# Adaptive Tracking Control for the Piezoelectric Actuated Stage Using the Krasnosel’skii-Pokrovskii Operator

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## Abstract

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## 1. Introduction

## 2. Hysteresis Modeling of the Piezoelectric Actuated Stage

## 3. Adaptive Tracking Control Law Design

## 4. Simulation and Experimental Results

#### 4.1. Simulation Results

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Structure chart of the Krasnosel’skii-Pokrovskii (KP) kernel and Preisach plane. (

**a**) KP kernel; (

**b**) Preisach plane.

**Figure 3.**Simulation results of the KP model with the gradient descent algorithm under the triangular wave voltage signal.

**Figure 4.**Simulation results of the KP model with the gradient descent algorithm under the sine wave voltage signal.

**Figure 6.**Experimental result of KP operator based on the gradient descent algorithm under the triangular wave voltage signal.

**Figure 7.**Experimental result of KP operator based on the gradient descent algorithm under the complex wave signal.

**Figure 8.**Estimated parameters $\widehat{\theta}$ and ${\widehat{g}}_{i}$ of the proposed adaptive controller for the triangular wave signal. (

**a**) convergent curve of $\widehat{\theta}$; (

**b**) convergent curve of $\widehat{{g}_{i}}$.

**Figure 9.**Experimental results of the piezoelectric actuated stage with the triangular wave signal. (

**a**) tracking curve of the piezoelectric actuated stage; (

**b**) tracking error.

**Figure 10.**Estimates results of the parameters $\widehat{\theta}$ and ${\widehat{g}}_{i}$. (

**a**) convergent curve of $\widehat{\theta}$; (

**b**) convergent curve of $\widehat{{g}_{i}}$.

**Figure 11.**Comparative results of the piezoelectric actuated stage with the complex wave signal. (

**a**) tracking curve of the piezoelectric actuated stage; (

**b**) tracking error.

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**MDPI and ACS Style**

Xu, R.; Tian, D.; Wang, Z.
Adaptive Tracking Control for the Piezoelectric Actuated Stage Using the Krasnosel’skii-Pokrovskii Operator. *Micromachines* **2020**, *11*, 537.
https://doi.org/10.3390/mi11050537

**AMA Style**

Xu R, Tian D, Wang Z.
Adaptive Tracking Control for the Piezoelectric Actuated Stage Using the Krasnosel’skii-Pokrovskii Operator. *Micromachines*. 2020; 11(5):537.
https://doi.org/10.3390/mi11050537

**Chicago/Turabian Style**

Xu, Rui, Dapeng Tian, and Zhongshi Wang.
2020. "Adaptive Tracking Control for the Piezoelectric Actuated Stage Using the Krasnosel’skii-Pokrovskii Operator" *Micromachines* 11, no. 5: 537.
https://doi.org/10.3390/mi11050537