Mathematical Problems in Engineering
Volume 3 (1997), Issue 4, Pages 287-328

Identification of hysteretic control influence operators representing smart actuators part I: Formulation

H. T. Banks,1 A. J. Kurdila,2,3 and G. Webb2

1Center for Research in Scientific Computation, Department of Mathematics, North Carolina State University, Raleigh 27695-8205, NC, USA
2Department of Aerospace Engineering, Texas A&M University, College Station, 77843, TX, USA
3Department of Mathematics, Texas A&M University, College Station, 77843, TX, USA

Received 9 May 1996

Copyright © 1997 H. T. Banks et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A large class of emerging actuation devices and materials exhibit strong hysteresis characteristics during their routine operation. For example, when piezoceramic actuators are operated under the influence of strong electric fields, it is known that the resulting input–output behavior is hysteretic. Likewise, when shape memory alloys are resistively heated to induce phase transformations, the input–output response at the structural level is also known to be strongly hysteretic. This paper investigates the mathematical issues that arise in identifying a class of hysteresis operators that have been employed for modeling both piezoceramic actuation and shape memory alloy actuation. Specifically, the identification of a class of distributed hysteresis operators that arise in the control influence operator of a class of second order evolution equations is investigated. In Part I of this paper we introduce distributed,hysteretic control influence operators derived from smoothed Preisach operators and generalized hysteresis operators derived from results of Krasnoselskii and Pokrovskii. For these classes, the identification problem in which we seek to characterize the hysteretic control influence operator can be expressed as an ouput least square minimization over probability measures defined on a compact subset of a closed half-plane. In Part II of this paper, consistent and convergent approximation methods for identification of the measure characterizing the hysteresis are derived.