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Problem 6.11 PP
A normalized second-order system with a damping ratio ^ = 0.5 and an additional zero is given by
s /a + \G(s) =
j ^ + j + r
Use Matlab to compare the Mpfrom the step response of the system fora = 0.01,0.1,1.10, and 
100 with the Mrfrom the frequency response of each case. Is there a correlation between Mr and 
Mp?
Step-by-step solution
step 1 of 15
Step 1 of 15
Peak overshoot K ) is the maximum peak value of the transient response.
Peak overshoot K ) is generally related to the damping of the system expressed as.
Step 2 of 15
Consider the seconder order system for a s 0.01 •
G(s)> 0.017 + 1
5 * + J + l
1005+1
5*+S + l
Write the MATLAB code for the step response of the transfer function. 
» num=[100 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» step(sys);
» hold on
Step 3 of 15
Consider the seconder order system for a s 0.1 •
5*+S + l 
105 + 1
5*+5 + l
Write the MATLAB code for the step response of the transfer function. 
» num=[10 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» step(sys)
Step 4 of 15
Consider the seconder order system for a s 1.
G(j).
i+ 1
_L
5*+5 + l 
5 + 1
5*+5 + l
Write the MATLAB code for the step response of the transfer function. 
» num=[1 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» step(sys)
Step 5 of 15
Consider the seconder order system for a s 10 •
5*+5 + l
0.15 + 1
5*+5 + l
Write the MATLAB code for the step response of the transfer function. 
» num=[0.1 1];
»den=[1 1 1]:
» sys=tf(num,den):
» step(sys);
Step 6 of 15
Consider the seconder order system for a s 100 •
*7 + 1
u ^5 ;«
inn
5*+5 + l
0.015 + 1
5*+5 + l
Write the MATLAB code for the step response of the transfer function. 
» num=[0.01 1];
»den=[1 1 1]:
» sys=tf(num,den):
» step(sys);
Step 7 of 15
Step response for various values of a is,
Therefore, peak overshoots for various values of a has been determined.
Step 8 of 15
The maximum value of the frequency-response magnitude is referred to as resonant peak . 
Frequency response resonant peak is related to the closed loop frequency response. 
Resonant-peak magnitude is generally related to the damping of the system expressed as.
For ^ = 0.5. the resonant peak magnitude is,
________1
2 ( 0 .5 ) ^ l - (0 .5 )*
= 1.1547
J14, = -
Step 9 of 15
Consider the seconder order system for q s 0.01 •
*7 + 1
G {s). 0.01
5*+5 + l
1005 + 1
5*+5 + l
Write the MATLAB code for the frequency response of the transfer function. 
» num=[100 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» bode(sys)
» hold on
Step 10 of 15
Consider the seconder order system for j s 0.1 ■
5*+5 + l
105 + 1
5*+5 + l
Write the MATLAB code for the frequency response of the transfer function. 
» num=[10 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» bode(sys):
Step 11 of 15
Consider the seconder order system for a s 1.
G(j).
i+ 1
_L
5*+5 + l 
5 + 1
5*+5 + l
Write the MATLAB code for the frequency response of the transfer function. 
» num=[1 1]:
»den=[1 1 1]:
» sys=tf(num,den):
» bode(sys):
Step 12 of 15
Consider the seconder order system for a s 10 •
0 { s ) - T o* '
5*+5 + l
0.15 + 1
5*+5 + l
Write the MATLAB code for the frequency response of the transfer function. 
» num=[0.1 1];
»den=[1 1 1]:
» sys=tf(num,den):
» bode(sys):
Step 13 of 15
Consider the seconder order system for a s 100 •
*7 + 1
G {S ): 100
5*+5 + l
0.015 + 1
5*+5 + l
Write the MATLAB code for the frequency response of the transfer function. 
» num=[0.01 1];
»den=[1 1 1]:
» sys=tf(num,den):
» bode(sys):
Step 14 of 15
Bode plot response of for various values of a is.
Therefore, resonant peak for various values of a has been determined.
Step 15 of 15
Tabulate the peak overshoot and resonant peak value for various values of a . 
Table 1
a Mr " r
0.01 98.8 54.1
0.1 9.93 4.94
1 1.46 0.30
10 1.16 0.16
100 1.15 0.16
As a is Increased, the resonant peak in frequency response Is decreases. This leads to 
expect extra peak overshoot M p in transient response.
From the Table 1, No Significant change in the frequency response is observed from a = 10 
onwards. Similarly significant change In the transient response is observed at a = 0.01,0.1 and
a - \
Therefore, the resonant peak M ,. in frequency response and peak overshoot response M p in 
transient response are correlated at a = 0.01,0.1 and a = l-