Identification of hysteretic control influence operators representing smart actuators, Part II: Convergent approximations

In a previous paper the authors investigated the lower semicontinuity properties of two generalizations of the classical Preisach operator the smoothed Preisach operator and the Krasnoselskii/Pokrovskii (KP) integral hysteresis operators. In particular, it was demonstrated that the output least squares identification problem for the KP operator is well-posed over compact subsets of the Preisach plane. The identification of the hysteretic control influence operator was shown to be equivalent to the identification of a measure in the space of probability measures taken with the weak* topology. In this paper, a consistent and convergent approximation scheme is introduced for this class of integral hysteresis operator. The Galerkin approximation scheme is shown to be function space parameter convergent. A numerical example is presented that illustrates aspects of the theory derived in this paper.