## 3 Experimental Validation  
### Set-Up and Model Identification  
This section details an evaluation of the model performance considering measurements performed on an experimental apparatus. As a system-oriented approach, the objective of this experimental validation is to asses to what extent the model provides a flexible tool to be implemented in a wide variety of control systems for instance. In order to experimentally investigate the validity of the proposed model, a piezoelectric stack actuator (PICMA(r) P-888.51 from PI(tm) Ceramic GmbH, Lederhose, Germany, with an internal capacitance of 6 uF and 0-100 V unipolar driving voltage) has been considered. The actuator, that is left in stress-free conditions, is driven by a dSPACE(tm), Inc, Paderborn, Germany, DS2003/DS2102 acquisition/control system through an amplifying module (PI(tm) Ceramic GmbH, Lederhose, Germany, E-503). Strain is monitored using a 120.3 \(\Omega\) strain gauge (BQ120-3 CA, Donghua Testing Technology, Jingjiang, China) with a sensitivity of 1.085 mV\(\cdot\mu\epsilon^{-1}\) connected to a signal conditioning module (Donghua Testing Technology, Jingjiang, China, DH 3840). Finally, an oscilloscope (Tektronix(tm), OR, Etats-Unis, TDS 1002) is used for additional monitoring of actuator driving voltage. Experimental set-up schematic is depicted in Figure 6.  
Experiments consisted in varying applied voltage magnitude (50 and 100 V) and frequency (1, 5, 10, 20, and 40 Hz). Exponential function has been considered for the quasi-static model:  
\[ \begin{array}{rcl}h\left(\left|V-V_{shift}\right|\right)&=&\gamma\exp\left(- \frac{\left|V-V_{shift}\right|}{V_{c}}\right)\\ &\Rightarrow&\alpha=\alpha_{0}+\gamma\exp\left(-\frac{\left|V-V_{shift}\right| }{V_{c}}\right)\end{array}  \tag{10}\]  
as it relates the hysteresis behavior of the considered actuator with better accuracy than the linear model [27]. In Equation (10), \(\gamma\) is the maximal variation of the strain derivative-voltage derivative coefficient, and \(V_{c}\) refers to the voltage coefficient allowing the control of the saturation occurrence of the function \(h\). Identification procedure first consisted in characterizing static model parameters (\(\alpha_{0}\), \(\gamma\), and \(V_{c}\)) by considering steady state at low frequency (1 Hz). More precisely, the procedure consisted in identifying these parameters through the experimentally evaluated value of \(\alpha\), as depicted in Figure 7. It can be noted that the parameter for \(\alpha_{0}\) has been found to 6 \(\mu\)e\(\cdot\)V\({}^{-1}\), which is quite far from the manufacturer's datasheet (8.33 \(\mu\)e\(\cdot\)V\({}^{-1}\)). This is explained by the fact that transient effect still arises at this frequency. Hence, next procedures will need to include a refinement for \(\alpha_{0}\). Then, still at the same frequency, transient response (strain shift) was used for finding the values of transient regime fractional derivative order (\(\lambda_{t}\)) and time constant (\(\tau_{t}\)), as well as refining \(\alpha_{0}\) (Figure 8). Finally, high frequency excitation (5-40 Hz) permitted extracting the values of the harmonic regime fractional derivative order (\(\lambda_{d}\)) and time constant (\(\tau_{d}\)). Model parameters are listed in Table 1. It is interesting to note that the best value found for the harmonic regime fractional derivation order, \(\lambda_{d}\), is greater than 1, meaning that the associated behavior is between losses and kinetic energy, while one may expect it to be between potential energy and losses (i.e., \(0\leq\lambda_{d}\leq 1\)). However, the fractional derivation order associated to transient regime follows this expectation quite well, with a quite small value well denoting a slowly varying process with frequency.