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MEE3038
Exam Time Table Code MEE3038
Faculty of Engineering and Physical Sciences
Level 3
Examination for the Degree of Master of Engineering
Engineering Dynamics 3 Wednesday, 14th August 2019 9:30 AM – 11:30 AM
Examiners: Professor A Clare Dr A K Behera
Write on both sides of the answer paper
Answer ALL Questions
You have TWO HOURS to complete this paper
Q1.
a) Consider the inverted pendulum shown in Fig. Q1a. The objective here is to keep the
pendulum in the upright position, that is to keep Θ = 0, in the presence of disturbances. An optical encoder is used to measure the angle of the pendulum with respect to the vertical. Using a block diagram, indicate the general elements for controlling the pendulum.
[5 marks]
[10 marks]
Determine the transfer function
[10 marks]
1 1 0 0 0 1 10
c) A single input, single output system has the state space equations
0 0 2 1 1 0 0
MEE3038/2019
Fig.Q1a. Control of an inverted pendulum b) Find the Laplace transform of the following function:
1, 0 1 0, 1 2 2 , 2 3 0, 3
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Q2.
(a) Consider the system shown in Fig. Q2a(i). When subjected to a unit step input, the response
of the system is shown in Fig. Q2a(ii) below the system block diagram. From these details, determine the value of K and T.
Fig.Q2a(ii). Response of system to a step input (b) Determine the stability of a system with the closed loop transfer function
(c)
What is the unit step response, c(t), of the system shown by the block diagram in Fig. Q2c. What is the final value of this system to the step input?
10
2 3 6 53
Use the Routh-Hurwitz criterion to justify your answer.
MEE3038/2019
[10 marks]
Fig.Q2a(i). Block diagram of system for Q.2 (a)
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[5 marks]
[10 marks]
Fig.Q2c. Block diagram representation of a system for Q.2(c)
Q3.
(a) Find the steady-state errors for inputs of 5 u(t), 5t u(t), and 5t2 u(t) to the system shown in Fig.
Q3a. The function u(t) is the unit step.
Fig.Q3a. Block diagram representation of a system for Q.3(a)
(b) Consider a system whose step response is shown in Fig. Q3b. Using Ziegler-Nichols tuning rules, determine gains for a PI controller for this system that works with unity negative feedback. What is the steady state error for this system with the PI control based on the final value from the closed-loop transfer function?
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MEE3038/2019
[15 marks]
0 1 0 0
0 0
1 5 6 1
1 0 00
END OF EXAMINATION
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1 0
MEE3038/2019
Fig.Q3b. Step response of a system for Q.3(b)
(c) Consider the system with the state space equations:
It is desired to place the closed loop poles at s = -2 + j4, s = -2 – j4 and s = -10. Determine the appropriate state feedback matrix to enable this placement.
[15 marks]
[20 marks]
ANNEXURE A. TABLE OF LAPLACE TRANSFORMS
1sin 1
2
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MEE3038/2019
f(t)
L[f(t)] = F(s)
f(t)
L[f(t)] = F(s)
1
(1)
aeat −bebt a−b
(18)
eat f(t)
F(s – a)
(2)
teat
(19)
U(t-a)
e−as s
(3)
tneat
(20)
f(t–a)U(t–a)
e–as F(s)
(4)
eat sin kt
(21)
δ(t)
1
(5)
eat cos kt
(22)
δ(t–a)
e –as
(6)
eat sinh kt
(23)
tn f(t)
1
(7)
eat cosh kt
(24)
f'(t)
sF(s) – f(0)
(8)
tsinkt
(25)
fn(t)
snF(s)-s(n-1)f(0)-…-f(n-1)(0)
(9)
tcoskt
(26)
F(s)G(s)
(10)
t sinh kt
(27)
tn(n=0,1,2,…)
n! sn+1
(11)
t cosh kt
(28)
tx (x≥-1 ε R)
Γ(x+1) sx+1
(12)
arctan
(29)
sin at
a
s2 + a2
(13)
√ /
√ √
(30)
cos at
s
s2 + a2
(14)
2√ /
√
(31)
(15)
erfc(√
√
(32)
sinh at
a
s2 −a2
(16)
1
(s − a)(s − b)
(33)
cosh at
s
s2 −a2
(17)
1
1
!
2
ANNEXURE B. ZIEGLER-NICHOLS TUNING RULES 1. S-SHAPED TRANSFER FUNCTIONS
2. USING CRITICAL GAIN AND CRITICAL PERIOD
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TYPE OF CONTROLLER
KP
TI
TD
P
∞
0
PI
.
.
0
PID
.
2L
0.5L
TYPE OF CONTROLLER
KP
TI
TD
P
∞
0
PI
.
0
PID
0.5
0.125
. .
.
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