A useful application of the pole-zero plot is to assess the stability of a syste

Question

A useful application of the pole-zero plot is to assess the stability of a system as the internal gain (e.g., the K in a transfer function) is changed. One case in which an internal gain may he changed is a feedback system, illustrated below, It is often the case that a desired response speed can only be achieved using a marginally or entirely unstable forward gain circuit (H(s)). In order for such a circuit to be useful, we must change the transfer function such that the system is stable using feedback gain (G(s)). Note: The "error" term (e(t)) may he characterized by E(s) = X(s) - G(s)Y(s). Derive the transfer function Y(s)/X(s) in terms of G(s)and H(s).

Explanation / Answer

Y(s) = H(s) E(s)

from the Diagram we interpreted above equation then

Given E(s) = X(s) - G(s) Y(s)   

substitute in above equation we get

Y(s) = H(s) [ X(s) - G(s) Y(s) ]

Y(s) = X(s) H(s) - H(s) G(s) Y(s)

after algebraic manipulation

Y(s) + G(s) H(s) Y(s) = X(s) H(s)

Y(s) ( 1 + G(s) H(s) ) = X(s) H(s)

Y(s) / X(s) = H(s) / (1 + G(s) H(s))

thus

transfer function given by

T.F = Transfer Function = H(s) / ( 1+ G(s)H(s) )    interms of G(s) and H(s)

thus ans

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