# neural network and fuzzy logic

October 22, 2018

UNIVERSITY OF
TECHNOLOGY, SYDNEY
FACULTY OF ENGINEERING AND
INFORMATION TECHNOLOGY

49275 NEURAL NETWORKS AND FUZZY
LOGIC

ASSIGNMENT 1

QUESTION
ONE [ Perceptron Dichotomiser] [ 50 marks ]

Two
perceptron dichotomisers are trained to recognise the following classification
of six patterns x with known class membership d.

0.8

0.2

0.9

0.2

1.0

0.0

0.5

0.7

0.7

0.8

0.2

0.1

x1

, x2

, x3

, x4

, x5

, x6

0.0

0.3

0.8

0.5

0.3

0.9

0.3

0.2

0.7

0.6

0.1

d1 .jpg”>1.jpg”>, d2 .jpg”> 1.jpg”>, d3 .jpg”>.jpg”>1.jpg”>, d4 .jpg”> 1.jpg”>, d5 .jpg”>1.jpg”>, d6 .jpg”> 1.jpg”>
.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>
1.1
The
first dichotomiser is a discrete perceptron as shown in Figure 1.1. Assign

-1 to all augmented inputs. For the
training task of this dichotomiser, the fixed correction rule is used, with an
arbitrary selection of learning constant 0.05 and the initial weight vector

0.0976
0.8632 w10.3296

0.3111 0.2162

Assume that the above training set may
need to be recycled if necessary, calculate the final weight vector. Show that
this weight vector provides the correct classification of the entire training
set. Plot the pattern error curve and the cycle error curve for 10 cycles (60
steps).

[ 25 marks ]

1.2
The
second dichotomiser is a continuous perceptron with a bipolar logistic

activation
function z

f2

(v)

1

ev

as shown in Figure 1.2.
Assign 1 to

1

e v

all augmented inputs. For the training
task of this dichotomiser, the delta training rule is used with an arbitrary
selection of learning constant 0.5

with the same initial weight
vector w1 in Question
1.1.

1

Assuming that
the above training set may need to be recycled if necessary,

calculate
the weight vector w7
after one cycle and the weight vector w301
after 50 cycles. Obtain the cycle error at the end of each cycle and plot the

cycle
error curve. How would the weight vectors w7
and w301 classify the
entire training set? Discuss your results.

[ 25 marks ]

Note:The following
formulae may be used to calculate the pattern error curve andthe cycle
error curve. There are 6 patterns in this question, i.e. P=6.

Pattern
error: Ep

1 (dp zp )2

2

Cycle
error: Ec

1

P

P

(dp zp )2

E p

2 p 1

p 1

.gif”>.gif”>

Figure 1.1 Discrete Perceptron Classifier Training
.gif”>.gif”>

Figure 1.2 Continuous Perceptron Classifier
Training

2

QUESTION
TWO [ 50 marks ]

2.1 [Flight Simulation] [ 15 marks ]

A new jet aircraft are subjected to
intensive flight simulation studies before they are tested under actual flight
conditions. In these studies, an important relationship is that between the
mach number (percent of the speed of sound) and the altitude of the aircraft.
This relationship is important to the performance of the aircraft and has a
definite impact in making flight plans over populated areas. If certain mach
levels are reached, breaking the sound barrier (sonic booms) can result in
human discomfort and light damage to glass enclosures on the earth’s surface.

Current rules of thumb establish crisp
breakpoints for the conditions which cause performance changes in aircrafts,
but in reality these breakpoints are fuzzy, because other atmospheric
conditions such as the humidity and temperature also affect breakpoints in
performance. For this problem, suppose the flight test data can be
characterised as “near” or “approximately” or “in the region of” the crisp
database breakpoints.

Define a
universe

of

aircraft

speeds

near

the

speed

of

sound as

X 0.725,0.730,0.735,0.740,0.745,0.750,0.755

mach, and a fuzzy set

M
for
the

speed “near
mach 0.74” where

0

0.25

0.75

1

0.75

0.25

0

M

0.725

0.730

0.735

0.740

0.745

0.750

0.755

and define a
universe of altitudes asY8350,8400,8450,8500,8550,8600,8650 m, and a fuzzy
set A for the altitude “approximately 8,500 m”, where

0

0.3

0.6

1

0.6

0.3

0

A

8400

8450

8500

8550

8600

8650

8350

2.1.1 Construct the relation R M
A

[5 marks]

2.1.2
For another aircraft speed, sayM1
for the speed “in the region of mach 0.74” where

0

0.5

0.8

1

0.6

0.2

0

M1

0.725

0.730

0.735

0.740

0.745

0.750

0.755

determine the
corresponding altitude fuzzy set A1a M1

Rusing the max-

min composition.

[5 marks]

2.1.3
For
the speed “in the region of mach 0.74” where

M1

0

0.5

0.8

1

0.6

0.2

0

0.725

0.730

0.735

0.740

0.745

0.750

0.755

determine the
corresponding altitude fuzzy set A1b M1 R using the sum-

product composition.

[5 marks]

3

2.2 [Laser
Beam Alignment] [ 35 marks ]

Fuzzy logic is used to control a
two-axis mirror gimball for aligning a laser beam using a quadrant detector.
Electronics sense the error in the position of the beam relative to the centre
of the detector and produces two signals representing the x and y direction
errors. The controller processes the error information using fuzzy logic and
provides appropriate control voltages to run the motors which reposition the
beam. The fuzzy logic controller for this system is shown in Figure 2.1.

To represent the error input to the
controller, a set of linguistic variables is chosen to represent 5 degrees of
error, 3 degrees of change of error, and 5 degrees of armature voltage.
Membership functions are constructed to represent the input and output values’
grades of membership as shown in Figure 2.2. The rule set in the form of
“Fuzzy Associative Memories” is shown in Figure 2.3.

The
controller gains are assumed to be GE
1, GCE 1, GU 1.

2.2.1
If
the Mean of Maximum (MOM) defuzzification strategy (sum-product inference) is
used with the fire strengthi of the i-th
rule calculated from

Ei ( e )
. CEi ( ce)
calculate the
defuzzified output voltages of this fuzzy controller at a particular instant.
The error and the change of error at this instant aree3.20 andce0.47.

[10 marks]

2.2.2
If
the Centre of Area (COA) defuzzification strategy (max-min inference) is used
with the fire strengthi of the i-th rule calculated from

min(
Ei ( e
), CEi ( ce))
calculate
the corresponding defuzzified output voltage at a particular instant when the
error and the change of error aree3.20 andce0.47 .

[25 marks]
.gif”>.gif”>.gif”>.gif”>.gif”>.gif”>

Figure 2.1 Fuzzy logic control system

4

.gif”>

Figure 2.2 Membership functions of a laser beam
alignment system
.gif”>

Figure 2.3 Fuzzy Associative Memories

H. T. Nguyen

March 2014

5

MARKING SCHEME

Assignment 1:

Neural Networks and Fuzzy Logic

Student Name:
____________________

Mark: ___________

Requirement

Criteria

Comment

Standard
“Declaration of

At front of report,
completed and

Yes/n

Originality” cover page

signed

o

as provided by the

Faculty

Question 1

Presentation

/25

Perceptron Dichotomiser

Final weight vector

1.1 Discrete Perceptron

Correct classification

Pattern error curve

Cycle error curve

Calculation/software code

1.2 Continuous

Presentation

/25

Perceptron

W(7)

W(301)

Cycle error curve

Classification after nc=1

Classification after nc=50

Software code

Section 2

Presentation

/15

2.1 Flight Simulation

RelationR M
A

Altitude Fuzz Set

A1a M1 R(max-min)

Altitude Fuzz Set

A1b M1 R(sum-

product)

Calculation/software code

2.2 Laser Beam

Presentation (MOM)

/35

Alignment

Fuzzification E

Fuzzification CE

Defuzzified output voltage

Calculation/software code

Presentation (COA)

Fuzzification E
& CE

Total Area

Total Moment

Defuzzified output voltage

Calculation/software code

6

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