Closed-Form Mathematical Representations of Interval Type-2 Fuzzy Logic Systems with Application to Inverted Pendulum Stabilization

References

• Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning. Inf Sci. 1975;8:43–80. doi:10.1016/0020-0255(75)90036-5.
   Web of Science ®Google Scholar
• Mendel J. Type-2 fuzzy sets and systems: an overview. IEEE Comput Intell Mag. February 2007;2:20–29. doi:10.1109/MCI.2007.357235.
   Web of Science ®Google Scholar
• Wu D, Tan WW. A simplified architecture for type-2 FLSs and its application to nonlinear control. in Proc. IEEE Conf. On Cybern. and Intell. Syst., Singapore, Dec. 2004, pp. 485–490. doi:10.1109/ICCIS.2004.1460463.
   Google Scholar
• Karnik NN, Mendel JM. Operations on type-2 fuzzy sets. Fuzzy Sets Syst. [Internet]. 2001 [cited 2019 Aug 30]; 122:327–348. Available at: https://www.sciencedirect.com/science/article/pii/S0165011400000798.
   Web of Science ®Google Scholar
• Mendel JM, John RIB. Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst. Apr. 2002;10(2):117–127. doi:10.1109/91.995115.
   Web of Science ®Google Scholar
• Castro JR, Castillo O, Melin P. An Interval Type-2 Fuzzy Logic Toolbox for Control Applications. 2007 IEEE International Fuzzy Systems Conference [Internet]. IEEE; 2007 [cited 2019 Aug 30]. p. 1–6. Available from: http://ieeexplore.ieee.org/document/4295341/.
   Google Scholar
• Mendel JM. Advances in type-2 fuzzy sets and systems. Inf Sci. 2007;177:84–110. doi:10.1016/j.ins.2006.05.003.
   Web of Science ®Google Scholar
• Liang Q, Karnik NN, Mendel JM. Connection admission control in ATM networks using survey-based type-2 fuzzy logic systems. IEEE Trans Syst Man Cybern C (Appl Rev). Aug 2000;30(3):329–339. doi:10.1109/5326.885114.
   Web of Science ®Google Scholar
• Wu D, Mendel JM. Recommendations on designing practical interval type-2 fuzzy systems. Eng Appl Artif Intell [Internet]. 2019 [cited 2019 Aug 30]; 85:182–193. Available at: https://www.sciencedirect.com/science/article/abs/pii/S0952197619301514.
   Web of Science ®Google Scholar
• Huang J, Ri M, Wu D, et al. Interval type-2 fuzzy logic modeling and control of a mobile two-wheeled inverted pendulum. IEEE Trans Fuzzy Syst [Internet]. 2018 [cited 2019 Aug 31]; 26:2030–2038. Available at: https://ieeexplore.ieee.org/document/8060588/.
   Web of Science ®Google Scholar
• Hassanzadeh HR, Akbarzadeh-T M-R, Akbarzadeh A, et al. An interval-valued fuzzy controller for complex dynamical systems with application to a 3-PSP parallel robot. Fuzzy Sets Syst [Internet]. 2014 [cited 2019 Aug 30]; 235:83–100. Available at: https://www.sciencedirect.com/science/article/pii/S016501141300081X.
   Web of Science ®Google Scholar
• Hagras H, Callaghan V, Colley M, et al. Creating an ambient-intelligence environment using embedded agents. IEEE Intell Syst [Internet]. 2004 [cited 2019 Aug 30]; 19:12–20. Available at: http://ieeexplore.ieee.org/document/1363729/.
   Web of Science ®Google Scholar
• Yao B, Hagras H, Lepley JJ, et al. An evolutionary optimization based interval type-2 fuzzy classification system for human behavior recognition and summarisation. 2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC) [Internet]. IEEE; 2016 [cited 2019 Aug 30]. p. 004706–004711. Available at: http://ieeexplore.ieee.org/document/7844974/.
   Google Scholar
• Castro JR, Rodríguez-Díaz A. A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks. Inf Sci [Internet]. 2009 [cited 2019 Aug 30]; 179:2175–2193. Available at: https://www.sciencedirect.com/science/article/pii/S002002550800426X.
   Web of Science ®Google Scholar
• Acampora G, Alghazzawi D, Hagras H, et al. An interval type-2 fuzzy logic based framework for reputation management in Peer-to-Peer e-commerce. Inf Sci [Internet]. 2016 [cited 2019 Aug 31]; 333:88–107. Available at: https://www.sciencedirect.com/science/article/pii/S0020025515008257.
   Web of Science ®Google Scholar
• Karnik NN, Mendel JM, Liang Q. Type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst [Internet]. 1999 [cited 2019 Aug 30]; 7:643–658. Available at: http://ieeexplore.ieee.org/document/811231/.
   Web of Science ®Google Scholar
• Mendel JM. Uncertain rule-based fuzzy systems. [Internet]. Cham: Springer International Publishing; 2017 [cited 2019 Aug 30]. Available at: http://link.springer.com/10.1007978-3-319-51370-6.
   Google Scholar
• Karnik NN, Mendel JM. Centroid of a type-2 fuzzy set. Inf Sci [Internet]. 2001 [cited 2019 Aug 30]; 132:195–220. Available at: https://linkinghub.elsevier.com/retrieve/pii/S002002550100069X.
   Web of Science ®Google Scholar
• Mendel JM, Liu F. Super-exponential convergence of the karnik–mendel algorithms for computing the centroid of an interval type-2 fuzzy set. IEEE Trans Fuzzy Syst [Internet]. 2007 [cited 2019 Aug 30]; 15:309–320. Available at: http://ieeexplore.ieee.org/document/4142760/.
   Web of Science ®Google Scholar
• Khanesar MA, Khakshour AJ, Kaynak O, et al. Improving the speed of center of sets type reduction in interval type-2 fuzzy systems by eliminating the need for sorting. IEEE TransFuzzy Syst [Internet]. 2017 [cited 2019 Aug 30]; 25:1193–1206. Available at: http://ieeexplore.ieee.org/document/7551199/.
   Web of Science ®Google Scholar
• Wu D. Approaches for reducing the computational cost of interval type-2 fuzzy logic systems: overview and comparisons. IEEE Trans Fuzzy Syst [Internet]. 2013 [cited 2019 Aug 30]; 21:80–99. Available at: http://ieeexplore.ieee.org/document/6208856/.
   Web of Science ®Google Scholar
• Han S, Liu X. Global convergence of Karnik–Mendel algorithms. Fuzzy Sets Syst [Internet]. 2016 [cited 2019 Aug 30]; 283:108–119. Available at: https://www.sciencedirect.com/science/article/pii/S0165011415001281.
   Web of Science ®Google Scholar
• Wu D. An overview of alternative type-reduction approaches for reducing the computational cost of interval type-2 fuzzy logic controllers. 2012 IEEE International Conference on Fuzzy Systems [Internet]. IEEE; 2012 [cited 2019 Aug 30]. p. 1–8. Available at: http://ieeexplore.ieee.org/document/6251242/.
   Google Scholar
• Runkler TA, Chen C, John R. Type reduction operators for interval type–2 defuzzification. Inf Sci [Internet]. 2018 [cited 2019 Aug 30]; 467:464–476. Available at: https://www.sciencedirect.com/science/article/pii/S0020025518306285.
   Web of Science ®Google Scholar
• Ontiveros E, Melin P, Castillo O. High order α-planes integration: A new approach to computational cost reduction of General Type-2 Fuzzy Systems. Eng Appl Artif Intell [Internet]. 2018 [cited 2019 Aug 31]; 74:186–197. Available at: https://www.sciencedirect.com/science/article/abs/pii/S0952197618301441.
   Web of Science ®Google Scholar
• Begian MB, Melek WW, Mendel JM. Stability analysis of type-2 fuzzy systems. 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence) [Internet]. IEEE; 2008 [cited 2019 Aug 30]. p. 947–953. Available at: http://ieeexplore.ieee.org/document/4630483/.
   Google Scholar
• Lam HK, Seneviratne LD. Stability analysis of interval type-2 fuzzy-model-based control systems. IEEE Trans Syst Man Cybern [Internet]. 2008 [cited 2019 Aug 30]; 38:617–628. Available at: http://ieeexplore.ieee.org/document/4522595/.
   PubMed Web of Science ®Google Scholar
• Castillo O, Aguilar L, Cázarez N, et al. Systematic design of a stable type-2 fuzzy logic controller. Appl Soft Comput [Internet]. 2008 [cited 2019 Aug 30]; 8:1274–1279. Available at: https://www.sciencedirect.com/science/article/pii/S156849460700124X.
   Web of Science ®Google Scholar
• Biglarbegian M, Melek WW, Mendel JM. On the stability of interval type-2 TSK fuzzy logic control systems. IEEE Trans Syst Man Cybern [Internet]. 2010 [cited 2019 Aug 30]; 40:798–818. Available at: http://ieeexplore.ieee.org/document/5299179/.
   PubMed Web of Science ®Google Scholar
• Coupland S, John R. Geometric type-1 and type-2 fuzzy logic systems. IEEE Trans Fuzzy Syst [Internet]. 2007 [cited 2019 Aug 30]; 15:3–15. Available at: http://ieeexplore.ieee.org/document/4088983/.
   Web of Science ®Google Scholar
• Coupland S, John R. A new and efficient method for the type-2 meet operation. 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04CH37542) [Internet]. IEEE; [cited 2019 Aug 30]. p. 959–964. Available at: http://ieeexplore.ieee.org/document/1375537/.
   Google Scholar
• Coupland S, John R. (1955). Fuzzy logic and computational geometry. RASC 2004 [Internet]. Nottingham, U.K.; 2004 [cited 2019 Aug 30]. p. 3–8. Available at: https://dora.dmu.ac.uk/handle/2086/951.
   Google Scholar
• Coupland S, John R. Towards More Efficient Type-2 Fuzzy Logic Systems. The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ ‘05. [Internet]. IEEE; [cited 2019 Aug 30]. p. 236–241. Available at: http://ieeexplore.ieee.org/document/1452399/.
   Google Scholar
• Coupland S. Type-2 Fuzzy Sets: Geometric Defuzzification and Type-Reduction. 2007 IEEE Symposium on Foundations of Computational Intelligence [Internet]. IEEE; 2007 [cited 2019 Aug 30]. p. 622–629. Available at: http://ieeexplore.ieee.org/document/4233971/.
   Google Scholar
• Coupland S, John R. A Fast geometric method for defuzzification of type-2 fuzzy sets. IEEE Trans Fuzzy Syst [Internet]. 2008 [cited 2019 Aug 30]; 16:929–941. Available at: http://ieeexplore.ieee.org/document/4505326/.
   Web of Science ®Google Scholar
• Coupland S, John R. Geometric Interval Type-2 Fuzzy Systems. EUSFLAT Conf. 2005. p. 449–454.
   Google Scholar
• Nie M, Tan WW. Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence) [Internet]. IEEE; 2008 [cited 2019 Aug 30]. p. 1425–1432. Available at: http://ieeexplore.ieee.org/document/4630559/.
   Google Scholar
• Yeh C-Y, Jeng W-HR, Lee S-J. An Enhanced type-reduction algorithm for type-2 fuzzy sets. IEEE Trans Fuzzy Syst [Internet]. 2011 [cited 2019 Aug 30]; 19:227–240. Available at: http://ieeexplore.ieee.org/document/5638621/.
   Web of Science ®Google Scholar
• Yager RR, Filev DP. Essentials of fuzzy modeling and control. New York. 1994;388.
   Google Scholar
• Weisstein EW. Function centroid [Internet]. Wolfram Research, Inc.; [cited 2019 Aug 30]. Available at: http://mathworld.wolfram.com/FunctionCentroid.html.
   Google Scholar
• Li J, John R, Coupland S, et al. On nie-tan operator and type-reduction of interval type-2 fuzzy sets. IEEE Trans Fuzzy Syst [Internet]. 2018 [cited 2019 Aug 30]; 26:1036–1039. Available at: https://ieeexplore.ieee.org/document/7849207/.
   Web of Science ®Google Scholar
• Passino KM, Yurkovich S. Fuzzy control. Addison-Wesley; 1998.
   Google Scholar
• Liang Q, Mendel JM. Interval type-2 fuzzy logic systems: theory and design. IEEE Trans Fuzzy Syst [Internet]. 2000 [cited 2019 Aug 30]; 8:535–550. Available at: http://ieeexplore.ieee.org/document/873577/.
   Web of Science ®Google Scholar
• Roose AI, Yahya S, Al-Rizzo H. Fuzzy-logic control of an inverted pendulum on a cart. Comput Electr Eng [Internet]. 2017 [cited 2019 Aug 30]; 61:31–47. Available at: https://www.sciencedirect.com/science/article/pii/S0045790617313654.
   Web of Science ®Google Scholar
• Ashraf Z, Roy ML, Muhuri PK, et al. Interval type-2 fuzzy logic system based similarity evaluation for image steganography. Heliyon. May 2020;6(5):e03771. https://doi.org/10.1016/j.heliyon.2020.e03771.
   Web of Science ®Google Scholar
• Shukla AK, Yadav M, Kumar S, et al. Veracity handling and instance reduction in big data using interval type-2 fuzzy sets. Eng Appl Artifi Intell. 2020;88:103315. https://doi.org/10.1016/j.engappai.2019.103315.
   Web of Science ®Google Scholar
• Cuevas F, Castillo O, Cortes-Antonio P. Optimal Design of Interval Type-2 Fuzzy Tracking Controllers of Mobile Robots Using a Metaheuristic Algorithm. In: Melin P, Castillo O, Kacprzyk J, editor. Recent Advances of Hybrid Intelligent Systems Based on Soft Computing. Studies in Computational Intelligence, vol 915. Cham: Springer; 2021. https://doi.org/10.1007/978-3-030-58728-4_18
   Google Scholar
• Chen C, Wu D, Garibaldi JM, et al. A Comprehensive study of the efficiency of type-reduction algorithms. IEEE Trans Fuzzy Syst. March 2020. doi:10.1109/TFUZZ.2020.2981002.
   Web of Science ®Google Scholar
• https://www.mathworks.com/matlabcentral/fileexchange/66578-interval-type-2-fuzzy-controller-simple-implementation
   Google Scholar
• Wu D, Mendel JM. Enhanced Karnik–Mendel algorithms. IEEE Trans Fuzzy Syst. 2009;17(4):923–934.
   Web of Science ®Google Scholar