Exercise 1:
a) Get the transfer function from R(s) to Y(s)
b) Get the transfer function from D(s) to Y(s)
c) Generate the system response; for K= 15, 25, 60, 100; due to a unit step input - r(t)
d) Generate the system response; for K= 15, 25, 60, 100; due to a unit step disturbance - d(t)
e) For each case find the percentage overshoot(%O.S.), rise time, settling time, steady state of y(t)
f) Compare the results of the two cases
g) Investigate the effect of changing the controller gain on the influence of the disturbance on the
system output
Solution
(a) The transfer function from R(s) to Y(s)
For K=15,
11𝑠+15
𝑠 +12𝑠+15
transfer function = 2
For K=25,
11𝑠+25
transfer function = 𝑠2 +12𝑠+25
For K=60,
11𝑠+60
transfer function = 𝑠2 +12𝑠+60
For K=100,
11𝑠+100
𝑠 +12𝑠+100
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transfer function = 2
2
(b) The transfer function from D(s) to Y(s)
For K=15,
1
𝑠 +12𝑠+15
transfer function = 2
For K=25,
1
transfer function = 𝑠2 +12𝑠+25
For K=60,
1
transfer function = 𝑠2 +12𝑠+60
For K=100,
1
transfer function = 𝑠2 +12𝑠+100
(c) The system response; for K=15,25,60,100; due to unit step input – r(t)
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MATLAB Program
3
Figure
(d) The system response; for K=15,25,60,100; due to unit step input – r(t)
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MATLAB Program
4
Figure
(e) The percent overload, rise time, settling time, steady state of y(t) due to a unit step input-r(t)
For K= 25,
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For K=15,
5
For K=60,
For K=100,
The percent overload, rise time, settling time, steady state of y(t) due to a unit step input-r(t)
K=25,
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K=15,
6
K=60,
K=100,
(f) Comparison between the unit step and unit step disturbance
Percent
Overshoot
K
Rise Time
Settling
Time
Steady
Time
Step Input
Disturban
ce
Input
Step
Input
Disturban
ce
Input
Step
Input
Disturban
ce
Input
Step
Input
Disturbanc
e
Input
1.7803
0
0.1842
1.5741
0.2867
2.8614
1
0.0667
25
5.7217
0
0.1587
0.8743
0.9433
1.5845
1
0.04
60
15.1172
2.1322
0.1181
0.3066
0.6471
0.6917
1
0.0167
100
22.0164
9.4773
0.0972
0.1856
0.6659
0.5943
1
0.01
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15
7
(g) In the unit step disturbance, Overshoot value is zero when K=15,25 and the system is overdamped.
When K=60,100, the system is underdamped. So the larger controller gain give the more
underdamped system
Exercise 2: Design of a Second order feedback system based on performances.
For the motor system given below, we need to design feedback such that the overshoot is
limited and there is less oscillatory nature in the response based on the specifications provided in the
table. Assume no disturbance (D(s)=0).
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Use MATLAB, to find the system performance for different values of Ka and find which value of the
gain Ka satisfies the design condition specified. Use the following table.
8
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Solution
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Figure
K
20
30
50
60
80
Percent
Overshoot
0
1.11758
7.6893
10.8433
16.2929
Settling Time
0.5835
0.3176
0.3790
0.3404
0.4038
For the given motor system, specifications for
(a) Percent overshoot < 8%
(b) Settling time < 1 s
The suitable value of Ka is 50.
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K-50 satisfies the design condition specified.
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11
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