*4.1. Compensation Results*

In order to verify the compensation control effect of the MSPI inverse model, the following experiment was performed on the experimental system from Figure 3, with the inverse model as an input:

(1) Adjust the laser interferometer. Connect computer, controller, and laser interferometer. The related software is turned on and waits for the measurements.

(2) Use the controller-related software to load the control voltage in the inverse model. Experiment 1 is carried out according to the voltage obtained by inverse model of Figure 15a, and the displacement is measured and recorded by the laser interferometer.

(3) Perform the experiment according to the voltage obtained by the inverse model in Figure 15b. Measure and record the displacement data by the laser interferometer. The interval should be the same as (2).

(4) Take several measurements.

(5) Check the equipment and turn it <sup>o</sup>ff. Process the experimental data.

Figure 16a is the measured single-ring linear voltage hysteresis feature compensation effect, while Figure 16b is the hysteresis compensation effect of the classical PI inverse model. The mean absolute deviation can be expressed as:

$$\varepsilon = \frac{1}{\eta} \sum\_{\mu=1}^{\eta} (\varepsilon\_{\mu} - \omega\_{\mu}) \tag{14}$$

where η is data quantity, <sup>ε</sup>μ are expected results, and <sup>ω</sup>μ are experimental results. Hence, the mean absolute deviation of the classical PI inverse model compensation control is 190.2 nm, the mean absolute deviation of the MSPI inverse model compensation control is 35.0 nm, and the nanopositioning accuracy is improved by 81.6%.

**Figure 16.** Single-ring linear voltage hysteresis compensation control effects: (**a**) classical Prandtl–Ishlinskii (PI) inverse model compensation and (**b**) MSPI inverse model compensation.

The reciprocating linear voltage hysteresis feature compensation effect is shown in Figure 17. The MSPI model in Figure 13c describes the hysteresis characteristics significantly better than the classical PI model description in Figure 7b; hence, the comparison is not made here. The mean absolute deviation of the MSPI inverse model compensation control of reciprocating linear hysteresis is 19.7 nm, and the positioning error is only 0.42%.

**Figure 17.** Reciprocating linear voltage hysteresis compensation control effect.

As predicted, the MSPI model is still flawed in its description of the end curve. Therefore, whether it is the reciprocating linear voltage hysteresis MSPI inverse model or the single-ring linear voltage hysteresis MSPI inverse model, the most significant error for both models is at the end of the compensation result. Accuracy can be improved without the end error or with additional segmentation modeling at the end.

In addition, in the two given examples, the MSPI model has more advantages when solving reciprocating linear voltage hysteresis compensation. Compared with the single-ring linear voltage hysteresis MSPI inverse model, the reciprocating linear voltage hysteresis MSPI inverse model has more marks and more segments, and hence the accuracy is improved by 43.7%.

## *4.2. Verification Tests*

In order to study whether the change of the voltage frequency affects the application of the MSPI model, verification experiment 1 is carried out to observe the hysteresis characteristics.

For example, it is observed that for single-ring linear voltage, the elongation speed of the piezoelectric ceramic displacement is a negative value and the shrinkage speed is a positive value. The amplitude of the triangular wave voltage is set to be the same as the amplitude of Figure 4a, which is 150V. Three sets of the speed time diagrams can be obtained by changing the voltage frequency. According to the period–frequency relationship *T* = 1*f* , the smaller the frequency is, the longer the triangular wave voltage period. The appropriate and easily observed frequency control period time is between 1 and 5 s.

Figure 18 shows the speed time diagrams of 1.0 Hz, 0.4 Hz, and 0.2 Hz voltage frequencies.

**Figure 18.** Characteristics of displacement velocity at different frequencies: (**a**) amplitude 150 V, frequency 1.0 Hz; (**b**) amplitude 150 V, frequency 0.4 Hz; (**c**) amplitude 150 V, frequency 0.2 Hz.

Although the output speeds are di fferent at di fferent frequencies and the maximum displacement time is shortened as the frequency increases, the same variation characteristics are maintained. If the speed is taken as an absolute value, the speed–time diagrams can find a similar relationship as with the *v* − *s*(*v*) diagram. Both the type I mark points and the type II mark points have been identified on the map in the same color as the *v* − *s*(*v*) diagram.

The hysteresis characteristic of the reciprocating linear voltage is the same. It can be seen that the di fferent voltage frequencies exhibit the same regularity for hysteresis characteristics, and thus the MSPI model is still e ffective.

Verification experiment 2 is then carried out. The MSPI model for another type of nanopositioning stage is used to observe the modeling e ffect.

The equipment is adjusted and connected according to the experimental steps in Section 2.2. The triangular wave voltages of 20 V and 15 V amplitudes are loaded as the reciprocating linear voltage inputs. The experimental data are measured and recorded for every 0.5 V.

Figure 19 shows the hysteresis characteristics of the experimental measurements and the modeling comparison between the classical PI model and MSPI model. The MSPI model has a higher description accuracy, which repeatedly proves that the MSPI model can be applied to di fferent hysteresis characteristics.

**Figure 19.** Comparison of the experimental measurements: (**a**) the classical PI model and (**b**) the MSPI model.
