cancellation controller transfer function design

Question

This chapter offers a free overview on cancellation controllers.

However, i did not get equation (6.4), how the transfer function of $G_R$ is determined?

Answer 1

You need to solve (6.3) for $G_R$, here is the step-by-step:

$G_w = \frac{G_R(z) G_P(z)}{1 + G_R(z) G_P(z)}$, multiply both sides by $1 + G_R(z) G_P(z)$ to get:

$G_R(z)G_w(z)G_P(z) - G_R(z)G_P(z) = -G_w$, now solve for $G_R(z)$:

$G_R(z) = -\frac{G_w(z)}{G_w(z)G_P(z) - G_P(z)} = -\frac{G_w(z)}{G_P(z)(G_w(z) -1)} = \frac{1}{G_P(z)}\frac{G_w(z)}{1 - G_w(z)}$

Answer 2

You solve equation (6.3) for $G_R$.