Alpha-Beta Filter

The alpha-beta filter can be viewed as a simplified version of the Kalman filter, where the state is represented by only two variables, one of which is the integral of the other. By drawing a simple analogy with physical systems, we can denote these variables as position $\mathbf {x}$ and velocity $\mathbf{v}$. Assuming that the velocity remains constant over a small time interval $\Delta T$, the a priori estimate (prediction) of the position at time $k$ is given by

$$\hat{\mathbf{x}}^{-}_k = \hat{\mathbf{x}}_{k-1} + \Delta T \mathbf{v}_{k-1}$$ (2.134)

while the velocity is always considered constant:

$$\hat{\mathbf{v}}^{-}_k = \hat{\mathbf{v}_{k-1}}$$ (2.135)

The output, however, is 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 referred to as the residual (a posteriori error estimate):

$$\mathbf{r}_k = \mathbf{x}_k - \hat{\mathbf{x}}^{-}_k$$ (2.136)

Let us define two parameters $\alpha$ and $\beta$ in order to obtain the posterior estimate as

$$\left\{ \begin{array}{rl} \hat{\mathbf{x}}_k & = \hat{\math... ...-}_k + \beta \frac{\mathbf{r}_k}{\Delta T} \end{array}\right.$$ (2.137)

In this way, an asymptotic observer for the position and velocity variables is obtained. Unlike the Kalman filter, the alpha-beta filter is a suboptimal filter where the parameters $\alpha$ and $\beta$ are tuned experimentally without any statistical validation. This approach is often supported by the fact that even in the Kalman filter, it is sometimes necessary to impose the noise matrices entirely empirically.