Takagi-Sugeno Fuzzy Inference - Parametric Fuzzy System -

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Takagi-Sugeno Fuzzy Inference- Parametric Fuzzy
System -

• Takagi and Sugeno introduced a new inference
  structure based on fuzzy sets theory. Such
  structure is either called a Takagi-Sugeno fuzzy
  inference system or a parametric fuzzy system. It
  has been demonstrated to function as a efficient
  model for systems that can be fully represented
  by their input / output relationships
• Like Mamdanis Rule-Based Fuzzy Systems,
  parametric fuzzy systems are also based on a rule
  base approach. But the rules consequents, instead
  of being formed by fuzzy relations, are linear
  parametric equations in terms of the inputs of
  the system.

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• Theoretically the consequents could be any
  function, even non-linear. However, linear
  functions have been employed most of the time.
  Adaptive training have been used for further
  non-linear capabilities
• Combination of a global rule-base description
  with local linear approximations by means of a
  linear regression model corresponding to a linear
  input-output model that one would use for
  describing the system locally.

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Parametric Fuzzy Controllers

• The parametric form of fuzzy rules has the
  following structure
• IF s1 S1i AND s2 S2 i THEN vouti a0i a1i
  s1a2i s2 api spi
• where si is an input variable vout is an output
  variable, Sji is a linguistic fuzzy membership
  function, and the coefficient set aji is the
  parameter set to be identified.

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• In the parametric method, the linear equation
  coefficients Aij are trained by the example data.
  This is comparable with the learning phase of a
  neural network !
• The linear equation outputs are then defuzzified,
  i.e., the weighted average of the consequents is
  evaluated by the respective membership values to
  determine the crisp output.

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• Example, consider two rules
• R1 IF x1 is BIG AND x2 is MEDIUM THEN u1
  x1-3x2
• R2 IF x1 is SMALL AND x2 is BIG THEN u2 4
  2x1
• mBIG(x1) 0.3 mBIG(x2) 0.35
• mSMALL(x1) 0.7 mMED(x2) 0.75
• Thus the weighted normalized sum one gets
• U 0.3(-176) 0.35(12)/(0.3 0.35) -74.77

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Power Electronics Problem

• Look at the current waveform for a three-phase
  rectifier and by measuring the width and height
  of the current pulses try to figure out the RMS
  total current value

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• Assume that a table of values, gained either from
  measurements or simulations, is available for a
  two-input (W and H) single-output (Is) system.
  The task is to find linear segments of the output
  function by fitting a straight line on those of
  its values which correspond to the fuzzy inputs
  defined by linguistic membership functions.

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• The fuzzy parametric estimation algorithm shown
  can be summarized as follows
• Read the parameters W and H for an operating
  condition.
• Convert W to a normalized value after dividing
  by the Wmax value to get W(pu).
• Identify the interval in which W(pu) lies.
• Fuzzify by calculating the degree of membership
  ?1 and ?2, as shown
• Fire the two relevant rules and calculate Is1 -
  Is2, If1 - If2 and DPF1 - DPF2 from the linear
  equations using the known W and H.
• Defuzzify the crisp output by the weighted
  average (C-o-A) method.

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Linear Filtering with Parametric Fuzzy Systems

• The beauty of Parametric Fuzzy Systems is the
  ability of representing very complex systems and
  to embedded linear controllers.
• Linear controllers are linear filters as

• This equation describes a filter of n-th order
  whose coefficients dependent on the state vector

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• The parametric fuzzy controller permits a smooth
  transition between the individual controllers
  with the following compact representation

• which expressed linguistically says that
• IF state vector x has the property Li
• THEN apply controller fi(x)

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Training Parametric Fuzzy Systems

• In order to train the parametric fuzzy system it
  is necessary to have input-output data sets for
  the system to be modeled.
• Fuzzy partitions are then created for the input
  variables of the data sets, i. e., fuzzy
  membership functions are defined to fill in the
  universe of discourse of the input variables.
• Then, the combination of these membership
  functions for each input variable form the
  antecedent part of the fuzzy rule base.

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• Equations below show the input-output expressions
  for a general parametric fuzzy system that uses a
  set of n fuzzy rules and k inputs.

where

• Given a set of input-output data represented by
  x1j, x2j,, xkj, yj, (j1,2,,m), the consequent
  parameters can be estimated by RLS

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• Let X (m x n.(k1) matrix), Y (m vector) and
• P (n.(k1) vector) be

• Then the parameter vector P can be calculated by

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• The parameter vector can be recursively estimated
  by a stable-state Kalman filter like

• where xi are lines of the matrix X. The initial
  values of P0 e S0 are set as follows, with ?
  being a large initial value and I the identity
  matrix.

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• The following Matlab programs were developed to
  implement Parametric Fuzzy Models
• Init_TSModel.m
• Adjust_TSModel.m
• Run_TSModel.m
• A test was made with a straight line data set
  (Adj1Reta.m) as well as a two straight lines data
  set (Adj2Reta.m).
• The TS system is configured with rectangular
  membership functions covering the entire input
  space, one at a time, only two rules and 4
  parameters. This will allow the system to learn
  the exact parameters of the straight lines used
  to generate the data sets.

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• Estimation of a system with two straight lines

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• Actual parameters
• a1 0.2
• b1 -5.0
• a2 -0.7
• b2 1.5
• Estimated Parameters
• a1 0.1727
• b1 -5.6977
• a2 -0.6289
• b2 1.6174

gt Discontinuity is very hard to
capture gt Convex membership functions will help
to smooth out the discontinuity gt System has
good convergence !
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Approximation of a Nonlinear Curve

• Function approximated
• Y x3 1
• Fuzzy partition with 6 terms, yielding 12
  parameters to be estimated.
• Average Error of 1 .

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Approximation of a Nonlinear Surface

• Function approximated
• Z 1 (x2 y2)
• Fuzzy partition with 10 terms, yielding 300
  parameters to be estimated.
• Average error of 2 .

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Performance Criteria

• Accuracy The accuracy of the parametric approach
  is generally superior to the rule based approach
  for the same number of rules. Of course, accuracy
  can be improved by a larger number of membership
  functions and a correspondingly larger number of
  rules.
• Response time One outstanding advantage of fuzzy
  estimation is a very fast that the response time
  is very fast when compared to conventional
  hardware or software estimation. This is because
  the fuzzy method tends to estimate the output
  instantaneously from the input pattern. The
  reason is that fuzzy systems are memoryless
  input-output mapping systems and pattern
  recognizers like a neural network.
• Robustness Robustness means the systems
  relative insensitivity to both external and
  internal disturbances. The fact that a system can
  be updated regularly on line in real time assures
  that the system is rapidly adapted to the latest
  changes that occurred.

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Comparison between rule based and parametric
fuzzy approaches

• Rule based fuzzy approach is more suitable for
  acquiring and implementing expert human operator
  knowledge, while the parametric fuzzy approach is
  best used when input/output numerical data are
  available.
• Parametric fuzzy approach yields a better
  estimation accuracy because it is a hybrid of
  rule based fuzzy and numerical components. The
  rule based fuzzy approach requires no training,
  while the parametric fuzzy approach requires
  linear coefficient adjustment performed by
  statistical multi-linear procedures.

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• Parametric fuzzy algorithm is inherently
  adaptive, because the coefficients Aij can be
  altered for system tuning. Thus a real-time
  adaptive implementation of the parametric
  approach is feasible by dynamically changing the
  linear coefficients by means of a recursive
  least-square algorithm repeatedly on a recurrent
  basis
• Adaptive versions of the rule-based approach ,
  changing the rule weights (Degree of Support) or
  the membership functions recurrently is possible.
  
• Disadvantage of the parametric fuzzy approach is
  the loss of the linguistic formulation of output
  consequents, sometimes important for industrial
  plant/process control environment