Some approximation properties of adaptive fuzzy systems with variable universe of discourse

In the last 20 years, while most research on fuzzy approximation theory has focused on nonadaptive fuzzy systems, little work has been done on adaptive fuzzy systems. This paper introduces an algorithm for adaptive fuzzy systems with Variable Universe of Discourse (VUD). By means of contraction–expansion factors, universe of discourse can be modified online, and fuzzy rules can reproduce automatically to adapt to the modified universe of discourse. Thus, dependence on the size of initial rule base is greatly reduced. Using Stone–Weierstrass theorem, VUD adaptive fuzzy systems are proved to be universal approximators with two-order approximation accuracy. In addition, the convergence properties of approximation error are discussed, and a sufficient condition is presented to partition universe of discourse and to calculate the size of rule base. An example is also given to illustrate the approximation power of VUD adaptive fuzzy systems.