Brief paper: Why are some hysteresis loops shaped like a butterfly?

Authors:
Bojana Drinčić;Xiaobo Tan;Dennis S. Bernstein
Affiliations:
Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140, United States;Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, United States;Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140, United States
Venue:
Automatica (Journal of IFAC)
Year:
2011

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Abstract

The contribution of this paper is a framework for relating butterfly-shaped hysteresis maps to simple (single-loop) hysteresis maps, which are typically easier to model and more amenable to control design. In particular, a unimodal mapping is used to transform simple loops to butterfly loops. For the practically important class of piecewise monotone hysteresis maps, we provide conditions for producing butterfly-shaped maps and examine the properties of the resulting butterflies. Conversely, we present conditions under which butterfly-shaped maps can be converted to simple piecewise monotone hysteresis maps to facilitate hysteresis compensation and control design. Examples of a preloaded two-bar linkage mechanism and a magnetostrictive actuator illustrate the theory and its utility for understanding, modeling, and controlling systems with butterfly-shaped hysteresis.