Question 11.7: A block diagram for internal model control, a control techni...

Using block diagram algebra

\begin{aligned}&Y=G_{d} D+G_{p} U\quad\quad\quad (1) \\&U=G_{c}\left[Y_{s p}-\left(Y-\widetilde{G}_{p} U\right)\right] \quad\quad\quad (2)\end{aligned}

From (2), \quad U=\frac{G_{c} Y_{s p}-G_{c} Y}{1-G_{c} \widetilde{G}_{p}}

Substituting for U in Eq. 1

\left[1+G_{c}\left(G_{p}-\tilde{G}_{p}\right)\right] Y=G_{d}\left(1-G_{c} \widetilde{G}_{p}\right) D+G_{p} G_{c} Y_{s p}

Therefore,

\frac{Y}{Y_{s p}}=\frac{G_{p} G_{c}}{1+G_{c}\left(G_{p}-\widetilde{G}_{p}\right)}

and

\frac{Y}{D}=\frac{G_{d}\left(1-G_{c} \tilde{G}_{p}\right)}{1+G_{c}\left(G_{p}-\tilde{G}_{p}\right)}