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
Raju GVS, Zhou J, Kisner RA (1991) Hierarchical fuzzy control. Int J Control 54(5):1201–1216
Raju GVS, Zhou J (1993) Adaptive hierarchical fuzzy controller. IEEE Trans Syst Man Cybernet 23:973–980
Wang LX (1998) Universal approximation by hierarchical fuzzy systems. Fuzzy Set Syst 93(1):223–230
Wang LX (1999) Analysis and design of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 7(5):617–624
Chen W, Wang LX (2000) A note on universal approximation by hierarchical fuzzy systems. Inf Sci 123:241–248
Takagi T, Sugeno M (1985) Fuzzy identification of system and its applications to modeling and control. IEEE Trans Syst Man Cybernet 15:116–132
Combs WE, Andrews JE (1998) Combinatorial rule explosion eliminated by a fuzzy rule configuration. IEEE Trans Fuzzy Syst 6:1–11
Wang XZ, Hong JR (1999) Learning optimization in simplifying fuzzy rules. Fuzzy Sets Syst 106(3):349–356
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
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
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
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
Wang XZ, Aamir R, Aimin F (2015) Fuzziness based sample categorization for classifier performance improvement. J Intell Fuzzy Syst 29:1185–1196
Liu PY, Li HX (2000) Approximation of generalized fuzzy systems to integrable functions. Sci China Ser E 30(5):413–423
Liu PY, Li HX (2001) Analyses for Lp-norm approximation capability of generalized Mamdani fuzzy systems. Inf Sci 138(2):195–210
Liu PY, Li HX (2005) Hierarchical T-S fuzzy system and its universal approximation. Inf Sci 169(3):279–303
Zeng XJ, John AK (2005) Approximation capabilities of hierarchical fuzzy systems. IEEE Trans Fuzzy Syst 13(5):659–672
Ricardo J, Campello GB, Wagner C (2006) Hierarchical fuzzy relational models: linguistic interpretation and universal approximation. IEEE Trans Fuzzy Syst 14(3):446–453
Yuan XH, Li HX, Yang X (2013) Fuzzy system and fuzzy inference modeling method based on fuzzytransformation. Acta Electron Sin 41(4):674–680
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
Wang DG, Song WY, Li HX (2015) Approximation properties of ELM-fuzzy systems for smooth functions and their derivatives. Neurocomputing 149:265–274
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
Vassilis SK, Yannis AP (2009) On the monotonicity of hierarchical sum-product fuzzy systems. Fuzzy Sets Syst 160(24):3530–3538
Abdolreza M, Mohammad R (2010) A novel hierarchical clustering combination scheme based on fuzzy similarity relations. IEEE Trans Fuzzy Syst 18(1):27–39
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
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
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
Wang GJ, Duan CX (2012) Generalized hierarchical hybrid fuzzy systems and their universal approximation. Control Theory Appl 29(5):673–680
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
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
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
Du XY, Zhang NY (2004) Equivalence analysis of binary-tree-type hierarchical fuzzy system. J Tsinghua Univ 44(7):33–36
Zhang XY, Zhang NY (2007) Universal approximation of general binary-tree-type hierarchical fuzzy systems. J Tsinghua Univ 47(1):37–41
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
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
Wang LX (2003) A course in fuzzy systems and control (Chinese Version). Tsinghua University Press, Beijing
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This work has been supported by National Natural Science Foundation China (Grant No. 61374009).
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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