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UNIVERSITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEERING AND INFORMATION TECHNOLOGY
49275 NEURAL NETWORKS AND FUZZY LOGIC
ASSIGNMENT 1
QUESTION ONE [ Perceptron Dichotomiser Training ] [ 50 marks ]
Two perceptron dichotomisers are trained to recognise the following classification of six patterns x with known class membership d.
.10 .00 .50 .80 1x ,
.90 .31 .10 .20 2x ,
.30 .30 .70 .90 3x ,
.20 .80 .70 .20 4x ,
.70 .50 .80 .01 5x ,
.60 .30 .20 .00 6x
d1 1 , d2 1 , d3 1 , d4 1 , d5 1 , d6 1
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 variable correction rule is used with the coefficient .5 1 and the initial weight vector
.64750 .47640 .84360 .58390 .23090 1w
Assuming 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 f v
e e
v v 2 1 1 ( ) as shown in Figure 1.2. Assign 1 to all augmented inputs. For the training task of this dichotomiser, the delta training rule is used with an arbitrary selection of learning constant .25 0 with the same initial weight vector 1 w in Question 1.1.
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Assuming that the above training set may need to be recycled if necessary, calculate the weight vector 7 w after one cycle and the weight vector 301 w after 50 cycles. Obtain the cycle error at the end of each cycle and plot the cycle error curve. How would the weight vectors 7 w and 301 w classify the entire training set? Discuss your results. [ 25 marks ]
Note: The following formulae may be used to calculate the pattern error curve and the cycle error curve. There are 6 patterns in this question, i.e. P=6.
Pattern error: 2 () 2 1 ppp zdE Cycle error: P p
p
P
p ppc EzdE 11 2() 2 1
Figure 1.1 Discrete Perceptron Classifier Training
Figure 1.2 Continuous Perceptron Classifier Training
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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 .755,0.750,0.745,0.740,0.735,0.730,0.7250X mach, and a fuzzy set M for the speed “near mach 0.74” where
and define a universe of altitudes as ,8650,8600,8550,8500,8450,84008350Y m, and a fuzzy set A for the altitude “approximately 8,500 m”, where
2.1.1 Construct the relation AM R [5 marks]
2.1.2 For another aircraft speed, say 1 M for the speed “in the region of mach 0.74” where
determine the corresponding altitude fuzzy set RMA 11 a using the maxmin composition. [5 marks]
2.1.3 For the speed “in the region of mach 0.74” where
determine the corresponding altitude fuzzy set RMA 11 b using the sumproduct composition. [5 marks]
8650 0
8600 .30
8550 .60
8500 1
8450 .60
8400 .30
8350 0
A
.7550 0
.7500 .250
.7450 .750
.7400 1
.7350 .750
.7300 .250
.7250 0
M
.7550 0
.7500 .20
.7450 .60
.7400 1
.7350 .80
.7300 .50
.7250 0
1M
.7550 0
.7500 .20
.7450 .60
.7400 1
.7350 .80
.7300 .50
.7250 0
1M
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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 GCE GU 1, 1, 1 .
2.2.1 If the Mean of Maximum (MOM) defuzzification strategy (sum-product inference) is used with the fire strength i of the i-th rule calculated from i E CE ii e ce ( ). ( ) calculate the defuzzified output voltages of this fuzzy controller at a particular instant. The error and the change of error at this instant are .20 3e and .470ce . [10 marks]
2.2.2 If the Centre of Area (COA) defuzzification strategy (max-min inference) is used with the fire strength i of the i-th rule calculated from i E CE ii e ce min( ( ), ( )) calculate the corresponding defuzzified output voltage at a particular instant when the error and the change of error are .20 3e and .47 0ce .
[25 marks]
Figure 2.1 Fuzzy logic control system
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Figure 2.2 Membership functions of a laser beam alignment system
Figure 2.3 Fuzzy Associative Memories
H. T. Nguyen March 2017
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MARKING SCHEME
Assignment 1: Neural Networks and Fuzzy Logic
Student Name: ____________________ Mark: ___________
Requirement Criteria Comment Standard “Declaration of Originality” cover page as provided by the Faculty At front of report, completed and signed Yes/no Question 1 Perceptron Dichotomiser 1.1 Discrete Perceptron Presentation Final weight vector Correct classification Pattern error curve Cycle error curve Calculation/software code /25
1.2 Continuous Perceptron
Presentation )7(W )301(W Cycle error curve Classification after nc=1 Classification after nc=50 Software code
/25
Question 2 2.1 Flight Simulation
Presentation Relation R=MxA Relation R=AoE (max-min) Relation R=AoE (sumproduct) Calculation/software code
/15
2.2 Laser Beam Alignment
Presentation (MOM) 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
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