*4.1. Experimental Setup*

As shown in Figure 11a, the experimental platform consists of a computer, a USB-6259BNC (from National Instruments, Austin, TX, USA) data acquisition card, a 1-D piezoelectric micro-motion platform, and a piezoelectric servo controller E-625.CR (from Piezomechanik, München, Germany). The P-622.1CD (from Piezomechanik, München, Germany) has a maximum stroke of 200 μm and a built-in capacitive displacement sensor. The E-625.CR has a piezoelectric amplifier and displacement acquisition module. Its voltage amplification factor is 10 and the sensitivity of the displacement acquisition module is 20 μm/V. The USB-6259BNC has multiple 16-bit digital-to-analog converters and 16-bit analog-to-digital converters and cooperates with the host computer to realize the real-time control of the micro-motion platform. Figure 11b shows the process block diagram of the experimental system.

**Figure 11.** Experimental system. (**a**) Experimental platform; (**b**) process block diagram.

#### *4.2. Asymmetric Hysteresis Description Results and Discussion*

To experimentally validate the PMPI model, the first step is to identify the parameters of PMPI model. Model type and parameter identification both affect the accuracy of hysteretic modeling. Many identification algorithms [28–30] have been proposed to obtain model parameters, such as least square method (LSE), particle swarm optimization (PSO), and differential evolution (DE) algorithm. However, ensuring that the identified parameters are the global optimal solutions is a challenging task. In this section, the hybrid algorithm Nelder–Mead differential evolution (NM-DE) [31], based on differential evolution and simplex algorithm, is used to identify the parameters of the PMPI model. The NM-DE algorithm takes into account both global and local search capabilities, and has the advantages of fast convergence and high accuracy.

It should be noted that the larger the number of operators, the more accurately the model can describe the hysteresis in theory. Table 1 shows the relationship between the number of operators n, identification errors, and run time, where the runtime reflects indirectly the computation. From this Table, it can be observed that modest increase in the number of operator can improve the accuracy of the model, but further increase in the number of operator show no significant improvement in the accuracy of model, the identification errors are almost at the same level when n = 10,20,30. In addition, increase in the number of operators will increase the run time (computation) which further affects the real time performance of compensation. We select *n* = 10 for the case studies. As mentioned above, the weighting coefficient *p*(*ri*) approaches 0 as *ri* becomes larger. The weighting coefficient *p*(*ri*) can be expressed as *p*(*ri*) = <sup>α</sup>1*e*<sup>−</sup>α<sup>2</sup> *r*i . This form reduces the number of parameters to be identified and greatly reduces the identification burden.


**Table 1.** The relationship between the number of operators *n*, identification error, and run time.

To demonstrate the superiority of PMPI model in characterizing asymmetric hysteresis, comparison of the three models PI, Gu-PI, and PMPI was carried out. The number of operators *n* is set to be 10, the thresholds are the same, and the parameters of models are the optimal values obtained after repeated identifications. The comparison experiments were carried out respectively in two cases (Case 1 and Case 2) as shown in Figure 12. The input–output curves of the three models appear to coincide because of the small modeling error. To directly reflect the superiority of PMPI model in hysteresis modeling accuracy, Figure 13a,b show respectively the modeling errors of the three models PI, Gu-PI, PMPI in two cases. In order to evaluate the accuracy of hysteresis model and quantify the modelling error, the maximum absolute error (MAE), the mean absolute deviation (MAD), and the root-mean-square error (RMSE) are defined as follows.

$$\begin{cases} \text{MAE} = \max\_{1 \le i \le N} [\mathfrak{g}(i) - \mathfrak{y}(i)] \\ \text{MRE} = \frac{\text{MAE}}{\text{y\_{\text{max}}}} \times 100\% \\ \text{MAD} = \frac{1}{N} \sum\_{i=1}^{N} [\mathfrak{g}(i) - \mathfrak{y}(i)] \\ \text{RMSE} = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} [\mathfrak{Y}(i) - \mathfrak{y}(i)]^2} \end{cases} \tag{21}$$

where *N* is the number of samples, *y*(*i*) is the real measured displacement, *y*(*i*) is the model predicted displacement, and *y*max is the maximum measured displacement. Among them, the MAE and MRE are used to evaluate local accuracy, and the MAD and RMSE are used to evaluate global accuracy.

**Figure 12.** Two cases of comparative experiment. (**a**) Case 1: the initial loading curve is not considered; (**b**) Case 2: the initial loading curve is considered.

The modeling error evaluation results of the three models in two cases are respectively listed in Tables 2 and 3. It can be seen from Table 2 that the prediction errors of the Gu-PI and PMPI model are significantly lower than that of the PI model in Case 1, and the MRE of prediction are only 0.968% and 0.698%. The result shows that the Gu-PI and PMPI model have obvious advantages in characterizing asymmetric hysteresis in Case 1. However, compared with PI and Gu-PI model, the accuracy of the PMPI model is significantly improved in Case 2, and the MAE of prediction is reduced by 83.3%. This is due to the lack of accuracy of Play operator in describing the displacement near zero voltage on the descending edge. This deficiency of Play operator shows that the local accuracy of Gu-PI model is approximately equal to PI model. The M-Play operator significantly improves the flexibility and accuracy of PMPI model. If the initial loading curve is not considered in hysteresis compensation, the compensator must make PEAs run for a period of time in advance, which will undoubtedly increase the burden of the compensator. In summary, Case 1 has high modeling accuracy, but it will increase the burden of the compensator. Case 2 has slightly low modeling accuracy, but the compensator has no such concern. The proposed PMPI model has superior modeling ability for hysteresis asymmetry in both cases.

**Figure 13.** Performance comparison of three modeling methods in two cases. (**a**) Modeling errors in Case 1; (**b**) modeling errors in Case 2.


**Table 2.** Comparison of three model errors in Case 1.

**Table 3.** Comparison of three model errors in Case 2.


#### *4.3. Hysteresis Compensation Results and Discussion*

Table 4 lists the identified parameters of the PMPI model, the parameters of PMPI model satisfy the condition (14) in the range (0 10). Therefore, the I-M compensator is globally stable. To verify the effectiveness of the I-M compensator, the tracking experiment with periodic sinusoidal references with *yr* = 50 + 50sin(2πt−<sup>π</sup>/2) is conducted. Figure 14a shows the comparison of the desired and actual trajectory. After compensation, the actual displacement can track the desired trajectory well, and no tracking loss occurs. Figure 14b shows the tracking errors, defined as the difference between the desired and actual trajectory. The MAE is 1.07 μm, the MRE is 1.07%, and the MAD is less than 0.4 μm. It is worth mentioning that, because of the existence of modeling uncertainty, the tracking errors appear periodic in periodic tracking experiments, which can be seen as systematic error, which can be eliminated by closed-loop control. To more intuitively reflect the compensation effect, Figure 14c shows the relationship between the desired and actual displacements. After compensation, the input–output shows an approximate linear relationship. The error is one order of magnitude less than that without any control, which shows that the I-M compensator can well suppress the hysteresis characteristics of PEAs.


**Table 4.** The identified parameters of PMPI model.

(**b**) (**c**)

Time(s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0




Trackinng error(

ΐm)

0.0

0.5

1.0

1.5

**Figure 14.** Periodic sinusoidal reference tracking experiment. (**a**) Trajectory tracking; (**b**) tracking error; (**c**) the relationship between desired and actual displacement.

0 20 40 60 80 100

Desired trajectory(ΐm)

0

To further verify the effectiveness of I-M compensator, a tracking experiment of frequency conversion attenuated triangular wave is performed. Figure 15 shows the results of this tracking experiment. It can be seen that the I-M compensator still has good tracking performance in tracking complex trajectory. The MRE is 1.18%, which is slightly larger than the ones of periodic sinusoidal. The experimental result further demonstrates the effectiveness of the I-M controller in hysteresis compensation.

**Figure 15.** Frequency conversion attenuated triangular wave reference tracking experiment. (**a**) Trajectory tracking; (**b**) tracking error; (**c**) relationship between desired and actual displacement.
