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EE2023 Systems Assignment

创建日期：2024-06-14 16:34:39

所属分类：计算机computer science

标签： EE2023

Systems Assignment

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EE2023 Systems Assignment

Instructions
1. The is an open-book take-home assignment
2. Write your name and student number in the spaces provided above.
3. Answer all questions in this booklet.
4. Important guidelines are given on Page 2.
5. Write your answer to each question in the space provided immediately below the question in this
booklet. Submission in any other form will be returned with a mark of zero.
6. Sign your declaration at the bottom of this page.
7. Rename the completed booklet which contains your answers using the filename
your_student_number.pdf (e.g. A0123456J.pdf)
and upload it via the CANVAS Systems Assignment file submission system.
8. If you are unable to write directly on this softcopy booklet, then you should print it out on A4
paper, write on the hardcopy booklet, scan the completed booklet into a single pdf file, and name
and upload the file as per instructions given in item (7) above.
Declaration
• I understand what plagiarism is and am aware of the University’s policy in this regard.
• I declare that this submission is my own original work.
• I have not used work previously produced by another student or any other person to hand
in as my own.
• I have not allowed, and will not allow, anyone to copy my work with the intention of
passing it off as his or her own work.
Signature
Page 2 of 10
IMPORTANT GUIDELINES
(a) In your final answers, all parameters and variables should be substituted with their numeric
values if they are known. And all evaluable numeric expressions must be evaluated to 3
significant figures.
For example if the answer is
( ) ( ) ( ) ( )( )2 2 1030.36 10 sin sin log (1)0.12341.88 16 A Cz t B C t Kt CA L B A B = + ⋅ − − +  
where 24750, 2, 100A B C= = = , then after substitution and evaluation, we have
( ) ( ) ( ) ( ) ( )340000.0273 sin 111.243 4.04 10 sin 2 (2)3054.15z t t KtL −= + − ×− 
Only correct answers of the form shown in (2) will earn full credit.
(b) Do not miscalculate the values of the system parameters Lm and L from your student
(matriculation) number because if your miscalculated values match those of another student,
then your submission can be construed as plagiarized work.
(c) Sketches of functions need not be drawn to scale but must be adequately labeled.
(d) Your answers must be unambiguous, neatly organized and legibly written in the spaces
provided in this question booklet.
Please follow the above guidelines to the letter to avoid penalty.
Page 3 of 10
The ASSIGNMENT
Consider the system in Figure 1 which shows a crane hoisting and moving a load which is attached to the
end of a rope. The cart is supposed to deliver the load from one place to another. The objective is to move
the cart in a way which does not cause the rope and its load to swing too much. This assignment explores
the modelling of such a system and leads up to the design of a control system for the cart such that the
load does not swing wildly.
Although the actual system is highly nonlinear, if the rope is considered to be stiff with a fixed length L
and the angle φ is assumed small, the system can be modelled using the following differential equations:

( ) ( )
2
2 =
Lad x t g t
dt
φ
( ) ( ) ( )
2
2 = −
T
T T L
d x t
m f t m g t
dt
φ
( ) ( ) ( )= −La T Lx t x t x t
( ) ( )Lx t L tφ=
where Lm is the mass of the load, Tm is the mass of the cart, Tx and Lx are displacements as defined in
Figure 1, φ is the rope angle with respect to the vertical, and Tf is the force applied to the cart. In this
problem, the mass of the cart, Tm can be considered a constant. For simplicity, you may choose 1=Tm
unit mass while the gravitational constant can be approximated as 210 m s=g . The mass of the load,
Lm and the length of the rope, L are parameters which vary for each hoisting problem.
In this assignment, generate Lm and L using your student number as follows:
( )
( )
10 unit mass
1 unit length
• = +
• = +
Lm C
L D
where C and D may be derived by reading the last 4 digits of your student number. Equate the first two
digits to C and the remaining two to D. For example, if your student number is A0123456J, then choose
34C = and 56D = .
Figure 1: Schematic of crane hoisting a load.
Tf
Tx
φ
Lm
Lx Lax
L
Tm
Page 4 of 10
Enter the values of , , and LC D m L into the boxes below
C = D = Lm = L =
and proceed to answer the following questions.
(1) Obtain the transfer function, ( )( )
Φ
T
s
V s
where ( ) ( ){ }T TV s v t= L and ( ) ( ){ }s tφΦ = L are the Laplace
transforms of the cart velocity, ( )Tv t and the rope angle, ( )tφ respectively.
Page 5 of 10
(2) Assume that the cart is driven at a constant velocity, 0 10 m s=v , i.e. ( ) ( )10Tv t u t= where ( )u t is
the unit step function. Derive and sketch the angle of sway of the rope as the cart moves.
Page 6 of 10
(3) Under the condition in part (2) above, show that the load will sway with a frequency of
0 rad sg Lω = .
(4) Find the transfer function, ( )( )
T
T
V s
F s
, from the applied force to the cart's velocity where
( ) ( ){ }T TV s v t= L and ( ) ( ){ }T TF s f t= L are the Laplace transforms of the cart velocity, ( )Tv t

and the force, ( )Tf t respectively.

Page 7 of 10

(5) Analyze what happens to the velocity of the cart if a constant force is applied to it. Sketch the
trajectory of the velocity. You may assume the constant force to be 1 unit.
Page 8 of 10
(6) Suppose a crane controller is designed as shown in Figure 2, to automatically modulate the cart
velocity in a manner which ensures that the load does not swing wildly when the cart is driven at
the constant velocity, 0v i.e. ( ) ( ){ }0TV s v u t= L where ( )u t is the unit step function. K is the
controller parameter to be tuned in order to achieve the control objective.
(a) Derive the final transfer function ( )( )
Φ
T
s
V s
, of the crane cart plus controller shown in Figure 2.
(b) Design a suitable value of K such that the resulting response of the rope sway is non-
oscillatory.
(c) Sketch the trajectory of the rope sway when the cart is driven at a constant velocity of
0 2 m s=v . You may assume that the initial angle of the rope is zero.
(d) Explain qualitatively how the control system works. How does the control parameter K affect
the quality of control?
Figure 2: Control Configuration for the Crane.

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