Abstract
Recently, various adaptive fuzzy control schemes have been proposed to deal with nonlinear systems with poorly understood dynamics by using the parameterized fuzzy approximator. However, all of the adaptive fuzzy control systems have been designed with the Gaussian membership functions so that in such a system there is only one kind of shape of membership function that appears to symmetric bell-shaped curve. Namely, it is unable to specify asymmetric membership functions, which are important in practical applications. In this paper, first, we use the product of two Sigmoidal functions to express membership function, which will refer to as Sigmoidal membership function and is able to specify various shapes of membership function such as asymmetric triangular, trapezoid, either open left or right and so on. Next, by using the first order Taylor's expansion, we present an approximation theorem concerning the properties of the fuzzy approximator with the Sigmoidal membership function so that we can stably tune all parameters, which appear either linearly or nonlinearly in the fuzzy approximator, in our adaptive fuzzy control system. Also it is shown the proposed fuzzy adaptive controller guarantees tracking error, between outputs of considered system and desired values, to be asymptotically in decay.
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