Neuro-Fuzzy Control

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# Neuro-Fuzzy Control - PowerPoint PPT Presentation

Neuro-Fuzzy Control. Adriano Joaquim de Oliveira Cruz NCE/UFRJ adriano@nce.ufrj.br. 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

NCE/UFRJ

Neuro-Fuzzy Systems
• Usual neural networks that simulate fuzzy systems
• Introducing fuzziness into neurons
ANFIS architecture
• Adaptive Neuro Fuzzy Inference System
• Neural system that implements a 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 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
• 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
• 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

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
• 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
• 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
• 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
• 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
• The single node in this layer calculates the overall output as a summation of all incoming signals.
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
• The structure is not unique.
• For instance layers 3 and 4 can be combined or weight normalisation can be performed at the last layer.

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
• The function f can be written as
• There is a hybrid learning algorithm based on the least-squares method and gradient descent.
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.