Solving singleton type-2 fuzzy relation equations based on semi-tensor product of matrices

The problem of solving singleton Type-2 fuzzy relation equations is considered. First, using Zadeh´s Extension Principle, in which the t-norm is set to be minimum t-norm, Type-1 fuzzy relation is extended to Type-2 fuzzy relation, the corresponding Type-2 fuzzy relation equation (T2 FRE) is then established. Second, the solution of singleton T2 FRE is split into two parts as primary sub-matrix and secondary sub-matrix, then the semi-tensor product of matrices and the algebra form of fuzzy logic functions are used to analyze the structure of primary sub-matrix. Meanwhile the minimum fuzzy inference is applied to reveal the conditions that the secondary sub-matrix should satisfy. Based on this, an algorithm to solving singleton T2 FREs is developed. Finally, some numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.