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ELEC3004 Control Systems

创建日期：2024-06-17 15:20:59

所属分类：电气工程electric engineering

标签： ELEC3004

Control Systems

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ELEC3004: Signals, Systems and Control

Problem Set 3: Control Systems
Note: This assignment is worth 20% of the final course mark. Please submit your answers
as a single pdf file via Gradescope, including your name and student number. Solutions,
including equations, should be typed. Explain your solutions as if you are trying to teach
a peer. Demonstrate your insight and understanding. Answering an entire question
with bare equations, lone numbers or without any explanation is not acceptable. Marks
may be reduced if an answer is of poor quality, demonstrates little effort or significant
misunderstanding. If you have used code in your solution, please include it in your report.
Questions
Question 1. Stability Analysis (15 marks)
A closed-loop digital feedback control system has the following transfer function:
C(z)
R(z)
=
z2 − z + 0.4
z3 − 1.8z2 + 0.3z + 0.64
(a) What is the characteristic equation (P (z) = 0) for this system? (1 mark)
(b) Evaluate the first three conditions of the Jury stability test for this system. (5 marks)
(c) Construct the Jury stability table for this system. (7 marks)
(d) Is the system stable? (2 marks)
1
ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 2. Root Locus (20 marks)
The feedback system shown below includes a proportional controller with gain K and a
zero-order hold (ZOH):
Figure 1: Block diagram of system for Question 2 and Question 3.
(a) Use the z-transform tables to show that the open-loop transfer function is given
by (4 marks)
G(z) =
K [(3T + exp (−3T )− 1) z + (1− exp (−3T )− 3T exp (−3T ))]
9 (z − 1) (z − exp (−3T ))
(b) Evaluate the transfer function for the following three sampling periods, and using
the rlocus function in MATLAB, plot the root locus for each sampling period.
i) T = 2 s (3 marks)
ii) T = 3 s (3 marks)
iii) T = 5 s (3 marks)
(c) For each sampling period, compute the critical gain value K for stability using the
root locus magnitude condition. (6 marks)
(d) What is the relationship between sampling period and critical gain value? (1 marks)
2
ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 3. Bode Plots (35 marks)
For the system shown in Figure 1, assume now that the sampling period is T = 0.5 s.
(a) Compute G(z) (2 marks)
(b) Using the bilinear transformation:
z =
1 +
(
T
2
)
w
1− (T
2
)
w
show that
G(w) =
K (0.06757w2 − 1.446w + 4.703)
w2 + 2.541w
(1)
(6 marks)
(c) G(w) can be approximated by
G(w) ≑
1.88K
(
1− w
17.4
) (
1− w
4
)
w
(
1 + w
2.5
) (2)
Use Equation (2) to fill in Table 1 with the slope contribution from each pole and
zero to the Bode magnitude plot. (10 marks)
Frequency (rad/s)
0.01 (Start: 2.5 (Start: 4 (Start: 17.4 (Start:
Description Pole at 0) Pole at -2.5) Zero at 4) Zero at 17.4)
Pole at 0
Pole at -2.5
Zero at 4
Zero at 17.4
Total slope (dB/dec)
Table 1: Bode magnitude plot slope contribution from each pole and zero in Equation (2).
3
ELEC3004: Signals, Systems and Control Sem 1, 2024
(d) Use Equation (2) to fill in Table 2 with the slope contribution from each pole and
zero to the Bode phase plot. (8 marks)
Frequency (rad/s)
0.25 (Start: 0.4 (Start: 1.74 (Start: 25 (End: 40 (End: 174 (End:
Description Pole at 2.5) Zero at 4) Zero at 17.4) Pole at -2.5) Zero at 4) Zero at 17.4)
Pole at -1
Zero at 10
Zero at -300
Total slope (deg/dec)
Table 2: Bode phase plot slope contribution from each pole and zero in Equation (2).
(e) Given a controller gain K = 1, approximate the Bode log-magnitude for frequency
ν = 0.01 rad/s from Equation (2). (3 marks)
(f) Using the bode function in MATLAB, generate the bode plot for the original transfer
function G(w) from Equation (1) with a gain of K = 1. (Hint: you can use this to
check your answers for (c)-(e).) (2 marks)
(g) Annotate your bode plot in (f) to show whether or not the system has a finite gain
margin. If you conclude that the system has a finite gain margin, what is it (rounded
down to the nearest dB)? If not, why not? (2 mark)
(h) Annotate your bode plot in (f) to show whether or not the system has a finite phase
margin. If you conclude that the system has a finite phase margin, what is it (to the
nearest 5◦)? If not, why not? (2 mark)
4
ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 4. State-space Analysis (15 marks)
Evaluate whether the following systems are:
i) Controllable
ii) Observable
(a)
x1(k + 1) = 0.6x2(k) + u(k)
x2(k + 1) = 1.1x1(k)− 0.7x2(k) + 0.3u(k)
y(k) = x1(k)
(4 marks)
(b) [
x1 (k + 1)
x2 (k + 1)
]
=
[
0.2 −0.8
0 −0.3
] [
x1(k)
x2(k)
]
+
[
1
0
]
u(k)
y(k) =
[
1 −0.4] [x1(k)
x2(k)
]
(4 marks)
(c) x1 (k + 1)x2 (k + 1)
x3 (k + 1)
 =
 0.1 1 0−0.9 0 0
0.2 0.2 −0.5
x1(k)x2(k)
x3(k)
+
 00.1
1
u(k)
y(k) =
[−0.1 1 0]
x1(k)x2(k)
x3(k)

(7 marks)
5
ELEC3004: Signals, Systems and Control Sem 1, 2024
Question 5. Design via Pole Placement (15 marks)
Consider the following pulse transfer function of a feedback control system:
Y (z)
U(z)
=
0.8z + 0.1
z2 − 1.2z − 0.7
Where y(k) is the corresponding output vector, u(k) is the input vector and x(k) is the
state vector.
(a) Derive the controllable canonical form of the state-space representation using the
direct programming method and write the A, B, C and D matrices. (5 marks)
(b) Is the system controllable? (2 marks)
(c) Is the system observable? (2 marks)
(d) Determine a suitable state feedback gain matrix K =
[
k1 k2
]
such that the system
will have closed-loop poles at z = 0.3± j0.4. (6 marks)

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