An improved shape-preserving affine Takagi-Sugeno controller using splines

First-order Takagi-Sugeno (TS) fuzzy systems are characterized by a multivariate fuzzy partition in the operating domain and a local affine function within each fuzzy region. This fuzzy inference method presents some undesirable properties in the regions where the membership functions of the antecedent fuzzy partitions overlap. The output function of a SISO affine TS system is in general a non-convex combination of the corresponding piecewise affine functions and the local derivative of the output is not bounded by the derivatives of the corresponding consequent affine functions. These characteristics can influence negatively the stability and robustness of the corresponding fuzzy system. In this paper, to overcome these drawbacks, SISO affine TS models defined on trapezoidal fuzzy partitions are transformed into equivalent zero-order TS models that are defined on triangular fuzzy partitions. Moreover, to improve the smoothness and continuity order of the output function, a B-spline convolution filter is applied to transform the corresponding C-0 triangular fuzzy partition into a C-m spline fuzzy partition (phi-spline fuzzy partition). The obtained spline fuzzy partition preserves the output function in the cores of the initial fuzzy intervals. This technique is directly generalized to a MISO system by the corresponding tensor product. Some examples are given to practically demonstrate the capability of the approach.