Digital Control
IMEC 4301P |
50950
August 29th, 2023
Instructor: Héctor A. Pacheco Martínez
Lab Team: Andrés Alemán, Héctor Moore, Ricardo Villanueva, Sonja Henttinen.
Laboratory #1 Experiment 4.1
Objectives
To evaluate the effect of pole and zero location upon the time response of first- and second-order systems.
Minimum Required Software Packages
MATLAB, Simulink, and the Control System Toolbox.
Prelab
Problem 1
Given the transfer function
, evaluate settling time and rise time for the following values of a:
1,2,3,4. Also, plot the poles.
Answer:
t1 = tf(1,[1 1]);
t2 = tf(1,[1 2]);
t3 = tf(1,[1 3]);
t4 = tf(1,[1 4]);
step(t1,t2,t3,t4)
legend('a = 1','a = 2','a = 3','a = 4')
1
st1 = stack(1,t1,t2,t3,t4);
pzplot(st1(:,:,1),st1(:,:,2),st1(:,:,3),st1(:,:,4))
legend('a = 1','a = 2','a = 3','a = 4')
grid
2
T1 = stepinfo(t1)
T1 = struct with fields:
RiseTime: 2.1970
TransientTime: 3.9121
SettlingTime: 3.9121
SettlingMin: 0.9045
SettlingMax: 0.9993
Overshoot: 0
Undershoot: 0
Peak: 0.9993
PeakTime: 7.3222
T2 = stepinfo(t2)
T2 = struct with fields:
RiseTime: 1.0985
TransientTime: 1.9560
SettlingTime: 1.9560
SettlingMin: 0.4523
SettlingMax: 0.4997
Overshoot: 0
Undershoot: 0
Peak: 0.4997
PeakTime: 3.6611
T3 = stepinfo(t3)
T3 = struct with fields:
RiseTime: 0.7323
TransientTime: 1.3040
SettlingTime: 1.3040
3
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
0.3015
0.3331
0
0
0.3331
2.4407
T4 = stepinfo(t4)
T4 = struct with fields:
RiseTime: 0.5493
TransientTime: 0.9780
SettlingTime: 0.9780
SettlingMin: 0.2261
SettlingMax: 0.2498
Overshoot: 0
Undershoot: 0
Peak: 0.2498
PeakTime: 1.8306
Problem 2
Given the transfer function
.
2.a
Evaluate precent overshoot, settling time, peak time, and rise time for the following values:
plot the poles.
Answer:
Gs2a = tf(25, [1 4 25]);
step(Gs2a)
4
,
. Also,
poles2a = pole(Gs2a)
poles2a = 2×1 complex
-2.0000 + 4.5826i
-2.0000 - 4.5826i
pzplot(Gs2a)
grid
5
gs2a = stepinfo(Gs2a)
gs2a = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.2930
1.6819
1.6819
0.9065
1.2537
25.3741
0
1.2537
0.6908
2.b
Calculate the values of a and b so that the imaginary part of the poles remains the same but the real part is
increased two times over that of Prelab 2a, and repeat Prelab 2a.
Answer:
Real part is increased by 2 times:
6
Gs2b = tf(37, [1 8 37]);
step(Gs2b)
poles2b = pole(Gs2b)
poles2b = 2×1 complex
-4.0000 + 4.5826i
-4.0000 - 4.5826i
pzplot(Gs2b)
grid
7
gs2b = stepinfo(Gs2b)
gs2b = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.3294
0.9882
0.9882
0.9041
1.0644
6.4396
0
1.0644
0.6908
2.c
Calculate the values of a and b so that the imaginary part of the poles remains the same but the real part is
decreased by one half over that of Prelab 2a, and repeat Prelab 2a.
Answer:
Real part is decreased by half:
8
Gs2c = tf(22, [1 2 22]);
step(Gs2c)
poles2c = pole(Gs2c)
poles2c = 2×1 complex
-1.0000 + 4.5826i
-1.0000 - 4.5826i
pzplot(Gs2c)
grid
9
gs2c = stepinfo(Gs2c)
gs2c = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.2608
3.6352
3.6352
0.7465
1.5037
50.3661
0
1.5037
0.6908
Problem 3
3.a
For the system of Prelab 2a, calculate the values of a and b so that the real part of the poles remains the same
but the imaginary part is increased two times over that of Prelab 2a, and repeat Prelab 2a.
Answer:
Real part is increased by 2 times:
10
Gs3a = tf(84, [1 4 84]);
step(Gs3a)
poles3a = pole(Gs3a)
poles3a = 2×1 complex
-2.0000 + 8.9443i
-2.0000 - 8.9443i
pzplot(Gs3a)
grid
11
gs3a = stepinfo(Gs3a)
gs3a = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.1341
1.8533
1.8533
0.7559
1.4946
49.4636
0
1.4946
0.3454
3.b
For the system of Prelab 2a, calculate the values of a and b so that the real part of the poles remains the same
but the imaginary part is increased four times over that of Prelab 2a, and repeat Prelab 2a.
Answer:
Real part is increased by 4 times:
12
Gs3b = tf(340, [1 4 340]);
step(Gs3b)
poles3b = pole(Gs3b)
poles3b = 2×1 complex
-2.0000 +18.3303i
-2.0000 -18.3303i
pzplot(Gs3b)
grid
13
gs3b = stepinfo(Gs3b)
gs3b = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.0615
1.9131
1.9131
0.4965
1.7097
70.9674
0
1.7097
0.1704
Problem 4
4.a
For the system of Prelab 2a, calculate the values of a and b so that the damping ratio remains the same but the
natural frequency is increased two times over that of Prelab 2a, and repeat Prelab 2a.
Answer:
For a LTI second ordersystem without zeros:
14
&
&
Natural frequency is increased by two times:
Gs4a = tf(100, [1 8 100]);
step(Gs4a)
poles4a = pole(Gs4a)
poles4a = 2×1 complex
-4.0000 + 9.1652i
-4.0000 - 9.1652i
pzplot(Gs4a)
grid
15
damp(Gs4a)
Pole
Damping
Frequency
(rad/seconds)
Time Constant
(seconds)
-4.00e+00 + 9.17e+00i
-4.00e+00 - 9.17e+00i
4.00e-01
4.00e-01
1.00e+01
1.00e+01
2.50e-01
2.50e-01
gs4a = stepinfo(Gs4a)
gs4a = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.1465
0.8409
0.8409
0.9065
1.2537
25.3741
0
1.2537
0.3454
4.b
For the system of Prelab 2a, calculate the values of a and b so that the damping ratio remains the same but the
natural frequency is increased four times over that of Prelab 2a, and repeat Prelab 2a.
Answer:
Natural frequency is increased by two times:
16
Gs4b = tf(400, [1 16 400]);
step(Gs4b)
poles4b = pole(Gs4b)
poles4b = 2×1 complex
-8.0000 +18.3303i
-8.0000 -18.3303i
pzplot(Gs4b)
grid
17
damp(Gs4b)
Pole
Damping
Frequency
(rad/seconds)
Time Constant
(seconds)
-8.00e+00 + 1.83e+01i
-8.00e+00 - 1.83e+01i
4.00e-01
4.00e-01
2.00e+01
2.00e+01
1.25e-01
1.25e-01
gs4b = stepinfo(Gs4b)
gs4b = struct with
RiseTime:
TransientTime:
SettlingTime:
SettlingMin:
SettlingMax:
Overshoot:
Undershoot:
Peak:
PeakTime:
fields:
0.0733
0.4205
0.4205
0.9065
1.2537
25.3741
0
1.2537
0.1727
Problem 5
Briefly describe the effects on the time response as the poles are changed in each of the Prelabs 2, 3, and 4.
Answer:
Lab
18
Problem 1
Using Simulink, set up the systems of Prelab 1 and plot the step response of each of the four transfer functions
on a single graph by using the Simulink LTI Viewer (See Appendix E.6 online for tutorial). Also, record the
values of settling time and rise time for each step response.
For your Simulink, please provide a screenshot that clearly shows the system.
Problem 2
Using Simulink, set up the systems of Prelab 2. Using the Simulink LTI Viewer, plot the step response of each of
the three transfer functions on a single graph. Also, record the values of pervent overshoot, settling time, peak
time, and rise time for each step response.
For your Simulink, please provide a screenshot that clearly shows the system.
Problem 3
Using Simulink, set up the systems of Prelab 2a and Prelab 3. Using the Simulink LTI Viewer, plot the step
response of each of the three transfer functions on a single graph. Also, record the values of percent overshoot,
settling time, peak time, and rise time for each step response.
For your Simulink, please provide a screenshot that clearly shows the system.
Problem 4
Using Simulink, set up the systems of Prelab 2a and Prelab 4. Using the Simulink LTI Viewer, plot the step
response of each of the three transfer functions on a single graph. Also, record the values of percent overshoot,
settling time, peak time, and rise time for each step response.
For your Simulink, please provide a screenshot that clearly shows the system.
Postlab
Problem 1
For the first-order systems, make a table of calculated and experimental values of settling time, rise time, and
pole location.
Problem 2
For the second-order systems of Prelab 2, make a table of caluclated and experimental values of percent
overshoot, settling time, peak time, rise time, and pole location.
Problem 3
For the second-order systems of Prelab 2a and Prelab 3, make a table of calculated and experimental values of
percent overshoot, settling time, peak time, rise time, and pole location.
Problem 4
19
For the second-order systems of Prelab 2a and Prelab 4, make a table of calculated and experimental values of
percent overshoot, settling time, peak time, rise time, and pole location.
Problem 5
Discuss the effects of pole location upon the time response for both first- and second-order systems. Discuss
any discrepancies between your calculated and experimental values.
20