I am learning control systems on my own and am having some trouble understanding how Richard C. Dorf and Robert H. Bishop found \$S^{T}_{G}\$ in Modern Control Systems (12th edition):
The problem:
The corresponding answer:
Where does, the very first equation, $$\frac{1}{1+GH(s)}$$ come from? No matter what I try I never find this equation. Could someone please explain and show how he got this equation? It doesn't seem to be the equation of the entire system because – according to what I tried – this equation should be $$T=\frac{RG_CG}{1+HG_CG}$$
I know how he found \$S^G_\tau\$:
$$S^G_\tau = \frac{\tau}{G} \frac{\partial G}{\partial \tau}$$
$$\frac{\tau}{\frac{100}{\tau s+1}} \frac{\partial}{\partial{\tau}}(\frac{100}{\tau s+1})$$
$$S^G_\tau = \frac{-\tau s}{s\tau +1}$$
Best Answer
Start with the two expressions:
$$T=\frac{G_c G }{1+ H G_c G}, \ \ \ \ \ \ \ \ \ \ \ \ S_G^T=\frac{\partial T}{\partial G}\frac{G}{T} $$
Compute the first term:
$$ \frac{\partial T}{\partial G} = \frac{G_c }{1+H G_c G}- \frac{H G_c^2 G }{(1+HG_c G)^2}= \frac{G_c+H G_c^2 G-H G_c^2 G }{(1+H G_c G)^2}=\frac{G_c }{(1+H G_c G)^2}$$
And substitute values to get the final expression:
$$ S_G^T= \frac{G_c }{(1+H G_c G)^2}\frac{G}{\frac{G_c G }{1+ H G_c G}} = \frac{1}{1+H G_c G}$$