Question

# H4 R(s) C(s) G1 G2 G3 G4 H2 H3 H1 ...

Find the total transfer function of the system based on the block diagram of a multi-loop control system given in the figure.

Transcribed: H4 C(s) R(s) G1 G2 G3 G4 H2 H3 H1

The given block diagram is:

Moving the take off point between to the right of ${G}_{4}$. So the modified diagram is

Solving the feedback loop (1)

$T{F}_{1}=\frac{{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}}$

The modified block diagram is:

Solving the feedback loop (2)

$\begin{array}{rcl}T{F}_{2}& =& \frac{\frac{{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}}}{1-\frac{{G}_{3}{G}_{4}{H}_{3}}{1+{G}_{3}{G}_{4}{H}_{2}}}\\ & =& \frac{{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}}\end{array}$

The modified block diagram is:

Solving the feedback loop (3)

$\begin{array}{rcl}T{F}_{3}& =& \frac{\frac{{G}_{2}{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}}}{1-\frac{{G}_{2}{G}_{3}{G}_{4}\left({H}_{4}}{{G}_{4}}\right)}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}}}\\ & =& \frac{{G}_{2}{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}-{G}_{2}{G}_{3}{H}_{4}}\end{array}$

The modified block diagram is:

Solving the feedback loop (3)

$\begin{array}{rcl}T{F}_{4}& =& \frac{\frac{{G}_{1}{G}_{2}{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}-{G}_{2}{G}_{3}{H}_{4}}}{1+\frac{{G}_{1}{G}_{2}{G}_{3}{G}_{4}{H}_{1}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}-{G}_{2}{G}_{3}{H}_{4}}}\\ & =& \frac{{G}_{1}{G}_{2}{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}-{G}_{2}{G}_{3}{H}_{4}+{G}_{1}{G}_{2}{G}_{3}{G}_{4}{H}_{1}}\end{array}$

Therefore, the total transfer function of the block diagram is

$\begin{array}{rcl}TF& =& \frac{C\left(s\right)}{R\left(s\right)}=\frac{{G}_{1}{G}_{2}{G}_{3}{G}_{4}}{1+{G}_{3}{G}_{4}{H}_{2}-{G}_{3}{G}_{4}{H}_{3}-{G}_{2}{G}_{3}{H}_{4}+{G}_{1}{G}_{2}{G}_{3}{G}_{4}{H}_{1}}\end{array}$

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