I
/
UNIVERSITI SAINS MALAYSIA
First Semester Examination
2015/2016 Academic Session
December 2015 / January 2016
EEE 350 - CONTROL SYSTEM
[SISTEM KAWALAN]
Duration 3 hours
[Masa : 3 jam]
( ) pages and Appendices
Please check that this examination paper consists of SEVENTEEN 17
This examination paper
ONE (1) pages of printed material before you begin the examination.
version from page
consist of two versions, The English version and Malay version. The English
EEN ( 17).
TWO (2) to page EIGHT (8) and Malay version from page NINE (9) to page SEVENT
( )
dan
Sila pastikan bahawa kertas peperiksaan ini mengandungi TUJUH BELAS 17 muka surat
Lampiran SATU (1) muka surat bercetak sebelum anda memulakan peperiksaan ini. Kertas
peperiksaan ini mengandungi dua versi, versi Bahasa Inggeris dan Bahasa Melayu. Versi
Bahasa Inggeris danpada muka surat DUA (2 ) sehingga muka surat LAPAN (8) dan versi
Bahasa Melayu daripada muka surat SEMBILAN (9) sehingga muka surat TUJUH BELAS (17).
Instructions: This question paper consists of SIX ( 6 ) questions. Answer FIVE (5) questions.
All questions carry the same marks.
fArahan: Kertas soalan ini mengandungi ENAM ( 6 ) soalan. Jawab LIMA (5) soalan. Semua
soalan membawa jumlah markah yang sama]
Answer to any question must start on a new page.
[Mulakan jawapan anda untuk setiap soalan pada muka surat yang barn]
“In the event of any discrepancies, the English version shall be used”.
[ Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa
Inggeris hendaklah diguna pakai]
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FNGL 1SH
1.
(a)
VERSION
What is the main difference between open-loop and closed-looP systems
?
(5 larks)
"
(b)
Discuss two advantages of closed-loop systems.
^
Figure 1.1 shows an inverting operational amplifier circuit where
(10 marks)
Vt
is the input
and V0 is the output. Assuming the ideal case (i.e the amplifier has infinite open
loop gain, infinite input impedance and zero output impedance), find the
function between the input and the output in terms of
Rx , R2 , C, and C2
transfer
(40 marks)
(d)
Determine the poles and zeros of the transfer function obtained in part (c).
(15 marks)
(e)
From the answer in (c), determine whether it is a P, PI, PD or PID controller.
Explain your answer.
(10 marks)
* /V
-AA
2
c,
.
rW
C2
t
^oW
Figure 1.1 : An inverting operational amplifier circuit as a controller.
...31-
[EEE 350]
-3-
Suppose the controller in Figure 1.1 is cascaded with another circuit as shown in
Figure 1.2. Assuming ideal case, find the transfer function from K, to
of
,
, R2 , R2 , R4 , C and C2
Vr in terms
.
(20 marks)
R2
,
C
C2
*
r-A/W
rVW
4
*
3
vM
AAArVl
"
Vy (s )
Figure 1.2: Cascaded operational amplifier circuits.
2.
A closed-loop system is shown in Figure 2.1 where C( s) is the controller and G( s ) is
the plant. The response of the system to a unit step when C( s ) = 1 is shown in Figure
2.2.
CW
U (s)
G(S)
Figure 2.1: Closed-loop system
Y(s)
1
[EEE 350]
-4-
Slep Response
Time (seconds)
Figure 2.2 : Step response of the closed-loop system in Figure 2.1
( (a)
.
What is the type of the system when C( s) = 1 ? Explain your answer.
(5 marks)
(b)
The maximum amplitude of the response is 0.966 at time=1.57s. Estimate the
transfer function of the closed-loop system (round the final values to 1 decimal
place). You may use the following approximations:
<
*
TF
7
e
=
• Mp
#
'
n
p
3.74
• ts »—
where
( peak overshoot)
( peak time)
( ± 2% settling time)
£ is the damping ratio and con
is the natural frequency.
(50 marks)
V
...5/-
’ y>
-5(c)
[EEE 350]
From your calculations in (b), locate the poles of the closed-loop transfer function
in the s-plane.
(20 marks)
Suppose C( s ) \ s a proportional controller with gain K . Suggest the value of K
such that the steady-state error is reduced to 0.1. (Hint: You may need to find the
transfer function of G( s ) first).
(25 marks)
An open-loop system is represented by the transfer function:
-
0
G
(a)
2
2s + 45 + 8
2
_
Let y(r) be the output of the system. Determine
>>(/)
when the input is a unit
step, and sketch its time response.
(40 marks)
(b)
Suppose the plant G ( s) is put in a unit feedback and a proportional controller K
is included to improve the-p rformance of the system. What is the largest value
of K (to two((JecimaL plaees) such that the damping ratio crfthe closed-loop
,
^
system is at least 0.5? Use the approximation
'
V
VA
o. V
where Mp is the peak overshoot and
^
is the damping ratio.
(25 marks)
Vs
*1
I
\
...6/-
K
I
>
\A
r
1«
•
-
-u
- u,\
i.
H
1
t
n
\
0+0 ~ ( _ 0 - ( -'i
( >
- -o
l
-
-
')
i
'
t
1 -0
[EEE 350]
-6-
- Lo
-\
Using the final value theorem, find the steady-state value of the closed-loop
system when K is the maximum value obtained in part (b).
(10 marks)
(c)
l
i-
-
\
\
wl
^
-
-i
WO
“
(d)
Suppose the design specifications are
•
the damping ratio is not lower than 0.5, and
•
the steady-state error is less than 20%
Explain whether these specifications can be satisfied at once.
(15 marks)
(e)
Suggest a modified structure of the controller to achieve zero steady-state error
for a step input, and prove your answer.
-t
t
.
IN'V
V -\
°
4.
(a)
'
(10 marks)
State the Routh-Hurwitz criterion in determining the stability of a linear system.
(10 marks)
-i - L
Use the Routh-Hurwitz criterion to find how many poles of the following
closedloop system, 7(s), are in the right half plane, in the left half plane, and on
the joaxis. Tabulate the summary of the pole locations in an appropriate table
(b)
\
o
~ I
-
l -n
-)
i u\
.
t
t
^
s3 +7 s2
s
o
6
- 21J +10
+ s - 6s* - s 2 - s + 6
-
1
I
H
-
c
l M
- -
5
i ~ t>
-l
- b -4
o
* -
T (s ) =
I
-
*
i-vit
(40 marks)
LI - <-
^
-i
-\
-\
...71-
-
i
i
W
7
[EEE 350]
-7-
having a process described as follow.
A unity feedback control system is
)
J
)
G( ) =
*
K
s(s +
lX- + 2X5 + 5)
y
the
If Routh-Hurwitz criterion is required to analyse
stability of the system
Identify the range of K for stability.
marginally stable.
Determine the value of K when the system is
marginally stable.
Find the frequency of oscillation when the system is
when the system is
Obtain the actual location of the closed-loop poles
s'
marginally stable.
Sketch the location of the poles on the s-plane.
*"'!} - 'S"\
V
(50 marks)
Consider a closed loop control system with
5
K.(s - 4S + 20 )
<?W = (s‘+ + 5)
lX^
5^ -t 3-s
;
-^
H(s) = l
°*^
'
^,
4
-t o i
Plot the root locus for the system.
State your subsequent steps in sketching it.
There is only 1 correct breakpoint out of these three possible roots
location:
,
( o- = -6.23; a2 = -2.53; <r3 = -0.53 )
Choose the correct break point and state whether
break away point. Label it on your root locus plot.
break-In or
-
(50 marks)
...SI-
a
-8-
\> »
[EEE 350]
Locate the exact point where the root locus crosses the 0.45 damping ratio line.
Hint: Use angle criterion to validate your exact point.
(30 marks)
C
<d)
Calculate the corresponding gain K at the crossing point (root locus
crosses the
0.45 damping line) in 5(b).
(10 marks)
i 1e)
Find the range of K within the system is stable
. Hint: locate the jco-axis crossing
by estimating the point from your root
locus sketch and calculate the
corresponding gain K at that point.
(10 marks)
6.
(a)
Name 2 advantages of frequency response techniques over
the root locus.
(10 marks)
(b)
Consider the transfer function;
«
C
Sketch the Bode diagram.
=
8
4> +l)
!
(40 marks)
(c)
From the Bode diagram obtained in 6(b), find
frequency at which the angle of G( ja> ) is -180°.
(/ G),)! ,
^
where a>x is the
(20 marks)
(d)
.
Find the exact value of |G(/ I)| and compare with the value that was found in
^
6(c).
(30 marks)
...9/-
V
APPENDIX
LAMPIRAN
[EEE 350]
IMPORTANT LAPLACE TRANSFORM PAIRS
/
Impulse function: £(/ )
Step function: u(t )
«
F( s )
1
1
s
e~al
1
s+ a
co
2
s + co2
s
2
s + co2
n\
sin cot
cos col
tn
sn +1
skF { s ) - / 7(0- )- sk -2 f (O
'
/
(t)
W=
ym
s
cos cot
<
= e c v sin
“
Vi - c
1-
1
(coJ\ - C t ); C < 1
2
[
2
= e ^ sin a)^ ^JyC t + <p ]\
JK
1
°°
co
( s + a )2 + co2
s+a
( ,s + a )2 + co 2
e “' sintftf
e
s
'
<>j = cos <£ ;
C, < 1
s 2 + 2£cons + col
s{s 2 + 2£cons + col
)
- / ( W ) (O )
'