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• First Name: Last Name: Student Number: ------ -------- -------
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
FINAL EXAMINATION, DECEMBER 19, 2017, 2:00PM
DURATION: 2.5 hours
MIE404H1 F- Control Systems I
Exam Type: D
Examiner - A. Bilton
Aids Allowed: • Non-programmable calculator • Slides and notes taken from lecture • Supplementary notes from Blackboard • Ruler • Text - Franklin, Powell, Emami-Naeini. Feedback Control of Dynamic Systems
Instructions: • Do not open the exam until instructed to do so. • Answer in the space provided. If you need an extra space, use the extra paper attached to the back of the exam and put a note you are using this extra space. • Put your name and student number on the top of each page, including this one. • Write neatly and clearly. If I can't read it, I can't give you marks for it. • Show your work, I can't give you part marks without it. • A list of basic formulas is provided at the back. You may remove this from the exam and use for reference. • There are 4 questions total. • The marks for each question are shown to help you allocate your time.
Question Number Marks Available 1 23 2 14 3 38 4 25
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Problem 1 (23 Marks)
You've recently joined a company making blimp drones, similar to the one Prof B recently saw during her recent hockey game. The drone is used to get images from around the arenas and stadiums during games. Unfortunately, the drone has roll oscillations, making it challenging to get steady images.
The company designed their control system using a proportional controller. The roll dynamics of this system are outlined below, where a is a parameter defined by the size of a roll stabilizer.
r + Gfs)=K
s+a <fJ.. ~ ~ ~
r r r
s2+0.25as r
A~ -
a) What is the closed loop transfer function. Leave both Kand a as parameters. (3 Marks)
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b) The blimp was originally designed with a=1. The controller was selected with K=1. Forthese choice of parameters, what are the locations of the closed loop poles. Explain why the drone has large roll oscillations. (4 Marks)
c) You have been tasked with removing the roll oscillations from the system. Unfortunately, the controls team does not want to change their software and you are limited to changing the stabilizer. Draw a root locus showing the closed-loop pole locations for the system for varying values of a~ 0. For the root locus, define any asymptotes, departure angles, break-in/break-away points, or imaginary axis crossings. (Hint: the characteristic equation in root locus form with 1 +aG(s)=O, and regular sketching rules apply for G(s)). Show all relevant calculations and show your final root locus on the paper on page 5. (12 Marks)
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d) Based on your root locus inc), what range of values of a could you choose (with K=J) to remove oscillations for step inputs in the desired roll angle. (4 Marks)
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Problem 2 (14 Marks)
You've been tasked with designing a speed controller for DC motor, which is driving a small robot. You do not know the properties of the motor and the robot well, but have been told that the relationship between the motor speed (and robot speed) and the input voltage can be represented using the following first order transfer function.
n(s) K
V(s) TS+ 1 a) Since you are unable to determine the transfer function analytically, you do an
experiment to determine the parameters. You apply a step input the voltage of 5V at t=0.1 seconds and see the following response.
5 t----+---+--t--,:::--1•-t•---+--t-----i----+--t-----i u (U
.!!? l4t----+---+--r------------------
"'O (U (U
c% 3
(1l C
g2t----+---+-I--+---+---+--+---+---+--+---! 1!l &
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Offset=O Time (seconds)
Determine the values of Kand r. An accuracy of 0.5 rad/sec and 0.02 seconds is sufficient. (Hint: you can use the steady-state value and the derivative at the time of the step input to determine the parameters). (6 Marks)
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b) Assume K=2 and r=0.06 (note, these are different values than you may obtain in part a)) a Pl controller is added to speed up the response of the motor to have a settling time (2%) less than 0.1 seconds and limit the steady-state error due to the step input in speed to less than 0.01 rad/sec. What would you choose for the values of Kp and Ki to meet these requirements. (8 Marks)
r K +Kls----K- i---,-B_ P i rs+ 1
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Problem 3 (38 Marks)
Consider a unity feedback system as shown in the figure below.
~ G,(s) ~IG,(s)l I Y► With G (s) = s+so
P s(s+0.1)(s+10)
a) Plot the bode diagrams for Gc(s)=1. Perform any necessary computations and then put your final plot on the following page. (8 Marks)
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b) From this bode diagram, estimate the gain and phase margins for this system. Is this feedback system stable with Gc(s)=1? (4 Marks)
c) Sketch the root locus for this system. For the root locus, define any asymptotes, departure angles, break-in/break-away points, or imaginary axis crossings. (8 Marks)
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I • • l,lr. ~ □ _ .. ,a ... .
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d) For the root locus above, how to does the value of K where the poles cross the imaginary-axis relate to the gain margin? (2 Marks)
e) It is desired to get a phase margin of 45° to limit any overshoot for step inputs using a proportional controller only. Determine the controller. Using the bode plot to do this approximately is fine. (6 Marks)
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f) Assuming that the speed of the response of the system is adequate, but the system is not able to track with enough accuracy. What type of controller might you implement to improve the tracking performance (of ramps) with a Kv >100 without affecting the phase margin. Determine the values for the controller. You can use the graph paper below to complete your design if needed. (10 Marks)
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Problem 4 (25 Marks)
Consider the feedback system shown in the figure below to control the orientation (8) of a satellite. The feedback uses low bandwidth orientation sensor in feedback.
d
r s+5 0 1 ·0 G
C s2+3s+2 s
1
s+l
a) Assuming the disturbance is 0, what is the transfer between the orientation and the reference? Leave Ge as a variable. (4 Marks)
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b) Assuming the reference is 0, what is the transfer function between the error (e) and the disturbance (d). Leave Ge as a variable.(5 Marks)
c) Assuming the disturbance d is a step function with a maximum value of 0.5, show if it is possible to use a proportional controller (Ge =K) to keep the absolute value of the error less than 0.02 radians and have the system be closed-loop stable from r to 8? (8 Marks)
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d) If it's not possible to stabilize the system and maintain a small absolute value of the error(< 0.02 radians) due to the disturbance using a proportional controller, try a PD controller (Ge =Kp+Kds). What range of values of Kp and Kd will satisfy the error constraint and result in a stable closed-loop system be closed-loop stable from r to 8? (8 Marks)
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EXTRA PAPER
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EXTRA PAPER
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■ ·•. a• •. r.
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EXTRA PAPER
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Basic MIE404 Formulas
Laplace Transforms - General Rules
f(t,)
J+9
rxf (o' ER)
q(t,) = {' f('r) d, · lo
f(o.t), o > 0
e"'J(t:)
1.iX!
F(s) = f(t)e_., dt. • 0
F+G
o:F
.~F(,) - f(O)
d' ,t-1f .i·r( ·) k-'J(·o) .J•-2 ~(O'J . 'o) -< ' ~ - s - " di: - .. , - dt''-1 l
( '( ·) _ F(s) _r,-;--s
F(s - 11)
Common Laplace Transforms
f(t)
1
8
(5(k)
t
t"' k~O
Id'
Cal
coswt
sin wt
cos(wt + ¢)
e-a,t cos wt
e-a.t sin wt
F(s)
1
s
l
Sk
1
s 2
1 sk+l
1 s - (I,
s = s2 +w2
1/2 1/2 --,-+ . S - .JW S + .JW
w l/2j l/2j ----s2 +w2 S - .JW S + jw
s cos¢- wsin¢ s2 +w2
s+a ( )
') . ., s +c1, - + w-
w
(s+a)2+w2
1
1F8532C2·C661-4FBC·99EB-9467189994BC
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First Order Systems
Transfer Function G(s) = ~ s+a
Step Response in Time Domain y(t) = 1 - e-at, t ~ 0
Time Constant -r = 1 / a
Rise Time (10-90%) Tr = 2·2
a
Settling Time (within 2% of steady state) Ts=~ a
Second Order Systems
2
Transfer Function G (s) = 2 /n 2 · s +2 WnS+Wn
Step Response in Time Domain y(t) = 1- e-at (coswdt + :d sinwdt), wd = wn.J1- ( 2 , rJ = (wn
Percent Overshoot %OS = exp (- .j~:(2 ) x 100
ln(o.02.ji-(2 ) 4 Settling Time (within 2% of steady state) Ts = - ( ::::: -
7 - approximation valid for O < ( < 0.9
. Wn ,wn
Rise Time (10-90%) Tr ::::: 1.a Wn
PeakTimeTP = F-f2=.!E.... Wn 1-( Wd
Damping relationship to Phase Margin<;= ::a for PM<70°
Final Value Theorem
f(t = oo) = limsF(s) S➔O
DC Gain
GUw, w = O) = lim G(s) S➔O
2
. . .
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