neuro-fuzzy control

Neuro-Fuzzy Control

Adriano Joaquim de Oliveira Cruz

NCE/UFRJ

[email protected]

*@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 2

Neuro-Fuzzy SystemsNeuro-Fuzzy Systems

Usual neural networks that simulate Usual neural networks that simulate fuzzy systemsfuzzy systems

Introducing fuzziness into neuronsIntroducing fuzziness into neurons


ANFIS architectureANFIS architecture

Adaptive Neuro Fuzzy Inference Adaptive Neuro Fuzzy Inference SystemSystem

Neural system that implements a Neural system that implements a Sugeno Fuzzy model.Sugeno Fuzzy model.


Sugeno Fuzzy ModelSugeno Fuzzy Model

A typical fuzzy rule in a Sugeno fuzzy model A typical fuzzy rule in a Sugeno fuzzy model has the formhas the form

If If xx is is AA and and yy is is BB then then zz = = f(x,y)f(x,y) AA and B and B are fuzzy sets in the antecedent.are fuzzy sets in the antecedent. z=f(x,y)z=f(x,y) is a crisp function in the consequent.is a crisp function in the consequent. Usually Usually zz is a polynomial in the input is a polynomial in the input

variables variables xx and and yy.. When When zz is a first-order polynomial the system is a first-order polynomial the system

is called a first-order Sugeno fuzzy model.is called a first-order Sugeno fuzzy model.


Sugeno Fuzzy ModelSugeno Fuzzy Model

x y

x y

w1

w2

A2

A1 B1

B2

z1=p1x+q1y+r1

z2=p2x+q2y+r2

21

2211

wwzwzw

z


Sugeno First Order ExampleSugeno First Order Example

If If xx is is smallsmall then then yy = 0.1 = 0.1xx + 6.4 + 6.4 If If xx is is medianmedian then then yy = -0.5 = -0.5xx + 4 + 4 If If xx is is largelarge then then y y = = xx – 2 – 2

Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Neuro-Fuzzy and Soft ComputingSoft Computing


Comparing Fuzzy and CrispComparing Fuzzy and Crisp

-10 -5 0 5 100

0.2

0.4

0.6

0.8

1

X

Mem

bers

hip

Gra

des small medium large

(a) Antecedent MFs for Crisp Rules

-10 -5 0 5 100

2

4

6

8

X

Y

(b) Overall I/O Curve for Crisp Rules

-10 -5 0 5 100

0.2

0.4

0.6

0.8

1

X

Mem

bers

hip

Gra

des small medium large

(c) Antecedent MFs for Fuzzy Rules

-10 -5 0 5 100

2

4

6

8

X

Y

(d) Overall I/O Curve for Fuzzy Rules


Sugeno Second Order ExampleSugeno Second Order Example

If If xx is is smallsmall and and yy is is smallsmall then then zz = - = -xx + + yy +1 +1 If If xx is is smallsmall and and yy is is largelarge then then zz = - = -yy + 3 + 3 If If xx is is largelarge and and y y is is smallsmall then z = - then z = -xx + 3 + 3 If If xx is is largelarge and and yy is is largelarge then then zz = = xx + + yy + 2 + 2

Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Neuro-Fuzzy and Soft ComputingSoft Computing


Membership FunctionsMembership Functions

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

X

Mem

bers

hip

Gra

des Small Large

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Y

Mem

bers

hip

Gra

des Small Large


Output SurfaceOutput Surface


ANFIS ArchitectureANFIS Architecture

Output of the Output of the iith node in the th node in the ll layer is layer is denoted as denoted as OOl,il,i

A1

A2

B1

B2

x

y

Layer 1 Layer 3Layer 2 Layer 4 Layer 5

x y

x yw1

w2

1w

2w

11 fw

22 fw

fO1,2


ANFIS Layer 1ANFIS Layer 1

Layer 1: Node function isLayer 1: Node function is

xx and and yy are inputs. are inputs. AAii and and BBii are labels (e.g. small, large). are labels (e.g. small, large). (x)(x) can be any parameterised membership can be any parameterised membership

function.function. These nodes are adaptive and the These nodes are adaptive and the

parameters are called premise parameters.parameters are called premise parameters.

4,3for)(

or2,1for),(

2,1

,1

iyO

ixO

i

i

Bi

Ai



Every node output in this layer is defined as:Every node output in this layer is defined as:

T is T-norm operator.T is T-norm operator. In general, any T-norm that perform fuzzy In general, any T-norm that perform fuzzy

AND can be used, for instance minimum and AND can be used, for instance minimum and product.product.

These are fixed nodes.These are fixed nodes.

2,1for),(T)(,2 iyxwOii BAii



The ith node calculates the ratio of the ith The ith node calculates the ratio of the ith rule’s firing strength to the sum of all rules’ rule’s firing strength to the sum of all rules’ firing strengthfiring strength

Outputs of this layer are called normalized Outputs of this layer are called normalized firing strengths.firing strengths.

These are fixed nodes.These are fixed nodes.

2,1for21

,3

iww

wwO iii



Every Every iith node in this layer is an adaptive th node in this layer is an adaptive node with the functionnode with the function

Outputs of this layer are called normalized Outputs of this layer are called normalized firing strengths.firing strengths.

ppii, , qqii and and rrii are the parameter set of this node are the parameter set of this node

and they are called consequent parameters.and they are called consequent parameters.

2,1for)(,4 iryqxpwfwO iiiiiii



The single node in this layer calculates The single node in this layer calculates the overall output as a summation of all the overall output as a summation of all incoming signals.incoming signals.

i

ii

iii

ii w

fwfwO 1,5



Every Every iith node in this layer is an th node in this layer is an adaptive node with the functionadaptive node with the function

Outputs of this layer are called Outputs of this layer are called normalized firing strengths.normalized firing strengths.

2,1for)(,4 iryqxpwfwO iiiiiii


Alternative StructuresAlternative Structures

The structure is not unique.The structure is not unique.

For instance layers 3 and 4 can be For instance layers 3 and 4 can be combinedcombined or w or weight normalisation can eight normalisation can be performed at the last layer.be performed at the last layer.


Alternative Structure cont.Alternative Structure cont.

A1

A2

B1

B2

x

y

w1

w2

fO1,2

11 fw

12 fw

ii fw

iw

Layer 1 Layer 2 Layer 5Layer 3

x y

x y

Layer 4


Training AlgorithmTraining Algorithm

The function f can be written asThe function f can be written as

There is a hybrid learning algorithm There is a hybrid learning algorithm based on the least-squares method and based on the least-squares method and gradient descent.gradient descent.

)()( 22221111

221

21

21

1

ryqxpwryqxpwf

fww

wf

www

f


Example Example

Modeling the functionModeling the function

Input range [-10,+10]x[-10,+10]Input range [-10,+10]x[-10,+10] 121 training data pairs121 training data pairs 16 rules, with four membership functions 16 rules, with four membership functions

assigned to each input.assigned to each input. Fitting parameters = 72; 24 premise and 48 Fitting parameters = 72; 24 premise and 48

consequent parameters.consequent parameters.

xyyx

z)sin()sin(


Initial and Final MFsInitial and Final MFs

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0

0.2

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0.8

1

input1

Deg

ree

of m

embe

rshi

p

Initial MFs on X

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0

0.2

0.4

0.6

0.8

1

input2

Deg

ree

of m

embe

rshi

p

Initial MFs on Y

-10 -5 0 5 10

0

0.2

0.4

0.6

0.8

1

input1

Deg

ree

of m

embe

rshi

p

Final MFs on X

-10 -5 0 5 10

0

0.2

0.4

0.6

0.8

1

input2

Deg

ree

of m

embe

rshi

p

Final MFs on Y


Training DataTraining Data

-100

10

-100

10

0

0.5

1

X

Training data

Y -100

10

-100

10

0

0.5

1

X

ANFIS Output

Y

0 50 1000

0.05

0.1

0.15

0.2

epoch number

root

mea

n sq

uare

d er

ror

error curve

0 50 1000.1

0.15

0.2

0.25

0.3

0.35

epoch number

step

siz

e

step size curve

neuro-fuzzy control

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