Study of a Play-like operator D. B. Ekanayake, R. V. Iyer * , Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1024, USA Abstract In this paper, we consider a play-like hysteresis operator defined by an n th order rate-independent differential system. We investigate the properties of the operator for n = 1 and n = 2. We show that the operator for n = 2 satisfies a one-step wiping out property. This result can be extended to show that the n th order operator satisfies an (n - 1)-th step wiping out property. Thus the new family of operators fall between the first-order differential equation models that do not satisfy any wiping-out properties and the Preisach-type operator that can show, in general, a countably infinite-step wiping out property. We will show that the “backlash- like” operator defined by Su, Stepanenko, Svoboda and Leung (SSSL) is a special case of our operator for n = 1. Key words: hysteresis, Duhem operators, play-like operators, PKP-type operators 1. Introduction Hysteresis representation by integral operators or dif- ferential equations dates back to Duhem’s model in 1897 [3]. These operators reflect the observation that hysteresis curves for physical systems are monotone except when the input changes direction. Most differential equation hystere- sis operators are first–order differential equations [3–6]. Fig. 1. Relationship between hysteresis models. Figure 1 shows the relation between some of the various hysteresis models. Most differential equation models are rate-independent and satisfy the Volterra property. Hence, they are general hysteresis operators (Prop 2.2.9 [2]). Ad- vantages of using a differential equation operator include simplicity of implementation and a limited number of pa- * Corresponding author: e-mail ram.iyer@ttu.edu rameters. However, unlike operators of the Preisach type, the output trajectories of these rate–independent operators do not depend on the previous input extrema, and hence, they do not satisfy Madelung’s rules 1 [2,3]. One such first-order differential equation hysteresis oper- ator was proposed by Bouc in 1964 [4,5], with the primary objective of describing forced vibrations of a hysteretic sys- tem under periodic excitation. Another first-order hystere- sis operator, called “backlash–like”, was introduced for the purpose of avoiding the inversion of hysteresis nonlinearity in an adaptive controller design [1]. We refer to this op- erator as the SSSL–play operator. We will show that the SSSL operator has a serious limitation in parameter selec- tion. This limitation is overcome by our generalized n th or- der play–like operator with the additional benefit that the robust, non-inversion type, adaptive controller defined in [1] is also applicable to this operator without change. In this paper, we consider a play–like operator and the construction of a Preisach-Krasnoselskii-Pokrovskii (PKP) type operator using an n th order differential system. We investigate the cases where n = 1 and n = 2 and show that 1 (i) Any curve C 1 emanating from a turning point A of the input- output graph is uniquely determined by the coordinates of A. (ii) If any point B on the curve C 1 becomes a new turning point, then the curve C 2 originating at B leads back to the point A. (iii) If the curve C 2 is continued beyond the point A, then it coincides with the continuation of the curve C which led to the point A before the C 1 - C 2 cycle was traversed. Preprint submitted to Elsevier 21 July 2007