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Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H/sup infinity/ control theory, and linear matrix inequalities
K. Tanaka, T. Ikeda, H. O. Wang
vol. 4, no. 1, pp. 1-13, Feb. 1996

This paper presents stability analysis for a class of uncertain nonlinear systems and a method for designing robust fuzzy controllers to stabilize the uncertain nonlinear systems, First, a stability condition for Takagi and Sugeno`s fuzzy model is given in terms of Lyapunov stability theory. Next, new stability conditions for a generalized class of uncertain systems are derived from robust control techniques such as quadratic stabilization, H/sup infinity/ control theory, and linear matrix inequalities. The derived stability conditions are used to analyze the stability of Takagi and Sugeno`s fuzzy control systems with uncertainty which can be regarded as a generalized class of uncertain nonlinear systems, The design method employs the so-called parallel distributed compensation, important issues for the stability analysis and design are remarked. Finally, three design examples of fuzzy controllers for stabilizing nonlinear systems and uncertain nonlinear systems are presented.

An approach to fuzzy control of nonlinear systems: stability and design issues
H. O. Wang, K. Tanaka, M. F. Griffin
vol. 4, no. 1, pp. 14-23, Feb. 1996

Presents a design methodology for stabilization of a class of nonlinear systems. First, the authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" is employed. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, the stability analysis and control design problems can be reduced to linear matrix inequality (LMI) problems. Therefore, they can be solved efficiently in practice by convex programming techniques for LMIs. The design methodology is illustrated by application to the problem of balancing and swing-up of an inverted pendulum on a cart.

ID3-derived fuzzy rules and optimized defuzzification for handwritten numeral recognition
Chi Zheru, Yan Hong
vol. 4, no. 1, pp. 24-31, Feb. 1996

Presents a technique to produce fuzzy rules based on the ID3 approach and to optimize defuzzification parameters by using a two-layer perceptron. The technique overcomes the difficulties in a conventional syntactic approach to handwritten character recognition, including problems of choosing a starting or reference point, scaling, and learning by machines. The authors` technique provides: a way to produce meaningful and simple fuzzy rules; a method to fuzzify ID3-derived rules to deal with uncertain, noisy, or fuzzy data; and a framework to incorporate fuzzy rules learned from the training data and those extracted from human recognition experience. The authors` experimental results on NIST Special Database 3 show that the technique out-performs the straight forward ID3 approach. Moreover, ID3-derived fuzzy rules can be combined with an optimized nearest neighbor classifier, which uses intensity features only, to achieve a better classification performance than either of the classifiers. The combined classifier achieves a correct classification rate of 98.6% on the test set.

H/sup infinity / tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach
Sen Chen Bor, Hsiang Lee Ching, Chan Chang Yeong
vol. 4, no. 1, pp. 32-43, Feb. 1996

A fuzzy logic controller equipped with a training (adaptive) algorithm is proposed in this work to achieve H/sup infinity / tracking performance for a class of uncertain (model free) nonlinear single-input single-output (SISO) systems with external disturbances. An attempt is also made to create a bridge between two important control design techniques, i.e., H/sup infinity / control design and fuzzy control design, so as to supply H/sup infinity / control design with more intelligence and fuzzy control design with better performance. The perfect matching of parameters in an adaptive fuzzy logic system is generally deemed impossible. Therefore, a desired tracking performance cannot be guaranteed in the conventional adaptive fuzzy control systems. In this study, the influence of both fuzzy logic approximation error and external disturbance on the tracking error is attenuated to a prescribed level. Both indirect and direct adaptive fuzzy controllers are employed to treat this H/sup infinity / tracking problem. The authors` results indicate that arbitrarily small attenuation level can be achieved via the proposed adaptive fuzzy control algorithm if a weighting factor of control variable is adequately chosen. The proposed design method is also useful for the robust tracking control design of the nonlinear systems with external disturbances and a large uncertainty or unknown variation in plant parameters and structures. Furthermore, only smooth control signals are needed via the proposed control designs. Two simulation examples are given finally to illustrate the performance of the proposed methods. Computer simulation results confirm that the effect of both the fuzzy approximation error and external disturbance on the tracking error can be attenuated efficiently by the proposed method.

Approximation accuracy analysis of fuzzy systems as function approximators
Jun Zeng Xiao, M. G. Singh
vol. 4, no. 1, pp. 44-63, Feb. 1996

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.

Generalized defuzzification strategies and their parameter learning procedures
Jiang Tao, Li Yao
vol. 4, no. 1, pp. 64-71, Feb. 1996

Defuzzification is a procedure of crucial importance for fuzzy systems because a final crisp output (control) action is required in many theoretical and practical applications. The choice of defuzzification strategy, therefore, can directly affect the success of such applications. Among the existing strategies, neither the center of area (COA) nor the mean of maximum (MOM) emerges as the better defuzzification strategy. A compromise strategy that combines the two methods may offer a synergetic solution. In this paper, the authors introduce two new objective defuzzification strategies, Gaussian distribution transformation-based defuzzification (GTD) and polynomial transformation-based defuzzification (PTD), which are based on a discrete universe of discourse. Both strategies can perform better than the existing strategies and the PTD strategy offers a generalized defuzzification tool for a wide class of practical problems. Both strategies include the COA and MOM strategies as special cases, and both are based on parameter learning processes using the extended Kalman filter as their iterative improvement algorithms on sample database containing fuzzy sets and the associate defuzzified values. The proposed parameter learning procedures are capable of either off-line or on-line processing.

FLC design for bounded separable functions with linear input-output relations as a special case
B. Armstrong, USA.
vol. 4, no. 1, pp. 72-9, Feb. 1996

A systematic procedure is presented for designing a knowledge base which exactly implements a specified bounded separable function in fuzzy logic. The design of a fuzzy logic control (FLC) for local linear control is a special case of the result. Examples, including controller design for a nonlinear process control application, are presented.

Comments on Singh and Zeng: "Approximation theory of fuzzy systems-SISO case"
F. Watkins, Jun Zeng Xiao, M. G. Singh
vol. 4, no. 1, pp. 80-1, Feb. 1996

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 to 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.