Energy formulation for Preisach models

Preisach models formulated in terms of density or measure-based expansions have proven highly successful for characterizing hysteresis and constitutive nonlinearities in materials where the underlying physics is difficult to quantify. This provides a rich mathematical framework for characterizing nonlinear material behavior as well as a framework which facilitates either full or approximate inversion for linear control design. However, the lack of an energy basis for Preisach representations yields models which often have a large number of parameters and are difficult to update to accommodate changing operating conditions (e.g., temperature) since the model parameters are not correlated with physical quantities. Moreover, it is difficult in general to incorporate the frequency-dependence exhibited by essentially all smart materials without resorting to vector-valued parameters or measures which much be identified throughout the range of operation for the system. In this paper, we develop an energy formulation for Preisach models through consideration of appropriate Gibbs and Helmholtz free energy representations. This permits the incorporation of frequency and temperature-dependence in the underlying basis, rather than in parameters identified for a specific system which expands significantly the flexibility of the technique.