Construction of Shape-preserving Affine Takagi-Sugeno Systems via Linear-rational Spline Fuzzy Partitions

The output function of standard affine Takagi-Sugeno systems presents shape failures and does not have continuous derivatives in the corresponding output function. These characteristics give an undesirable behaviour in many practical applications such us fuzzy control or fuzzy modelling. In order to avoid these problems, this paper presents a new method for deriving a shape-preserving affine Takagi-Sugeno model, using a standard trapezoidal fuzzy partition specification. Each univariate trapezoidal fuzzy partition is primarily transformed into an equivalent triangular one. An adaptive first-order symmetrical B-spline filter is later applied on each triangular partition in order to derive a mixed linear-rational spline fuzzy partition. The obtained model retains monotony, convexity and positivity of the corresponding control points defined on the cores of the original fuzzy partition. These properties are due to the characteristics of the linear-rational spline partitions obtained. Finally, a simple example is given to practically show the usefulness of the approach.