Sufficient conditions to impose derivative constraints on MISO Takagi-Sugeno fuzzy logic systems

A fuzzy logic system (FLS) can be completely described through its behavioral function. If the analytical expression of this function is known, then the system synthesis consists in the realization of a structure implementing it in the most convenient way. So, the problem of the FLS design can be considered as a function approximation problem. Many methods have been proposed in literature to minimize a certain error parameter. Nevertheless, sometimes the specifications of the problem need some further analytical properties of the approximating function. This paper analyzes multiple-input-single-output (MISO) FLSs with polynomial membership functions inspecting the case of two input variables. In particular, it will be shown what characteristics such systems should have in order to respect the constraints imposed on the derivatives of the implemented function, such as their continuity up to a desired order and even their value in the grid points of the input space. Furthermore, we prove that these systems can uniformly approximate, under certain conditions, any multivariate function with a desired approximation accuracy both on the target function and its partial derivatives. Proper illustrative examples will show the behavior of these FLSs comparing their performance with other methods proposed in literature.