[?] Transform a function into a certain form that is easier to read?

Questions
1

I would like to transform a function into a certain form. So I can read the coefficients more easily later.
For example, if I have the following function:

G=((z-1)/z)*(((K*z)/(z-1))-K/(z-Exp[-T/TS]))

Bild1.png

I would like to bring this into the form

G=((b1*z^-1)/(1+a1*z^-1))

Bild2.png
Bild2.png358×145 1.86 KB

In general terms, the form looks like this

Bild3.png

Thanks for your help
Sebastian

2

Try this, with x=1/z

In[1]:= PadeApproximant[(z - 1)/
   z ((K z)/(z - 1) - K/(z - E^(-T/Ts))) /. z -> 1/x, {x, 0, {1, 1}}]

Out[1]= (K - E^(-(T/Ts)) K x)/(1 + E^(-(T/Ts)) (-1 + E^(T/Ts)) x)

or this

In[6]:= PadeApproximant[(z - 1)/
    z ((K z)/(z - 1) - K/(z - E^(-T/Ts))) /. z -> 1/x, {x, 
   0, {1, 1}}] /. x -> z^(-1)

Out[6]= (K - (E^(-(T/Ts)) K)/z)/(1 + (E^(-(T/Ts)) (-1 + E^(T/Ts)))/z)