Twelve considerations in choosing between Gaussian and trapezoidal membership functions in interval type-2 fuzzy logic controllers

Interval type-2 fuzzy logic controllers (IT2 FLCs) have been attracting great research interests recently. There are many decisions to be made in designing an IT2 FLC. One of them is to determine which membership function type to use, e.g., Gaussian or trapezoidal. There have not been comprehensive studies on this problem so far. In this paper we present 12 considerations in choosing between Gaussian and trapezoidal membership functions for an IT2 FLC, including representation, construction, optimization, adaptiveness, novelty, analytical structure, continuity, monotonicity, stability, robustness, computational cost, and control performance. It can help practitioners select the appropriate membership function type in IT2 FLC design, and researchers identify new research opportunities on IT2 FLCs. Our study shows that each MF type has its own advantages: Gaussian IT2 FLCs are simpler in design because they are easier to represent and optimize, always continuous, and faster for small rulebases, whereas trapezoidal IT2 FLCs are simpler in analysis.

[1]  Jerry M. Mendel,et al.  Stability analysis of type-2 fuzzy systems , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[2]  Dongrui Wu,et al.  Type-2 FLS Modeling Capability Analysis , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[3]  Dongrui Wu,et al.  Comparison and practical implementation of type-reduction algorithms for type-2 fuzzy sets and systems , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[4]  Y. Wang,et al.  A Type-2 Fuzzy Switching Control System for Biped Robots , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[5]  Jianqiang Yi,et al.  On the properties of SIRMs connected type-1 and type-2 fuzzy inference systems , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[6]  Jianqiang Yi,et al.  Stability analysis of SIRMs based type-2 fuzzy logic control systems , 2010, International Conference on Fuzzy Systems.

[7]  Jin S. Lee,et al.  Parameter conditions for monotonic Takagi-Sugeno-Kang fuzzy system , 2002, Fuzzy Sets Syst..

[8]  Dongrui Wu,et al.  Approaches for Reducing the Computational Cost of Interval Type-2 Fuzzy Logic Systems: Overview and Comparisons , 2013, IEEE Transactions on Fuzzy Systems.

[9]  H. Ying.  General interval type-2 Mamdani fuzzy systems are universal approximators , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[10]  M. Melgarejo,et al.  A Fast Recursive Method to Compute the Generalized Centroid of an Interval Type-2 Fuzzy Set , 2007, NAFIPS 2007 - 2007 Annual Meeting of the North American Fuzzy Information Processing Society.

[11]  Jianqiang Yi,et al.  Analysis and design of monotonic type-2 fuzzy inference systems , 2009, 2009 IEEE International Conference on Fuzzy Systems.

[12]  Chi-Hsu Wang,et al.  Dynamical optimal training for interval type-2 fuzzy neural network (T2FNN) , 2003, SMC'03 Conference Proceedings. 2003 IEEE International Conference on Systems, Man and Cybernetics. Conference Theme - System Security and Assurance (Cat. No.03CH37483).

[13]  Dongrui Wu,et al.  Computationally Efficient Type-Reduction Strategies for a Type-2 Fuzzy Logic Controller , 2005, The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ '05..

[14]  Jerry M. Mendel,et al.  On the Stability of Interval Type-2 TSK Fuzzy Logic Control Systems , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[15]  Fatah Chetouane,et al.  Fuzzy Control of a Gyroscopic Inverted Pendulum , 2010 .

[16]  H. Hagras,et al.  Type-2 FLCs: A New Generation of Fuzzy Controllers , 2007, IEEE Computational Intelligence Magazine.

[17]  Neeraj Gupta,et al.  Comparative Analysis of Fuzzy Power System Stabilizer Using Different Membership Functions , 2010 .

[18]  B. Bose,et al.  Evaluation of membership functions for fuzzy logic controlled induction motor drive , 2002, IEEE 2002 28th Annual Conference of the Industrial Electronics Society. IECON 02.

[19]  Oscar Castillo,et al.  Systematic design of a stable type-2 fuzzy logic controller , 2008, Appl. Soft Comput..

[20]  W.W. Tan,et al.  A simplified architecture for type-2 FLSs and its application to nonlinear control , 2004, IEEE Conference on Cybernetics and Intelligent Systems, 2004..

[21]  Dongrui Wu.  An overview of alternative type-reduction approaches for reducing the computational cost of interval type-2 fuzzy logic controllers , 2012, 2012 IEEE International Conference on Fuzzy Systems.

[22]  E. H. Mamdani,et al.  An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller , 1999, Int. J. Man Mach. Stud..

[23]  Dongrui Wu,et al.  GENETIC LEARNING AND PERFORMANCE EVALUATION OF TYPE-2 FUZZY LOGIC CONTROLLERS , 2006 .

[24]  Hani Hagras,et al.  Interval Type-2 Fuzzy Logic Congestion Control for Video Streaming Across IP Networks , 2009, IEEE Transactions on Fuzzy Systems.

[25]  F. Chung-Hoon Rhee.  Uncertain Fuzzy Clustering: Insights and Recommendations , 2007 .

[26]  Ebrahim Mamdani,et al.  Applications of fuzzy algorithms for control of a simple dynamic plant , 1974 .

[27]  M. Sami Fadali,et al.  Stability Analysis and Control of Discrete Type-1 and Type-2 TSK Fuzzy Systems: Part II. Control Design , 2011, IEEE Transactions on Fuzzy Systems.

[28]  Yuanli Cai,et al.  Advantages of the Enhanced Opposite Direction Searching Algorithm for Computing the Centroid of An Interval Type‐2 Fuzzy Set , 2012 .

[29]  Chee Peng Lim,et al.  Optimization of Gaussian fuzzy membership functions and evaluation of the monotonicity property of Fuzzy Inference Systems , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[30]  Ioan Dumitrache,et al.  FUZZY CONTROL OF AUTONOMOUS MOBILE ROBOT , 2010 .

[31]  Jerry M. Mendel,et al.  On the robustness of Type-1 and Interval Type-2 fuzzy logic systems in modeling , 2011, Inf. Sci..

[32]  Jerry M. Mendel,et al.  Type-2 fuzzy logic systems , 1999, IEEE Trans. Fuzzy Syst..

[33]  Jerry M. Mendel,et al.  Perceptual Computing: Aiding People in Making Subjective Judgments , 2010 .

[34]  Robert Ivor John,et al.  The collapsing method of defuzzification for discretised interval type-2 fuzzy sets , 2009, Inf. Sci..

[35]  Jerry M. Mendel,et al.  Connection admission control in ATM networks using survey-based type-2 fuzzy logic systems , 2000, IEEE Trans. Syst. Man Cybern. Part C.

[36]  Jerry M. Mendel,et al.  Enhanced Karnik--Mendel Algorithms , 2009, IEEE Transactions on Fuzzy Systems.

[37]  D Wu,et al.  AN INTERVAL TYPE-2 FUZZY LOGIC SYSTEM CANNOT BE IMPLEMENTED BY TRADITIONAL TYPE-1 FUZZY LOGIC SYSTEM , 2011 .

[38]  M. Gorzałczany,et al.  Decision making in signal transmission problems with interval-valued fuzzy sets , 1987 .

[39]  Mohammad Biglarbegian,et al.  Robustness of interval type-2 fuzzy logic systems , 2010, 2010 Annual Meeting of the North American Fuzzy Information Processing Society.

[40]  Hak-Keung Lam,et al.  Stability Analysis of Interval Type-2 Fuzzy-Model-Based Control Systems , 2008, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[41]  Woei Wan Tan,et al.  A simplified type-2 fuzzy logic controller for real-time control. , 2006, ISA transactions.

[42]  Gerardo M. Mendez,et al.  Hybrid learning for interval type-2 fuzzy logic systems based on orthogonal least-squares and back-propagation methods , 2009, Inf. Sci..

[43]  M. Sami Fadali,et al.  Stability Analysis and Control of Discrete Type-1 and Type-2 TSK Fuzzy Systems: Part I. Stability Analysis , 2011, IEEE Transactions on Fuzzy Systems.

[44]  Jia Zeng,et al.  Type-2 fuzzy hidden Markov models and their application to speech recognition , 2006, IEEE Transactions on Fuzzy Systems.

[45]  Dongrui Wu,et al.  A type-2 fuzzy logic controller for the liquid-level process , 2004, 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542).

[46]  Juan Luis Castro,et al.  Fuzzy logic controllers are universal approximators , 1995, IEEE Trans. Syst. Man Cybern..

[47]  Milos Manic,et al.  Uncertainty-Robust Design of Interval Type-2 Fuzzy Logic Controller for Delta Parallel Robot , 2011, IEEE Transactions on Industrial Informatics.

[48]  Woei Wan Tan,et al.  Towards an efficient type-reduction method for interval type-2 fuzzy logic systems , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[49]  Haibo Zhou,et al.  Deriving the input-output mathematical relationship for a class of interval type-2 mamdani fuzzy controllers , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[50]  Yeong-Hwa Chang,et al.  Simplified type-2 fuzzy sliding controller for wing rock system , 2012, Fuzzy Sets Syst..

[51]  J Gayathri Monicka,et al.  Performance Evaluation of Membership Functions on Fuzzy Logic Controlled AC Voltage Controller for Speed Control of Induction Motor Drive , 2011 .

[52]  Shie-Jue Lee,et al.  An Enhanced Type-Reduction Algorithm for Type-2 Fuzzy Sets , 2011, IEEE Transactions on Fuzzy Systems.

[53]  Jerry M. Mendel,et al.  Encoding Words Into Interval Type-2 Fuzzy Sets Using an Interval Approach , 2008, IEEE Transactions on Fuzzy Systems.

[54]  Witold Pedrycz,et al.  Type-2 Fuzzy Logic: Theory and Applications , 2007, 2007 IEEE International Conference on Granular Computing (GRC 2007).

[55]  Woei Wan Tan,et al.  Analytical Structure and Characteristics of Symmetric Karnik–Mendel Type-Reduced Interval Type-2 Fuzzy PI and PD Controllers , 2012, IEEE Transactions on Fuzzy Systems.

[56]  Jianqiang Yi,et al.  The monotonicity and convexity of unnormalized interval type-2 TSK Fuzzy Logic Systems , 2010, International Conference on Fuzzy Systems.

[57]  Dongrui Wu,et al.  Interval Type-2 Fuzzy PI Controllers: Why They are More Robust , 2010, 2010 IEEE International Conference on Granular Computing.

[58]  Jerry M. Mendel,et al.  On the Continuity of Type-1 and Interval Type-2 Fuzzy Logic Systems , 2011, IEEE Transactions on Fuzzy Systems.

[59]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[60]  Robert Ivor John,et al.  Geometric Type-1 and Type-2 Fuzzy Logic Systems , 2007, IEEE Transactions on Fuzzy Systems.

[61]  Woei Wan Tan,et al.  Derivation of the analytical structure of symmetrical IT2 fuzzy PD and PI controllers , 2010, International Conference on Fuzzy Systems.

[62]  Mohammad Narimani,et al.  LMI-based stability conditions for interval type-2 fuzzy-model-based control systems , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[63]  Hao Ying,et al.  Derivation and Analysis of the Analytical Structures of the Interval Type-2 Fuzzy-PI and PD Controllers , 2010, IEEE Transactions on Fuzzy Systems.

[64]  Hiroaki Ishii,et al.  On the Monotonicity of Fuzzy-Inference Methods Related to T–S Inference Method , 2010, IEEE Transactions on Fuzzy Systems.

[65]  Dongrui Wu,et al.  Genetic learning and performance evaluation of interval type-2 fuzzy logic controllers , 2006, Eng. Appl. Artif. Intell..

[66]  Hao Ying,et al.  Interval Type-2 Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators , 2009, NAFIPS 2009 - 2009 Annual Meeting of the North American Fuzzy Information Processing Society.

[67]  Jianqiang Yi,et al.  A Novel Type-Reduction Method for Interval Type-2 Fuzzy Logic Systems , 2008, 2008 Fifth International Conference on Fuzzy Systems and Knowledge Discovery.

[68]  Jerry M. Mendel,et al.  Enhanced Interval Approach for Encoding Words Into Interval Type-2 Fuzzy Sets and Its Convergence Analysis , 2012, IEEE Transactions on Fuzzy Systems.

[69]  Jerry M. Mendel,et al.  Uncertainty bounds and their use in the design of interval type-2 fuzzy logic systems , 2002, IEEE Trans. Fuzzy Syst..

[70]  M. Melgarejo,et al.  Improved iterative algorithm for computing the generalized centroid of an interval type-2 fuzzy set , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[71]  Jerry M. Mendel,et al.  Computing derivatives in interval type-2 fuzzy logic systems , 2004, IEEE Transactions on Fuzzy Systems.

[72]  H. Zhou,et al.  A Method for Deriving the Analytical Structure of a Broad Class of Typical Interval Type-2 Mamdani Fuzzy Controllers , 2013, IEEE Transactions on Fuzzy Systems.

[73]  Dongrui Wu,et al.  On the Fundamental Differences Between Interval Type-2 and Type-1 Fuzzy Logic Controllers , 2012, IEEE Transactions on Fuzzy Systems.

[74]  Jerry M. Mendel,et al.  Equalization of nonlinear time-varying channels using type-2 fuzzy adaptive filters , 2000, IEEE Trans. Fuzzy Syst..