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Rule number and approximation of the hybrid fuzzy system based on binary tree hierarchy

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Abstract

To effectively avoid internal rule explosion of a fuzzy system or computer memory overflow caused by increased input variables, a hybrid fuzzy system is established by unifying the Takagi–Sugeno and the Mamdani fuzzy systems based on a binary tree hierarchical method. This method can greatly reduce the total number of rules within the system. Firstly, a calculation formula of the total number of rules for the hybrid fuzzy system is given, by comparing with other layered systems, the total number of rules based on the binary tree hierarchy has the largest decline. Secondly, a new K-integral norm is redefined by introducing a K-quasi-subtraction operator. Using the piecewise linear function the approximation capability of the hybrid fuzzy system after hierarchy to a kind of integrable functions is studied. Finally, the binary tree hierarchical structure expressions of the hybrid fuzzy system are given through two simulation examples.

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References

  1. Raju GVS, Zhou J, Kisner RA (1991) Hierarchical fuzzy control. Int J Control 54(5):1201–1216

    Article  MathSciNet  MATH  Google Scholar 

  2. Raju GVS, Zhou J (1993) Adaptive hierarchical fuzzy controller. IEEE Trans Syst Man Cybernet 23:973–980

    Article  Google Scholar 

  3. Wang LX (1998) Universal approximation by hierarchical fuzzy systems. Fuzzy Set Syst 93(1):223–230

  4. Wang LX (1999) Analysis and design of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 7(5):617–624

    Article  Google Scholar 

  5. Chen W, Wang LX (2000) A note on universal approximation by hierarchical fuzzy systems. Inf Sci 123:241–248

  6. Takagi T, Sugeno M (1985) Fuzzy identification of system and its applications to modeling and control. IEEE Trans Syst Man Cybernet 15:116–132

  7. Combs WE, Andrews JE (1998) Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans Fuzzy Syst 6:1–11

    Article  Google Scholar 

  8. Wang XZ, Hong JR (1999) Learning optimization in simplifying fuzzy rules. Fuzzy Sets Syst 106(3):349–356

    Article  MathSciNet  MATH  Google Scholar 

  9. Ying H (1998) Sufficient conditions on uniform approximation of multivariate functions by general Takagi-Sugeno fuzzy systems with linear rule consequent. IEEE Trans Syst Man Cybernet 28: 515–520

  10. Yin TK (2004) A characteristic-point-based fuzzy inference system aimed to minimize the number of fuzzy rules. IEEE Trans Actions Fuzzy Syst 12(2):250–273

  11. Wang XZ, Hong JR (1998) On the handling of fuzziness for continuous-valued attributes in decision tree generation. Fuzzy Sets Syst 99(3):283–290

    Article  MathSciNet  MATH  Google Scholar 

  12. Tsang ECC, Wang XZ, Yeung DS (2000) Improving learning accuracy of fuzzy decision trees by hybrid neural networks. IEEE Trans Fuzzy Syst 8(5):601–614

    Article  Google Scholar 

  13. Wang XZ, Aamir R, Aimin F (2015) Fuzziness based sample categorization for classifier performance improvement. J Intell Fuzzy Syst 29:1185–1196

    Article  MathSciNet  Google Scholar 

  14. Liu PY, Li HX (2000) Approximation of generalized fuzzy systems to integrable functions. Sci China Ser E 30(5):413–423

  15. Liu PY, Li HX (2001) Analyses for Lp-norm approximation capability of generalized Mamdani fuzzy systems. Inf Sci 138(2):195–210

    Article  MATH  Google Scholar 

  16. Liu PY, Li HX (2005) Hierarchical T-S fuzzy system and its universal approximation. Inf Sci 169(3):279–303

    Article  MathSciNet  MATH  Google Scholar 

  17. Zeng XJ, John AK (2005) Approximation capabilities of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 13(5):659–672

    Article  Google Scholar 

  18. Ricardo J, Campello GB, Wagner C (2006) Hierarchical fuzzy relational models: linguistic interpretation and universal approximation. IEEE Trans Fuzzy Syst 14(3):446–453

    Article  Google Scholar 

  19. Yuan XH, Li HX, Yang X (2013) Fuzzy system and fuzzy inference modeling method based on fuzzytransformation. Acta Electron Sin 41(4):674–680

    Google Scholar 

  20. Wang DG, Song WY, Shi P, Li HX (2013) Approximation to a class of non-autonomous systems by dynamic fuzzy inference marginal linearization method. Inf Sci. 245:197–217

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang DG, Song WY, Li HX (2015) Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives. Neurocomputing 149:265–274

    Article  Google Scholar 

  22. Moon GJ, Thomas S (2009) A method of converting a fuzzy system to a two-layered hierarchical fuzzy system and its run-time efficiency. IEEE Trans Fuzzy Syst 17(1):93–103

    Article  Google Scholar 

  23. Vassilis SK, Yannis AP (2009) On the monotonicity of hierarchical sum-product fuzzy systems. Fuzzy Sets Syst 160(24):3530–3538

    Article  MathSciNet  MATH  Google Scholar 

  24. Abdolreza M, Mohammad R (2010) A novel hierarchical clustering combination scheme based on fuzzy similarity relations. IEEE Trans Fuzzy Syst 18(1):27–39

    Article  Google Scholar 

  25. Zsofia L, Robert B, Bart DS (2011) Sequential stability analysis and observer design for distributed T-S fuzzy systems. Fuzzy Sets Syst 174(1):1–30

    Article  MathSciNet  Google Scholar 

  26. Luo MN, Sun FC, Liu HP (2013) Hierarchical structured sparse representation for T-S fuzzy systems identification. IEEE Trans Fuzzy Syst 21(6):1032–1043

    Article  Google Scholar 

  27. Chen CH (2013) Design of TSK-type fuzzy controllers using differential evolution with adaptive mutation strategy for nonlinear system control. Appl Math Comput 219(15):8277–8294

    MathSciNet  MATH  Google Scholar 

  28. Wang GJ, Duan CX (2012) Generalized hierarchical hybrid fuzzy systems and their universal approximation. Control Theory Appl 29(5):673–680

    Google Scholar 

  29. Wang GJ, Li XP, Sui XL (2014) Universal approximation and its realization of generalized Mamdani fuzzy system based on K-integral norms. Acta Autom Sin 40(1):143–148

  30. Wang GJ, Song WW, Han QJ (2015) Generalized hybrid fuzzy system based on consequent direct link type-hierarchy and its integral norm approximation. Control Decis 30(10):1742–1750

    MATH  Google Scholar 

  31. Tao YJ, Wang HZ, Wang GJ (2015) Approximation ability and its realization of the generalized Mamdan fuzzy system in the sense of Kp-integral norm. Acta Electron Sin 43(11):2284–2291

    Google Scholar 

  32. Du XY, Zhang NY (2004) Equivalence analysis of binary-tree-type hierarchical fuzzy system. J Tsinghua Univ 44(7):33–36

  33. Zhang XY, Zhang NY (2007) Universal approximation of general binary-tree-type hierarchical fuzzy systems. J Tsinghua Univ 47(1):37–41

    MathSciNet  MATH  Google Scholar 

  34. Yang Y, Wang GJ, Yang YQ (2013) Reducing the number of inference rules for generalized hybrid fuzzy systems based on binary tree type hierarchy. Control Theory Appl 30(6):765–772

    MathSciNet  Google Scholar 

  35. Wang GJ, Li XP (2011) Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms. Sci China Inf Sci 54(11):2307–2323

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang LX (2003) A course in fuzzy systems and control (Chinese Version). Tsinghua University Press, Beijing

    Google Scholar 

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Acknowledgements

This work has been supported by National Natural Science Foundation China (Grant No. 61374009).

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Correspondence to Guijun Wang.

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This work has been supported by National Natural Science Foundation China (Grant No. 61374009).

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Wang, G., Yang, Y. & Li, X. Rule number and approximation of the hybrid fuzzy system based on binary tree hierarchy. Int. J. Mach. Learn. & Cyber. 9, 979–991 (2018). https://doi.org/10.1007/s13042-016-0622-z

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  • DOI: https://doi.org/10.1007/s13042-016-0622-z

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