Prescott, Arizona Campus
Department of Electrical and Computer Engineering
EE 402 Control Systems Laboratory
Fall Semester 2016
Lab Section 51PC
Thursday 1:25 – 4:05 pm
King Eng. Bldg. Rm 122
Lab Instructor:
Dr. Stephen Bruder
Lab 05
Second Order System Modeling
Date Experiment Performed:
Thursday, Feb 25 2016
Instructor’s Comments:

Comment #1

Comment #2
Date Report Submitted:
Day, Date, Month, 2016
Group Members:
Student # 1 Name & Email
Student # 2 Name & Email
Grade:
?? / 90
EE 402 Control Systems Lab
TABLE OF CONTENTS
Spring 2016
PAGE
1.
Abstract ................................................................................................................................................. 3
2.
Introduction ........................................................................................................................................... 3
3.
Theory and Experimental Methods ....................................................................................................... 4
4.
5.
1.a.
Open-Loop Analysis ..................................................................................................................... 4
1.b.
Closed-Loop Analysis ................................................................................................................... 5
Equipment and Procedures.................................................................................................................... 7
4.i.
Open-Loop Response .................................................................................................................... 7
4.ii.
Closed-Loop Response ................................................................................................................. 9
Results and Discussion ....................................................................................................................... 10
5.i.
Open-Loop Response .................................................................................................................. 10
5.ii.
Closed-Loop Response ............................................................................................................... 10
6.
Conclusion .......................................................................................................................................... 11
7.
References ........................................................................................................................................... 11
LIST OF TABLES
PAGE
Table 1 Comparison of estimated DC motor parameters from open-loop response ................................... 10
Table 2 Comparison of estimated DC motor parameters from closed-loop response ................................ 10
Table 3 Comparison of estimated DC motor parameters ............................................................................ 10
LIST OF F IGURES
PAGE
Figure 1 A simplified open-loop model of the DC motor ............................................................................. 4
Figure 2 Simulink model (left) and unit step response (right) ...................................................................... 5
Figure 3 A simplified unity feedback closed-loop configuration ................................................................. 5
Figure 4 Updated Simulink model (left) and unit step response (right)........................................................ 6
Figure 5 A generic underdamped 2nd order unit-step response ..................................................................... 7
Figure 6 Open-loop model: Simulation vs real-hardware ............................................................................. 8
Figure 7 Open-loop response: Motor angle - simulation vs real-hardware................................................... 8
Figure 8 Closed-loop model: Simulation vs real-hardware .......................................................................... 9
Figure 9 Closed-loop response: Motor angle - simulation vs real-hardware ................................................ 9
Figure 10 Improved closed-loop response: Motor angle - simulation vs real-hardware ............................ 11
LIST OF SYMBOLS
Names of Students in the Group
PAGE
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EE 402 Control Systems Lab
Spring 2016
1. ABSTRACT
A stand-alone summary of the report – the work done and results observed.
 The abstract should address:
o
What is the objective of the experiment?
o
What type of experiment is performed to achieve the objective?
o
What are the major results of the experiment?
o
What conclusions can be made from these results?

Write this after you have finished all of the other sections!!

The abstract should be less than a page and closer to a ½ page in length.
2. INTRODUCTION
In the last lab we applied a step input to the Quanser QUBE-Servo and measured its speed response.
Using this data, a 1st order model of the motor of the form G ( s)  K / ( s  a) was developed. In this
lab we will turn our attention to the motor’s shaft angle, as the output of interest, and the effect of closing
a unity feedback loop on the motor’s response. Furthermore, by measuring characteristics (e.g.,
overshoot and time to peak) of the close-loop response we will attempt to estimate parameters of the
open-loop transfer function.
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EE 402 Control Systems Lab
Spring 2016
3. THEORY AND EXPERIMENTAL METHODS
In the last lab we saw that the DC motor could be “reasonably” well approximated by a 1st order
transfer function G ( s ) 
s( s)
K

, when the input was considered to be the armature voltage ( ea )
Ea ( s ) s  a
and the output the motor’s speed ( 
 ). If we now consider the motor’s shaft angle as the output of
interest, our transfer function becomes G ( s) 
( s)
K
. Note that this is now a 2nd order

Ea ( s) s  s  a 
system!!
1.a.
Question 1.
Open-Loop Analysis
Is this system, G ( s) , stable, marginally stable, or unstable? ________________
QUBE DC Motor
Ea ( s )
K
sa
speed
1
s
( s )
angle
Figure 1 A simplified open-loop model of the DC motor
Task 1. Considering motor’s shaft angle as the output, referring to Figure 1, determine the symbolic
form of the system’s unit step response.
1

?
 (t )  L1  G ( s)   L1   
s

?
Question 2.
(1)
Referring to Eqn. (1), what is the slope of the system’s unit step response after the
transient response has died away? Slope 
?
?
Create a simulink model akin to that of Figure 1, add a unit step input (in Volts) and a scope to capture
the output angle (in radians). Use the “real hardware” values for K and a (i.e., G physical ( s ) ) which you
obtained in Lab 4 (see Eqn. 1.4)1. Paste a copy of your model in Figure 2. Run your simulation for two
seconds and insert a copy of the resulting step response into Figure 2.
1
Every motor in the Lab will have slightly different parameters.
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EE 402 Control Systems Lab
Spring 2016
Figure 2 Simulink model (left) and unit step response (right)
1.b.
Closed-Loop Analysis
By applying closed-loop feedback we can dramatically change the behavior of a system. Figure 3
shows the application of a unity gain feedback loop to the original system.
QUBE DC Motor
Ea ( s )
+
K
sa

-
speed
1
s
( s )
angle
Unity Feedback
Figure 3 A simplified unity feedback closed-loop configuration
Question 3.
What is the transfer function of the closed-loop system? Please ensure that the
coefficient of the s2 term in the denominator is 1 (i.e., a monic polynomial).
GCL ( s ) 
Question 4.
( s )
?
 2
Ea ( s ) s  ? s  ?
(2)
Using your values for K and a from Lab 04, is this system, GCL ( s ) , stable, marginally
stable, or unstable?? _______________
Update your Simulink model to perform unity closed-loop feedback, as shown in Figure 3.
Paste a copy of your updated Simulink model into Figure 4. Run your simulation for two seconds
(step time = 0!!) and insert a copy of the resulting step response into Figure 2.
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EE 402 Control Systems Lab
Spring 2016
Figure 4 Updated Simulink model (left) and unit step response (right)
Recall that the transfer function of a generic underdamped 2nd order system can be described as
G( s) 
n2
s 2  2n s  n2
(3)
with a corresponding unit step time response of (see textbook page 173) of
c(t )  1 

1

ent cos( nt   )
(4)

.
 1  2 


where   1   2 and   tan 1 

Task 2. By comparing Eqn. (3) with Eqn. (2) determine the expression for the following quantities in
terms of  and n :
K ?
(5)
a ?
(6)
By observing the undamped 2nd order step response (see Figure 5) and recalling that the overshoot is
only a function of the damping ration (textbook page 176)
%OS  100 e /
1 2
(7)
we can solve for the damping ration as
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EE 402 Control Systems Lab
Spring 2016
 
 ln(%OS /100)
(8)
 2  ln 2 (%OS /100)
Furthermore, knowing the time to peak ( T p ) and the damping ratio defines the natural frequency ( n )
via
TP 

n 1   2
.
(9)
Figure 5 A generic underdamped 2nd order unit-step response
4. EQUIPMENT AND PROCEDURES
Starting with your Simulink model from Lab 04, or otherwise, develop a Simulink model to drive the
QUBE-Servo motor with a unit step input and now plot angle in radians, as your output (NOT speed).
4.i. Open-Loop Response
Place your previously developed open-loop simulation model (see Figure 2) in parallel with the realhardware, as shown in Figure 6. Run the model for 2 seconds.
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EE 402 Control Systems Lab
Spring 2016
Figure 6 Open-loop model: Simulation vs real-hardware
Export the scope date to MATLAB and plot (legend, title, … ) the two open-loop unit step responses
in Figure 7.
Figure 7 Open-loop response: Motor angle - simulation vs real-hardware
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EE 402 Control Systems Lab
Spring 2016
4.ii. Closed-Loop Response
Place your previously developed closed-loop simulation model (see Figure 3) in parallel with the realhardware, as shown in Figure 8. Run the model for 2 seconds.
Figure 8 Closed-loop model: Simulation vs real-hardware
Export the scope date to MATLAB and plot (legend, title, … ) the two closed-loop unit step responses
in Figure 9.
Figure 9 Closed-loop response: Motor angle - simulation vs real-hardware
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EE 402 Control Systems Lab
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5. RESULTS AND DISCUSSION
As is often the case theory and practice do not often exactly coincide. In this section we will use the
data collected from the open-loop and closed-loop unit-step responses, for both the simulated model and
real-hardware, to estimate the parameters of appropriate transfer function models.
5.i. Open-Loop Response
Assuming the motor’s open-loop transfer function to be of the form G ( s) 
( s)
K
,

Ea ( s) s  s  a 
referring to Figure 7, determine which parameters of this model can you estimate from the real-hardware
response and the simulated response. List the parameters and values in Table 1.
Table 1 Comparison of estimated DC motor parameters from open-loop response
Parameter
From Simulated Response
From Real-Hardware Response
Measured
??
Calculated
??
5.ii. Closed-Loop Response
Assuming the motor’s closed-loop transfer function to be of the form which you determined in Eqn.
(2), referring to Figure 9, decide which parameters of this model can you estimate from real-hardware
response and simulated response. You may use Table 2 to guide you, however, it is not the only valid
approach.
Calculated
Measured
Table 2 Comparison of estimated DC motor parameters from closed-loop response
Parameter
From Simulated Response
From Real-Hardware Response
Mp
OS (in%)
Tp

n
K
a
Compare the values for the K and a parameters (real-hardware case) which you calculated in this Lab
vs the values obtained during Lab 04 using Table 3.
Table 3 Comparison of estimated DC motor parameters
Lab# / Parameter
K
a
Lab 04
Lab 05
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EE 402 Control Systems Lab
Spring 2016
6. CONCLUSION
What can you say (qualitatively) about the differences between the simulated and real DC motor
responses (both open- (Figure 7) and closed-loop (Figure 9) cases)?
Question 5.
BONUS QUESTION (+5%): Referring to Figure 9, how would you improve the match
between simulation and real-hardware? Implement your suggested changes to the simulation
model, regenerate the plot, and past into Figure 10 to hopefully illustrate the improvement.
Figure 10 Improved closed-loop response: Motor angle - simulation vs real-hardware
7. REFERENCES
[1] www.quanser.com
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