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Matlab代写-ELEC3114
时间：2022-07-09
ELEC3114: Control Systems T2 2022
Created by Dr. Arash Khatamianfar, T2 2022. Page 1 of 5
ELEC3114: Assignment A
Preliminary Notes
Please read the following information very carefully.
• This assignment is worth 15% of your total course mark.
• From the time of its release, you have 1 week to complete the assignment (please see Moodle for the
exact due date as it may have already been extended beyond 1 week).
• You are required to provide complete workings and derivations analytically.
o You are required to provide MATLAB code and/or Simulink schematics in addition to your analytical
solution for any question that comes with a sign (M/S).
• MATLAB can be used to plot the time domain responses or to find the roots of a 3rd-order polynomial (or
higher order ones). Unless you are specifically asked to analytically calculate the time-domain function of
the responses. You can use Control System Toolbox functions to generate time-domain responses.
• Use the best sample solutions from last year for best practices in completing this assignment (your
submission must be legible, i.e. clear enough to read).
• You must submit all of your solutions before the due date on Moodle.
• This assignment is marked out of 100 plus some bonus marks, which will be scaled down to 13 for your
course mark.
How to Submit
• Convert your hand-written analytical solutions and workings (paper-based or digital) for each question
into separate PDF files, plus any added pictures of the graphs, MATLAB codes, and Simulink blocks
into a single PDF file. Having separate files for your solutions and notes will attract 10% penalty mark.
• You must provide the original MATLAB codes/script and Simulink files in addition to having copies of
the in the solution of each question
• Name the .pdf, .m, and .slx files using your zID and surname as below.
o Surname_z1234567_q1.pdf, Surname_z1234567_q2.pdf, Surname_z1234567_q3.pdf
o Surname_z1234567_matlab.m (for MATLAB scripts all in one file)
o Surname_z1234567_Simulink_qxx.slx
Use qxx to refer to the question number, e.g., Surname_z1234567_Simulink _q12.m for Q1.2 if you
wish to do Q1.2 in Simulink.
• Upload your files in the submission box under Assignments section in Moodle.
NOTE: Not following the file naming convention will attract 10% penalty mark.
Late Submission Policy
UNSW has a standard late submission penalty of 5% per day for all assessments where a penalty applies.
This is capped at five days (120 hours) from the assessment deadline, after which a student cannot submit
an assessment, and no permitted variation. For late submission you do not need to ask for special
consideration. Extension due to special circumstances with legitimate reason/s has be to submitted through
Special Consideration.
Final Advice:
DO NOT PLAGIARIZE AND/OR COLLUDE. This includes paying someone to solve the questions for you or
you ask a friend to hand over their solutions to you and you try to write them in a different way. Sharing high-
level ideas is ok though in a collaborative way. But the biggest losers are those who hand over their solutions
to other students to help them get easy marks without any effort. I want you to learn something from this
course and not waste your money, and you have to be honest with yourself about it. It is more honourable to
not get a full mark rather than getting caught cheating. The amount of distress and trauma that you can bring
upon yourself if UNSW Academic Integrity unit contacts you regarding cheating is just too much to bear (I
have seen it firsthand how students are traumatised by this, whether wrongfully or rightfully accused of
cheating). So just don’t do it, that’s all I ask!!
ELEC3114: Assignment A
Page 2 of 5
The Impulse response of a 2nd-order system with unknown paramaters is given in Fig. 1.
0.5 1 1.5 2 2.5 3
Time (s)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
O
u
tp
u
t
S
ig
n
a
l
t1 = 0.27 s
M1 = 1.31
t2 = 1.77 s
M2 = 0.082
t2t1
M2
M1
Fig. 1. Impulse response of a 2nd-order system.
Q1.1. [20] If the transfer function of the unknown system is given as below, determine the paramaters , ,
and parametrically in terms of the first two peak values 1 and 2, and 1 and 2. Then use the
numerical values to find the transfer function () and verify your solution by providing the MATLAB
plot of the impulse response with those measurements (up to 2% numerical precision error is
acceptable) (M/S).
() =
()
()
=
2 + 2 + 2
Q1.2. [10] Find an expression for the output response () if the following input signal is applied to your
identified () as shown in Fig. 2. (Hint: Use the properties of the Laplace transform and the fact that
() is an LTI system.)
5 10
Time (s)
0
10
12.5
In
p
u
t
S
ig
n
a
l
15
0
Fig. 2. Input signal applied to the identified transfer function ().
Q1.3. [5] (Bonus) Simulate the response of () for the input signal in Q1.2 and compare it with the plot of
the time-domain function, (), of the output response from Q1.2 (M/S).
ELEC3114: Control Systems Lab Manual
Page 3 of 5
Consider the transfer function () from Q1.1 in the closed-loop system shown in Fig. 3.
R(s) +
Gc(s)
E (s) Y(s)
H(s)
KD s
Td (s)
+ U(s)
s2 + 2a s + w
2
K
s
KP (s + z)
Process G(s)
+
+UC (s)
Fig. 3. Closed-loop system for Q2.
Q2.1. [5] Find the closed-loop transfer function from the reference signal () to () parametrically in terms
of , , and , i.e., ()/() when () = 0. (Do NOT use the numerical values of from your
solution in Q1.1). Then replace the transfer functions with their paramaters as shown in Fig. 3. Assume
the known paramaters are , , and and the unknown paramaters are , , and .
Q2.2. [2] Determine a condition on based on , , and such that ()/(1 + ()()) is a 2nd-order
critically damped system.
Q2.3. [15] Using the condition from Q2.2, find a condition on and based on and such that the closed-
loop system undergoes a pole-zero cancellation and also behaves like a 2nd-order critically damped
system. Determine the closed-loop settling time. (Hint: Using coefficient matching, equate the desired
characteristic equation to the closed-loop characteristic equation while applying the given conditions of
pole-zero cancellation and critically damped behaviour).
Q2.4. [8] Show that the final value of the output due to a unit step disturbance will approach zero, that is if
() = 0 and () = 1/ then → 0 as → ∞. (You do not need to have solved Q2.2 and Q2.3 to
solve this question.)
Q2.5. [5] Use = 10, = 2, and = √20 to find numerical values of , , and . Verify your results for
both Q2.3 (assume unit step response) and Q2.4 via simulation (M/S).
Q2.6. [5] (Bonus) Is it possible to still have the closed loop system behave like a 2nd-order critically damped
system without pole-zero cancellation? You may support your answer with both analytical calculations
as well as an explanation.
The dynamic equation of motion for an air levitation system is given as below with the schematics of the
system shown in Fig. 4. The objective is to balance the height of the ball inside a tube via the air flow blown
by a DC fan from the bottom, in an upward direction. The height of the ball is measured with respect to a
position sensor attached at the top of the tube. There is an adjustable nozzle allowing for the air to escape
from the top of the tube. This means that at any constant input voltage, the steady state air flow speed and
the ball position will depend on the diameter of the nozzle. The speed of the air flow is and the mass of the
ball is with the gravitational acceleration denoted as . The parameter is the product of the drag coefficient
, air density , and the surface area of the ball exposed to the upward air flow divided by 2, =
1
2
. The
drag force is then given by = ( − ̇)
2
.
̈ = ( − ̇)
2
−
ELEC3114: Assignment A
Page 4 of 5
mg
fd
+vi
vf
Sensor and
nozzle
ze
D z
ball
Fig. 4. Air levitation system with DC electric fan.
Q3.1. [10] Assume a constant air flow of is applied to bring the ball to a specific height (operating point)
. Derive the linearised state space model of the system at this operating point ( , ). Note that this
operating point is obtained at steady state, which is when the velocity and acceleration of the ball are
zero ̇ = ̈ = 0. (Hint: Apply the operating/equilibrium point condition to find a required relation between
and input offset ).
Q3.2. [3] Find the eigenvalues of the linearised system, and then obtain the state transition matrix.
Q3.3. [5] Assume the transfer function of the DC fan is modelled as a simple first-order system with a time
constant , a DC gain of , input voltage , and as it output. Combine the DC fan transfer function
with the linearised models from Q3.1 and obtain a 3rd-order linear model with Δ as the input (small
changes in the input voltage of the DC fan) and Δ as the output (small changes in the position of the
ball)
Q3.4. [10] Derive a 3rd-order state space model by choosing the state variables as 1 = Δ, 2 = Δ̇, 3 = Δ
with Δ as the input and 1 = Δ as the output. Then use the transfer function from Q3.3 to find a
different state space representation of this system using the same input and output.
Q3.5. [7] Use Simulink to build both the original nonlinear air levitation system and the linearised model, with
the DC fan voltage as the input. Compare the step responses of both systems starting with a small
magnitude of the step voltage, all the way to a unit step voltage. Explain the differences between the
behaviour of these systems and the impact of linearisation (M/S).
Note: You need to first bring the nonlinear system to its operating point by calculating and applying a
constant voltage that results in an input offset (equilibrium/operating) air speed . If you have
calculated the correct and subsequently correct , the position of the ball will reach a negative
equilibrium value when the DC fan is connected in simulation. If you disconnect the fan and directly
apply the correct , the ball position will be zero right from the beginning. This is expected because
the reference coordinate for measuring the output in the equations is considered at the equilibrium,
whereas in reality, the actual position is measured by the sensor (as shown in Fig. 4). We can add a
feasible initial output offset as our chosen equilibrium point via the initial condition of the integrator.
To compare the step response, apply a small step change around the input offset = + Δ on the
nonlinear model after the ball is settled (you can counteract the negative output offset due to the DC
fan by adding its equivalent positive value as the initial condition). Then, use the same timing of the
applied small changes in your nonlinear model and have a delayed step block with only Δ for the
linear model. This way you can nicely compare your results in a single plot.
ELEC3114: Control Systems Lab Manual
Page 5 of 5
The values of the parameters are available below.
% DC fan parameters
Kdf = 1.15; % (V.m/s) DC fan DC gain
tau = 0.375; % (s) DC fan time constant
% Tube and ball characteristics
m = 0.01 % (kg) Mass of the ball
Cd = 0.4; % Drag coefficient for a sphere in turbulent flow
Rho = 1.2; % (kg/m^3) Air density
A = 0.01; % (m^2) Frontal area of the ball
g = 9.81; % (m/s^2) Gravitational acceleration
--- End of Assignment A ---
Created by Dr. Arash Khatamianfar, T2 2022. Page 1 of 5
ELEC3114: Assignment A
Preliminary Notes
Please read the following information very carefully.
• This assignment is worth 15% of your total course mark.
• From the time of its release, you have 1 week to complete the assignment (please see Moodle for the
exact due date as it may have already been extended beyond 1 week).
• You are required to provide complete workings and derivations analytically.
o You are required to provide MATLAB code and/or Simulink schematics in addition to your analytical
solution for any question that comes with a sign (M/S).
• MATLAB can be used to plot the time domain responses or to find the roots of a 3rd-order polynomial (or
higher order ones). Unless you are specifically asked to analytically calculate the time-domain function of
the responses. You can use Control System Toolbox functions to generate time-domain responses.
• Use the best sample solutions from last year for best practices in completing this assignment (your
submission must be legible, i.e. clear enough to read).
• You must submit all of your solutions before the due date on Moodle.
• This assignment is marked out of 100 plus some bonus marks, which will be scaled down to 13 for your
course mark.
How to Submit
• Convert your hand-written analytical solutions and workings (paper-based or digital) for each question
into separate PDF files, plus any added pictures of the graphs, MATLAB codes, and Simulink blocks
into a single PDF file. Having separate files for your solutions and notes will attract 10% penalty mark.
• You must provide the original MATLAB codes/script and Simulink files in addition to having copies of
the in the solution of each question
• Name the .pdf, .m, and .slx files using your zID and surname as below.
o Surname_z1234567_q1.pdf, Surname_z1234567_q2.pdf, Surname_z1234567_q3.pdf
o Surname_z1234567_matlab.m (for MATLAB scripts all in one file)
o Surname_z1234567_Simulink_qxx.slx
Use qxx to refer to the question number, e.g., Surname_z1234567_Simulink _q12.m for Q1.2 if you
wish to do Q1.2 in Simulink.
• Upload your files in the submission box under Assignments section in Moodle.
NOTE: Not following the file naming convention will attract 10% penalty mark.
Late Submission Policy
UNSW has a standard late submission penalty of 5% per day for all assessments where a penalty applies.
This is capped at five days (120 hours) from the assessment deadline, after which a student cannot submit
an assessment, and no permitted variation. For late submission you do not need to ask for special
consideration. Extension due to special circumstances with legitimate reason/s has be to submitted through
Special Consideration.
Final Advice:
DO NOT PLAGIARIZE AND/OR COLLUDE. This includes paying someone to solve the questions for you or
you ask a friend to hand over their solutions to you and you try to write them in a different way. Sharing high-
level ideas is ok though in a collaborative way. But the biggest losers are those who hand over their solutions
to other students to help them get easy marks without any effort. I want you to learn something from this
course and not waste your money, and you have to be honest with yourself about it. It is more honourable to
not get a full mark rather than getting caught cheating. The amount of distress and trauma that you can bring
upon yourself if UNSW Academic Integrity unit contacts you regarding cheating is just too much to bear (I
have seen it firsthand how students are traumatised by this, whether wrongfully or rightfully accused of
cheating). So just don’t do it, that’s all I ask!!
ELEC3114: Assignment A
Page 2 of 5
The Impulse response of a 2nd-order system with unknown paramaters is given in Fig. 1.
0.5 1 1.5 2 2.5 3
Time (s)
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
O
u
tp
u
t
S
ig
n
a
l
t1 = 0.27 s
M1 = 1.31
t2 = 1.77 s
M2 = 0.082
t2t1
M2
M1
Fig. 1. Impulse response of a 2nd-order system.
Q1.1. [20] If the transfer function of the unknown system is given as below, determine the paramaters , ,
and parametrically in terms of the first two peak values 1 and 2, and 1 and 2. Then use the
numerical values to find the transfer function () and verify your solution by providing the MATLAB
plot of the impulse response with those measurements (up to 2% numerical precision error is
acceptable) (M/S).
() =
()
()
=
2 + 2 + 2
Q1.2. [10] Find an expression for the output response () if the following input signal is applied to your
identified () as shown in Fig. 2. (Hint: Use the properties of the Laplace transform and the fact that
() is an LTI system.)
5 10
Time (s)
0
10
12.5
In
p
u
t
S
ig
n
a
l
15
0
Fig. 2. Input signal applied to the identified transfer function ().
Q1.3. [5] (Bonus) Simulate the response of () for the input signal in Q1.2 and compare it with the plot of
the time-domain function, (), of the output response from Q1.2 (M/S).
ELEC3114: Control Systems Lab Manual
Page 3 of 5
Consider the transfer function () from Q1.1 in the closed-loop system shown in Fig. 3.
R(s) +
Gc(s)
E (s) Y(s)
H(s)
KD s
Td (s)
+ U(s)
s2 + 2a s + w
2
K
s
KP (s + z)
Process G(s)
+
+UC (s)
Fig. 3. Closed-loop system for Q2.
Q2.1. [5] Find the closed-loop transfer function from the reference signal () to () parametrically in terms
of , , and , i.e., ()/() when () = 0. (Do NOT use the numerical values of from your
solution in Q1.1). Then replace the transfer functions with their paramaters as shown in Fig. 3. Assume
the known paramaters are , , and and the unknown paramaters are , , and .
Q2.2. [2] Determine a condition on based on , , and such that ()/(1 + ()()) is a 2nd-order
critically damped system.
Q2.3. [15] Using the condition from Q2.2, find a condition on and based on and such that the closed-
loop system undergoes a pole-zero cancellation and also behaves like a 2nd-order critically damped
system. Determine the closed-loop settling time. (Hint: Using coefficient matching, equate the desired
characteristic equation to the closed-loop characteristic equation while applying the given conditions of
pole-zero cancellation and critically damped behaviour).
Q2.4. [8] Show that the final value of the output due to a unit step disturbance will approach zero, that is if
() = 0 and () = 1/ then → 0 as → ∞. (You do not need to have solved Q2.2 and Q2.3 to
solve this question.)
Q2.5. [5] Use = 10, = 2, and = √20 to find numerical values of , , and . Verify your results for
both Q2.3 (assume unit step response) and Q2.4 via simulation (M/S).
Q2.6. [5] (Bonus) Is it possible to still have the closed loop system behave like a 2nd-order critically damped
system without pole-zero cancellation? You may support your answer with both analytical calculations
as well as an explanation.
The dynamic equation of motion for an air levitation system is given as below with the schematics of the
system shown in Fig. 4. The objective is to balance the height of the ball inside a tube via the air flow blown
by a DC fan from the bottom, in an upward direction. The height of the ball is measured with respect to a
position sensor attached at the top of the tube. There is an adjustable nozzle allowing for the air to escape
from the top of the tube. This means that at any constant input voltage, the steady state air flow speed and
the ball position will depend on the diameter of the nozzle. The speed of the air flow is and the mass of the
ball is with the gravitational acceleration denoted as . The parameter is the product of the drag coefficient
, air density , and the surface area of the ball exposed to the upward air flow divided by 2, =
1
2
. The
drag force is then given by = ( − ̇)
2
.
̈ = ( − ̇)
2
−
ELEC3114: Assignment A
Page 4 of 5
mg
fd
+vi
vf
Sensor and
nozzle
ze
D z
ball
Fig. 4. Air levitation system with DC electric fan.
Q3.1. [10] Assume a constant air flow of is applied to bring the ball to a specific height (operating point)
. Derive the linearised state space model of the system at this operating point ( , ). Note that this
operating point is obtained at steady state, which is when the velocity and acceleration of the ball are
zero ̇ = ̈ = 0. (Hint: Apply the operating/equilibrium point condition to find a required relation between
and input offset ).
Q3.2. [3] Find the eigenvalues of the linearised system, and then obtain the state transition matrix.
Q3.3. [5] Assume the transfer function of the DC fan is modelled as a simple first-order system with a time
constant , a DC gain of , input voltage , and as it output. Combine the DC fan transfer function
with the linearised models from Q3.1 and obtain a 3rd-order linear model with Δ as the input (small
changes in the input voltage of the DC fan) and Δ as the output (small changes in the position of the
ball)
Q3.4. [10] Derive a 3rd-order state space model by choosing the state variables as 1 = Δ, 2 = Δ̇, 3 = Δ
with Δ as the input and 1 = Δ as the output. Then use the transfer function from Q3.3 to find a
different state space representation of this system using the same input and output.
Q3.5. [7] Use Simulink to build both the original nonlinear air levitation system and the linearised model, with
the DC fan voltage as the input. Compare the step responses of both systems starting with a small
magnitude of the step voltage, all the way to a unit step voltage. Explain the differences between the
behaviour of these systems and the impact of linearisation (M/S).
Note: You need to first bring the nonlinear system to its operating point by calculating and applying a
constant voltage that results in an input offset (equilibrium/operating) air speed . If you have
calculated the correct and subsequently correct , the position of the ball will reach a negative
equilibrium value when the DC fan is connected in simulation. If you disconnect the fan and directly
apply the correct , the ball position will be zero right from the beginning. This is expected because
the reference coordinate for measuring the output in the equations is considered at the equilibrium,
whereas in reality, the actual position is measured by the sensor (as shown in Fig. 4). We can add a
feasible initial output offset as our chosen equilibrium point via the initial condition of the integrator.
To compare the step response, apply a small step change around the input offset = + Δ on the
nonlinear model after the ball is settled (you can counteract the negative output offset due to the DC
fan by adding its equivalent positive value as the initial condition). Then, use the same timing of the
applied small changes in your nonlinear model and have a delayed step block with only Δ for the
linear model. This way you can nicely compare your results in a single plot.
ELEC3114: Control Systems Lab Manual
Page 5 of 5
The values of the parameters are available below.
% DC fan parameters
Kdf = 1.15; % (V.m/s) DC fan DC gain
tau = 0.375; % (s) DC fan time constant
% Tube and ball characteristics
m = 0.01 % (kg) Mass of the ball
Cd = 0.4; % Drag coefficient for a sphere in turbulent flow
Rho = 1.2; % (kg/m^3) Air density
A = 0.01; % (m^2) Frontal area of the ball
g = 9.81; % (m/s^2) Gravitational acceleration
--- End of Assignment A ---
