Control block diagram with sampling output

Question

I'm learning control theory and in my book it shows an example of this block diagram, the output

I'm quite confused because the last step I got is , I can't get the same answer after transfer this answer to Z domain.

Please help me

Answer 1

Let \$\small G_3(s)=\left[ G_1(s)\rightarrow sampler \rightarrow G_2(s)\right]\$, and \$\small C(s)=\$ output signal of \$\small G_2(s)\$; then the loop equation is: $$\small C(s)= \frac{R(s)\:G_3(s)}{1+H(s)\:G_3(s)}$$

Taking z-transforms:

\$\small R(s)G_3(s)\rightarrow RG_3(z)\rightarrow RG_1(z)\:G_2(z)\$ (noting that \$\small RG_1\$ and \$\small G_2\$ may be separated due to the intervening sampler);

\$\small H(s)G_3(s)\rightarrow HG_1G_2(z)\$ (noting that \$\small HG_1\$ and \$\small G_2\$ cannot be separated since \$\small HG_1\$ is not a signal)

Hence: $$\small C(z)= \frac{RG_1(z)\:G_2(z)}{1+HG_1G_2(z)}$$