Approximation theory of fuzzy systems-SISO case

In this paper, the approximation problem of SISO fuzzy systems is discussed. Based on the fact that fuzzy systems can be represented by a linear combination of fuzzy basic functions (FBF's), we first give a systematic and detailed analysis of FBF's and present five properties of FBF's: structure similarity and compatibility between the membership functions and FBF's, complementarity and less fuzziness of FBF's, and composition of fuzzy systems. These properties provide a clear picture of the shape and features of FBF's. Based on these properties of FBF's, we obtain some basic approximation properties of fuzzy systems: basic approximation property, uniform approximation property, uniform convergent property and universal approximation property. These results reveal the basic approximation mechanism of fuzzy systems and verify a few basic ideas in the design of fuzzy systems     

Approximation properties of fuzzy systems generated by the mininference

This paper discusses the approximation properties of fuzzy systems generated by the min inference. Firstly, the paper analyzes the properties of fuzzy basis functions (FBFs); Then based on the properties of FBPs, several basic approximation properties concerning approximation mechanisms, uniform approximation bounds, uniform convergency, and universal approximation are obtained. Further, the similarity and difference between the fuzzy systems generated by the product inference and by the min inference are discussed.     

Kernel shapes of fuzzy sets in fuzzy systems for function approximation

The shapes of if-part fuzzy sets affect the approximating capability of fuzzy systems. In this paper, the fuzzy systems with the kernel-shaped if-part fuzzy sets are built directly from the training data. It is proved that these fuzzy systems are universal approximators and their uniform approximation rates can be estimated in the single-input-single-output (SISO) case. On the basis of these rates, the relationships between the approximating capability and the shapes of if-part fuzzy sets are developed for the fuzzy systems. Furthermore, the sinc functions that serve as input membership functions are proved to have the almost best approximation property in a particular class of membership functions. The theoretical results are confirmed from the simulation data. In addition, the estimations of the uniform approximation rates are extended to the multi-input-single-output (MISO) case.     

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.     

Approximation capability of fuzzy systems using translations anddilations of one fixed function as membership functions

This paper reports on a related study on approximation theory of fuzzy systems. First, some basic principles are presented to construct membership functions. Then, an approach is proposed to form membership functions by using translations and dilations of one fixed function (called a basis function) which is very similar to that in wavelets analysis. The properties of this type of membership function reflect the advantages of the given approach. Finally, it is proved that fuzzy systems based on such membership functions are universal approximators under certain mild conditions on the basis function. This conclusion expands the family of fuzzy systems which can be universal approximators     

K-积分模意义下折线模糊神经网络的泛逼近性

为克服模糊数运算的复杂性引入折线模糊数的定义,利用折线模糊数的优良性质获得了两个重要不等式,并给出实例说明折线模糊数的逼近能力有效.其次,引进K-拟可加积分和K-积分模概念,在折线模糊数空间满足可分性的基础上,借助于模糊值简单函数和模糊值Bernstein多项式研究了若干函数空间的稠密性问题,获得了可积有界模糊值函数类依K-积分模构成完备可分的度量空间.最后,在K-积分模意义下讨论了四层正则折线模糊神经网络对模糊值简单函数的泛逼近性,进而得到该网络对可积有界函数类也具有泛逼近性.该结果表明正则折线模糊神经网络对连续模糊系统的逼近能力可以推广为对一般可积系统的逼近能力.     

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．     

基于自适应模糊逻辑系统的一类混沌系统同步控制

针对一类带有未知函数和干扰的混沌系统,进行了基于自适应模糊逻辑系统的自适应同步控制器的设计。首先基于模糊逼近原理,通过对该混沌系统中未知函数的输入输出进行采样,根据采样数据信息设计出具有参数自适应功能的Mamdani型模糊逻辑系统;然后利用该模糊逻辑系统给出一种带有参数自适应的驱动响应同步控制器设计方法;最后通过数值仿真算例表明了该方法的有效性。     

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.     

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．     

基于模糊推理的变步长LMS自适应滤波算法

LMS算法是一种基于最速下降法的最小均方误差自适应滤波算法.为了提高LMS算法的收敛速度,依据模糊控制原理,推导出一种结构简单的步长与误差的非线性函数关系,进而得出一种新的变步长LMS自适应滤波算法（FVSLMS）,该算法结构简单,易于实现.在理论上,根据万能逼近定理,用FVSLMS算法可以以任意精度逼近步长与误差的非线性函数关系,因此它可以作为以误差调节步长的变步长LMS算法的一类统一形式.最后,通过计算机仿真说明了FVSLMS算法具有较好的收敛性能.     

Some approximation properties of adaptive fuzzy systems with variable universe of discourse

In the last 20 years, while most research on fuzzy approximation theory has focused on nonadaptive fuzzy systems, little work has been done on adaptive fuzzy systems. This paper introduces an algorithm for adaptive fuzzy systems with Variable Universe of Discourse (VUD). By means of contraction-expansion factors, universe of discourse can be modified online, and fuzzy rules can reproduce automatically to adapt to the modified universe of discourse. Thus, dependence on the size of initial rule base is greatly reduced. Using Stone-Weierstrass theorem, VUD adaptive fuzzy systems are proved to be universal approximators with two-order approximation accuracy. In addition, the convergence properties of approximation error are discussed, and a sufficient condition is presented to partition universe of discourse and to calculate the size of rule base. An example is also given to illustrate the approximation power of VUD adaptive fuzzy systems.     

Approximation accuracy analysis of fuzzy systems as functionapproximators

This paper establishes the approximation error bounds for various classes of fuzzy systems (i.e., fuzzy systems generated by different inferential and defuzzification methods). Based on these bounds, the approximation accuracy of various classes of fuzzy systems is analyzed and compared. It is seen that the class of fuzzy systems generated by the product inference and the center-average defuzzifier has better approximation accuracy and properties than the class of fuzzy systems generated by the min inference and the center-average defuzzifier, and the class of fuzzy systems defuzzified by the MoM defuzzifier. In addition, it is proved that fuzzy systems can represent any linear and multilinear function and explicit expressions of fuzzy systems generated by the MoM defuzzified method are given     

Approximation Capabilities of Hierarchical Fuzzy Systems

Derived from practical application in location analysis and pricing, and based on the approach of hierarchical structure analysis of continuous functions, this paper investigates the approximation capabilities of hierarchical fuzzy systems. By first introducing the concept of the natural hierarchical structure, it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters. Then, based on Kolmogorov's theorem, it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property. Further, the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed     

一类模糊神经网络的函数逼近能力

根据多元Fourier变换理论提出一种多元函数的积分变换方法.据此讨论一类模糊 神经网络作为函数逼近器时的逼近误差与其结构关系,得到模糊神经网络的逼近误差与其隐 含层的节点数成反比的结论.并论证了模糊神经网络的函数逼近精度与输入变量数无关.     

模糊安全性和活性

形式规约使用形式语言构建所开发的软硬件系统的规约,刻画系统的模型和性质。其中,性质规约中的分支时间规约对于系统验证有着非常重要的作用。在经典情形下,系统性质规约是基于二值逻辑的,不能描述不一致或不确定的信息。因此,将其推广到模糊逻辑背景下,有助于对模糊系统进行形式验证。文中首先给出了性质规约中分支时间属性在模糊背景下的形式化定义,重点研究了其中的安全性和活性;然后,定义了两种闭包操作,从而产生了4种类型的属性,即泛安全性、泛活性、存在安全性和存在活性;最后,证明了每个分支时间属性,或是存在安全性和存在活性的交,或是泛安全性和泛活性的交,或是存在安全性和泛活性的交。     

Comments on Singh and Zeng: “Approximation theory of fuzzysystems-SISO case” [and reply]

The author comments on the paper by Singh and Zeng (see ibid., vol.2, no.2, p.162-76, 1994). He states that every bounded function f: R→R has an exact representation as an additive fuzzy system. If f is not constant, one fuzzy set and two rules define the system. Otherwise, a single rule suffices. This result shows that the approximation properties of one-input fuzzy systems derive solely from interpolation between output extrema. The basis for the interpolation at any point is the value of the input fuzzy sets at that point. In reply Singh and Zeng state that in the comments by Watkins, it is proven that every SISO function can be exactly represented by a fuzzy system, which implies that fuzzy approximation (i.e., to approximate functions by fuzzy systems) is unnecessary or moot. However, they state that this conclusion is invalid because his presented representation scheme does not meet the basic requirements in the applications of fuzzy systems and is impractical     

Approximation of fuzzy-valued functions by regular fuzzy neural networks and the accuracy analysis

In this paper, it is shown that four-layer regular fuzzy neural networks can serve as universal approximators for the sendograph-metric-continuous fuzzy-valued functions. The proof is constructive. We propose a principled method to design four-layer regular fuzzy neural neural network to approximate the target functions. In the previous work, a step function is used as the activation function. To improve the approximation accuracy, in the present work, we also consider using a semi-linear sigmoidal function as the activation function. Then it shows how to design the regular fuzzy neural networks (RFNNs) when the activation functions are the semi-linear sigmoidal function and the step function, respectively. After analyze the approximation accuracy of these two classes of RFNNs, it is found that the former has a much better performance than the latter in approximation accuracy. This conclusion also holds when the target functions satisfy other types of continuity. So the results in this paper can also be used to improve the related work. At last, we give a simulation example to validate the theoretical results.     

Subsystem inference representation for fuzzy systems based uponproduct-sum-gravity rule

A new representation which expresses a product-sum-gravity (PSG) inference in terms of additive and multiplicative subsystem inferences of single variable is proposed. The representation yields additional insight into the structure of a fuzzy system and produces an approximate functional characterization of its inferred output. The form of the approximating function is dictated by the choice or polynomial, sinusoidal, or other designs of subsystem inferences. With polynomial inferences, the inferred output approximates a polynomial function the order of which is dependent on the numbers of input membership functions. Explicit expressions for the function and corresponding error of approximation are readily obtained for analysis. Subsystem inferences emulating sinusoidal functions are also discussed. With proper scaling, they produce a set of orthonormal subsystem inferences. The orthonormal set points to a possible “modal” analysis of fuzzy inference and yields solution to an additive decomposable approximation problem. This work also shows that, as the numbers of input membership functions become large, a fuzzy system with PSG inference would converge toward polynomial or Fourier series expansions. The result suggests a new framework to consider fuzzy systems as universal approximators     

Sufficient conditions on uniform approximation of multivariatefunctions by general Takagi-Sugeno fuzzy systems with linear ruleconsequent

We have constructively proved a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems to be universal approximators. The systems use any types of continuous fuzzy sets, fuzzy logic AND, fuzzy rules with linear rule consequent and the generalized defuzzifier. We first prove that the TS fuzzy systems can uniformly approximate any multivariate polynomial arbitrarily well, and then prove they can also uniformly approximate any multivariate continuous function arbitrarily well. We have derived a formula for computing the minimal upper bounds on the number of fuzzy sets and fuzzy rules necessary to achieve the prespecified approximation accuracy for any given bivariate function. A numerical example is furnished. Our results provide a solid-theoretical basis for fuzzy system applications, particularly as fuzzy controllers and models