Identication of fuzzy function via interval analysis

A number of techniques have been introduced to construct fuzzy models from measured data. One of the most common is the use of mathematical parametric models. In this paper, a new approach based on interval analysis is proposed to compute guaranteed estimates of suitable characteristics of the set of all values of the fuzzy parameter vector such that the error between experimental data and the model outputs belongs to some predefined feasible set. Subpavings consisting of boxes with nonzero width are used to encompass all the acceptable values of the parameter vector. The problem of estimating the parameters of the model is viewed as one of set inversion, which is solved in an approximate but guaranteed way with the tools of interval analysis. The estimation task is formulated here as a constrained optimization problem. Our approach emphasizes the use of interval mathematics for fuzzy representation, which is especially suited to nonlinear models, a situation where most available methods fail to provide any guarantee on the results. An algorithm is proposed, which makes it possible to obtain all fuzzy parameter vectors that are consistent with the data. Properties of this algorithm are established and illustrated on a simple example.