## 1 Introduction  
Thanks to their high integration potential and energy conversion abilities, piezoelectric materials have been of rising interest over the last decades for application in actuation systems [1, 2], such as medical probes [3], inkjet devices [4], or car injectors to name just a few examples. However, when actuated at high electric field, such actuators exhibit nonlinear hysteretic effects that may compromise their correct operation. Therefore, properly designing actuators and their associated control system necessitates good understanding and model of such a hysteresis response. Many hysteresis models are available in the literature [5], mainly for magnetic devices but with possible extension to piezoelectric systems, which can be possibly divided into two kinds of approaches. The first one consists in numerical models, where hysteresis is constructed from many elementary units, called \"hysterons\", that are combined together to shape the global actuator's hysteresis response. Such approaches include the well-known Preisach model [6, 7] along with its derivatives such as Krasnosel'Skii-Pokrovskii (KP) technique [8; 9], as well as Maxwell-Slip methods and associated dry friction models [10; 11]. Usually based on phenomenological approach, such models are effective in finely relating the piezoactuator's behavior (including minor loops for instance), but are computationally and memory-intensive, therefore preventing their use in lightweight embedded systems. On the other hands, mathematical techniques, based on explicit or implicit formulations, offer hysteresis models with reduced parameter sets and possibly physical interpretation of their construction. This class includes Jiles-Atherton [12; 13], Bouc-Wen [14; 15], or Duhem [16; 17] approaches for instance. However, proper parameter identification as well as stability and faithful representation of particular hysteretic behaviors may raise some issues when considering these techniques. However, most of the previous models, at least in their initial formulations, where dedicated to quasi-static regime and did not included transient and harmonic behaviors of the actuators, which can however yield significant changes in the electromechanical response. Therefore, works aiming at enhancing such models by including time-dependent components have been proposed in order to provide more accurate representations of hysteretic behavior of piezoelectric or magnetic elements. This includes, for instance, modification of the Bouc-Wen [18], Jiles-Atherton [19], or dry friction [20] approaches, but also numerical techniques such as Preisach approaches [21]. In addition to dynamical effects such as creep or losses (potentially with nonfractional terms [22; 23; 24; 25]), transient stage can also be related to quasi-static regime, for instance through, stabilization and accommodation that relate the partial repolarization due to irreversible polarization process that occurs in ferroelectric materials [26]. In that purpose, this paper proposes an enhancement of a simple mathematical approach inspired by the butterfly loop of piezoelectric materials previously introduced by the authors [27]. Compared to previously exposed models, this approach provides a simple and compact formulation of the hysteretic operator, allowing lightweight implementation with respect to hysteron-based models and part of mathematically-based ones, as well as efficient hysteretic behavior representation including for instance minor loops. This model is besides mainly dedicated to devices while the large majority of other approaches target material characteristics. However, this model, considering the coefficient relating the derivative of the strain with respect to the input voltage derivative, has so far been developed for quasi-static regime without any inclusion of dynamic effects directly within the model (hybrid approach was proposed in [28]).