Question

In the following block diagram, R(s) and D(s) are two inputs. The output Y(s) is expressed as Y(s) = G$_1$(s)R(s) + G$_2$(s)D(s).
G$_1$(s) and G$_2$(s) are given by 

• A. $G_1(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)} \;\;$ and $\;\; G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$
• B. $G_1(s)=\dfrac{G(s)}{1+G(s)+H(s)} \;\;$ and $\;\; G_2(s)=\dfrac{G(s)}{1+G(s)+H(s)}$
• C. $G_1(s)=\dfrac{G(s)}{1+G(s)+H(s)} \;\;$ and $\;\; G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$
• D. $G_1(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)} \;\;$ and $\;\; G_2(s)=\dfrac{G(s)}{1+G(s)+H(s)}$

Hint:

Write node equations from the summers and substitute forward: outer loop \(R \to (.) - Y\), inner loop subtracts \(H(s)Y(s)\) before \(G(s)\). Collect \(Y\) terms on the left to read off the closed-loop gains to each input.

Answer 1

The Correct Option is A

Let the first summer output be $e_1 = R(s) - Y(s)$ (outer negative feedback).
Let the second summer output be $e_2 = e_1 + D(s) - H(s)Y(s)$ (inner negative feedback through $H$).
The plant output is $Y(s) = G(s)\,e_2$.
Substitute: $Y = G\,[\, (R - Y) + D - H Y \,] = G(R + D) - G(1+H)Y$.
Bring $Y$ terms together: $Y\,[\,1 + G(1+H)\,] = G(R + D)$.
Therefore, \[ Y = \frac{G}{1+G+G H}\,R \;+\; \frac{G}{1+G+G H}\,D, \] so $G_1(s)=G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}$.
\[ \boxed{G_1(s)=G_2(s)=\dfrac{G(s)}{1+G(s)+G(s)H(s)}} \]