Neural-Network-Based Fuzzy Logical Control and Decision System 主講人虞台文

Neural-Network-Based Fuzzy Logical Control and Decision System

主講人虞台文

Content Introduction Basic Structure of Fuzzy Systems Connectionist Fuzzy Logic Control and

Decision Systems Hybrid Learning Algorithm Example: Fuzzy Control of Unmanned Vehicle


Introduction

Reference

Chin-Teng Lin and C. S. George Lee, “Neural-network-based

fuzzy logic control and decision system,” IEEE Transactions

on Computers, Volume: 40 , Issue: 12 , Dec. 1991, Pages:132

0 – 1336.

Neural-Network & Fuzzy-Logic Systems

Neural-Network Systems– Highly connected PE’s (distributive representation)– Learning capability (Learning from examples)– Learning result is hardly interpretable– Efficient in pattern matching, but inefficient in computation

Fuzzy-Logic Systems– Inference based on human readable fuzzy rules– Linguistic-variable based fuzzy rules– Fuzzy rules from experienced engineers– Fuzzification before inference– Inference using compositional rule– Defuzzification before output

Neural-Network & Fuzzy-Logic Systems

Neural-Network Systems– Highly connected PE’s (distributive representation)– Learning capability (Learning from examples)– Learning result is hardly interpretable– Efficient in pattern matching, but inefficient in computation

Fuzzy-Logic Systems– Inference based on human readable fuzzy rules– Linguistic-variable based fuzzy rules– Fuzzy rules from experienced engineers– Fuzzification before inference– Inference using compositional rule– Defuzzification before output

The construction of fuzzy rule base & the

determination of membership functions are

subjective.

The construction of fuzzy rule base & the

determination of membership functions are

subjective.

Back-propagation learning algorithm is efficient if the appropriate network structure is used.However, the determination of the appropriate network structure is difficult.

Back-propagation learning algorithm is efficient if the appropriate network structure is used.However, the determination of the appropriate network structure is difficult.

Neuro-Fuzzy Systems

NeuralNetwork + Fuzzy

LogicGood for learning.

Not good for human to interpret its internal representation.

• Supervised leaning• Unsupervised learning• Reinforcement learning

Human reasoning scheme.

Fuzzy rules and membership functions are subjective.

• Readable Fuzzy rules• Interpretable

Neuro-Fuzzy Systems


LogicGood for learning.

Not good for human to interpret its internal representation.

• Supervised leaning• Unsupervised learning• Reinforcement learning

Human reasoning scheme.

Fuzzy rules and membership functions are subjective.

• Readable Fuzzy rules• Interpretable

A neuro-fuzzy system is a fuzzy system that uses a learn

ing algorithm derived from or inspired by neural netwo

rk theory to determine its parameters by processing da

ta samples.




ta samples.

Neuro-Fuzzy Systems


Logic




ta samples.




ta samples.

fuzzy sets and fuzzy rules

fuzzy sets and fuzzy rules


Basic Structure of Fuzzy Systems

Basic Structure of Fuzzy Systems

X Y( ) X ( ) YFuzzifierFuzzifier Inference

Engine

Inference Engine DefuzzifierDefuzzifier

Fuzzy Knowledge Base


Fuzzifier


Engine




FuzzifierFuzzifier

Converts the crisp input to a linguistic variable using the membership functions stored in the fuzzy knowledge base.

Inference Engine


Engine




Inference Engine

Inference Engine

Using If-Then type fuzzy rules converts the fuzzy input to

the fuzzy output.

Defuzzifier


Engine




Converts the fuzzy output of the inference engine to crisp using membership functions analogous to the ones used by the fuzzifier.

DefuzzifierDefuzzifier


Fuzzy Knowledge BaseFuzzy Knowledge Base




Engine


Information storage for1. Linguistic variables definitions.2. Fuzzy rules.

Input/Output Vectors


Engine




2

1, ,

1 1 2, , , ,, , , ,, i

i

i

i ii i i

kx

kx x xi

ixi

pxU M MT Tx MT

X

2

1, ,

21 1, , , ,, ,, , , i i

i ii ii iil

yi yq

yi

y yylT Ty MU T M M

Y

Fuzzy Rules

2

1, ,

1 1 2, , , ,, , , ,, i

i

i

i ii i i

kx

kx x xi

ixi

pxU M MT Tx MT

X

2

1, ,

21 1, , , ,, ,, , , i i

i ii ii iil

yi yq

yi

y yylT Ty MU T M M

Y

1 2MIMO MIMO MIMO, , , nR R R R

MIMO: multiinput and multioutput.

1

1

1

IMO 1M : and and

IF

THEN

is

is

and is and is

q

pxi

y q y

p x

y T y

T xR x T

T

1 1MIMO : qp yx

ix yT TTR T

Fuzzy Rules

1 2MIMO MIMO MIMO, , , nR R R R

MIMO: multiinput and multioutput.

1

1

1

IMO 1M : and and

IF

THEN

is

is

and is and is

q

pxi

y q y

p x

y T y

T xR x T

T

1 1MIMO : qp yx

ix yT TTR T

MIMO MISO

Fuzzy Reasoning

1 2

1 11 21

1 is is: IF TH and N is E yx xx T x y TR T

1 2

1 21 22


1 2

2 11 23


1 2

2 21 24


X DeffuzzifierDeffuzzifier y

Fuzzy Reasoning

1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24



Rule Firing Strengths

1

11( )xM x

1

11( )xM x

1

21( )xM x

1

21( )xM x

2

12( )xM x

2

22( )xM x

2

12( )xM x

2

22( )xM x

2

1, ,

1 1 2, , , ,, , , ,, i

i

i

i ii i i

kx

kx x xi

ixi

pxU M MT Tx MT

X

2

1, ,

21 1, , , ,, ,, , , i i

i ii ii iil

yi yq

yi

y yylT Ty MU T M M

Y

1 =

2 =

3 =

4 =

1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24



1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24



Fuzzy Sets of Decisions

1

11( )xM x

1

11( )xM x

1

21( )xM x

1

21( )xM x

2

12( )xM x

2

22( )xM x

2

12( )xM x

2

22( )xM x

2

1, ,

1 1 2, , , ,, , , ,, i

i

i

i ii i i

kx

kx x xi

ixi

pxU M MT Tx MT

X

2

1, ,

21 1, , , ,, ,, , , i i

i ii ii iil

yi yq

yi

y yylT Ty MU T M M

Y

1 =

2 =

3 =

4 =

1 ( )yM w

2 ( )yM w

3( )yM w

4 ( )yM w

1

2

3

4

1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24




1

11( )xM x

1

11( )xM x

1

21( )xM x

1

21( )xM x

2

12( )xM x

2

22( )xM x

2

12( )xM x

2

22( )xM x

2

1, ,

1 1 2, , , ,, , , ,, i

i

i

i ii i i

kx

kx x xi

ixi

pxU M MT Tx MT

X

2

1, ,

21 1, , , ,, ,, , , i i

i ii ii iil

yi yq

yi

y yylT Ty MU T M M

Y

1 =

2 =

3 =

4 =

1 ( )yM w

2 ( )yM w

3( )yM w

4 ( )yM w

1

2

3

4

11

1 ( ) ( )y yM w M w

22

2 ( ) ( )y yM w M w

33

3( ) ( )y yM w M w

44

4 ( ) ( )y yM w M w

1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24




11

1 ( ) ( )y yM w M w

22

2 ( ) ( )y yM w M w

33

3( ) ( )y yM w M w

44

4 ( ) ( )y yM w M w

1 2 3 4( ) ( ) ( ) ( ) ( )y y y y yM w M w M w M w M w

( )yM w

1 2

1 11 21


1 2

1 21 22


1 2

2 11 23


1 2

2 21 24



Defuzzification Decision Output

11

1 ( ) ( )y yM w M w

22

2 ( ) ( )y yM w M w

33

3( ) ( )y yM w M w

44

4 ( ) ( )y yM w M w

( )yM w

DeffuzzifierDeffuzzifier

or( )( )

( )( )

j y jy j

y jy j

w M wwM w dwy

M wM w dw

General Model of Fuzzy Controller and Decision Making System


Connectionist Fuzzy Logic Control and Decision Systems

The Architecture

Layer 1input

linguistic nodes

Layer 2input term nodes

Layer 3rule

nodes

Layer 4Output term node

Layer 5output

linguistic nodes

1x 2x nx

1y 1ymy ˆmy

1x 2x nx

1y 1ymy ˆmy

The Architecture

Layer 1input

linguistic nodes


Layer 3rule

nodes


Layer 5output

linguistic nodes

Fuzzifier

Inference Engine

Defuzzifier

1x 2x nx

1y 1ymy ˆmy

The Architecture

Layer 1input

linguistic nodes


Layer 3rule

nodes


Layer 5output

linguistic nodes Fully

Connected

FullyConnected

1x 2x nx

1y 1ymy ˆmy

The Architecture

Layer 1input

linguistic nodes


Layer 3rule

nodes


Layer 5output

linguistic nodes

antecedent

consquent

Basic Structure of Neurons

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

1 1,n ,et-input ; ),( ,k kp

k kpu wuf w

Layer k

output ( )kio a f

Layer 1 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

1if u

a f1x 2x nx

1y 1ymy ˆmy

1x 2x nx

1y 1ymy ˆmy

Layer 2 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

2 2

2

( )( , )

i

i ijjx ij ij

ij

u mf M m

fa ecenter

width

1x 2x nx

1y 1ymy ˆmy

Layer 3 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

3 3 31 2min , , , pf u u u

a f

1x 2x nx

1y 1ymy ˆmy

Layer 4 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

4 4

1

p

i ii

f w u

min(1, )a f

Down-Up Mode

{0, 1}

Layer 4 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

Up-Down Mode

5 2

2

( )( , )

i

i ijjy ij ij

ij

u mf M m

fa ecenter

width

1x 2x nx

1y 1ymy ˆmy

Layer 5 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

Up-Down Mode

if ya f

1x 2x nx

1y 1ymy ˆmy

Layer 5 Neurons()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

()f

()a

1ku 2

ku kpu

kio

1kw

2kw

kpw

Down-Up Mode

5

5

( ) ij

i

i ij

ij

umf

u

a f

1x 2x nx

1y ˆmy2y ˆmy


Hybrid Learning Algorithm

Initialization

1 2( ) ( ) ( )nT x T x T x rule nodes

1x 2x nx

1y 1y2y 2y

1x 2x nx

1y 1ymy ˆmy

Initialization

1 2( ) ( ) ( )nT x T x T x rule nodes

Two-Phase Learning Scheme

Self-Organized Learning Phase– Unsupervised learning of the membership func

tions.– Unsupervised learning of the rulebase.

Supervised Learning Phase– Error back-propagation for optimization of the

membership functions.

Unsupervised Learning of the Membership Functions

Step 1: First estimation of the membership function’s centers using Kohonen’s learning rule.

Step 2: The widths of the membership functions are estimated from the widths using a simple mathematical formula.

Note that the membership functions calculated are far from ideal but this is only a pre-estimation in order to create the rulebase.

1x 2x nx

1y 1ymy ˆmy

1x 2x nx

1y 1ymy ˆmy





1 | ( )|

( ) ( ) min ( ) ( )winner ii T x

x t m t x t m t

( 1) ( ) ( ) ( ) ( )winner winner winnerm t m t t x t m t

fo( r1) ( ) i i i winnerm t m t m m

Winner-take-all:

1x 2x nx

1y 1ymy ˆmy





1

22

1

1

2nearest

ni j

i j N i

m mE r

i closesti

m m

r

N-nearest-neighbors

Minimize

1-nearest-neighbors

r : overlay parameter

Unsupervised Learning of the Rulebase

Method:• Competitive Learning + Learn-if-win• Deletion of rule nodes• Combination of rule nodes

4 3( )ij j ij iw t o w o Learn-if-win:

1x 2x nx

1y 1ymy ˆmy

1x 2x nx

1y 1ymy ˆmy

Example of Combination of Rule Nodes

Supervise Learning Phase

Error back-propagation for optimization of the membership functions.

1x 2x nx

1y 1ymy

ˆmy 21

2ˆ( ) ( )E y t y t

Ew

w

( 1) ( )E

w t w tw

LearningRate



212

ˆ( ) ( )E y t y t Ew

w

E E f

w f w

( 1) ( )E

w t w tw

()f

()a

wHow w effects E?

How w effects f?

How f effects E?



212


w

E E f

w f w

( 1) ( )E

w t w tw

()f

()a

w

aE f

fa w

How f effects E?

How f effects

a?

How f effects

a?How a effects

E?

How a effects

E?

How w effects E?

How w effects f?



212


w

E E f

w f w

( 1) ( )E

w t w tw

()f

()a

w

aE f

fa w

error backpropagation

Learning Layer 5 Neurons

212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

y y

1x 2x nx

, iim 55 ˆa fy

55

5

( ) ii

i i

im uf

u

5

5

5

5i i

a

a

E E f

fm m

5

5

5

5i i

a

a

E E f

f

55 5

5

5

E E a

af f


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

5ˆ( ) ( )

a

Ey t y t

5

5

1a

f

55

5i

iii

iu

m

f

u

5

5

5 ( )i i

i

i

ii i

m u

u

a

5

5ˆ( ) ( ) ii

ii

uy t y t

u

5 5 5 5

25ˆ( ) ( )

i ii i

i

ii i

i

iu u u uy t y t

u

m m

ˆ( ) ( )y t y t

55 ˆa fy

55

5

( ) ii

i i

im uf

u

5

5

5

5i i

a

a

E E f

fm m

55 5

5

5

E E a

af f

5

5

5

5i i

a

a

E E f

f


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

y y

1x 2x nx

4 4 4

1

p

i j jj

f w u

4 4min(1, )iia f 55

ˆ( ) ( )E

y t y tf

5

5ˆ( ) ( )

Ey t y t

f

No need to learn.

No need to learn.4 {0,1}ijw

4 {0,1}ijw Error back-propagation only:

54

4 5 4ii i

E E f

f f f

5 5

4

4

4 4ii i

if f

af f

a

5

1 or 0

5iu

5 4

45i

ii

a

u

f

f


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

4 4 4

1

p

i j jj

f w u

4 4min(1, )iia f

Error back-propagation only:

5

5iu

1 or 0

5

55

( ) jj

j

j

j j

jm

uf

u

5

5

5

2

5

5

i j i j

j

j j

j

i jj j

ij

u

u

m

u

f m u

54

4 5 4ii i

E E f

f f f

5 5

4

4

4 4ii i

if f

af f

a

5 4

45i

ii

a

u

f

f

5

2

5

5

5

5

1

10

j jj ji j i ji j

jj j

u u

iu

i

m mu

u

5 5

52

5

5

5

0

1

1

i j i j

j

j jj j

j

i

j

ju u

iu

m m

i

u

u

1x 2x nx


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

55

4i

4i

3 3 3 3 31 2

3min , , , , jj p jf u au u f

No need to learn.

No need to learn.

3 {0,1}jkw 3 {0,1}jkw

Error back-propagation only:3

33 33

j

jj

j j

E E a

f a f

3 4j ja u

E E

1

4ju

44

4

40ij

i

w ji

f

u

E

f

14

i

4 {0,1}ijw 4 {0,1}ijw

4

4

0ij

iw

1x 2x nx


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

55

4i

4i

3j

3j

2

22 ( )k

kk

x mf

22 kf

ka e

, kkm

2

2

2

2k

k

k k k

ka

m m

fE E

fa

2

2

2

2k

k

k k k

ka f

fa

E E

3ku

2 3k ka u

E E

3 {0,1}jkw 3 {0,1}jkw

33

0jkw j

E

f

3

3

0jk

jw


212

ˆ( ) ( )E y t y t E E f

w f w

( 1) ( )E

w t w tw

aE f

fa w

2

2

2kf

k

kae

f

2

2

2( )k

k k

kf

m

x m

2

3

2( )2

k

k

k

kf x m

2

22 ( )k

kk

x mf

22 kf

ka e

2

2

2

2k

k

k k k

ka

m m

fE E

fa

2

2

2

2k

k

k k k

ka f

fa

E E

3ku

2 3k ka u

E E

33

0jkw j

E

f

3

3

0jk

jw

2

32

3

0

2( )k

jk

kfj

w k

xe

m

2

33

23

0

2( )k

jk

k

k

fj

w

x me


Example: Fuzzy Control of Unmanned Vehicle

The Fuzzy Car

0x 1x

2 ,x y

0x0x 1x1x

2 ,x y2 ,x y

The Fuzzy System Learned

0x0x 1x1x

2 ,x y2 ,x y

The Fuzzy Rules Learned

10 1

10

0

6

20

1IF and is is

is

and

TEHN

is

x xx

y

x T

y

xT

T

Tx10 1

10

0

6

20

1IF and is is

is

and

TEHN

is

x xx

y

x T

y

xT

T

Tx

The Membership Functions Learned

0x0x 1x1x

2 ,x y2 ,x y

The Membership Functions Learned

0x0x 1x1x

2 ,x y2 ,x y

Learning Curves

Learning rate 0.15

Error tolerance 0.01

Simulation

Documents

Neural-Network-Based Fuzzy Logical Control and Decision System 主講人 虞台文

Neural-Network-Based Fuzzy Logical Control and Decision System 主講人虞台文