Block Diagram Reduction

Question

i am trying to simplify this systems block diagram. I calculated something but I am not sure about it, is my reduction true? Thank you.

Answer 1

Doing away with the pesky s for the time being;

• $E=R-Y$
• $X=EF+K_1R-K_2Y$ $$\begin{align} Y&=GX\\ \\ &=G(EF+K_1R-K_2Y)\\ \\ &=G\left(F\left(R-Y\right)+K_1R-K_2Y\right)\\ &=GFR-GFY+GK_1R-GK_2Y \end{align}$$ Grouping all the Y's together; $$Y\left(1+GF+GK_2\right)=GR(F+K_1)$$ $$\frac{Y}{R}=\frac{G(F+K_1)}{1+G(F+K_2)}$$ The notationally-correct expression is hence; $$\frac{Y(s)}{R(s)}=\frac{K_1G(s)+F(s)G(s)}{1+K_2G(s)+F(s)G(s)}$$ The block-diagram reduction is done in 4 steps as follows: