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                 x 1  , x 2    , x 3   , x 4   , x 5   , x 6     0.0     0.3  0.8  0.5  0.3         0.9    0.3    0.2    0.7    0.6    0.1

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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 w 1 0.3296

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 logisticactivation function zf 2(v)1 e v as shown in Figure 1.2. Assign 1 to 1e  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 w 1  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 w 7  after one cycle and the weight vector w 301  after 50 cycles. Obtain the cycle error at the end of each cycle and plot the

cycle error curve. How would the weight vectors w 7  and w 301  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 PP  (dp  zp  )2 E  p  2  p 1p 1

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Figure 1.1  Discrete Perceptron Classifier Training

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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 ofaircraftspeedsnearthe speed ofsound as X 0.725,0.730,0.735,0.740,0.745,0.750,0.755mach, and a fuzzy setM  for the speed “near mach 0.74” where                   0  0.25 0.75  1  0.75 0.25 0   M                       0.7250.7300.7350.7400.7450.7500.755

and define a universe of altitudes asY 8350,8400,8450,8500,8550,8600,8650 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                 840084508500855086008650 8350        2.1.1 Construct the relation R  M A       [5 marks]

2.1.2 For another aircraft speed, sayM 1  for the speed “in the region of mach 0.74” where 0 0.5 0.8 1 0.6  0.2 0  M 1                 0.7250.7300.7350.7400.745 0.7500.755           determine the corresponding altitude fuzzy set A 1 a  M 1 R using the max- min composition.           [5 marks]

2.1.3 For the speed “in the region of mach 0.74” whereM 1  0 0.5 0.8 1 0.6 0.2 0                  0.7250.7300.7350.7400.7450.7500.755

determine the corresponding altitude fuzzy set A 1 b  M 1  R  using the sum-

product composition.

[5 marks]
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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 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

E i  ( e ) . CE i  ( 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( E i  ( e ), CE i  ( 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]

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Figure 2.1  Fuzzy logic control system
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Figure 2.2  Membership functions of a laser beam alignment system

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Figure 2.3  Fuzzy Associative Memories

H. T. Nguyen

March 2014

5MARKING SCHEME      Assignment 1:Neural Networks and Fuzzy Logic Student Name: ____________________  Mark: ___________        RequirementCriteria Comment  Standard “Declaration ofAt front of report, completed and Yes/n Originality” cover pagesigned   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     A 1 a  M 1  R (max-min)     Altitude Fuzz Set     A 1 b  M 1  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

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