第4卷第4期 智能系统学报 Vol.4 No.4 2009年8月 CAAI Transactions on Intelligent Systems Aug.2009 doi:10.3969/i.issn.1673-4785.2009.04.002 SIS0 Mamdan i模糊系统作为函数逼近器的必要条件 孙富春，杨晋，刘华平 (清华大学智能技术与系统国家重点实验室，北京100084) 摘要模糊系统已被证明是通用逼近器，但实现高精度通常需要大量规则.模糊系统满足给定精度的必要条件能 指导最优系统的构造，如输入输出模糊集、模糊规则的选取.研究了单输入单输出(SISO)Mamdani模糊系统在给定 逼近精度下作为函数逼近器的必要条件.由于通用型SIS0adni模糊系统在划分子区间单调，所以模糊系统的最 优配置是输入域的划分数至少为系统输出的单调性变化次数.当模糊系统满足给定逼近精度时，通过分析目标函数 的局部特性，基于目标函数的极点，建立了SIS0 amdanmi模糊系统的必要条件.更重要的是证明了现有的必要条件 仅仅是该文结论的一种特例.最后，使用数值实例来验证该文的结论，分析模糊系统作为函数逼近器的优劣. 关键词：模糊系统：必要条件：模糊规则：逼近精度 中图分类号：TP18文献标识码：：A文章编号：：1673-4785(2009)04-028807 Preconditions for SISO Mamdani fuzzy systems to perform as function approximators SUN Fu-chun,YANG Jin,LIU Hua-ping (State Key Laboratory of Intelligent Technology and System,Tsinghua University,Beiing 100084,China) Abstract:It has been proven that fuzzy systems are universal approximators.However,a large number of rules are usually needed for high accuracy.Knowledge of conditions necessary for a fuzzy system to have a given level of ac- curacy can provide guidance for design of an optimal system,such as selection of optimal input/output fuzzy sets and fuzzy rules.The necessary conditions for a single-input single-output(SISO))Mamdani fuzzy system to operate as a function approximator subject to a given precision were discussed.Since the general SISO Mamdani fuzzy sys- tem is monotonic at subintervals,its optimal configuration is when the number of division points is not less than the number of times its monotonicity changes.Thus by analyzing the local characteristics of the object function under the fuzzy system,necessary conditions for a SISO Mamdani fuzzy systems were obtained in accordance with the ex- trema of the object function.Furthermore,it was shown that the necessary conditions found in prior documents are only a special case of those described here.Finally,simulation examples were given to verify our conclusions and analyze the performance as well as the limitations of a fuzzy system as a function approximator. Keywords::fuzzy systems;necessary conditions;fuzzy rules;approximation accuracy 自1965年Zadch”提出模糊集合的概念以来，， 统主要有两大类：Mamdani模糊模型与Takagi--Suge 模糊理论及其应用技术得到了迅猛的发展.特别是 no(T-S))模糊模型.2类模型的主要区别是Mamdani 模糊系统具有直观、有效地表示非线性系统的优势， 模型的后件是单点模糊集，而TS模糊模型的后件 使得模糊理论在控制领域得到了大规模的应用，并 是线性函数.在许多应用中，涉及的模糊系统都可视 掀起了新的一轮研究热潮.目前，广泛的应用模糊系 为基于模糊规则的函数逼近器.近年来，有很多学者 对此展开了广泛而深入的研究，证明了多类模糊系 收稿日期：：2009-01-01· 基金项：国家自然科学基金资助项目(9071602L,60621062) 统是通用逼近器.还有学者建立了一些模糊系统作 通信作者：杨晋.Eail:yangjino6@mas.tsinghuaedu.cn. 为函数逼近器的充分条件和必要条件.而研究模糊SISO Mamdani模糊系统作为函数逼近器的必要条件 统主要有两大类: Mamdani模糊模型与Takagi-Suge￾no(T-S) 模糊模型.2类模型的主要区别是 Mamdani 模型的后件是单点模糊集,而 TS模糊模型的后件 是线性函数.在许多应用中,涉及的模糊系统都可视 为基于模糊规则的函数逼近器.近年来,有很多学者 对此展开了广泛而深入的研究,证明了多类模糊系 统是通用逼近器.还有学者建立了一些模糊系统作 为函数逼近器的充分条件和必要条件.而研究模糊 摘 要:模糊系统已被证明是通用逼近器,但实现高精度通常需要大量规则.模糊系统满足给定精度的必要条件能 指导最优系统的构造,如输入输出模糊集、模糊规则的选取.研究了单输入单输出(SISO) Mamdani模糊系统在给定 逼近精度下作为函数逼近器的必要条件.由于通用型 SISO Mamdani模糊系统在划分子区间单调,所以模糊系统的最 优配置是输入域的划分数至少为系统输出的单调性变化次数.当模糊系统满足给定逼近精度时,通过分析目标函数 的局部特性,基于目标函数的极点,建立了SISO Mamdanmi模糊系统的必要条件.更重要的是证明了现有的必要条件 仅仅是该文结论的一种特例.最后,使用数值实例来验证该文的结论,分析模糊系统作为函数逼近器的优劣. Preconditions for SISO Mamdani fuzzy systems to perform as function approximators Abstract:It has been proven that fuzzy systems are universal approximators. However,a large number of rules are usually needed for high accuracy. Knowledge of conditions necessary for a fuzzy system to have a given level of ac￾curacy can provide guidance for design of an optimal system,such as selection of optimal input/output fuzzy sets and fuzzy rules. The necessary conditions for a single-input single-output(SISO) Mamdani fuzzy system to operate as a function approximator subject to a given precision were discussed. Since the general SISO Mamdani fuzzy sys￾tem is monotonic at subintervals, its optimal configuration is when the number of division points is not less than the number of times its monotonicity changes. Thus by analyzing the local characteristics of the object function under the fuzzy system, necessary conditions for a SISO Mamdani fuzzy systems were obtained in accordance with the ex￾trema of the object function. Furthermore, it was shown that the necessary conditions found in prior documents are only a special case of those described here. Finally, simulation examples were given to verify our conclusions and analyze the performance as well as the limitations of a fuzzy system as a function approximator. 孙富春,杨 晋,刘华平 (清华大学智能技术与系统国家重点实验室,北京100084) 第4卷第4期 2009年8月 SUN Fu-chun,YANG Jin,LIU Hua-ping (State Key Laboratory of Intelligent Technology and System,Tsinghua University,Beiing 100084,China) 自1965年Zadch"提出模糊集合的概念以来, 模糊理论及其应用技术得到了迅猛的发展.特别是 模糊系统具有直观、有效地表示非线性系统的优势, 使得模糊理论在控制领域得到了大规模的应用,并 掀起了新的一轮研究热潮.目前,广泛的应用模糊系 doi: 10.3969/j.issn.1673-4785.2009.04.002 CAAI Transactions on Intelligent Systems 智 能 系 统 学 报 关键词: 模糊系统;必要条件;模糊规则;逼近精度 Keywords: 通信作者: 杨 晋.E-mail: yangjin06@ mails.tsinghua.edu.cn. fuzzy systems; necessary conditions; fuzzy rules; approximation accuracy Vol.4 No.4 Aug.2009 中图分类号: TP18 文献标识码: A 文章编号: 1673-4785(2009) 04-0288-07 收稿日期: 2009-01-01 基金项目: 国家自然科学基金资助项目(90716021,60621062)