# cancellation controller transfer function design

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?

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 $$G_R(z)$$:
$$G_R(z) = -\frac{G_w(z)}{G_w(z)G_P(z) - G_P(z)} = -\frac{G_w(z)}{G_P(z)(G_w(z) -1)} = \frac{1}{G_P(z)}\frac{G_w(z)}{1 - G_w(z)}$$
You solve equation (6.3) for $$G_R$$.