Elsevier

Information Sciences

Volume 180, Issue 16, 15 August 2010, Pages 2991-3005
Information Sciences

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

https://doi.org/10.1016/j.ins.2010.05.004Get rights and content

Abstract

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.

Introduction

Fuzzy controllers with Variable Universe of Discourse (VUD) have received extensive attention in the last twenty years due to their high control accuracy. In 1989, an idea of linear modification of Universe of Discourse (UD), which can be viewed as an early VUD method, was introduced in [14]. Later, the method of nonlinear modification of UD was discussed in [13], [15]. Further studies can be found in [7], [8] where some specific VUD algorithms based on nonlinear modification of UD were presented for the design of fuzzy controllers. In 2001, it was reported in [9] that Hongxin Li et al. succeeded in the experiment of controlling the simulation model of quadruple inverted pendulum with a VUD method. The next year, they advanced their experiment by applying the VUD method to a real quadruple inverted pendulum and also succeeded. As controlling a quadruple inverted pendulum was a great challenge at that time in the field of control science, these two achievements surprised many control experts. Since then, an increasing number of researchers have recognized the importance of VUD fuzzy systems.

Obviously, the success of inverted pendulum experiments alone is inadequate vis-à-vis the advancement of VUD fuzzy system theory. It is well known that the key reason for the broad application of BP neural networks is that BP networks were proved to be universal approximators, i.e., they can approach any nonlinear function with the given precision. Many researchers were quite interested in the same issues of fuzzy systems and some valuable research results were obtained. For example, universal approximation properties were investigated for several typical fuzzy systems, including Mamdani fuzzy system, T-S fuzzy system and Boolean fuzzy system [1], [2], [3], [5], [6], [10], [11], [12], [20], [21], [25], [26]. Some necessary and sufficient conditions of fuzzy approximators were discussed in [4], [10], [11], [21], [22], [23], [24], and some formulas were presented for computing the number of fuzzy rules that are needed to satisfy the given accuracy. Additionally, the approximation capability of hierarchical fuzzy systems was investigated in [27], [28]. These results played an important role in the theoretical analysis and the application design of fuzzy systems. However, nearly all these studies involved in fuzzy approximation focused not on the adaptive fuzzy systems, but on the nonadaptive fuzzy systems. As the VUD fuzzy controllers are high-performance adaptive fuzzy systems [7], [8], it is useful to study their approximation properties, which gives rise to two questions: Are the VUD fuzzy systems universal approximators? If so, how can they be designed to satisfy the given precision? The main objective of this work is to answer such questions to provide theoretical support in the further research of VUD fuzzy systems.

The paper is organized as follows: After the introduction, the basic structure of VUD adaptive fuzzy systems is presented in Section 2. Section 3 proves that VUD adaptive fuzzy systems are universal approximators. Section 4 discusses their convergence properties of approximation error and a sufficient condition on UD partition. Some conclusions are given in the final section.

Section snippets

Basic structure of VUD adaptive fuzzy systems

Before introducing VUD fuzzy systems, a general fuzzy system is considered first. Denote sample steps to be k = 0, 1, 2, …, input variables to be xk=[x1k,x2k,,xnk], output variable to be yk+1, the UD of input variables to be U = U1 × U2 ×  × Un  Rn, and the UD of the output variable to be V  R. A fuzzy rule base is listed as follows:R(j)(j=1,2,,m):Ifx1isA1jandx2isA2jandandxnisAnj,thenyisBj.With a singleton fuzzifier, product inference, a central average defuzzifier [18], the fuzzy system: xk  U   Rn  yk+1  V  R

Universal approximation properties of VUD adaptive fuzzy systems

The importance of evaluating the capacity of fuzzy systems in terms of function approximation is common knowledge. Hence, it is necessary to discuss the approximation properties of VUD adaptive fuzzy systems. There is no doubt that f(x) as shown in (10) is a static function in a sample step, i.e., t  [k, k + 1). Considering that the value of αi(xik) varies with xik, however, f(x) should be regarded as a dynamic function in the whole time domain. In this regard, this section deals next with the

Convergence properties of approximation error and sufficient condition on fuzzy partition

It has been proved, in the third section, that the fuzzy systems in the form of (10) are universal approximators, i.e., there exists VUD adaptive fuzzy systems that can approach any continuous function with arbitrary precision in any sample step k. However, only when approximation error is convergent, can Theorem 1 extend to all VUD fuzzy systems in the whole time domain because f(xk) shown in (10) is a dynamic function for k = 0, 1, 2, … If approximation errors are divergent, VUD fuzzy systems need

Conclusion

Based on previous works [7], [8], [9], an algorithm has been introduced for VUD adaptive fuzzy systems in this paper. Fuzzy rules can be reproduced online by imitating initial rules in this algorithm. Using the Stone–Weierstrass theorem, VUD adaptive fuzzy systems have been proved to be universal approximators. That is, they are capable of approaching any real continuous function with arbitrary accuracy. It has been also proved that the approximation error between such a fuzzy system and the

Acknowledgement

This work was supported in part by the National Natural Science Foundation of China under Grant 60874070, and in part by the National Research Foundation for the Doctoral Program of Higher Education of China under Grant 20070533131, and in part by the Planned Science and Technology Project of Hunan Province of China under Grant 2009GK3020.

The authors are very grateful to the editors and the anonymous referees for their constructive comments and suggestions that help in the quality improvement

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