578 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 4, AUGUST 2001 A Fast Approach for Automatic Generation of Fuzzy Rules by Generalized Dynamic Fuzzy Neural Networks Shiqian Wu, Meng Joo Er, Member, IEEE, and Yang Gao Abstract—In this paper, a fast approach for automatically generating fuzzy rules from sample patterns using generalized dynamic fuzzy neural networks (GD-FNNs) is presented. The GD-FNN is built based on ellipsoidal basis function and function- ally is equivalent to a Takagi–Sugeno–Kang fuzzy system. The salient characteristics of the GD-FNN are: 1) structure identifi- cation and parameters estimation are performed automatically and simultaneously without partitioning input space and selecting initial parameters a priori; 2) fuzzy rules can be recruited or deleted dynamically; 3) fuzzy rules can be generated quickly without resorting to the backpropagation (BP) iteration learning, a common approach adopted by many existing methods. The GD-FNN is employed in a wide range of applications ranging from static function approximation and nonlinear system iden- tification to time-varying drug delivery system and multilink robot control. Simulation results demonstrate that a compact and high-performance fuzzy rule-base can be constructed. Compre- hensive comparisons with other latest approaches show that the proposed approach is superior in terms of learning efficiency and performance. Index Terms—Ellipsoidal basis function (EBF), function approximation, fuzzy rule extraction, on-line self-organizing learning, Takagi–Sugeno–Kang (TSK) fuzzy reasoning. I. INTRODUCTION F UZZY logic is a key tool to express knowledge of domain experts so that valuable experience of human beings can be incorporated into controllers design and applied to handle real-life situations that the classical control approach finds dif- ficult or impossible to tackle. Fuzzy systems, as a model-free ap- proach, can approximate any continuous function on a compact set to any accuracy. It has been shown that fuzzy-logic-based modeling and control could serve as a powerful methodology for dealing with imprecision and nonlinearity efficiently [1], [2]. However, the conventional way of designing a fuzzy system has been a subjective approach. It is difficult for a designer, even a domain expert, to examine all the input–output data from a complex system so as to find appropriate number of rules to im- plement the fuzzy system. The main issues associated with a fuzzy system are 1) pa- rameter estimation, which involves determining the parameters of premises and consequences, and 2) structure identification, Manuscript received October 24, 2000; revised February 13, 2001. S. Wu is with the Centre for Signal Processing, Innovation Centre, Nanyang Technological University, Singapore 637722, Singapore. M. J. Er and Y. Gao are with the School of Electrical and Electronic Engi- neering, Nanyang Technological University, Singapore 639798, Singapore. Publisher Item Identifier S 1063-6706(01)06659-0. which concerns partitioning the input space and determining the number of fuzzy rules for a specific performance [3]. To acquire fuzzy rules, several paradigms have been developed to generate fuzzy rules from numerical training data [4]–[8]. In general, these approaches are simple and fast, i.e., they involve neither time-consuming iterative procedures nor a complicated rule-generation mechanism. The major drawbacks of these methods are that they are heuristic methods and need to predefine the membership function and the number of fuzzy rules. Recently, more attentions have been focused on fuzzy neural networks (FNNs) to acquire fuzzy rules based on the learning ability of neural networks [9]. The typical approach of FNNs is to build standard neural networks, which are designed to approximate a fuzzy algorithm or a process of fuzzy inference through the structure of neural networks [9]. The main idea is the following: assuming that some particular membership functions have been defined, we begin with a fixed number of rules by resorting to either the trial-and-error method [10]–[13] or expert knowledge [14], [15]. Next, the parameters are modified by the BP learning algorithm [10]–[12], [14], [15] or hybrid learning algorithm [13]. These FNNs can readily solve two problems of conventional fuzzy reasoning: 1) lack of systematic design for membership functions and 2) lack of adaptability for possible changes in the reasoning environment. These two problems intrinsically concern parameter estimation. Nevertheless, structure identification, such as partitioning the input and output space and determination of number of fuzzy rules, is still time-consuming. The reason is that, as shown in [16], the problem of determining the number of hidden nodes in neural networks can be viewed as the choice of the number of fuzzy rules. Different from the aforementioned FNNs, several adaptive paradigms have been presented whereby not only can the connection weights be adjusted but also the structure can be self-adaptive during learning [16]–[18]. In [17], a hierarchically self-organizing approach, by which the structure was identified by input–output pairs, was developed. An on-line self-constructing paradigm was proposed in [18]. The premise structure in [18] is determined by clustering the input via an on-line self-organizing learning approach. As to the consequent part, only a singleton value selected by a clustering method is assigned to each rule initially. Afterwards, some additional significant terms (input variables) selected via a pro- jection-based correlation measure for each rule will be added to the consequent part incrementally as learning progresses. Accordingly, the FNN proposed in [18] is inherently a modified Takagi–Sugeno–Kang (TSK) fuzzy system. As BP learning 1063–6706/01$10.00 © 2001 IEEE