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Digital Control Systems

创建日期：2024-04-11 14:53:18

所属分类：数学mathematics

标签： EMESTER

Digital Control Systems

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Digital Control Systems

Time allowed – 90 minutes
Candidates must answer ANY TWO questions

Each question carries equal marks (30). Marks shown in sub-
sections are indicative only. It is desirable to show the method of
calculation and the steps taken to achieve the results.

A datasheet of useful formula is provided.

A table of standard Z-transforms is provided

A sheet containing z-plane diagrams is provided

[Turn over
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Page 2 of 7

Question 1

System 1: The output response of a digital system is defined
by the following Z-transfer function:

() =
4
2−3.2+0.6

The sampling time, T = 0.1 seconds.

a) Using an appropriate inverse Z-transform technique,
determine the Z-transform series (limited to the first six terms)
for this digital signal.
[5 Marks]

System 2: The closed loop transfer function for a discrete control
system is;
() =
0.2
2−1.75+0.95

The sampling time, T = 0.1 seconds.

b) Using your knowledge of discrete control system analysis:

i. Determine the DC gain of the closed loop system.
ii. Determine the stability of the closed loop system.
[5 Marks]

System 3: The closed loop transfer function for a discrete control
system is;
() =
2+0.5+0.75
4−0.63+0.42+0.2+0.9

The sampling time, T = 0.1 seconds.

[EEE3018]

Page 3 of 7

c) Using your knowledge of discrete control system analysis,
determine the stability of the closed loop system.
[5 Marks]

System 4: Consider Figure 1. Here, a continuous system plant,
P(s), is to be controlled by a digital PI (Proportional-Integral)
controller. The sampling time for the system, T = 0.1 seconds.

C(z)
y(t) r(t)
ZOH
T
+
-
P(s)
e(t) u(t)

() =
2
3
)(


s
sP T = 0.1 seconds
Figure 1

d) Including the effect of the zero order hold (ZOH), derive a z-
domain function P(z) relating the intermediate signal U to
output Y.
[5 Marks]

e) Determine the discrete closed loop transfer function for this
control system, as a function of the PI controller gains;
proportional gain Kp, and integral gain Ki. (see datasheet)
[5 Marks]

f) The controller is then configured as an Integral (I) only
controller (Kp = 0). Determine the range of values of Ki for
which the digital control system will remain stable. (You are
free to use any appropriate method of analysis)
[5 Marks]

[End of Question]
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Page 4 of 7

Question 2

Consider Figure 2. This shows an analogue control system with
continuous controller, C(s), and continuous system plant,
comprising of PA(s) and PB(s).

PB(s)
y(t) r(t)
-
+
C(s) PA(s)
u(t)

() =
2
(+6)
() =
4(+2)
(+8)
() =
1
(+2)

Figure 2

The control system is to be upgraded and the continuous
controller is to be replaced with a digital equivalent controller,
C(z), as shown in Figure 3. The sampling time of the digital
system T = 0.05 seconds.

PB(s)
y(t) r(t)
-
+
C(z) PA(s)
T
e(t)
ZOH
u(k)

?=(z)C () =
4(+2)
(+8)
() =
1
(+2)

Figure 3

a) What is the Laplace transfer function for the Zero Order Hold
(ZOH) block shown in Figure 3?
[3 Marks]

b) Draw a diagram of the frequency response of the Zero Order
Hold block shown in Figure 3. Clearly label the axes of the
diagram.
[3 Marks]
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Page 5 of 7

c) Including the effect of the Zero Order Hold, derive a z-domain
function P(z) relating the intermediate signal U to output Y.
[6 Marks]
d) Using a pole zero mapping technique, obtain an equivalent z-
domain digital transfer function C(z) for the controller.
[6 Marks]
e) Find the discrete closed loop transfer function relating the
output to input, Y(z)/R(z).
[6 Marks]

f) The system poles can be transformed from the s-plane to the
z-plane via the transform: =

i) Consider Figure 4. This shows the root locus for a
constant damping factor in the s-plane. Draw a simple
sketch to show the shape of the equivalent root locus in
the z-plane.

ii) Consider Figure 5. This shows the root locus for a
constant real component in the s-plane ( = − +
√1 − 2. Draw a simple sketch to show the shape of
the equivalent root locus in the z-plane

Re s
Im s
Root locus for
constant 
Im s
Re s
0
Root locus
for constant
 n
 n
Figure 4 Figure 5

[6 Marks]
[End of Question]
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Page 6 of 7

Question 3
System 1: Consider the z-plane diagram shown in Figure 6. This
shows the pole/zero locations for a discrete closed loop control
system; O = zero location, X = pole location. The sampling time
of the discrete control system is 0.1 seconds.

Figure 6

a) Given the information in Figure 6, estimate the closed
loop transfer function for the discrete closed loop control
system.
[3 Marks]
b) Estimate the frequency of oscillation in the output
response of this discrete system.
[3 Marks]
c) Estimate the following transient parameters:

i. Peak Overshoot
ii. settling time (to 2%)
[4 Marks]
X
X
O
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Page 7 of 7

System 2: A discrete closed loop control system must satisfy
the following design specification:

i. Sampling time = 0.1 sec.
ii. Mp ≤ 16 % overshoot
iii. settling time ts ≤ 1.8s
iv. natural frequency ≥ 3.6 rads/s

d) Indicate on a z-plane diagram (separate sheet provided),
the acceptable areas for the closed loop poles to be
located in order to satisfy the design specification.
Clearly show all the calculation work involved in reaching
your final answer.
[12 Marks]

e) Consider Figure 7. This shows a digital control system
with digital controller, C(z), and continuous system plant,
P(s). Sample time, T = 0.1 second.

() =

(−1)
() =
1.2
(+4.8)

Figure 7

If K = 5, determine whether this control system will meet
the design specification in d). You are free to use any
appropriate method of analysis to validate your answer.

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