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This paper presents a constructive Takagi-Sugeno fuzzy modeling method for a general class of nonlinear systems. This method is particularly suitable for stability analysis based on piecewise quadratic Lyapunov functions. The modeling error is appropriately inserted into the model and an algorithm is proposed to automatically determine the model parameters to keep the modeling error smaller than a desired upper bound. Based on the constructed fuzzy model, exponential stability analysis is performed and the stability constraints are transformed into linear matrix inequalities. Modeling error is also included in the stability analysis to validate the results for the original nonlinear system. The way to utilize the modeling method and stability analysis to systematically find a Lyapunov function for a nonlinear system is demonstrated via an example and the potential capability of the method in estimating the domain of attraction is discussed.