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Problem 6.49PP
The open-loop transfer function of a unity-feedback system is
C(*) = -j(«/5-M)(j/50-H)‘
(a) Use Bode plot sketches to design a lag compensator for G(s) so that the closed-loop system 
satisfies the following specifications;
(i) The steady-state error to a unit-ramp reference input is less than 0.01.
(ii) PM >40“.
(b) Verify and refine your design by using Matlab.
Step-by-step solution
step 1 of 5
(a)
The open loop transfer function of a unity feedback system is.
C M -
Determine the open-loop gain K that gives a phase margin of 40«. 
The steady state error to a unit ramp input is.
s l im j -
i
1
1 .0.01
/ : 2 io o
Take X = 100
F ) F )
Step 2 of 5
Draw the Bode plot of the uncompensated system using MATLAB for K — 5- 
num=5;
den=conv([1/5 1 0],[1/50 1]);
sys=tf(num,den);
margin(sys)
BodeDiafmn
Gm - 20.S dB (it IS.S n d h ) . Pm - 47.4 deg (at 3.92 n d ^ )
Step 3 of 5
The value of K equal to 5 gives a phase margin of 47.4® at a cross-over frequency of 
3.92rad/s
The compensator transfer function is.
D{s) = a — — - « > 1 
' ' a r f+ 1
The low frequency gain must be raised by a factor of 20, so the lag compensation should have 
a equal to 20.
Choose the comer frequency * — to be one octave to one decade below the cross-over
frequency, that is, (0.2)(3.92) = 0.784 rad/s
0.784
T
r = i . 3
Step 4 of 5
Determine the other corner frequency.
1
aT
1
■(20)(1 .3 )
_ J ^
° 2 6
The desired compensator is.
“i-i-K Sri)
Thus, the desired compensator is
l,5 0 0 jU .0 0 2 6 j + U
Step 5 of 5
(b)
Verify the design using MATLAB. 
num=100*[1.3 1];
den=conv([1/5 1 0],conv([1/50 1].[26,1]));
sys=tf(num,den);
margin(sys)
Get the MATLAB output for the Bode plot.
BadeMagraM
Increasing the gain does not result in further increase of phase margin. 
Hence, the design is verified and cannot be refined.