# Compensation Method for the Nonlinear Characteristics with Starting Error of a Piezoelectric Actuator in Open-Loop Controls Based on the DSPI Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Mechanical Design

#### 2.2. Experimental Equipment

## 3. Causes of Starting Error

#### 3.1. Start-Up Error Characteristic 1

#### 3.2. Start-Up Error Characteristic 2

#### 3.2.1. Causes of Voltage-Affected Starting Error

## 4. Modeling

#### 4.1. CPI Model

#### 4.2. Start-Up Error Characteristic 3

#### 4.3. Data-Separated Prandtl-Ishlinskii Model

_{1}, a

_{2}, a

_{3}… a

_{x}. The displacement value of the data collected in the second cycle is b

_{1}, b

_{2}, b

_{3}… b

_{x}. The displacement value of the data collected in the third cycle is c

_{1}, c

_{2}, c

_{3}… c

_{x}. The displacement value of the data collected in the nth cycle is n

_{1}, n

_{2}, n

_{3}… n

_{x}.

_{1}+ c

_{1}+…+ n

_{1})/(n − 1), (b

_{2}+ c

_{2}+…+ n

_{2})/(n − 1), (b

_{3}+ c

_{3}+…+ n

_{3})/(n − 1), … (b

_{n}+ c

_{n}+…+ n

_{n})/(n − 1). After averaging, the new displacement value of the second cycle is d

_{1}, d

_{2}, d

_{3}… d

_{x}. As shown in Figure 10, the voltage-displacement curves of the first cycle data and the new second cycle data obtained from the data reclassification process are presented.

#### 4.4. Compensated Control and DSPI Inverse Model

## 5. Experiments and Discussion

#### 5.1. Results of the DSPI Model

#### 5.2. Results of the DSPI Inverse Model

#### 5.3. Model Comparison

- Adjust the laser interferometer. Connect the computer, laser interferometer, and controller. The relevant software is opened and waiting for a measurement;
- Using the software to make the controller CPI inverse model loaded with the control voltage. The experimental voltage obtained by the CPI inverse model was used for experiment one, and the displacement was measured and recorded at equal time intervals using a laser interferometer. The experimental results are shown in Figure 21;
- Experiment 2 is performed according to the voltage obtained from the DSPI inverse model, and the displacement is measured and recorded at equal time intervals using a laser interferometer. The time interval described in step 3 is kept the same as in 2;
- Change the time interval to complete multiple measurements;
- Check the apparatus and shut it down, and process the final experimental data.

#### 5.4. Another Experimental Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Three-dimensional schematic: (

**a**) one-degree-of-freedom nanopositioning platform; (

**b**) piezoelectric ceramic stacking.

**Figure 2.**Nano platform experimental system: (

**a**) system schematic diagram; (

**b**) system build diagram.

**Figure 3.**Schematic diagram of the change of electric domains in piezoelectric ceramics during the polarization process: (

**a**) before polarization; (

**b**) during polarization; (

**c**) after polarization.

**Figure 4.**Experimental loading voltage: (

**a**) first cycle voltage after start-up; (

**b**) the first three cycles of voltage; (

**c**) the first seven cycles of voltage.

**Figure 5.**Experimental voltage-displacement curves: (

**a**) voltage-displacement curve of the first cycle after start-up; (

**b**) voltage-displacement curve of the first three cycles; (

**c**) voltage-displacement curve of the first seven cycles.

**Figure 12.**Voltage–hysteresis tangent slope plots: (

**a**) cycle 1 v-s(v) plot; (

**b**) cycle 1 slope data separation plot.

**Figure 13.**Voltage–hysteresis tangent slope plots: (

**a**) new second cycle v-s(v) plot; (

**b**) new second cycle slope data separation plot.

**Figure 15.**DSPI model: (

**a**) modeling of the first cycle data hysteresis characteristics; (

**b**) local enlargement.

**Figure 16.**DSPI model: (

**a**) modeling of the new second cycle data hysteresis characteristics; (

**b**) local enlargement.

**Figure 17.**DSPI model: (

**a**) modeling of the hysteresis characteristics after merging the first and new second cycle data; (

**b**) local enlargement.

**Figure 18.**DSPI inverse model: (

**a**) the first cycle; (

**b**) the new second cycle; (

**c**) overall modeling of the DSPI inverse model.

**Figure 21.**Comparison of the compensation control effect between the DSPI and CPI inverse models: (

**a**) CPI inverse model; (

**b**) DSPI inverse model; (

**c**) local enlargement of (

**b**).

**Figure 25.**Time–displacement–error curves of DSPI Model and MSPI Model: (

**a**) overall diagram; (

**b**) local enlargement.

The Given Voltage | Start-Up Error Value |
---|---|

0–150–0 V | 2.117105 μm |

0–100–0 V | 1.411403 μm |

0–50–0 V | 0.827996 μm |

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**MDPI and ACS Style**

An, D.; Li, J.; Li, S.; Shao, M.; Wang, W.; Wang, C.; Yang, Y. Compensation Method for the Nonlinear Characteristics with Starting Error of a Piezoelectric Actuator in Open-Loop Controls Based on the DSPI Model. *Micromachines* **2023**, *14*, 742.
https://doi.org/10.3390/mi14040742

**AMA Style**

An D, Li J, Li S, Shao M, Wang W, Wang C, Yang Y. Compensation Method for the Nonlinear Characteristics with Starting Error of a Piezoelectric Actuator in Open-Loop Controls Based on the DSPI Model. *Micromachines*. 2023; 14(4):742.
https://doi.org/10.3390/mi14040742

**Chicago/Turabian Style**

An, Dong, Ji Li, Songhua Li, Meng Shao, Weinan Wang, Chuan Wang, and Yixiao Yang. 2023. "Compensation Method for the Nonlinear Characteristics with Starting Error of a Piezoelectric Actuator in Open-Loop Controls Based on the DSPI Model" *Micromachines* 14, no. 4: 742.
https://doi.org/10.3390/mi14040742