Synthesis of Havriliak-Negami functions for time-domain system identification

Abstract

Fractional differentiation models have proven their usefulness in representing high dimensional systems with only few parameters. Generally, two elementary fractional functions are used in time-domain identification: Cole-Cole and Davidson-Cole functions. A third elementary function, called Havriliak-Negami, generalizes both previous ones and is particularly dedicated to dielectric systems. The use of this function is however not very popular in time-domain identification because it has no simple analytical impulse response. The only synthesis method of Havriliak-Negami elementary functions proposed in the literature is based on diffusive representation which sets restrictive conditions on fractional orders. A new synthesis method, with no such restrictions, is developed in this paper. For that purpose Havriliak-Negami function is first split into a Davidson-Cole function and a complementary one. Both functions are then synthesized in a limited frequency band using poles and zeros recursive distribution developed by Oustaloup (1995). As an example, this Havriliak-Negami function is used for a thermal system modeling.