**3. Modeling**

This section describes the process of establishing the MSPI model in three parts. In the first, part the curve of the classical PI model is obtained to describe the hysteresis characteristics. The second part analyzes the specific problem of the classical PI model's inner-loop hysteresis description, and defines the voltage–slope curve corresponding to the hysteresis rate tangent to establish the MSPI model. The third part uses a threshold method to judge whether it is the mark point. The segmentation of the mark points gives the curve described by the MSPI model and its inverse model.

#### *3.1. Play Operator and Classical Prandtl–Ishlinskii Model*

The classical PI model is a weighted superposition of a finite number of Play operators. The Play operator is shown in Figure 6a. When the input signal is *<sup>x</sup>*(*k*), the Play operator expression with the threshold *r* is:

$$p(k) = \max[\mathbf{x}(k) - r, \min[\mathbf{x}(k) + r, p(k-1)]]\tag{7}$$

where 0 = *k*0 < *k*1 <...< *ks* is the appropriate division on the input signal interval, *k* ∈ [0, *ks*]. When *k* = 0, *p*(−<sup>1</sup>) is the initial value. In the PI model of the piezoelectric effect hysteresis problem, the initial voltage is usually 0 without displacement, so *p*(−<sup>1</sup>) = 0. Here, *p*(*k*) is the output of the input signal.

**Figure 6.** Play operator models: (**a**) full-sided operator model (**b**) and single-sided operator model.

The voltage supplied by the voltage driver is positive, and hence the PI model is usually modeled with a single-sided Play operator. As shown in Figure 6b, when the operator inputs *x*(*k*) ≤ *r*, the operator outputs *p*(*k*) = 0; when the operator inputs *r* < *x*(*k*) ≤ *<sup>x</sup>*(*ks*), the unweighted operator has a slope of 1, so the operator outputs *p*(*k*) = *x*(*k*) − *r*. The operator shown in Figure 6b outputs *p*(*k*) = *x*(*ks*) when the input decreases from *x*(*k*) to *x*(*ks*) − 2*r* and outputs *p*(*k*) = *x*(*k*) + *r* when the input decreases from *x*(*ks*) − 2*r* to 0. The operator may have no *p*(*k*) = *x*(*k*) + *r* output and a part of *p*(*k*) = *<sup>x</sup>*(*ks*), when the threshold *r* is increased or the input *x*(*ks*) is decreased. In this case, the specific characteristics of the operator should be considered.

A finite number of Play operators are superimposed according to the weighting of the above output characteristics, and a PI model is obtained to describe the hysteresis of the nanopositioning stage. The equation is:

$$P[\mathbf{x}(k)] = \theta \mathbf{o} \cdot \mathbf{x}(k) + \sum\_{i=1}^{n} \theta\_i \cdot p\_i(k) \tag{8}$$

where *<sup>P</sup>*[*x*(*k*)] is the corresponding PI model output for the operator input *<sup>x</sup>*(*k*). Here, θ0 is a positive value, *pi*(*k*) is the output of the *i*th operator that has a threshold *ri* and a corresponding weight θ*i*.

The more times the PI model is superimposed, the smoother the model contour is and the closer it is to the piezoelectric hysteresis characteristic curve. However, the accuracy of the voltage–displacement characteristics obtained from the experiments is limited, so the number of operators used for the superposition should be realistic.

Figure 7 shows the modelling of the PI model to display the single-ring linear voltage hysteresis characteristic and reciprocating linear voltage hysteresis characteristic of the second section. The modeling results show that the classical PI model describes the hysteresis characteristics well under single-ring linear voltage, but the accuracy under the reciprocating linear voltage is comparatively poor. The main reason is that the hysteresis characteristics of the reciprocating linear voltage are

more complicated, and the hysteresis rates between the inner loop and the outer loop are different. Meanwhile, the classical PI model cannot describe the Madelung principle of the hysteresis inner loops.

**Figure 7.** PI modeling: (**a**) modeling of single-ring linear voltage hysteresis characteristics and (**b**) modeling of reciprocating linear voltage hysteresis characteristics.

#### *3.2. Hysteresis Tangent Line and Slope*

In order to further improve the description accuracy of hysteresis characteristics, the hysteresis rates must be studied in depth. The hysteresis rates corresponding to the voltage–displacement characteristic curve are the weighted superposition of the Play operators in the PI model at that point. The weight θ*i* of the *i*th operator depends on the angle α between the tangent line of the hysteresis loop at that point and the v-axis. The angle α, as shown in Figure 8a, is not exactly the same in each tangent on the hysteresis loop. The hysteresis rate of the reciprocating linear voltage is even more complicated. As shown in Figure 8b, the tangent at approximately similar positions of the inner loop and outer loop tends to have different hysteresis rates.

**Figure 8.** Tangent of the hysteresis loop: (**a**) the hysteresis rate corresponding to each point on the same hysteresis loop is different; (**b**) the hysteresis rates between the inner loop and outer loop of the hysteresis are different. Note: v = voltage; y = displacement.

The voltage–displacement data can approximate the characteristic curve, thereby establishing a *<sup>v</sup>*–*y* coordinate system. If *j* is the *j*th data obtained by the experiment, the hysteresis loop passes through the point *vj*, *yj*. The equation for the hysteresis rate tangent *l*tan(*v*) at *v* is defined as:

$$l\_{\tan}(v) : y = s(v) \cdot v + t(v) \tag{9}$$

where *s*(*v*) is the hysteresis rate tangent slope at *v* and *<sup>t</sup>*(*v*) is the hysteresis rate's tangent intercept at *v*.

In a single-ring linear voltage hysteresis characteristic curve, the voltage *v* corresponds to two hysteresis tangent lines in the linear boost phase and the linear back phase, respectively. Similarly, in the reciprocating linear voltage hysteresis characteristic curve, the *v* value is likely to correspond to a plurality of hysteresis tangent lines; for example, the hysteresis rate tangent number in Figure 8b corresponding to *v* is as shown in Figure 9.

**Figure 9.** The voltage value corresponding to hysteresis characteristics has more than one tangent. Note: v = voltage; y = displacement.

The slope of the hysteresis tangent can reflect the trend of hysteresis at this point. The hysteresis tangent slope *s*(*v*) can be expressed as:

$$s(v) = \frac{\delta y}{\delta v} = \frac{y\_{j+1} - y\_j}{v\_{j+1} - v\_j} \tag{10}$$

where *vj*, *yj* and *vj*+1, *yj*+<sup>1</sup> are adjacent data and satisfy the equation min(*vj*, *vj*+<sup>1</sup>) ≤ *v* < max(*vj*, *vj*+<sup>1</sup>).

The voltage–slope diagram describes the characteristics of the hysteresis rate at any voltage. The different input linear voltage leads to varied hysteresis rate tangent regulation. Figure 10 is the *v* − *s*(*v*) diagram of experimental data for the single-ring linear voltage and the reciprocating linear voltage, respectively. Both groups of data evidently show segmentation in the *v* − *s*(*v*) diagram.

**Figure 10.** Voltage hysteresis rate tangent slope diagrams: (**a**) single-ring linear voltage and (**b**) reciprocating linear voltage.

Due to the fact that the piezoelectric hysteresis characteristic generally has a segmentation variation rule and there are obvious jump points between the segments, a segmented PI model is used to model it.

#### *3.3. Mark-Segmented Prandtl–Ishlinskii Model*

The voltage–slope diagram embodies the change of the hysteresis rate. For reciprocating hysteresis, plenty of turning points appear at the critical edge of boost phases and back phases. Compared with the hysteresis under single-ring linear voltage [24], reasonable identification I required for all mark points in order to fulfill the demands of complex hysteresis segmentation. Meanwhile, data with continuous and similar variation laws should be modeled in the same segment. Therefore, a mark-segmented PI (MSPI) model is proposed.

To identify the segmentation mark point, the threshold ϕ is set in the *v* − *s*(*v*) diagram. The threshold ϕ is directly proportional to the quantity of experimental data, which of the minimum data amount is always 8–10 times of the average data difference. When the *m*th hysteresis rate tangent slope value segmen<sup>t</sup> satisfies *<sup>s</sup>*(*vm*+<sup>1</sup>) − *<sup>s</sup>*(*vm*), then *<sup>s</sup>*(*vm*+<sup>1</sup>) − *s*(*vm* ) ≥ ϕ; *<sup>s</sup>*(*vm*+<sup>1</sup>) − *s*(*vm*) is defined as the hysteresis rate jump segment. Data (*vm*+1, *ym*+<sup>1</sup>) is defined as the type I mark point.

Therefore, the single-ring linear voltage characteristic experimental curve can be segmented to find one type I mark point, which divides the hysteresis characteristic into 2 segments, as shown in Figure 11a; the reciprocating linear voltage characteristic experimental curve finds five type I mark points, and the data is divided into 6 segments, as shown in Figure 11b.

**Figure 11.** Type I mark points associated with threshold ϕ: (**a**) single-ring linear voltage *v* − *s*(*v*) characteristic diagram and (**b**) reciprocating linear voltage *v* − *s*(*v*) characteristic diagram.

The segmentation data is required to select an appropriate single-sided Play operator according to its approximate hysteresis characteristics or according to the concavity and convexity. In most cases, the condition for selecting the single-sided Play operator satisfies *s* (*v*) < 0 or *s* (*v*) > 0,where *s* (*v*) is the differential coefficient of *<sup>s</sup>*(*v*). If there are still some cases where the segmentation data satisfies both the abovementioned conditions at the same time, then it needs to be divided by the segmentation marker point at *s* (*v*) = 0, which is defined as a type II mark point. The *v* − *s*(*v*) diagram obtained from the experimental data is not derivable, and the maximum or minimum value can be used as the type II mark point. In the single-ring linear voltage hysteresis *v* − *s*(*v*) diagram shown in Figure 12a, one type II mark point is found, and a total of two segmentation mark points divide the curve into three segments. The reciprocating linear voltage hysteresis characteristic *v* − *s*(*v*) diagram shown in Figure 12b finds one type II mark point, and the total number of segments is 7. Eventually, each segmen<sup>t</sup> selects a single-sided Play operator by characteristics.

**Figure 12.** Type II mark point associated with *s* (*v*): (**a**) single-ring linear voltage *v* − *s*(*v*) diagram and (**b**) reciprocating linear voltage *v* − *s*(*v*) diagram.

The placement of the segmentation points is special because they participate in the modeling in both the segments that are divided by themselves. As shown in Figure 13a, the two segmentation points participate in the fitting of the three segments. The MSPI model with single-ring linear voltage hysteresis has good connectivity at the segmentation point. Figure 13b amplifies the MSPI model at one of the mark-segmented points.

**Figure 13.** The mark-segemented Prandtl–Ishlinskii (MSPI) model: (**a**) modeling of single-ring linear voltage hysteresis; (**b**) modeling diagram (a) of partial amplification; (**c**) modeling of reciprocating linear voltage hysteresis characteristics; (**d**) modeling diagram (c) of partial amplification.

Similarly, the five segmentation points of the MSPI model of the reciprocating linear voltage shown in Figure 13c participate in the fitting of the six segments. The hysteresis inner-loop MSPI model of the reciprocating linear voltage is enlarged and shown in Figure 13d. The inner-loop hysteresis characteristic can hence be accurately described.

During the modeling process, the slope of the MSPI model at the end is often larger than the tangent slope of the hysteresis rate that is caused by the forced zeroing of the end of the Play operator. This problem can be solved by ignoring the self-property of the superposition end, and by adding end segmentation and modeling according to its specific hysteresis characteristics.
