436462 AIM, Sem2 2005

1
436-462 AIM, Sem2 2005

• Fuzzy Systems
• Prepared by Nalin Wickramarachchi and Saman
  Halgamuge
• Presenter Saman K. Halgamuge

2
Fuzzy Inference Systems (FIS)

• Inspiration lexical imprecision in natural
  language reasoning

price of crude oil which has edged higher in
recent weeks after being remarkably stable
through much of the year, may fluctuate as much
as a dollar a barrel in the months ahead, but
abrupt changes are not likely, many analysts
believe.

• Almost all our everyday reasoning is approximate
  in nature.

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FIS Inspiration

• Exploit the tolerance for imprecision.
• High precision entails high cost.
• park the car
• park the car 10cm from the curb
• High precision entails low tractability
• reduce the precision of information to make a
  complex problem more tractable

4
FIS Applications

• Replacement of human operator by a FIS
• Sendai subway (Hitachi), Elevator control
  (Hitachi, Toshiba)
• Nuclear reactor control (Hitachi)
• Automobile transmission (Nissan, Subaru, Honda)
• Video image stabilisation (Canon, Minolta)
• Replacement of human expert by a FIS
• medical diagnosis
• Securities
• Fault diagnosis
• Credit worthiness

5
Fuzzy Sets and Fuzzy Logic

• A fuzzy set is an extension of an ordinary
  (crisp) set.
• Fuzzy set allows partial membership an element
  may partially belong to a set.
• The membership of elements in a crisp set can be
  described by the characteristic function ?A(x)
  where

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Fuzzy Sets and Fuzzy Logic
?
?
1.0
1.0
0
0
x
x
X
X
A
A
Crisp set A
Fuzzy set A
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Fuzzy Sets and Fuzzy Logic

• Fuzzy set A is characterised by its membership
  function ?A(x)
• The universe of discourse X may be continuous or
  discrete.
• Example young people

8
Fuzzy Linguistic Terms
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Fuzzy Sets and Fuzzy Logic

• Membership function
• can take a variety of functional forms
  triangular, trapezoidal, Gaussian, bell shaped
• usually subjective in design no theoretical
  basis for its shape
• so, need to be fine tuned on trial and error
  basis when used in inference systems
• integrated neuro-fuzzy systems can automate the
  fine-tuning process

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Fuzzy Operations

• Union operation (OR)
• Intersection operation (AND)
• Complement operation (NOT)

A
B
A
B
A
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Fuzzy Rule Base

• Fuzzy rules express knowledge
• if X is small and Y is large then Z is medium
• if X is large and Y is not very small then Z is
  low
• if speed is medium and distance is large then
  brake pressure is medium
• if A is very large and B is nearly zero then
  Cp1Ap2Bp0

12
Fuzzy Rule Base

• Fuzzy rules can be formulated
• from human experts knowledge or experience
• by statistical analysis of numerical data
  obtained from experimentation
• through neuro-fuzzy optimisation (learning)
  process ANFIS (adaptive neuro-fuzzy inference
  system), FuNe (neuro-fuzzy learning network with
  rule generation)

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Example Canon Auto Focus

• Camera measures the distance to 3 spots in the
  view frame to discern where the object of
  interest is located.
• Px possibility of object of interest is at X.

L
C
R
if C is near then PC is high if L is near then PL
is high if R is near then PR is high if L is far
and C is medium and R is near then PC is high if
R is far and C is medium and L is near then PC is
high
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FIS
Defuzzifier
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FIS Mamdani Procedure

• ith rule if x1 is A1i and and xn is Ani then y
  is Bi
• Determine the degree of membership of each input
  to different fuzzy terms Aji
• Determine the strength of each rule antecedent
• Determine the contribution of each rule
• Rule aggregation
• Defuzzification

j runs on each fuzzy term i runs on each rule
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Mamdani Fuzzy Inference

• Single rule with single antecedent
• Rule if x is A then y is B
• Fact x is A
• Inference y is B
• Graphical Representation

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Mamdani Fuzzy Inference

• Single rule with multiple antecedent
• Rule if x1 is A and x2 is B then y is C
• Fact x1 is A and x2 is B
• Inference y is C
• Graphical Representation

A
A
C
B
B
?
X1
Y
X2
min
A
B
C
Y
X1
X2
x1 is A
Y is C
X2 is B
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Mamdani Fuzzy Inference

• Multiple rules with multiple antecedent
• Rule 1 if x1 is A1 and x2 is B1 then y is C1
• Rule 2 if x1 is A2 and x2 is B2 then y is C2
• Fact x1 is A and x2 is B
• Inference y is C
• Graphical Representation (next slide)
• not all rules will produce a non-zero output
• in practice, only a few rules will fire (give
  non-zero output) out of many rules in the rule
  base.

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Mamdani Fuzzy Inference

• Graphical Representation

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Defuzzification

• Produces a crisp output that best represents the
  output fuzzy set (C) from the FIS.

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Takagi-Sugeno (TS) FIS

• Instead of fuzzy consequent, TS type rules have
  crisp output as a function of inputs.
• Powerful and easy to use for process control
  applications.
• TS-FIS fuse local system models together by
  interpolating between them.
• TS-FIS can be easily optimised.

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Takagi-Sugeno (TS) FIS

• Rule i if x1 is A1 and and xn is An then y is
  bi where
• Step 1 and 2 are the same as Mamdani method
• Step 3 not required (contribution is bi)
• Step 4 (rule aggregation) gives crisp output

1st order TS-FIS
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FIS Optimisation

• Why optimise?
• trial and error tuning is laborious
• it can be impossibly complicated if number of
  input parameters are large (100s or more)
• far too many parameters to tune number of rules,
  membership functions, rule consequents
• arbitrariness of FIS is eliminated
• if not optimised, the FIS performance is not
  optimum

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FIS Optimisation

• Optimisation methods
• Adaptive Neuro-Fuzzy systems
• multilayer perceptron neural networks
  (ANFIS,FuNeI)
• RBFN
• evolutionary techniques (genetic algorithms)
• clustering methods (cluster analysis, Hard
  C-means, Fuzzy C-means)

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FIS Optimisation

• Optimisation can be viewed as knowledge
  discovery.
• Optimisation Learning
• Resulting model describes input-output
  relationships qualitatively and quantitatively.
• Easy validation and verification of FIS

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Adaptive Neuro-Fuzzy Systems

• Introduction
• Neurons that can justify their actions !
• Artificial brains that can grow !
• Synergistic Combination Neuro-Fuzzy
• Functional Equivalence
• Structure Adapting Neuro-Fuzzy
• Multi Layer Perceptron Neuro-Fuzzy
• Radial Basis Function Neuro-Fuzzy

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Directed Object Refinement

• Finding structures in data with class labels
• traffic sign recognition
• robot path selection
• protein motif classification
• Learning from available data
• Interface with expert view on input data
• Aim association of new data to class labels

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Classifier type Fuzzy System
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Generalised Fuzzy Systems
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References

• Fuzzy Fundamentals, Earl Cox, pp 58-61, IEEE
  Spectrum October 1992
• Introduction to Computational Intelligence Use
  of Neural Networks in Building Fuzzy Controllers,
  Saman Halgamuge, (Subject web site)
• Inverted Pendulum Multimedia Software associated
  with Introduction to Fuzzy Systems, Bart Kosko
• Neural Networks in Designing Fuzzy Systems for
  Real World Applications, Saman Halgamuge, FSS
  1994, (Subject Web site)
• My contact saman_at_unimelb.edu.au