Affine Takagi-Sugeno fuzzy modelling algorithm by fuzzy c-regression models clustering with a novel cluster validity criterion

Abstract

An effective approach is developed to establish affine Takagi-Sugeno (T-S) fuzzy model for a given nonlinear system from its input–output data. Firstly, the fuzzy c-regression model (FCRM) clustering technique is applied to partition the product space of the given input–output data into hyper-plan-shaped clusters. Each cluster is essentially a basis of the fuzzy rule that describes the system behaviour, and the number of clusters is just the number of fuzzy rules. Particularly, a novel cluster validity criterion for FCRM is set up to choose the appropriate number of clusters (rules). Once the number of clusters is determined, the consequent parameters of each IF-THEN rule are directly obtained from the functional cluster representatives (affine linear functions). The antecedent fuzzy sets of each IF-THEN fuzzy rule are acquired by projecting the fuzzy partitions matrix U onto the axes of individual antecedent variable to obtain point-wise defined fuzzy sets and to approximate these point-wise defined fuzzy sets by normal bell-shaped membership functions. Additionally, a check and repartition algorithm is suggested to prevent the inappropriate premise structure where separate regions of data shared the same regression model. Finally, the gradient descent algorithm is included to adjust the fuzzy model precisely. An affine T-S fuzzy model with compact IF-THEN rules could thus be generated systematically. Several simulation examples are provided to demonstrate the accuracy and effectiveness of the affine T-S fuzzy modelling algorithm.