What type of z transform is this? I am trying to understand this equation:



It comes from:

$Force = mass * acceleration$

$F(t) = m * a(t)$

$F(s) = m * (s^2 * y(s) - s*y0 - y0)$, where $y0=0$

$F(s) = m * s^2 * y(s)$

$y(s) =  F(s)/(m*s^2)$

Then I don't understand what type of substitution they are using for s. It appears they substitute:

$s =  (1-z^{-1})/T$ 

And then the last $z^{-1}$ is just to indicate they are delaying the whole thing one sample for another reason.

What type of s substitution is this called?
Have a look at these notes.

In particular, this slide:



shows that using a backward difference model sets up the approximation:
$$
\frac{d}{dt} \approx \frac{z - 1}{Tz} = \frac{1 - z^{-1}}{T}
$$

There are many ways to approximate the continuous-time derivative with discrete-time difference operations. That's all that substitution is doing: choosing one way of making the approximation.