Difference equation system identification
Note that for a small sampling interval $T$, $\big(d[k+1]-d[k]\big)/T$ is a good approximation for the velocity. So if you fit $au[k]+b$ to a given set of measurements $v[k]$, it is valid to conclude
$$d[k+1]=d[k]+T\big(au[k]+b\big)\tag{1}$$
In the text you refer to they might have normalized $T$, so it changes the units without changing the values of $a$ and $b$.