Question 13.5: Find the z-transform of the system shown in Figure 13.10(a).

The objective of the problem is to proceed in an orderly fashion, starting with the block diagram of Figure 13.10(a) and reducing it to the one shown in Figure 13.10(ƒ).

One operation we can always perform is to place a phantom sampler at the output of any subsystem that has a sampled input, provided that the nature of the signal sent to any other subsystem is not changed. For example in Figure 13.10(b), phantom sampler S4 can be added. The justification for this, of course, is that the output of a sampled-data system can only be found at the sampling instants anyway, and the signal is not an input to any other block.

Another operation that can be performed is to add phantom samplers S2 and S3 at the input to a summing junction whose output is sampled. The justification for this operation is that the sampled sum is equivalent to the sum of the sampled inputs, provided, of course, that all samplers are synchronized.

Next, move sampler S1 and G(s) to the right past the pickoff point, as shown in Figure 13.10(c). The motivation for this move is to yield a sampler at the input of G(s)H(s) to match Figure 13.9(b). Also, G(s) with sampler S1 at the input and sampler S4 at the output matches Figure 13.9(a). The closed-loop system now has a sampled input and a sampled output.

G(s)H(s) with samplers S1 and S3 becomes GH(z), and G(s) with samplers S1 and S4 becomes G(z), as shown in Figure 13.10(d). Also, converting R^{*} (s) to R(z) and C^{*} (s) to C(z), we now have the system represented totally in the z-domain.

The equations derived in Chapter 5 for transfer functions represented with the Laplace transform can be used for sampled-data transfer functions with only a change in variables from s to z. Thus, using the feedback formula, we obtain the first block of Figure 13.10(e). Finally, multiplication of the cascaded sampled-data systems yields the final result shown in Figure 13.10(ƒ).