Let $L(s)$ be the transfer function of a system $L.$ Consider the diagram below.

Let $N(s)$ denote the transfer function from $q$ to $z.$ Let $S(s)$ denote the transfer function from $q$ to $p.$

$($a$)$ Derive $N(s)$ and $S(s)$ in terms of $L(s).$

$($b$)$ Derive $L(s)$ in terms of $N(s).$

$($c$)$ Derive $L(s)$ in terms of $S(s).$

Application: As we have learned, the gain$-$margin and time$-$delay margin can be computed from a frequency$-$response plot of the open$-$loop system $L($which is often $PC).$ In this problem, we see that through simple algebra$/$arithmetic$,$ the value of $L$ can be inferred from certain closed$-$loop frequency responses, namely $N$ or $S.$ In many industries, a working, stable closed$-$loop system is available for experimentation, so that frequency$-$response of $N$ and$/$or $S$ can be determined experimentally. With that data in hand, the value of $L(j)$ can be solved for, and then the margin analysis carried out. In this manner, the margins are determined from closed$-$loop frequency$-$response experiments, rather than from differential equation models of $P$ and $C.$