Closed-loop transfer function: Difference between revisions

Revision as of 18:48, 5 February 2020

Revision as of 18:51, 5 February 2020

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: <math>Z(s) =X(s)-H(s)Y(s) </math>

Now, plug the second equation into the first to eliminate Z(s):

: <math>Y(s) = G(s)(X(s)-H(s)Y(s)) = G(s)X(s) - G(s)H(s)Y(s)</math>

: <math>Y(s)+ = G(s)H[X(s)Y-H(s) = G(s)XY(s)]</math>

Move all the terms with Y(s) to the left hand side, and keep the term with X(s) on the right hand side:

: <math>Y(s)(1+G(s)H(s)) = G(s)X(s)</math>

: <math>\Rightarrow \dfrac{Y(s)}{X+G(s)H(s)Y(s)} = \dfrac{G(s)}{1+G(s)HX(s)}</math>

Therefore,

: <math>Y(s) = (1+G(s)(X(s)-H(s)Y(s)) = G(s)X(s) - G(s)H(s)Y(s)</math>

: <math>\Rightarrow \dfrac{Y(s)(1+G}{X(s)H(s))} = \dfrac{G(s)X}{1+G(s)H(s)}</math>

==See also==