Neuro-Fuzzy Control

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

• Usual neural networks that simulate fuzzy systems
• Introducing fuzziness into neurons

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ANFIS architecture

• Adaptive Neuro Fuzzy Inference System
• Neural system that implements a Sugeno Fuzzy
  model.

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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.
• zf(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.

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Sugeno Fuzzy Model
z1p1xq1yr1
m
m
A1
B1
w1
y
x
m
m
B2
A2
w2
y
x
z2p2xq2yr2
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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

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Comparing Fuzzy and Crisp
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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

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Membership Functions
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Output Surface
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ANFIS Architecture

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

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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.

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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.

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ANFIS Layer 3

• The ith node calculates the ratio of the ith
  rules firing strength to the sum of all rules
  firing strength
• Outputs of this layer are called normalized
  firing strengths.
• These are fixed nodes.

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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.

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ANFIS Layer 5

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

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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.

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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.

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Alternative Structure cont.
Layer 1
Layer 2
Layer 5
Layer 3
Layer 4
x
y
A1
w1
S
x
P
A2
f
O1,2
B1
P
y
w2
B2
x
y
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Training Algorithm

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

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Example

• Modeling the function
• Input range -10,10x-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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Initial and Final MFs
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Training Data