Output of block diagram in Z-Domain ob block diagram

Question

I want to calculate Y(Z)[Z-Domain] for the given system with matlab. See picture of the block diagram

I have tried to solve it with matlab. I have written the following matlab code.

clear all
clc
s = tf('s');
% model Gp(s)
Gp = 2/s;
H = s;
% close-loop transfer function in continuous domain
Tc = feedback(Gp,H);
% model Y(s)
Y = Tc*(2/s);
%% continuous to discrete transformation
T = 1; % sampling time 1 sec
Y_z = c2d(Y,T,'zoh')

As solution I get:

\begin{equation} \begin{gathered} 0.6667 \mathrm{z}+0.6667 \\ ----------- \\ \mathrm{z}^{\wedge} 2-2 \mathrm{z}+1 \end{gathered} \end{equation}

Can anybody have a look to my calculation, because I am not sure if I have done this right with matlab.

Answer 1

First steps may be:

With \$\small T=1\$, the sampler and ZOH have the Laplace transform: $$ G_{zoh}(s)=\small\frac{1-e^{-s}}{s}$$. Combining this with \$G_p(s)\$, and partitioning appropriately:

$$G_{zoh}(s)G_p(s) =\small\frac{(1-e^{-s})}{s}\times\frac{2}{s}=(1-e^{-s})\times \frac{2}{s^2}$$

Taking the ZTF (call this \$ \small G(z)\$)

$$G(z)=\small\left(\frac{z-1}{z}\right)\small\frac{2z}{(z-1)^2} $$

etc ...