Short note: Obtaining a pulse transfer function

The pulse transfer function $G(z) = Y(z)/X(z)$ relates the z-transform of the sampled output to the z-transform of the sampled input, with zero initial conditions. The general procedure is:

1. Draw the block diagram with all samplers. Name the input of each sampler (for example $E$, $U$). Name the outputs as starred signals $E^*$, $U^*$. Add a fictitious sampler at the output if the output is continuous.
2. Write the s-domain equations. Express each sampler input and the output in terms of the starred signals and the input $R(s)$, e.g. $U(s) = G_1(s)E^*(s)$.
3. Take the starred transform of each equation. Use $[G(s)X^*(s)]^* = G^*(s)X^*(s)$. Elements not separated by a sampler must be combined first: $[G_1G_2X^*]^* = (G_1G_2)^*X^*$, and this is not equal to $G_1^*G_2^*X^*$.
4. Find each pulse transfer function using partial fractions or residues: $$G(z) = \mathcal{Z}\big[G(s)\big] = \sum \text{Res}\left[G(s)\frac{z}{z-e^{Ts}}\right]$$ With a ZOH, $G(z) = (1-z^{-1})\,\mathcal{Z}\left[G_p(s)/s\right]$.
5. Solve the algebraic equations for $C(z)$ and form $C(z)/R(z)$. If $R(s)$ enters through a continuous block before the first sampler (as $GR(z)$), only $C(z)$ can be found. In that case no pulse transfer function exists.

Example: Unity feedback with a sampler on the error, a ZOH and plant $G_p(s)$. Then $E^* = R^* - (G_hG_p)^*E^*$, which gives $\dfrac{C(z)}{R(z)} = \dfrac{G(z)}{1+G(z)}$ with $G(z) = (1-z^{-1})\mathcal{Z}[G_p/s]$.