## 4 Numerical Example: Application to Linearized Control  
Based on the previous model that includes static, transient, and harmonic regimes, it is proposed in this section to illustrate the interest in the exposed approach for providing a mean for efficiently and accurately controlling a piezoelectric actuator with respect to a given strain command. This illustrative example will be numerically evaluated by considering an experimentally validated model of the same actuator used in the previous section and exposed in [28], using a Bouc-Wen model for relating the transient regime and a fourth-order transfer function for the harmonic behavior (Figure 13). Note that compared to the work in [28], the harmonic transfer function has been converted to Laplace domain through bilinear transform as:  
\[\frac{S(s)}{S_{stat}(s)}=\frac{92144s^{3}-68696\times 10^{3}s^{2}+114816\times 10^{6 }s+432\times 10^{9}s}{148196s^{3}+154912\times 10^{3}s^{2}+123824\times 10^{6 }s+432\times 10^{9}} \tag{11}\]  
with \(S\) the actuator output strain and \(S_{stat}\) the output of the quasi-static model (made of linear, system-level, and Bouc-Wen blocks). Also, to accommodate with the slight discrepancies between the hybrid model in [28] and actual experimental device, the values of \(\lambda_{t}\), \(\tau_{t}\), and \(\tau_{d}\) have been marginally changed to 0.2, 32 \(\times\)\(10^{-5}\) s and 2.2 \(\times\)\(10^{-3}\) s, respectively (note that for the transient-related fractional  
Figure 11: Experimental driving voltage with varying amplitude. Figure 12: Experimental and predicted response for varying voltage amplitude. transfer function, this changes the coefficient of the frequency-dependent term from 0.19 s\({}^{0.15}\) to 0.2 s\({}^{0.2}\)). All the other parameters are kept the same as per Table 1.  
The principles of the proposed control scheme lie in a compensation approach as exposed in [38]. More precisely, based on the model related in this study (Figure 2), the expression of the strain \(S\) in the Laplace domain yields:  
\[S(s)=\alpha_{0}V(s)G_{t}(s)G_{d}(s)+H\left(V(s)G_{t}(s)G_{d}(s)\right) \tag{12}\]  
with \(H(. )\) the hysteresis operator (defined by the function \(h\) and voltage memory \(V_{shift}\), but not including the linear part given by \(\alpha_{0}\)). Therefore, considering that \(S_{com}\) is the desired output strain, the corresponding required voltage \(V_{req}\) can be found from Equation (12) as  
\[V_{req}(s)=\frac{1}{\alpha_{0}}{G_{t}}^{-1}(s){G_{d}}^{-1}(s)\left[S_{com}(s)- H\left(V_{req}(s)G_{t}(s)G_{d}(s)\right)\right] \tag{13}\]  
The corresponding implementation for the control is depicted in Figure 14 (the fractional derivative blocks being also part of the FOMCON toolbox [34; 35]), along with the actuator model of [28] presented in Figure 13 and a direct control path consisting of the inversion of the sole linear part. Note that compared to Equation (12), model reduction was performed by including the compensation loop before the inverse transient and harmonic functions, and thus removing the associated direct functions before the hysteresis block. Figure 13: Considered model of the piezoelectric actuator for the control assessment. Figure 15 shows the system response under sine excitation at various frequencies. Therefore, it can be seen that the compensation approach including the proposed model allows significantly limiting the hysteretic response of the actuator and provides a mean of allowing a good and accurate control from a desired strain. Additionally, it can be observed that the proposed approach also addresses well the strain drift, as the strain bias at zero voltage appearing in the direct control case is significantly suppressed when using the compensation technique. Slightly lower and larger strains can be noted however at low and high frequencies respectively, which can however be attributed to slight deviation of the actuator model compared to the real one [28]. Figure 14: Schematic of the application example to linearized control. Figure 15: Control using compensation in sine case.