Piecewise parametric polynomial fuzzy sets

We present a scheme for tractable parametric representation of fuzzy set membership functions based on the use of a recursive monotonic hierarchy that yields different polynomial functions with different orders. Polynomials of the first order were found to be simple bivalent sets, while the second order polynomials represent the typical saw shape triangles. Higher order polynomials present more diverse membership shapes. The approach demonstrates an enhanced method to manage and fit the profile of membership functions through the access to the polynomials order, the number and the multiplicity of anchor points as wells as the uniformity and periodicity features used in the approach. These parameters provide an interesting means to assist in fitting a fuzzy controller according to system requirements. Besides, the polynomial fuzzy sets have tractable characteristics concerning the continuity and differentiability that depend on the order of the polynomials. Higher order polynomials can be differentiated as many times as the order of the polynomial less the multiplicity of the anchor points. An algorithmic optimization approach using the steepest descent method is introduced for fuzzy controller tuning. It was shown that the controller can be optimized to model a certain output within small number of iterations and very small error margins. The mathematics generated by the approach is consistent and can be simply generalized to standard applications. The recursive propagation was noticed for its clarity and ease of calculations. Further, the degree of association between the sets is not limited to the neighbors as in traditional applications; instead, it may extend beyond. Such approach can be useful in dynamic fuzzy sets for adaptive modeling in view of the fact that the shape parameters can be easily altered to get different profiles while keeping the math unchanged. Hypothetically, any shape of membership functions under the partition of unity constraint can be produced. The significance of the mentioned characteristics of such modeling can be observed in the field of combinatorial and continuous parameter optimization, automated tuning, optimal fuzzy control, fuzzy-neural control, membership function fitting, adaptive modeling, and many other fields that require customized as well as standard fuzzy membership functions. Experimental work of different scenarios with diverse fuzzy rules and polynomial sets has been conducted to verify and validate our results.