*3.4. Inverse Control*

The MSPI model obtains an accurate approximation of the voltage–displacement correspondence. In order to achieve accurate compensation of the linear displacement, the displacement–voltage correspondence of the MSPI inverse model is used as a feedforward control. According to the compensation control principle [31], the MSPI inverse model is the inverse function of the hysteresis characteristic curve, as shown in Figure 14.

**Figure 14.** Hysteresis compensation control system diagram.

The classical PI model has an analytical inverse. The MSPI model is composed of separate PI models, so the inverse model equation is consistent with the PI inverse model. The equation is:

$$\begin{aligned} P^{-1}[p(k)] &= \theta\_0' \cdot p(k) + \sum\_{i=1}^n \theta\_i' \cdot \mathbf{x}\_i(k) = \theta\_0' \cdot p(k) + \\ \sum\_{i=1}^n \theta\_i' \cdot \max\{p(k) - r\_i', \min\{p(k) + r\_i', \mathbf{x}\_i(k-1)\}\} \end{aligned} \tag{11}$$

where *<sup>P</sup>*−<sup>1</sup>[*p*(*k*)] is the output corresponding to the PI inverse model operator input *p*(*k*). Here, *xi*(*k*) is the output of the *i*th operator, θ 0 = 1 θ0. The threshold and weight coe fficient of the inverse model are:

$$r'\_i = \sum\_{h=1}^i \theta\_h \cdot (r\_i - r\_h) \text{ } i = 1, 2, \dots, n \tag{12}$$

$$\Theta'\_i = -\frac{\Theta\_l}{\left(\Theta\_0 + \sum\_{h=1}^i \Theta\_h\right) \left(\Theta\_0 + \sum\_{h=1}^{i-1} \Theta\_h\right)} \quad i = 1, 2, \dots, n-1 \tag{13}$$

Figure 15 is an MSPI inverse model corresponding to Figure 13a,c. It can be seen that the MSPI inverse model has ideal connectivity between the segments.

**Figure 15.** The MSPI inverse models: (**a**) single-ring linear voltage hysteresis inverse model and (**b**) reciprocating linear voltage hysteresis inverse model.
