Consider the standard feedback block diagram shown in Figure 6-1. Recall that we could determine the performance or stability of the closed-loop system from the closed-loop transfer function
In the direct synthesis procedure, we select a desired closed-loop response, gCL(s), and, based on the known process, gp(s), find the controller gc(s) that will yield this response. Solving Equation (6.6) for gc(s), we obtain
By now you should be able to perform block diagram manipulations to find the relationship ...