## 2 Model Principles
### System-Level Model Basics  
The principles of the original hysteresis model for piezoelectric actuators lie in considering the coefficient \(\alpha\) linking the strain derivative \(dS\) to the voltage derivative \(dV\) as \[dS=\alpha dV \tag{1}\]  
Obviously, if \(\alpha\) is a constant, linear relationship holds. However, introducing voltage dependency in \(\alpha\) and inducing a butterfly shape of this coefficient as a function of the voltage leads to hysteresis in the voltage-strain plane. Specifically, this butterfly form is achieved by applying an effective voltage that equals the driving voltage \(V\) minus a memory voltage \(V_{shift}\), set to the driving voltage value when the later reaches a local extremum, or equivalently when its derivative cancels. The coefficient \(\alpha\) is then obtained through a constant value \(\alpha_{0}\), relating the linear part of the actuator, and a hysteretic part achieved through a predefined mathematical function \(h\):  
\[\alpha(V)=\alpha_{0}+h\left(V-V_{shift}\right) \tag{2}\]  
Note that the choice of the mathematical function allows obtaining different hysteresis shapes. Specifically, considering the absolute effective voltage \(\left|V-V_{shift}\right|\) in the function yields a symmetric hysteretic behavior, which will be assumed all along this study. Moreover, excluding the linear part \(\alpha_{0}\) from the hysteretic one represented by \(h\) leads to the required property \(h(0)=0\) for properly relating the small signal behavior. Considering the simple case of linear function such as \(\alpha(V)=\alpha_{0}+\gamma\left|V-V_{shift}\right|\) as an example yields plots depicted in Figure 1, where the driving voltage is a unipolar shifted sine wave. As the driving voltage and memory voltage are zero (position (1)), the increase in the driving voltage to position (2) also yields a linear increase of the coefficient \(\alpha\), so that the strain \(S\) evolves in a quadratic way. Once in position (2), corresponding to a driving voltage maximum position, the memory voltage is set to this maximal value, so that \(h\left(\left|V-V_{shift}\right|\right)=0\) and \(\alpha\) goes back to \(\alpha_{0}\) (position (3)). Then, the driving voltage decreases, leading to an increase of \(\left|V-V_{shift}\right|\) (as \(V_{shift}>V\)) and thus of \(\alpha\), until position (4) where the driving voltage reaches a minimum value (actually 0 V in this example) to which is set \(V_{shift}\), as in position (1) where \(\alpha=\alpha_{0}\). Therefore, this approach provides a quite simple and computationally efficient modeling method, as it only requires a few parameters (typically 1 for the low-field behavior--\(\alpha_{0}\)--and 1-2 for the function \(h\)) and only one memory (\(V_{shift}\)) while relating quite well the quasi-static behavior of actuators under various cases [27]. It also provides versatility and adaptivity as the function \(h\) can be any function  
Figure 1: Original model principle (a.u. refers to “arbitrary units”). (so that the strain variation is not necessarily quadratic), such as exponential or even Gaussian to get closer to material hysteresis loop [27]. One can note that in the particular case of linear function, the method is quite close to a differential form of Rayleigh hysteresis model [29], but this differential approach permits avoiding issues that arise in unipolar driving when considering the regular Rayleigh approach [30; 31]. It has to be recalled here that the model targeting applicative aspects, it mainly aims at relating actuator behaviors under lower excitation magnitude compared to the case of material characterization. In other words, it is considered that compared to the full hysteretic/butterfly shape of the material, only minor material loops are under consideration. Therefore, it is assumed that no significant domain switching occurs for instance.