PaperlayerView exact revision on arXiv

Source-linked AI summary

Recommendations on Designing Practical Interval Type-2 Fuzzy Systems

Dongrui Wu, Jerry Mendel

arXiv:1907.01697v1cs.AIcs.SYeess.SY

TL;DR

IT2 fuzzy-system design presents beginners with numerous choices, including output processing and type-reduction. The paper recommends practical starting choices, which it says have a high chance of producing systems that outperform T1 ones.

  • Problem

    IT2 design requires choices across fuzzification, membership functions, inference, t-norms, output processing, and optimization, with output processing and type-reduction especially difficult.

    Source: §I · p. 2

  • Method

    The authors recommend representative starting choices, including singleton fuzzification, simplest TSK rules, up to seven Gaussian FOUs per input, product t-norm, and specified output-processing and optimization approaches.

    Source: §IV · p. 13

  • Results

    The recommendations have a high chance of leading to IT2 fuzzy systems that can outperform T1 ones.

    Source: §IV · p. 13

  • Takeaways & Limitations

    The recommended choices are intended as starting points to lower learning barriers for IT2 beginners, not as universally best designs.

    Source: §I · p. 2, §IV · p. 13

    The authors do not claim their recommendations are always best; experienced researchers may design better systems using other choices.

    Source: §IV · p. 13

Abstract

from arXiv · show

Interval type-2 (IT2) fuzzy systems have become increasingly popular in the last 20 years. They have demonstrated superior performance in many applications. However, the operation of an IT2 fuzzy system is more complex than that of its type-1 counterpart. There are many questions to be answered in designing an IT2 fuzzy system: Should singleton or non-singleton fuzzifier be used? How many membership functions (MFs) should be used for each input? Should Gaussian or piecewise linear MFs be used? Should Mamdani or Takagi-Sugeno-Kang (TSK) inference be used? Should minimum or product $t$-norm be used? Should type-reduction be used or not? How to optimize the IT2 fuzzy system? These questions may look overwhelming and confusing to IT2 beginners. In this paper we recommend some representative starting choices for an IT2 fuzzy system design, which hopefully will make IT2 fuzzy systems more accessible to IT2 fuzzy system designers.

I. INTRODUCTION

IT2 fuzzy systems offer modeling and performance advantages, but their many design choices can overwhelm beginners. The paper recommends representative starting choices and explains how practical IT2 designs extend T1 system structures.

I. INTRODUCTION

  • Type-2 systems are argued to model membership-function uncertainty and words, and to outperform T1 systems in many applications.The authors’ design philosophy is to first build the best possible T1 system, moving to IT2 if it cannot perform satisfactorily.Source: §I · pp. 1–2
  • IT2 design involves many choices, especially in output processing and type-reduction, which can overwhelm beginners; the paper offers representative starting choices.The questions include fuzzifier, membership-function count and shape, inference, t-norm, and optimization choices.Source: §I · p. 2

II. CONSIDERATIONS FOR PRACTICAL T1 AND IT2 FUZZY SYSTEM DESIGNS

  • An IT2 system uses IT2 rather than T1 fuzzy sets in its rules and may add type-reduction before defuzzification, creating more design choices.Type-reduction reduces IT2 sets to T1 sets before defuzzification produces a crisp output.Source: §II · p. 2, §I · p. 2
  • The cited account links IT2 systems’ greater state-space sculpting than T1 systems—even at equal parameter counts—to their performance potential [36].Overlapping membership functions partition the input state space into nonlinear subsystems.Source: §II · pp. 2–3

A. Fuzzifier: Singleton or Non-Singleton

The section recommends practical choices for fuzzification and MF shape, balancing noise accommodation, simplicity, continuity, and ease of analysis. It favors singleton fuzzification and Gaussian MFs/FOUs as starting choices, with exceptions for noisy inputs and analysis needs.

A. Fuzzifier: Singleton or Non-Singleton

  • IT2 non-singleton fuzzifiers can output a T1 or IT2 fuzzy set, with IT2-valued fuzzification useful under non-stationary noise.Non-singleton fuzzifiers have performed better than singleton fuzzifiers in some such applications.Source: A. Fuzzifier: Singleton or Non-Singleton · p. 3
  • For very noisy measurements, consider non-singleton fuzzification; otherwise begin with singleton fuzzification in both T1 and IT2 systems for simplicity.The paper considers singleton systems in the sequel.Source: A. Fuzzifier: Singleton or Non-Singleton · p. 3

B. Rulebase: Gaussian or Piecewise Linear MFs?

  • T1 MF performance comparisons are application-dependent, so the authors do not prefer Gaussian or piecewise-linear shapes on performance grounds.Gaussian MFs always spread across input domains, while incomplete input-MF coverage can yield discontinuities.Source: B. Rulebase: Gaussian or Piecewise… · p. 4
  • Blurring both the mean and standard deviation creates a more general Gaussian FOU, but this approach is rarely used in practice.Source: B. Rulebase: Gaussian or Piecewise… · p. 4
  • Gaussian IT2 sets need three or four parameters, while piecewise-linear IT2 sets need nine; the Gaussian FOU with uncertain standard deviations may be easier to compute.That computational convenience does not establish better performance, and no comprehensive comparison is reported.Source: B. Rulebase: Gaussian or Piecewise… · pp. 4–5
  • With full UMF and LMF coverage, the example mapping is continuous; UMF-only coverage with an LMF gap yields jumps, while incomplete x2 UMF and LMF coverage yields gaps and jumps.The example uses the center-of-sets type-reducer.Source: B. Rulebase: Gaussian or Piecewise… · p. 5
  • For IT2 systems, Gaussian FOUs are recommended for simpler design, automatic continuity, and faster computation with small rulebases.Piecewise-linear FOUs are easier to analyze.Source: B. Rulebase: Gaussian or Piecewise… · p. 5

C. Rulebase: How Many MFs?

The number of MFs per input affects rulebase size, computation, and interpretability. The authors suggest using no more than seven MFs in each input domain.

C. Rulebase: How Many MFs?

  • Rule counts grow exponentially with the number of MFs per input, rapidly increasing computational cost.For two inputs with three MFs each, there are 9 rules; with nine MFs each, there are 81.Source: C. Rulebase: How Many MFs? · p. 5
  • As the number of rules increases, the interpretability advantage of fuzzy systems over black-box models diminishes.Source: C. Rulebase: How Many MFs? · p. 5
  • The authors suggest no more than 7 MFs per input domain to reduce computational cost and facilitate interpretation.They relate this threshold to the 7 ± 2 objects an average person can hold in working memory.Source: C. Rulebase: How Many MFs? · p. 6

D. Rulebase: Zadeh or TSK Rules

The section recommends simple TSK rule forms for both T1 and IT2 systems, while noting that IT2 ANFIS extensions have not yet produced an optimal solution with the cited type-reducer.

D. Rulebase: Zadeh or TSK Rules

  • For T1 systems, start with simplest TSK rules: under certain popular defuzzification methods, they are equivalent to Zadeh rules.TSK rules are described as more popular in practice for their simplicity and flexibility.Source: D. Rulebase: Zadeh or TSK Rules · p. 6
  • For IT2 systems, start with rule consequents that are functions of the inputs or single numbers, the simpler forms most popular in practice.The paper recommends these forms for their simplicity.Source: D. Rulebase: Zadeh or TSK Rules · p. 7
  • IT2 ANFIS extensions using the center-of-sets type-reducer had a switch-point mismatch and were not optimal; direct defuzzification is proposed as a possible remedy under investigation.The authors identify this as ongoing work.Source: D. Rulebase: Zadeh or TSK Rules · p. 7

E. Inference: Minimum or Product t-Norm

Minimum and product t-norms are both widely used in T1 and IT2 systems, with no evidence that one performs better. The authors recommend product for its computational and analytical convenience.

E. Inference: Minimum or Product t-Norm

  • Product t-norm is recommended for T1 and IT2 systems because gradient-based derivative computation and analytical-structure derivation are simpler.The minimum operation requires a test to select between its inputs.Source: E. Inference: Minimum or Product t-Norm · p. 7
  • Both minimum and product t-norms are widely used, and the paper reports no evidence that either is better.For IT2 systems, the corresponding t-norm is applied to LMFs for the left firing-interval endpoint and UMFs for the right endpoint.Source: E. Inference: Minimum or Product t-Norm · p. 7

F. Output Processing

Output computation depends on the rule type. The authors favor TSK-style weighted averages for T1 systems and a simplified TSK model with interval consequents for IT2 systems.

F. Output Processing

  • Mamdani T1 centroid defuzzification combines output fuzzy sets before finding their centroid; its union-computation complexity has significantly limited its adoption.Center-of-sets instead replaces each rule consequent with a crisp number before weighted averaging.Source: F. Output Processing · p. 7
  • For T1 systems, the authors recommend TSK rules with weighted-average output computation for simplicity and flexibility.The weights are the rules’ firing levels.Source: F. Output Processing · p. 7
  • For IT2 systems, the recommended approach is equivalent to center-of-sets type-reduction with each rule consequent replaced by its centroid, yielding simplified TSK interval consequents.The rule consequents are intervals [c_0, c0].Source: F. Output Processing · p. 8

1) Type-Reduction and Defuzzification:

The section describes type-reduction and defuzzification choices, recommends EIASC with language-dependent speed qualifications, and introduces direct defuzzification alternatives.

1) Type-Reduction and Defuzzification:

  • Center-of-sets type-reduction combines rule firing intervals and consequents into a T1 fuzzy set; the final defuzzified output follows once its endpoints are obtained.The paper identifies center-of-sets as the most commonly used type-reducer.Source: 1) Type-Reduction and Defuzzification · p. 8
  • EIASC is recommended for speed and simplicity, though the fastest algorithms differ by programming language.EIASC is fastest in Matlab, C, and Java; Enhanced KM and optimized DA* are fastest in R and Python.Source: 1) Type-Reduction and Defuzzification · p. 8

2) Direct Defuzzification:

  • Direct defuzzification bypasses type-reduction; the two most popular methods named are Nie-Tan and Begian-Melek-Mendel.The type-reduced set measures uncertainty propagated through computations but is almost never used in practical applications.Source: 2) Direct Defuzzification · p. 8

G. Optimization: Gradient-Based Methods or Evolutionary Computation Methods

Because manual tuning is difficult, the paper recommends evolutionary computation with a two-step strategy for IT2 optimization and gives practical starting choices for T1 and IT2 systems.

G. Optimization: Gradient-Based Methods or Evolutionary Computation Methods

  • Automatic optimization is usually needed because fuzzy systems have many parameters, and evolutionary computation is recommended for IT2 systems because derivatives can be difficult to compute.The difficulty is especially pronounced when LMF or UMF formulas test the location of their independent variable.Source: G. Optimization: Gradient-Based… · pp. 8–9
  • The recommended two-step strategy tunes an optimal T1 system first, then adds optimal FOUs; including the T1 system in the IT2 population guarantees performance at least as good as T1.The approach also reveals the IT2 system’s performance improvement over the T1 baseline.Source: G. Optimization: Gradient-Based… · p. 9
  • When the measured IT2 improvement is not significant, practitioners may prefer T1 for simplicity and speed.The paper recommends EC algorithms that can include the optimal T1 system in the population, such as genetic algorithms and particle swarm optimization.Source: G. Optimization: Gradient-Based… · p. 9

H. Summary

  • For T1 design, the authors recommend singleton TSK with at most seven Gaussian MFs per input, the product t-norm, and weighted-average defuzzification.They recommend ANFIS when applicable or EC when derivatives are difficult to compute.Source: H. Summary · p. 10
  • For practical IT2 design, the authors recommend singleton TSK, at most seven Gaussian FOUs per input, and the product t-norm.They recommend EC optimization using the two-step approach.Source: H. Summary · p. 10
  • IT2 output can be computed with EIASC for center-of-sets type-reduction followed by defuzzification, or directly with NT or BMM.The general procedure computes input membership intervals, rule firing intervals using the product t-norm, then combines firing intervals with rule consequents.Source: H. Summary · p. 10

I. Example

A PI fuzzy controller illustrates the recommended T1 and IT2 computations. Its T1 output and IT2 processing results are calculated, and the three IT2 output methods can yield different outputs.

I. Example

  • For input x′ = (˙e′, e′) = (−0.2, −0.3), the four T1 rule firing levels sum to −0.5491; controller output is ˙u = f1y1 + f2y2 + f3y3 + f4y4.Source: §I · p. 11
  • The T1 PI fuzzy controller’s Gaussian membership functions have centers at ±1 and standard deviations of 0.6; the IT2 example blurs that standard deviation to [0.5, 0.7].Source: §I · pp. 10–11
  • For the illustrated IT2 input, EIASC gives yl = −0.8846 and yr = 0.0058 before the final defuzzified output is computed.The four rule firing intervals are listed in Table VI.Source: §I · p. 11, Table VI · p. 12
  • The three IT2 output approaches produce different outputs for the same rules and MFs, and their optimal rules and MFs generally differ.The paper says designers should choose a method first and then tune rules and MFs for the chosen performance metric.Source: §I · p. 11

J. Software

The paper points beginners to Wu’s functions as a concise introduction to IT2 fuzzy systems by example.

J. Software

  • Wu’s functions are suggested as a beginner starting point because they concisely illustrate IT2 essentials through a simple example.The paper also names Mendel’s software, Taskin and Kumbasar’s open-source Matlab/Simulink toolbox, and Juzzy as available toolkits or functions.Source: J. Software · p. 12

III. MYTHS ABOUT IT2 FUZZY SYSTEMS

The paper addresses two myths about IT2 fuzzy-system performance, illustrating its clarifications with controllers and noting their broader applicability.

III. MYTHS ABOUT IT2 FUZZY SYSTEMS

  • The authors use IT2 fuzzy controllers to illustrate clarifications of two performance myths, which they say can extend to other IT2 applications.Source: §III · p. 12

A. Myth 1: Changing T1 Fuzzy Sets to IT2 Fuzzy Sets Automatically Improves Performance

IT2 systems do not automatically outperform T1 systems; the paper recommends beginner-oriented starting choices and stresses tuning across the operating conditions where a controller will be used.

A. Myth 1: Changing T1 Fuzzy Sets to IT2 Fuzzy Sets Automatically Improves Performance

  • Changing T1 fuzzy sets to IT2 fuzzy sets alone does not ensure better control; carefully designed FOUs and retuning may improve performance.Arbitrary FOUs almost never improve performance, and retuning can start from scratch or use a T1 controller as a baseline.Source: A. Myth 1: Changing T1 Fuzzy Sets to… · pp. 12–13

narios Guarantees Its Optimal Performance in an Unknown Scenario

  • Good performance across operating conditions requires considering a complete and diverse set of those conditions during controller tuning.The cited example reports consistent experimental and simulation results when the design included the relevant scenarios.Source: narios Guarantees Its Optimal… · p. 13

IV. CONCLUSIONS

  • The paper recommends singleton fuzzification, simplest TSK rules, ≤7 Gaussian FOUs per input, product t-norm, specified output processing, and two-step EC optimization as starting choices.Output processing can use EIASC for center-of-sets type-reduction followed by defuzzification, or NT or BMM directly.Source: §IV · p. 13
  • The authors say their recommendations have a high chance of producing IT2 systems that outperform T1 systems, while noting they are not always best.Experienced researchers may design better systems using other choices.Source: §IV · p. 13
Loading 1907.01697v1…