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This paper discusses the stability of Takagi-Sugeno fuzzy models via the so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. Based on a fuzzy Lyapunov approach, we gives the stability conditions for open-loop fuzzy systems. All the conditions derived here are represented in terms of linear matrix inequalities (LMIs) and contain upper bounds of the time derivative of premise membership functions as LMI variables. Hence, the treatment of the upper bounds play an important and effective role in system analysis and design. In addition, relaxed stability conditions are also derived by considering the property of the time derivative of premise membership functions. Several analysis and design examples illustrate the utility of the fuzzy Lyapunov approach.
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Tanaka et al. (2001) studied this question.
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