Neural Networks And Fuzzy System

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32

ONE [ The Generalised XOR Problem ] [ 50 marks ]

A neural network model with two input neurons, three hidden neurons in a single

hidden layer, and an output neuron is used to learn the decision surface of the wellknown

generalised XOR problem:

sgn( ) 1 2 d ï€½ x x

The input range is [-1,1]. The desired output value is either –1 (corresponds to logical

zero) or 1 (corresponds to logical one).

This generalised XOR problem has been utilised recently to compare the

effectiveness of various neural network learning algorithms. A simplified version is

used in this question. Eight input samples 1 2 7 8 x ,x ,ïŒ,x ,x and their corresponding

target vectors 1 2 7 8 d ,d ,ïŒ,d ,d in the training set are:

ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ© ï€½ 0.5242

0.7826

x1 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©

ï€ ï€½ 0.5377

0.9003

x2 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©

ï€

ï€ ï€½ 0.9630

0.0871

x3 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©ï€ ï€½ 0.7873

0.1795

x4 ,

ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ© ï€½ 0.5839

0.2309

x5 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©

ï€ ï€½ 0.0280

0.2137

x6 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©

ï€

ï€ ï€½ 0.1886

0.6475

x7 , ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©ï€ ï€½ 0.2076

0.6026

x8

ï› ï1 d1 ï€½ , ï› ï1 d2 ï€½ ï€ , d3 ï€½ ï› ï1 , ï› 1ï d4

ï€½ ï€ , ï›1ï d5 ï€½ , ï› 1ï d6 ï€½ ï€ , d7 ï€½ ï› ï1 , d8

ï€½ ï› ï ï€1

Assume that the network as shown in Figure 1 has 3 hidden neurons, 1 output neuron,

and all continuous perceptrons use the bipolar activation function f e

e

2

1

1 ( ) ï®

ï®

ï® ï€½ ï€

ï€«

ï€

ï€ .

Note that due to the necessary augmentation of inputs and of the hidden layer by one

fixed input, the trained network should have 3 input nodes, 4 hidden neurons, and 1

output neuron. Assign -1 to all augmented inputs.

1.1 Assume that the learning constant is ï¨ ï€½ 0.2, and the initial random output

layer weight matrix W( ) 1 and hidden layer weight matrix W ( ) 1 are

W (1) ï€½ ï› ï 0.3443 0.6762 ï€ 0.9607 0.3626

ïƒº

ïƒº

ïƒº

ïƒ»

ïƒ¹

ïƒª

ïƒª

ïƒª

ïƒ«

ïƒ©

ï€

ï€ ï€

ï€ ï€

ï€½

0.0056 0.3908 0.3644

0.6636 0.1422 0.6131

0.2410 0.4189 0.6207

Using the error back propagation training, calculate the next weight updates

W W ( ), ( ) 2 2 .

[ 25 marks ]

1.2 The above training set was trained with the same set of initial random output

layer weight matrix W( ) 1 and hidden layer weight matrix W ( ) 1 as above, and

a learning constant of ï¨ ï€½ 0.2. The training set was recycled when necessary.

Determine the final weight matrices W W (4001) f ï€½ and W W (4001) f ï€½ after

500 cycles. Plot the cycle error curve for this training exercise.

[ 20 marks ]

Consider two particular input vectors of the test set

ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ©

ï€ ï€½ 0.9803

0.6263

xt1 and ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ© ï€½ 0.0500

0.0700

xt 2

Classify these two test input vectors using the final weight matrices and

discuss the results.

[ 5 marks ]

QUESTION TWO [Truck-Backer Upper Control] [ 50 marks ]

Backing up a truck to a loading dock is a nonlinear control problem. The truck and

loading zone are shown in Figure 2.1. The truck position is exactly determined by the

three state variables ï¦, , x y where ï¦ is the angle of the truck with the horizontal.

Control to the truck is the angle ï± .

Only backing up is considered. The truck moves backward by a fixed unit distance

every stage. For simplicity, assume that there is enough clearance between the truck

and the loading dock such that y does not have to be considered as an input. The task

here is to design a control system, whose inputs are ï¦ ïƒŽ ï€90 270 0 20 ï‚° ï‚°ïƒŽ ,, , x and

whose output is ï± ïƒŽï€ ï‚° ï‚° 40 40 , such that the final stages will be ï€¨ , ï€© ï€½ ï€¨ ï€© 10,90ï‚° f f x ï¦ .

The dynamics of the truck backer-upper procedure can be approximated by:

ï› ï ï› ï ï› ï

ï› ï ï› ïï› ï

ïƒº

ïƒ»

ïƒ¹ ïƒª

ïƒ«

ïƒ© ï€« ï€½ ï€

ï€« ï€½ ï€« ï€« ï€

ï€« ï€½ ï€« ï€« ï€«

ï€

b

k k k

y k y k k k k k

x k x k k k k k

2sin[ ( )] ( 1) ( ) sin

( 1) ( ) sin ( ) ( ) sin ( ) cos ( )

( 1) ( ) cos ( ) ( ) sin ( ) sin ( )

1 ï± ï¦ ï¦

ï¦ ï± ï± ï¦

ï¦ ï± ï± ï¦

where b is the length of the truck. Assume that b ï€½ 4.

Fuzzy logic is required for this truck backer-upper control. In this simple fuzzy logic

controller, a set of linguistic variables is chosen to represent 5 degrees of truck angle

ï€¨ ï€© ï¦ error ï› ï ï€ï‚° ï‚° ï‚° ï‚° ï‚° 90 70 90 110 270 ,,, , , 5 degrees of truck position ï€¨ ï€© x error

ï› ï 0 7 10 13 20 mm m m m ,, , , , and 5 degrees of control angle

ï€¨ï± ï€© ï›ï€ ï‚°ï€ ï‚° ï‚° ï‚° ï‚° 40 10 0 10 40 , ,, , ï

as shown in Figure 2.2. The generic rule set in the form of “Fuzzy Associative

Memories” is shown in Figure 2.3.

The initial states of this truck are assumed to be ) ( (1), x(1), y(1)) (35 ,15m,15m ï¯ ï¦ ï€½ .

2.1 If the Centre of Area (COA) defuzzification strategy is used with the fire

strength ï¡i of the i-th rule calculated from

ï¡ i XX ï ï i i ï€½ min( ( ), ( )) x x 1 2 1 2

determine the defuzzified control angle ï± (1) and the next state

[ï¦(2), x(2), y(2)].

[ 20 marks ]

If the Mean of Maximum (MOM) defuzzification strategy is used with the fire

strength ï¡i of the i-th rule calculated from

( ). ( ) 1 1 2 2 x x X i X i

ï¡i ï€½ ï ï

determine the defuzzified control angle ï± (1) and the next state

[ï¦(2), x(2), y(2)]. Then continue and calculate ï± (2) and [x(3),ï¦(3), y(3)].

Write a computer program to calculate the system state vector

[x(k ï€«1),ï¦(k ï€«1), y(k ï€«1)]

and the defuzzified control angle ï± (k) for 100

consecutive sampling points. Plot the corresponding vertical truck position

y(k) against the horizontal truck position x(k) for these 100 sampling points.

Plot the defuzzified control angle ï± (k) for these 100 sampling points.

[ 20 marks ]

Find the dominant rule which contributes the highest fire strength to the

control action for the defuzzified control angle ï±(1) . If softer control action

(for slower response) is required, modify this dominant rule and recalculate

the new defuzzified control angle ) (1 * ï± and the next state vector

[ï¦(2), x(2), y(2)]. Using the modified FAM table, plot the corresponding

vertical truck position y(k) against the horizontal truck position x(k) for

these 100 sampling points. Plot the new defuzzified control angle ) (

* ï± k for

these 100 sampling points.

[ 10 marks ]

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