UNIVERSlTl
--·"--~~· ·#~--~-
TEKNOLOGI
-------·--
PETRONAS
COURSE MDB3013- CONTROL SYSTEMS
DATE 15 AUGUST 2018 (WEDNESDAY)
TIME 9:00AM - 12:00 NOON (3 HOURS)
INSTRUCTIONS TO CANDIDATES
1. Answer ALL questions in the Answer Booklet.
2. Begin EACH answer on a new page in the Answer Booklet.
3. Indicate clearly answers that are cancelled, if any.
4. Where applicable, show clearly steps taken in arriving at the solutions and
indicate ALL assumptions, if any.
5. DO NOT open this Question Booklet until instructed.
Note
1. There are TEN (10) pages in this Question Booklet including the cover
page and appendices.
ii. DOUBLE-SIDED Question Booklet.
Universiti Teknologi PETRONAS
MOB 3013
1. a. The vertical position, x(t) of the grinding wheel shown in FIGURE Q1a is
controlled by a closed-loop system. The input to the system is the desired
depth of grind, and the output is the actual depth of grind. The difference
between the desired depth and the actual depth drives the motor, resulting
in a force applied to the work. This force results in a feed velocity for the
grinding wheel.
x-axis servomotor
, Force transducer
r-axis servomotor
\ / "' "'
PMAC
motor
PC
Cupped control
grinding
wheel --
Workpiece
Eddy current
Shaft
probe
encoder
z-axis servo
motor
FIGURE Q1a
i. Draw a closed-loop functional block diagram for the grinding process,
showing the input, output, force, and grinder feed rate.
[6 marks]
ii. Propose efforts relating to answer in part a(i) in order to produce a
good controller for the system.
[4 marks]
2
MOB 3013
b. Reduce the block diagram shown in FIGURE Q1 b to a single block
representing the transfer function,
T(s) = C(s)
R(s)
for G1(s) =G2(s) =G3(s) =s and H1(s) =H2(s) =H3(s) =H4(s) = 1.
R(s) +
H2(s)
FIGURE Q1b
[15 marks]
MOB 3013
2. Effective control of insulin injections can result in better lives for diabetic
persons. Automatically controlled insulin injection by means of a pump and a
sensor that measures blood sugar can be very effective. A pump and injection
system has a feedback control as shown in FIGURE Q2. R(s) is the desired
blood-sugar level and Y(s) is the actual blood-sugar level.
Pump Human body
E(s) Y(s)
R(s)
+ _...
Insulin 1
r K Blood-sugar
s (s + 2)
- level
Sensor
FIGURE Q2
a. Discuss the typical design objectives of having feedback control
systems.
[5 marks]
b. Determine the closed-loop transfer function Y(s )/R(s ).
[5 marks]
c. Evaluate the natural frequency mn and damping ratio s when K = 1.
[5 marks]
d. Calculate the steady-state error when K = 1 for a unit step input.
[5 marks]
e. Propose a suitable gain K such that the overshoot in the blood-sugar
level is less than 10% for a unit step input.
[5 marks]
4
MOB 3013
3. An aircraft pitch control system includes a controller and a gyroscope to
automatically control the pitch of the aircraft based on its dynamics. A simplified
block diagram of the pitch control system for an aircraft is shown in FIGURE
03, where K is the gain. K should be as large as possible to reduce the effect
of wind disturbances.
Command Controller Aircraft dynamics
__:ti_
pitch angle
~- s(s+3\(s+10)
Pitch angle
K
Gyroscope I I r·------.--.J
FIGURE Q3
a. Determine the range of K to maintain a stable system using Routh
Stability Criterion.
[1 0 marks]
b. Select the poles of the closed-loop system forK in part (a) and discuss
the stability of the system.
[5 marks]
c. Evaluate the stability of the system if the transfer function of the
gyroscope is changed to _!_.
s
[10 marks]
MOB 3013
4. A controller is designed to improve performances of a mechanical system by
adding a gain K and a feedback function H(s) as shown in FIGURE Q4a. It is
selected that H(s) = 1.
R(s) + C(s)
K G(s)
-
H(s)
FIGURE Q4a
2
s+-3
a. For the system with C(s) =
s 2 (s+6)
1. Sketch the root locus by including calculations of asymptotes and
their angles, and points of break-in and breakaway.
[10 marks]
11. Determine K and the closed-loop transfer function where the three
closed-loop poles meet.
[5 marks]
1
b. The root locus of the system with C(s) = c ) is shown in
s+l (s+3)(s+S) 2
FIGURE Q4b. For designing the gain K, the system is approximated as
a second-order system.
1. If the system should yield a settling time of 8 seconds, predict K,
damping ratio S, frequency mn, and percentage of overshoot %OS
of the system.
[7 marks]
6
MOB 3013
11. When the system becomes a critically damped system, predict the
value of K.
[3 marks]
Root locus
/
1
/
(jj
~
-~
/
L
0 ------ -<;-~(----+-----><)"--------- ~ ..
ro
c
a,
"'
E
-1
1
-2
\\. \\
___
'· ·,'
'•i
1~
>.
•'-,
'· I
\. I
'· _I
0 2
Real Axis
FIGURE Q4b
-END OF PAPER-
7
MOB 3013
TIME RESPONSE APPENDIX A
Tr = 2.2; 4 r; = -ln(%0S 11 00)
a Ts =-
a
J;r2 + ln2(%0S /100)
Am n
G(s) =
s 2 + 2r;m n s + mn 2 Tp-
- rc
{On~l-r;2;
%OS= e -((n 1~1-:;2) xlOO
8
MOB 3013
APPENDIX 8
LAPLACE TRANSFORMS
Properties Theorem Time Domain s-domain
Definition L [f(t)] = F(s) = r f(t)e "dt
f--
8(t) 1
Inverse L- I [F(s)] = f(t) = I-
27fj
r•;r .
a-;r
F(s)e''ds 8(t-kN) e-kNs
Linearity L [kl(t)] = kF(s) Us( f) -
I
s
theorem
Linearity L LJ;(t) + / 2 (1)] = F;(s) + F 2 (s) t
n
s·
theorem
Scaling e-at
L [/(at)]= ~F ~
1 ('' J s+a
theorem
1 -e-at
~] = sF(s)- f(O
Differentiation a
---
L[ ) s(s +a)
theorem
Differentiation
theorem
l !l
L d
dr
2
2
= s F(s)- sf(O-)- l'(O )
· ·
te-at
----
(s
l
+ a) 2
Differentiation
theorem
l jl
L -Jn· n =s"F(s)-2.>·"kfk
df
n
k-I
1
(0-)
t2e-at 2
(s + a)
3
Integration [' F(s)
L fu/(r)dr = ----;-- J COSUJ( --
s
s2 + llJ2
theorem
Final value f(ro) =lim
,_.u
sF(s) sinmt llJ
52 + llJ 2
theorem
Initial value f(O ') = limsF(s) e-at cos UJt s+a
S----}-OC
(s+a)2+llJ2
theorem
-~-- _ _ _ L_ __ )_____
c:
MOB 3013
APPENDIX C
ROOT LOCUS
1. 1 + KG(s)H(s) =: 1 + K ;~;~ = 0
2. K= _ D(s)
N(s)
3. K = IID(s)ll
IIN(s)ll
4.
5. .
Break-In . dK
and breakaway po1nts:-
ds
= 0,-
d (D(s))
-
ds N(s)
= 0 or -d (N(s))
ds
-
D(s)
=0
6.
10
