High order α-planes integration: A new approach to computational cost reduction of General Type-2 Fuzzy Systems

Nowadays, there are different representations of Generalized Type-2 Fuzzy Sets that consider a non-uniform distribution of the uncertainty, for example, the Geometric approach, the Z-Slices method and the alpha-planes approximation. Each representation has advantages and disadvantages, however, the present work is focused on the alpha-planes representation, and this representation consists on realizing a horizontal discretization, the solution of each horizontal slice, and then the integration of these planes. Each horizontal slice results in an Interval Type-2 Fuzzy System, so, the computational cost is proportional to the discretization level, which means, is proportional to the number of alpha-planes used for modeling the Generalized Type-2 FS. The aim of this work is reducing the computational cost of Generalized Type-2 FS by a new approach of alpha-planes representation. In this paper the Newton Cotes quadrature for the alpha-planes integration is proposed, achieving in this way a high-level discrete integration compared with the conventional alpha-planes integration. The proposed approach aims at reducing the number of alpha-planes necessary to obtain a good approximation of Generalized Type-2 FS. In order to validate the proposed approach, a set of experiments was realized with a randomly generated Generalized Type-2 Fuzzy Sets, and they are realized with a different number of alpha-planes in order to compare the performance of the proposed approach with respect to the conventional approach. On the other hand, the proposed approach was also applied to a control problem, as an example of applications of the proposed approach to real-world problems.