This section first reviews the inversion of a generalized P–I model. Then, it presents the procedure and experimental results of directly developing an inverse generalized P–I model [

10] to precisely identify the known model parameters. Finally, we report the experimental validation and performance evaluation of hysteresis compensation achieved by the identified inverse generalized P–I model used as a feedforward controller.

#### 2.1. Inverse Generalized P–I Model

The complex input–output characteristics of an asymmetric hysteresis with dead zone and output saturation of the nonlinear target system

${\mathsf{\Phi}}^{*}$ illustrated in

Figure 1 should be linearized. To this end, it is critical to derive an exact inverse hysteresis model

${\mathsf{\Phi}}^{-1}$. We adopt a recently proposed method for direct construction of the inverse generalized P–I model [

10] instead of analytically deriving the inversion from an estimated model

$\mathsf{\Phi}$. We briefly describe the discrete version of the inverse generalized P–I model below. The output of the inverse model

${\mathsf{\Phi}}^{-1}[\xb7](k)$ is expressed as

with the classical play hysteresis operator

${\mathrm{R}}_{{q}_{j}}[\xb7](k)$ being defined as

where

$(N+1)$ play operators are considered,

${\gamma}_{l}$ and

${\gamma}_{r}$ are envelope functions, and

${q}_{j}$ and

${g}_{j}$ are finite sequences of thresholds and discrete weights, respectively, corresponding to the density function in the inverse model. Details of the derivation of this inverse model can be found in our previous study [

10]. The uncertain model parameters,

${q}_{j}$ and

${g}_{j}$, in the inverse threshold and density functions for

${\mathsf{\Phi}}^{-1}[\xb7](k)$ can be identified directly from input–output measurements using CNT-PSO [

16].

#### 2.2. Parameter Identification and Experimental Validation

We conducted experiments on the EHVS actuated by a proportional control valve, as shown in

Figure 2, to measure the output data when applying a triangle wave voltage input

$v(t)$ of frequency

$0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and amplitude

$7.5\phantom{\rule{0.166667em}{0ex}}\mathrm{V}$ for

$t\in [0,100]$ with sampling interval

${T}_{s}=0.1\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. The specifications of the experimental EHVS are listed in

Table 1.

Figure 3 shows the complex nonlinearities in the EHVS input–output characteristics, which indicate an asymmetric hysteresis with dead zone and output saturation between the input

$v(k)$ and measured output

${y}_{\mathrm{hyste}}(k)$.

Let

${\mathsf{\Phi}}^{*}$ denote the real nonlinear behavior in

Figure 3 of the EHVS in

Figure 2. The unknown parameters of inverse generalized P–I model

${\mathsf{\Phi}}^{-1}[\xb7](k)$ in (

1) and (

2) are directly estimated using the identification method in Reference [

10].

Figure 4 illustrates the schematic diagram in our identification procedure for direct construction of

${\mathsf{\Phi}}^{-1}[\xb7](k)$. For identifying inverse model

${\mathsf{\Phi}}^{-1}[\xb7](k)$ of the EHVS, the envelope, threshold, and density functions are given by

Therefore, the design parameter vector to be identified is ${\mathbf{X}}_{\mathrm{hyste}}:=({\mathbf{X}}^{q},{\mathbf{X}}^{g},{\mathbf{X}}^{\ell},{\mathbf{X}}^{r})\in {\mathbb{R}}^{12}$, where ${\mathbf{X}}^{\ell}=({a}_{{\ell}_{1}},{a}_{{\ell}_{2}},{a}_{{\ell}_{3}},{a}_{{\ell}_{4}})\in {\mathbb{R}}^{4}$, ${\mathbf{X}}^{r}=({a}_{{r}_{1}},{a}_{{r}_{2}},{a}_{{r}_{3}},{a}_{{r}_{4}})\in {\mathbb{R}}^{4}$, ${\mathbf{X}}^{q}={b}_{q}\in \mathbb{R}$, and ${\mathbf{X}}^{g}=({c}_{{g}_{0}},{c}_{{g}_{1}},{c}_{{g}_{2}})\in {\mathbb{R}}^{3}$.

Let the sampled output of inverse model

${\mathsf{\Phi}}^{-1}[\xb7](k)$ be represented by

${v}_{{\mathsf{\Phi}}^{-1}}(k)$ as shown in

Figure 4. Then, we define the objective function as

because

$v-{c}_{v}{v}_{{\mathsf{\Phi}}^{-1}}={c}_{v}(\frac{1}{{c}_{v}}v-{v}_{{\mathsf{\Phi}}^{-1}})$, where

${c}_{v}$$(>1)$ is a user-defined scaling factor. We select

${c}_{v}$ and

${c}_{y}$$(>1)$ in

Figure 4 to be 10 for the available maximum values to be scaled to norm 1.

Finally, we use CNT-PSO [

16] to solve optimization problem

${min}_{{\mathbf{X}}_{\mathrm{hyste}}}\mathcal{L}({\mathbf{X}}_{\mathrm{hyste}})$. The identification results of the unknown parameters in

${\mathbf{X}}_{\mathrm{hyste}}\in {\mathbb{R}}^{12}$ are summarized as follows:

The relation of input

${y}_{\mathrm{hyste}}(k)$ into the identified inverse model and the resulting output of

${\widehat{\mathsf{\Phi}}}^{-1}[{y}_{\mathrm{hyste}}](k):={c}_{v}\xb7{\mathsf{\Phi}}^{-1}[{y}_{\mathrm{hyste}}/{c}_{y}](k)$ (i.e., inverse hysteresis loop) is shown in

Figure 5, where inverse model

${\mathsf{\Phi}}^{-1}[\xb7](k)$ is given by (

1) and (

2) with estimated design parameters given in (

7)–(10). We implement the identified inverse generalized P–I model,

${\widehat{\mathsf{\Phi}}}^{-1}[\xb7](k)$, as the feedforward controller shown in

Figure 1 and verify the precision of the estimated model parameters.

Figure 6 shows the compensation results when applying triangle wave input

$r(k)$ with frequency of

$0.01\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ and amplitude of

$9.6\phantom{\rule{0.166667em}{0ex}}\mathrm{L}/\mathrm{min}$ to the EHVS compensated by

${\widehat{\mathsf{\Phi}}}^{-1}[\xb7](k)$ (i.e.,

${\mathsf{\Phi}}^{*}\circ {\widehat{\mathsf{\Phi}}}^{-1}[r](k)$).

However, the derived feedforward control lacks robustness, which is essential for the EHVS given the related low-frequency disturbances and high-frequency noise. Therefore, feedforward control should be combined with feedback control to increase robustness. In

Section 3, we present the parameter identification of a linear dynamics model excluding nonlinear components to design the corresponding feedback controller.