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Prévia do material em texto

Problem 3.26PP
For the unity feedback system shown in Fig., specify the gain K of the proportional controller so 
that the output y(t) has an overshoot of no more than 10% in response to a unit step.
Figure Unity feedback system
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Step-by-step solution
step 1 of 3
Step 1 of 3
The fbUowiiig is tfie b lock diagram a un ity feedback system:
Step 2 of 3
The transfer function o f th e system is.
K
r ( j ) s(s+ 2) 
R (s)= -
i ( i+ 2 )
K
~ s(s+ 2 )+ K
K
^ + 2 s + K
The tnaxiimim peak overshoot is»
W here ^ is the dan^nng ratio vrfudi lies between 0 1.
F or m axim um overshoot o f 10%,
M .
0 . 1 = 8 ^
Solve fo r ^
2 .3 0 2 6 =
Q 2 3 0 2 6 
-r ^ — = 0.7329
^ = 0 .T 3 2 9 4 \ - C
Square o n bo th sides,
^ = 0 . 7 3 2 9 ' ( l - ^ )
^ ’ = 0 .5 3 7 (1 -^ “) 
^ = 0 .5 3 7 -0 .5 3 7 ^ ^ 
1.537^^ = 0.537 
Sim plify fiirther to obtam ̂ 
^ 0-537 
^ 1.537
^ = 0 .3 4 9 4 
^ = 0 .5 9
Step 3 of 3 ^
Cotnpate ^ + 2 s + K w ife the standard second-order aqwessioii.
2 ^