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Neuro-Fuzzy Control. Adriano Joaquim de Oliveira Cruz NCE/UFRJ [email protected] Neuro-Fuzzy Systems. Usual neural networks that simulate fuzzy systems Introducing fuzziness into neurons. ANFIS architecture. Adaptive Neuro Fuzzy Inference System

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Neuro fuzzy control
Neuro-Fuzzy Control

Adriano Joaquim de Oliveira Cruz

NCE/UFRJ

[email protected]


Neuro fuzzy systems
Neuro-Fuzzy Systems

  • Usual neural networks that simulate fuzzy systems

  • Introducing fuzziness into neurons


Anfis architecture
ANFIS architecture

  • Adaptive Neuro Fuzzy Inference System

  • Neural system that implements a Sugeno Fuzzy model.


Sugeno fuzzy model
Sugeno Fuzzy Model

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

    If x is A and y is B then z = f(x,y)

  • A and B are fuzzy sets in the antecedent.

  • z=f(x,y)is a crisp function in the consequent.

  • Usually z is a polynomial in the input variables x and y.

  • When z is a first-order polynomial the system is called a first-order Sugeno fuzzy model.


Sugeno fuzzy model1
Sugeno Fuzzy Model

z1=p1x+q1y+r1

m

m

A1

B1

w1

y

x

m

m

B2

A2

w2

y

x

z2=p2x+q2y+r2


Sugeno first order example
Sugeno First Order Example

  • If x is small then y = 0.1x + 6.4

  • If x is median then y = -0.5x + 4

  • If x is large then y = x – 2

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



Sugeno second order example
Sugeno Second Order Example

  • If x is small and y is small then z = -x + y +1

  • If x is small and y is large then z = -y + 3

  • If x is large and y is small then z = -x + 3

  • If x is large and y is large then z = x + y + 2

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




Anfis architecture1

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

x

y

A1

w1

x

O1,2

P

N

A2

f

S

B1

P

N

y

w2

B2

x

y

ANFIS Architecture

  • Output of the ith node in the l layer is denoted as Ol,i


Anfis layer 1
ANFIS Layer 1

  • Layer 1: Node function is

  • x and y are inputs.

  • Ai and Bi are labels (e.g. small, large).

  • m(x) can be any parameterised membership function.

  • These nodes are adaptive and the parameters are called premise parameters.


Anfis layer 2
ANFIS Layer 2

  • Every node output in this layer is defined as:

  • T is T-norm operator.

  • In general, any T-norm that perform fuzzy AND can be used, for instance minimum and product.

  • These are fixed nodes.


Anfis layer 3
ANFIS Layer 3

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

  • Outputs of this layer are called normalized firing strengths.

  • These are fixed nodes.


Anfis layer 4
ANFIS Layer 4

  • Every ith node in this layer is an adaptive node with the function

  • Outputs of this layer are called normalized firing strengths.

  • pi, qi and ri are the parameter set of this node and they are called consequent parameters.


Anfis layer 5
ANFIS Layer 5

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


Anfis layer 51
ANFIS Layer 5

  • Every ith node in this layer is an adaptive node with the function

  • Outputs of this layer are called normalized firing strengths.


Alternative structures
Alternative Structures

  • The structure is not unique.

  • For instance layers 3 and 4 can be combined or weight normalisation can be performed at the last layer.


Alternative structure cont

S

/

Alternative Structure cont.

Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

x

y

A1

w1

S

x

P

A2

f

O1,2

B1

P

y

w2

B2

x

y


Training algorithm
Training Algorithm

  • The function f can be written as

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


Example
Example

  • Modeling the function

  • Input range [-10,+10]x[-10,+10]

  • 121 training data pairs

  • 16 rules, with four membership functions assigned to each input.

  • Fitting parameters = 72; 24 premise and 48 consequent parameters.




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