Calculate (10)/(1+F(s)G(s)) (Y(s))/(R(s))=(F(s)G(s))/(1+F(s)G(s))

Ralzereep9h 2022-10-21 Answered
We have a system with output given by
Y ( s ) R ( s ) = F ( s ) G ( s ) 1 + F ( s ) G ( s )
where F ( s ) G ( s ) = K s + 1 s 2 + s + 1
Let K=4 and R(s)=10/s. Using the final value theorem, calculate the steady state error of the system.
Now, I thought the error would be given by lim s 0 10 F ( s ) G ( s ) 1 + F ( s ) G ( s ) , but the solution says to calculate 10 1 + F ( s ) G ( s ) . Can anyone explain why?
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Answers (1)

Sauppypefpg
Answered 2022-10-22 Author has 23 answers
We have to find the steady state error
lim t ( r ( t ) y ( t ) )
According to the final value theorem
lim t ( r ( t ) y ( t ) ) = lim s 0 [ s ( R ( s ) Y ( s ) ) ]
The Laplace transform of the error is given by
R ( s ) Y ( s ) = R ( s ) F ( s ) G ( s ) 1 + F ( s ) G ( s ) R ( s ) = 1 1 + F ( s ) G ( s ) R ( s )
Hence, for a step reference with Laplace transform R ( s ) = 10 / s the steady state error value can be derived by
lim t ( r ( t ) y ( t ) ) = lim s 0 [ s 1 1 + F ( s ) G ( s ) 10 s ] = lim s 0 [ 10 1 + F ( s ) G ( s ) ]
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