10 - Sugeno-TSK Model

Fuzzy Rule-based Models

*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

A Classification of Fuzzy Rule-based models for function approximation

*Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999Fuzzy Rule-based ModelsNonAdditive Rule ModelsAdditive Rule ModelsMamdani Model(Mamdani)TSK Model(Takagi-Sugeno-Kang)Standard Additive Model (Kosko)Tsukamoto Model(Tsukamoto)

Mamdani model

*Fuzzy Logic - J.Yen, and R. Langari, Prentice Hall 1999Named after E.H. Mamdani who developed first fuzzy controller.

The inputs may be crisp or fuzzy numbers

Uses rules whose consequent is a fuzzy set, i.e. If x1 is Ai1 and and xn is Ain then y is Ci, where i=1,2 .M, M is the number of the fuzzy rules

Uses clipping inference

Uses max aggregation

Why TSK?Main motivationto reduce the number of rules required by the Mamdani modelFor complex and high-dimensional problemsdevelop a systematic approach to generate fuzzy rules from a given input-output data setTSK model replaces the fuzzy consequent, (then part), of Mamdani rule with function (equation) of the input variables*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

TSK Fuzzy Rule If x is A and y is B then z = f(x,y)Where A and B are fuzzy sets in the antecedent, andZ = f(x,y) is a crisp function in the consequence, e.g f(x,y)=ax+by+c.Usually f(x,y) is a polynomial in the input variables x and y, but it can be any function describe the output of the model within the fuzzy region specified by the antecedence of the rule.*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

First order TSK Fuzzy Model f(x,y) is a first order polynomial Example: a two-input one-output TSKIF x is Aj and y is Bk then zi= px+qy+r The degree the input matches ith rule is typically computed using min operator:wi = min(mAj(x), mBk(y)) *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

First-Order TSK Fuzzy Model (Cont) Each rule has a crisp outputOverall output is obtained via weighted average (reduce computation time of defuzzification required in a Mamdani model)z = Si wizi/ Si wiWhere Wi is matching degree of rule Ri (result of the if part evaluation)

To further reduce computation, weighted sum may be used, I.e. z = Si wizi


First-Order: TSK Fuzzy Model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Example #1: Single-input A single-input TSK fuzzy model can be expresses as

If X is small then Y = 0.1 X +6.4.If X is medium then Y = -0.5X +4.If X is large then Y = X-2.


Example #1: Non fuzzy rule set If small, medium. and large are non fuzzy sets , then the overall input-output curve is piecewise linear.*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Example #1: Fuzzy rule set If small, medium, and large are fuzzy sets (smooth membership functions) , then the overall input-output curve is a smooth one.*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Example #2 : Two-input A two-input TSK fuzzy model with 4 rules can be expresses as

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.


Example #2 : Two-input


Zero-order TSK Fuzzy Model When f is constant, we have a zero-order TSK fuzzy model (a special case of the Mamdani fuzzy inference system which each rules consequent is specified by a fuzzy singleton or a pre defuzzified consequent)Minimum computation time*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Summary: TSK Fuzzy Model Overall output via either weighted average or weithted sum is always crisp

Without the time-consuming defuzzification operation, the TSK (Sugeno) fuzzy model is by far the most popular candidate for sample-data-based fuzzy modeling.

Can describe a highly non-linear system using a small number of rules

Very well suited for adaptive learning.*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Tsukamoto Fuzzy ModelsThe consequent of each fuzzy if-then rule is represented by a fuzzy set with monotonical MFAs a result, the inferred output of each rule is defined as a crisp value induced by the rules firing strength.The overall output is taken as the weighted average of each rules output.*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Tsukamoto Fuzzy Models *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Example: Single-input Tsukamoto fuzzy model A single-input Tsukamoto fuzzy model can be expresses as

If X is small then Y is C1If X is medium then Y is C2If X is large then Y is C3


Example: Single-input Tsukamoto fuzzy model *Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Standard Additive Model (SAM) Introduced by Bart Kosko in 1996Efficient to computeSimilar to Mamdani model, butAssumes the inputs are crispUses the scaling inference method (prod.]Uses addition to combine the conclusions of rulesUses the centroid defuzzification technique


Standard Additive Model (SAM) IF x is Ai and y is Bi then z is Ci

then for crisp inputs x=x0 and y=y0

Z* = Centroid(Si mAi(x0) mBi(y0) mCi(z) )


Standard Additive Model (SAM) Z* = Centroid(Si mAi(x0) mBi(y0) mCi(z) )then Z* can be representedZ* = Si (mAi(x0) mBi(y0) ) Areai gi/Si (mAi(x) mBi(y) ) Areai

WhereAreai = mCi(z) dz, {Area of Ci}gi = z x mCi(z) dz / mCi(z) dz{Centroid of Ci}*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

Standard Additive Model (SAM) Main Advantage is the efficiency of its computation, i.e.

Both Areai and gi can be pre computed!*Neuro-fuzzy and Soft Computing - J.Jang, C. Sun, and, E. Mizutani, Prentice Hall 1997

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10 - Sugeno-TSK Model