Construction of fuzzy linguistic model

Using linguistic variables to describe the behavior of a hybrid system, which consists of a discrete event system and a continuous system, could make the design of the controller and verification of the system perform on an unified framework. In this paper, we show the construction of a fuzzy linguistic model from a given mathematical model of a physical system. By considering the state-space realization of the model, the system construction problem can be transformed into a function approximation problem, We propose to use projection theorem in obtaining an optimal fuzzy system which is the best approximation of a given nonlinear function in L₂(U) space. We show that the existence and uniqueness of the optimal solution is assured when the fuzzy basis functions (FBFs) are linearly independent. We demonstrate that the dependence of the basis functions can be examined by checking the condition of the Gram determinant associated to the FBFs. Finally, a method is proposed in converting the optimal coefficients into fuzzy sets to obtain a fuzzy linguistic model