模糊系统作为通用逼近器的10 年历程

总结了模糊系统作为通用逼近器在存在性、充分性和必要性3个方面所作过的主要理论研究，并分析了这些理论成果在工程上的若干应用，最后就该理论的未来发展方向作了大胆而理性的分析和预测。     

模糊系统万能逼近理论研究综述

模糊系统通用逼近理论是模糊理论研究的一个重要方向．目前，对模糊系统通用逼近性的研究已经取得了很大的进展．对模糊系统的通用逼近性、模糊系统作为通用逼近器的充分条件和必要条件以及模糊系统的逼近精度等方面的研究进行了较为详尽的综述，分析了各种分析方法的主要成果及其特点（包括优点和局限性），并指出了今后模糊系统通用逼近理论研究中有待解决的许多问题．     

特定Mamdani模糊系统的通用逼近性

特定Ｍａｍｄａｎｉ模糊系统是指采用模糊单点作为规则后件的多输入单输出Ｍａｍｄａｎｉ模糊系统。在每个输入变量的模糊子集满足一致笥以及录属函数连续且分段可微的条件下,证明了特定Ｍａｍ－ｄａｎｉ模糊系统是通用逼近器,在此基础上,进上步给出了特定Ｍａｍｄａｎｉ模糊系统一致逼近紧致集上任意连续实函数的充分条件。     

模糊系统的逼近精度和规则关系研究

从模糊系统的逼近精度角度来研究模糊系统的杂程度、逼近精度、控制规则数目之间的关系。通过修改逼近精度的边界而减少了控制规则,近而达到了简化系统复杂程度的目的。通过实例分析可以看出,所设计的模糊系统既达到了所要求的精度,同时又使规则数目大大地减少了。     

一类模糊系统的逼近问题

对于一类模糊系统，首次给出了它们可在紧论域上以任意要求的精度逼近任意C^1函数的构造性充分条件，同时导出了可用于计算所需模糊规则条数的显式公式，对于多项式函数情形，证明了所给结果比献[1]中结果保守性要小。     

线性T-S模糊系统作为通用逼近器的充分条件

作为模糊系统理论研究和实际应用的基础,对模糊系统通用逼近性的研究已经取得了很大的进展,但是只有为数不多的文献研究了模糊系统作为通用逼近器的充分条件.提出了线性T-S模糊系统以任意精度一致逼近紧致集上任意连续实函数的一个充分条件,并给出了数值示例.该示例表明提出的充分条件明显优于现有的其它结果.     

模糊系统逼近理论：现状与展望

近年来,各类模糊系统已被证明是万能函数逼 近器,它们能实现任意的非线性连续控制规律和动态模型．本文综述与分析了模糊系统逼近 理论研究的最新成果,并指出了进一步研究工作的方向．     

广义递阶Mamdani模糊系统及其泛逼近性

从解决模糊系统的“规则爆炸”问题出发,本文首先给出广义递阶M amdan i模糊系统的定义,然后证明其与具有中间变量的广义M amdan i模糊系统等价,并借助方形分片线性函数构造性的证明了在最大模和积分模意义下该系统是泛逼近器.最后仿真实例证实了该系统的有效性.     

正则模糊神经网络是模糊值函数的泛逼近器

通过分析多元模糊值Bernstein多项式的近似特性，证明了4层前向正则模糊神经网络（FNN）的逼近性能，该类网络构成了模糊值函数的一类泛逼近器，即在欧氏空间的任何紧集上，任意连续模糊值函数能被这类FNN逼近到任意精度，最后通过实例给出了实现这种近似的具体步骤。     

单体模糊神经网络的函数逼近能力

研究了单体模糊神经网络的函数逼近能力,由于在ＭＦＮＮｓ中神经元的基本运算由原来的积－和运算改为求极小－极大运算,网络的函数逼近性质发生了很大的改变。给出了单调传递函数的ＭＦＮＮｓ按序单调特性,连续映射定理以及非函数一致逼近定理,从而说明ＭＦＮＮｓ虽然能够保持连续映射,但不如原神经网络具有函数逼近能力。     

Necessary conditions on minimal system configuration for general MISO Mamdani fuzzy systems as universal approximators

Recent studies have shown that both Mamdani-type and Takagi-Sugeno-type fuzzy systems are universal approximators in that they can uniformly approximate continuous functions defined on compact domains with arbitrarily high approximation accuracy. In this paper, we investigate necessary conditions for general multiple-input single-output (MISO) Mamdani fuzzy systems as universal approximators with as minimal system configuration as possible. The general MISO fuzzy systems employ almost arbitrary continuous input fuzzy sets, arbitrary singleton output fuzzy sets, arbitrary fuzzy rules, product fuzzy logic AND, and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. Our necessary conditions are developed under the practically sensible assumption that only a finite set of extrema of the multivariate continuous function to be approximated is available. We have first revealed a decomposition property of the general fuzzy systems: A r-input fuzzy system can always be decomposed to the sum of r simpler fuzzy systems where the first system has only one input variable, the second one two input variables, and the last one r input variables. Utilizing this property, we have derived some necessary conditions for the fuzzy systems to be universal approximators with minimal system configuration. The conditions expose the strength as well as limitation of the fuzzy approximation: (1) only a small number of fuzzy rules may be needed to uniformly approximate multivariate continuous functions that have a complicated formulation but a relatively small number of extrema; and (2) the number of fuzzy rules must be large in order to approximate highly oscillatory continuous functions. A numerical example is given to demonstrate our new results.     

General SISO Takagi-Sugeno fuzzy systems with linear ruleconsequent are universal approximators

Takagi-Sugeno (TS) fuzzy systems have been employed as fuzzy controllers and fuzzy models in successfully solving difficult control and modeling problems in practice. Virtually all the TS fuzzy systems use linear rule consequent. At present, there exist no results (qualitative or quantitative) to answer the fundamentally important question that is especially critical to TS fuzzy systems as fuzzy controllers and models, “Are TS fuzzy systems with linear rule consequent universal approximators?” If the answer is yes, then how can they be constructed to achieve prespecified approximation accuracy and what are the sufficient renditions on systems configuration? In this paper, we provide answers to these questions for a general class of single-input single-output (SISO) fuzzy systems that use any type of continuous input fuzzy sets, TS fuzzy rules with linear consequent and a generalized defuzzifier containing the widely used centroid defuzzifier as a special case. We first constructively prove that this general class of SISO TS fuzzy systems can uniformly approximate any polynomial arbitrarily well and then prove, by utilizing the Weierstrass approximation theorem, that the general TS fuzzy systems can uniformly approximate any continuous function with arbitrarily high precision. Furthermore, we have derived a formula as part of sufficient conditions for the fuzzy approximation that can compute the minimal upper bound on the number of input fuzzy sets and rules needed for any given continuous function and prespecified approximation error bound, An illustrative numerical example is provided     

布尔型模糊系统逼近的充分条件

为了保证布尔模糊系统逼近定义在紧集上任意实值连续函数的逼近精度．给出一个估计布尔模糊系统的输入变量与输出变量各自需要构造的模糊集个数的公式,讨论如何设计布尔模糊系统．以便实现逼近任给的实值连续函数到需要的逼近精度．最后通过一个例子展示了如何设计布尔模糊系统来逼近所给的连续函数的具体方法．     

Sufficient and necessary conditions for Boolean fuzzy systems as universal approximators

A formula is first presented to compute the lower upper bounds on the number of fuzzy sets to achieve pre-specified approximation accuracy for an arbitrary multivariate continuous function. The necessary condition for Boolean fuzzy systems as universal approximators with minimal system configurations is then discussed. Two examples are provided to demonstrate how to design a Boolean fuzzy system in order to approximate a given continuous function with a required approximation accuracy.     

Comparison of necessary conditions for typical Takagi-Sugeno andMamdani fuzzy systems as universal approximators

Both Takagi-Sugeno (TS) and Mamdani fuzzy systems are known to be universal approximators. We investigate whether one type of fuzzy approximators is more economical than the other. The TS fuzzy systems are the typical two-input single-output TS fuzzy systems. We first establish necessary conditions on minimal system configuration of the TS fuzzy systems as function approximators. We show that the number of the input fuzzy sets and fuzzy rules needed by the TS fuzzy systems depend on the number and locations of the extrema of the function to be approximated. The resulting conditions reveal the strength of the TS fuzzy approximators. The drawback, though, is that a large number of fuzzy rules must be employed to approximate periodic or highly oscillatory functions. We then compare these necessary conditions with the ones that we established for the general Mamdani fuzzy systems in our previous papers. Results of the comparison unveil that the minimal system configurations of the TS and Mamdani fuzzy systems are comparable. Finally, we prove that the minimal configuration of the TS fuzzy systems can be reduced and becomes smaller than that of the Mamdani fuzzy systems if nontrapezoidal or nontriangular input fuzzy sets are used. We believe that all the results in present paper hold for the TS fuzzy systems with more than two input variables but the proof seems to be mathematically difficult. Our new findings are valuable in designing more compact fuzzy systems, especially fuzzy controllers and models which are two most popular and successful applications of the fuzzy approximators     

基于新模糊系统与T-S模糊系统的比较与研究

构造了一类新的模糊系统并应用它对非线性系统进行了辨识,而对新的模糊系统的性能作必要的分析研究。针对该系统进行分析,并将其与T-S模糊系统作比较,得出相关结论。     

TSK fuzzy function approximators: design and accuracy analysis

Fuzzy systems are excellent approximators of known functions or for the dynamic response of a physical system. We propose a new approach to approximate any known function by a Takagi-Sugeno-Kang fuzzy system with a guaranteed upper bound on the approximation error. The new approach is also used to approximately represent the behavior of a dynamic system from its input-output pairs using experimental data with known error bounds. We provide sufficient conditions for this class of fuzzy systems to be universal approximators with specified error bounds. The new conditions require a smaller number of membership functions than all previously published conditions. We illustrate the new results and compare them to published error bounds through numerical examples.     

A Pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems

In the last years, the numerous successful applications of fuzzy rule-based systems (FRBSs) to several different domains have produced a considerable interest in methods to generate FRBSs from data. Most of the methods proposed in the literature, however, focus on performance maximization and omit to consider FRBS comprehensibility. Only recently, the problem of finding the right trade-off between performance and comprehensibility, in spite of the original nature of fuzzy logic, has arisen a growing interest in methods which take both the aspects into account. In this paper, we propose a Pareto-based multi-objective evolutionary approach to generate a set of Mamdani fuzzy systems from numerical data. We adopt a variant of the well-known (2+2) Pareto Archived Evolutionary Strategy ((2+2)PAES), which adopts the one-point crossover and two appropriately defined mutation operators. (2+2)PAES determines an approximation of the optimal Pareto front by concurrently minimizing the root mean squared error and the complexity. Complexity is measured as sum of the conditions which compose the antecedents of the rules included in the FRBS. Thus, low values of complexity correspond to Mamdani fuzzy systems characterized by a low number of rules and a low number of input variables really used in each rule. This ensures a high comprehensibility of the systems. We tested our version of (2+2)PAES on three well-known regression benchmarks, namely the Box and Jenkins Gas Furnace, the Mackey-Glass chaotic time series and Lorenz attractor time series datasets. To show the good characteristics of our approach, we compare the Pareto fronts produced by the (2+2)PAES with the ones obtained by applying a heuristic approach based on SVD-QR decomposition and four different multi-objective evolutionary algorithms.     

On choosing models for linguistic connector words for Mamdani fuzzy logic systems

We examine ten antecedent connector models in the framework of a singleton or nonsingleton fuzzy logic system (FLS), to establish which models can be used. In this work, a usable connector model must lead to a separable firing degree that is a closed-form and piecewise-differentiable function of the membership function parameters and also the parameter characterizing that connector model. Our analysis shows that: for a singleton FLS where the Mamdani-product or Mamdani-minimum implication method is used, all ten antecedent connector models are usable; for a nonsingleton FLS where the Mamdani-product implication method is used, only one antecedent connector model is usable; and for a nonsingleton FLS where the Mamdani-minimum implication method is used, none of the ten antecedent connector models is usable. We also show, by examples, that the parameter of the antecedent connector model provides additional freedom in adjusting a FLS, so that the FLS has the potential to achieve better performance than a FLS that uses the traditional product or minimum t-norm for the antecedent connections.