$$\left\{ \begin{array}{rl} \hat{\mathbf{x}}_k & = \hat{\math... ...-}_k + \beta \frac{\mathbf{r}_k}{\Delta T} \end{array}\right.$$ (3.55)

This yields an asymptotic observer for the position and velocity variables. Unlike the Kalman filter, the alpha-beta filter is a suboptimal filter in which the parameters $\alpha$ and $\beta$ are tuned experimentally, without any statistical justification. This approach is usually justified by the fact that, even in the Kalman filter, the noise matrices sometimes have to be specified entirely empirically.