Fuzzy Basis Function, Universal Approximation and OLS Learning

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Fuzzy Basis Function, Universal Approximation and
OLS Learning

• to be presented in Fuzzy Systems Course by
• Neda Shahidi
• (n_shahidi_at_ce.sharif.edu)
• Sharif University of Technology
• Fall 2005

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Outline

• Linear Combinations of Fuzzy Basis Functions as
  Universal Approximators.
• Orthogonal Least Square Learning for Designing
  Fuzzy Systems based on Input-output Pairs.
• Design a Controller for Ball and Beam system

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Fuzzy System Parameter Tuning
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Why FBF?

• Linguistic fuzzy IF-THEN rules are naturally
  related to a FBF.
• Combining FBFs resulting from Linguistic fuzzy
  rules and FBFs resulting from input-output pairs.
• Linear learning algorithms such as OLS can be
  used instead of back-propagation.

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Fuzzy System

• Rj IF x1 is A1j and x2 is A2j andand xn is Anj,
  THEN z is Bj
• Specifications
• Multi input single output,
• Singleton fuzzyfiers, Gausian membership
  functions,
• Product inference, Centroid defuzzifiers

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Defining FBF
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FBF Properties

• Gaussian like FBF
• Sigmoidal like FBF

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Approximation

• Parameters

fixed
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Linear Regression Model
Experimental data
Linear regression model
Matrix form
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Linear Regression Model
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Gram-Schmidt Orthogonalization
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OLS (base)
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OLS (algorithm)

• For k1 to M do
• - Assume each input-output pare corresponds to
  one fuzzy rule, therefore P should have N
  columns w1 wN with initial value p1 pN
• - For each column, compute and
• - Select wk, the column with maximum error
  reduction ratio.
• - Orthogonalize wk respect to w1, , wk-1.
• end

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Ball and Beam System
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Controllers

• case 1 N200, M20
• case 2 N40, M20
• case 3 N40,M20

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Simulation Results
case 1 controller
case 2 controller
case 3 controller
A controller based on only linguistic rules
A controller uses input-output linearization
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References

• Wang L. X., Mendel J. M., Fuzzy Basis Functions,
  Universal Approximation, and Orthogonal
  Least-Square Learning, IEEE Trans. on Neural
  Networks, Vol. 3, No. 5, Sep. 92.
• Chen S., Cowan C.F.N., Grant P.M., Orthogonal
  Least Squares Learning Algorithm for Radial Basis
  Function Network, IEEE Trans. on Neural
  Networks, Vol. 2, No. 2, Mar. 91.
• Strang G., Linear Algebra and Its Applications,
  Academic Press Inc., New York, 1976.

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• Thank You
• This presentation is available at
• ce.sharif.edu/n_shahidi