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## PowerPoint Slideshow about ' Neuro-Fuzzy Control' - melina

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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 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 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.

/

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.

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