Kp-积分模意义下广义Mamdani模糊系统的逼近性能及其实现

doi:10.3969/j.issn.0372-2112.2015.11.022

电子学报 ›› 2015, Vol. 43 ›› Issue (11): 2284-2291.DOI: 10.3969/j.issn.0372-2112.2015.11.022

Kp-积分模意义下广义Mamdani模糊系统的逼近性能及其实现

陶玉杰¹, 王宏志¹, 王贵君²

1. 通化师范学院数学学院, 吉林通化 134002;
2. 天津师范大学数学科学学院, 天津 300387

收稿日期:2014-07-15 修回日期:2014-10-20 出版日期:2015-11-25
- 通讯作者:
- 王贵君
- 作者简介:
- 陶玉杰 女,1975年8月出生,副教授,吉林通化人.2004年获燕山大学理学硕士学位,现在通化师范学院数学学院任教.主要研究方向:模糊系统分析.E-mail:taotujie@126.com;王宏志 男,1975年6月出生,副教授,吉林梅河口人,2006年获吉林大学数学所理学硕士学位,现在通化师范学院数学学院任教.主要研究方向:模糊系统逼近分析.E-mail:whz-98@126.com
- 基金资助:
- 国家自然科学基金 (No.61374009)

Approximation Ability and Its Realization of the Generalized Mamdani Fuzzy System in the Sense of Kp-Integral Norm

TAO Yu-jie¹, WANG Hong-zhi¹, WANG Gui-jun²

1. College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134002, China;
2. College of Mathematics Sciences, Tianjin Normal University, Tianjin 300387, China

Received:2014-07-15 Revised:2014-10-20 Online:2015-11-25 Published:2015-11-25

摘要/Abstract

摘要：

利用积分模(度量)研究模糊系统对可积函数类的逼近性是人们普遍关注的方法,而基于K-拟算术运算诱导的Kp-积分模不仅是一维积分模的推广,而且是刻画p-次可积函数类的重要工具.本文通过引入拟减运算重新定义Kp-积分模,且在Kp-积分模意义下讨论分片线性函数对一类û_p-可积函数的逼近性,进而构造性地证明广义Mamdani模糊系统对û_p-可积函数类仍有逼近性.最后通过实例分析说明广义Mamdani模糊系统的逼近效果.结果表明广义Mamdani模糊系统可以按任意精度逼近一类û_p-可积函数.

关键词: K-拟算术运算, û_p-可积函数, Kp-积分模, 分片线性函数, 广义Mamdani模糊系统, 逼近性

Abstract:

Researching on the approximation of fuzzy system to integrable function class by means of the integral norm(a metric) is one method of common concern to the people.The Kp-integral norm induced by the K-quasi-arithmetic operations is not only a generalization for a one dimensional integral norm,but also an important tool to describe the p-integrable function classes.In this paper,the Kp-integral norm is redefined by introducing the quasi-subtraction operator.In the sense of the Kp-integral norm,the approximation of the piecewise linear functions to a kind of û_p-integrable functions is discussed.Then,we prove constructively that the generalized Mamdani fuzzy system has the approximation to a class of û_p-integrable functions.Finally,by a practical example the approximation effect of the generalized Mamdani fuzzy systems is illustrated.The results show that the generalized Mamdani fuzzy system can approximate a kind of û_p-integrable functions to arbitrary accuracy.

Key words: K-quasi-arithmetic operation, û_p-integrable function, Kp-integral norm, piecewise linear function, generalized Mamdani fuzzy systems, approximation

中图分类号:

TP183
O159

陶玉杰, 王宏志, 王贵君. Kp-积分模意义下广义Mamdani模糊系统的逼近性能及其实现[J]. 电子学报, 2015, 43(11): 2284-2291.

TAO Yu-jie, WANG Hong-zhi, WANG Gui-jun. Approximation Ability and Its Realization of the Generalized Mamdani Fuzzy System in the Sense of Kp-Integral Norm[J]. Acta Electronica Sinica, 2015, 43(11): 2284-2291.

参考文献

[1] Takagi T,Sugeno M.Fuzzy identification of system and its applications to modeling and control[J]. IEEE Transactions on Systems,Man and Cybern,1985,15(1):116-132.
[2] Kosko B.Fuzzy systems are universal approximators[J]. IEEE Transactions on Computers,1994,43(11):1329-1333.
[3] Buckley J J.Sugeno type controllers are universal controllers[J]. Fuzzy Sets and Systems,1993,53(3):299-303.
[4] Wang L X,Mendel J.Fuzzy basis functions,universal approximation and orthogonal least-squares learning[J]. IEEE Transactions on Neural Networks,1992,3(5):807-814.
[5] Wang L X.Universal approximation by hierarchical fuzzy systems[J]. Fuzzy Set and Systems,1998,93(1):223-230.
[6] Wang L X.Analysis and design of hierarchical fuzzy systems[J]. IEEE Transactions on Fuzzy Systems,1999,7(5):617-624.
[7] 刘普寅,李洪兴.广义模糊系统对于可积函数的逼近性[J]. 中国科学(E辑),2000,30(5):413-423. Liu Pu-yin,Li Hong-xing.Approximation of generalized fuzzy systems to integrable functions[J]. Science in China,Series E,2000,30(5):413-423.(in Chinese)
[8] Liu Pu-yin,Li Hong-xing.Analyses for L_p(μ)-norm approxi-mation capability of the generalized Mamdani fuzzy systems[J]. Information Sciences,2001,138(2):195-210.
[9] 曾珂,张乃尧,徐文立.线性T-S模糊系统作为通用逼近器的充分条件[J]. 自动化学报,2001,27(5):606-612. Zeng Ke,Zhang Nai-yao,Xu Wen-li.Sufficient condition for linear T-S fuzzy systems universal approximators[J]. Acta Automatica Sinica,2001,27(5):606-612.(in Chinese)
[10] 孙富春,杨晋,刘华平.SISO Mamdani模糊系统作为函数逼近器的必要条件[J]. 智能系统学报,2009,4(4):288-294. Sun Fu-chun,Yang Jin,Liu Hua-ping.Preconditions for SISO Mamdani fuzzy systems to perform as function approximators[J]. Transactions on Intelligent Systems,2009,4(4):288-294.(in Chinese)
[11] 曾珂,徐文立,张乃尧.特定Mamdani模糊系统的通用逼近性[J]. 控制与决策,2000,15(4):435-438. Zeng Ke,Xu Wen-li,Zhang Nai-yao.Universal approximationofspecial Mamdani fuzzy systems[J]. Control and Decision,2000,15(4):435-438.(in Chinese)
[12] 张宇卓,李洪兴.广义递阶Mamdani模糊系统及其泛逼近性[J]. 控制理论与应用,2006,23(3):449-454. Zhang Yu-zhuo,Li Hong-xing.Generalized hierarchical Mamdani fuzzy systems and their universal approximation[J]. Control Theory and Application,2006,23(3):449-454.(in Chinese)
[13] 袁学海,李洪兴,孙凯彪.基于参数单点模糊化方法的模糊系统及逼近能力[J]. 电子学报,2011,39(10):2372-2377. Yuan Xue-hai,Li Hong-xing,Sun Kai-biao.Fuzzy systems and their approximation capability based on parameter singleton fuzzifier methods[J]. Acta Electronica Sinica,2011,39(10):2372-2377.(in Chinese)
[14] 袁学海,李洪兴,杨雪.基于模糊变换的模糊系统和模糊推理建模法[J]. 电子学报,2013,41(4):674-680. Yuan Xue-hai,Li Hong-xing,Yang Xue.Fuzzy system and fuzzy inference modeling method based on fuzzy transformation[J]. Acta Electronica Sinica,2013,41(4):674-680.(in Chinese)
[15] 李洪兴,袁学海,王加银.Fuzzy系统的范数与Fuzzy系统的分类[J]. 中国科学(信息科学),2010,40(12):1596-1610. Li Hong-xing,Yuan Xue-hai,Wang Jia-yin.The normal numbers of the fuzzy systems and their classes[J]. Science China Information Science,2010,40(12):1596-1610.(in Chinese)
[16] 孟艳萍,谭艳华,李洪兴.自适应Fuzzy系统及其逼近性能分析[J]. 数学进展,2012,41(4):423-435. Meng Yan-ping,Tan Yan-hua,Li Hong-xing.Adaptive fuzzy systems and the analysis of approximation[J]. Advances in Mathematics,2012,41(4):423-435.(in Chinese)
[17] 王贵君,李晓萍.K-积分模意义下折线模糊神经网络的泛逼近性[J]. 中国科学(信息科学),2012,42(3):362-378. Wang Gui-jun,Li Xiao-ping.Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms[J]. Science China Information Science,2011,54(11):2307-2323.(in Chinese)
[18] 王贵君,段晨霞.广义分层混合模糊系统及其泛逼近性[J]. 控制理论与应用,2012,29(5):673-680. Wang Gui-jun,Duan Chen-xia.Generalized hierarchical hybrid fuzzy system and its universai approximation[J]. Control Theory and Application,2012,29(5):673-680.(in Chinese)
[19] 王贵君,李晓萍,隋晓琳.广义Mamdani模糊系统依K-积分模的泛逼近及其实现过程[J]. 自动化学报,2014,40(1):143-148. Wang Gui-jun,Li Xiao-ping,Sui Xiao-lin.Universal approxi-mation and its realize process of generalized Mamdani fuzzy system in K-integral norms[J]. Acta Automatica Sinica,2014,40(1):143-148.(in Chinese)
[20] 彭维玲.基于剖分模糊系统输入空间的多维分片线性函数的构造及逼近[J]. 系统科学与数学,2014,34(3):340-351. Peng Wei-ling.Structure and approximation of a multidimens-ional piecewise linear function based on the input space of su-bdvision fuzzy systems[J]. System Science and Mathematical Sciences,2014,34(3):340-351.(in Chinese)
[21] Sugeno M,Murofushi T.Pseudo-additive measures and integrals[J]. Math Anal Appl,1987,122:197-222.
[22] 王贵君,李晓萍.K-拟可加模糊数值积分的伪自连续及结构特征[J]. 应用数学学报,2010,33(1):66-77. Wang Gui-jun,Li Xiao-ping.The pseudo-autocontinuity and structural characteristics of K-quasi-additive fuzzy number valued integrals[J]. Acta Mathematicae Applicatae Sinica,2010,33(1):66-77.(in Chinese)

Kp-积分模意义下广义Mamdani模糊系统的逼近性能及其实现

Approximation Ability and Its Realization of the Generalized Mamdani Fuzzy System in the Sense of Kp-Integral Norm

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

编辑推荐

Metrics

[1]	董文强, 王贵君. 基于调控参数建模的混合模糊系统的逼近分析[J]. 电子学报, 2017, 45(5): 1158-1164.
[2]	袁学海, 李洪兴, 杨雪. 基于模糊变换的模糊系统和模糊推理建模法[J]. 电子学报, 2013, 41(4): 674-680.
[3]	袁学海;李洪兴;孙凯彪. 基于参数单点模糊化方法的模糊系统及其逼近能力[J]. 电子学报, 2011, 39(10): 2372-2377.