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# how can i find Open loop Transfer Function from Closed loop Transfer Function

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chiranjib biswal on 28 Oct 2018
Edited: Aquatris on 29 Oct 2018
A unity feedback system is having Closed loop transfer function T(s)=361/s^2 16s 361. how can i find Open loop Transfer Function and dc gain of the system using matlab

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### Answers (1)

Aquatris on 29 Oct 2018
Edited: Aquatris on 29 Oct 2018
From the basics of control theory you should know that for a given system G(s), the closed loop transfer function with unity feedback P(s) is;
P(s) = G(s)/(1+G(s));
From there you can do some algebra. Since it is a SISO case, I will not care about the order of multiplication;
1.P(s) = G(s)/(1+G(s))
2.P(s)*(1+G(s)) = G(s)
3.P(s)+P(s)G(s) = G(s)
4.P(s) = G(s)-P(s)G(s)
5.P(s) = G(s)*(1-P(s))
6.G(s) = P(s)/(1-P(s))
If you followed it so far, it should be easy to see that if
P(s) = 361/(s^2+16*s+361)
then;
G(s) = 361/(s^2+16*s)
To get the gain you can simply plug s=0*1i into the transfer function. Since G(s) has a pole at zero, DC gain of G(s) is infinity. Alternatively you can use bode() command to draw bode diagram of the G(s) and see the DC gain, or there is a dcgain() command in Matlab.

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