# Hysteresis Modelling and Feedforward Control of Piezoelectric Actuator Based on Simplified Interval Type-2 Fuzzy System

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## Abstract

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## 1. Introduction

## 2. Simplified Interval Type-2 Fuzzy System

#### 2.1. Basic Concepts

#### 2.2. Model

- The fuzzy system uses singleton fuzzifier and direct defuzzifier. Without type-reduction, the output of the fuzzy system is analytically computed via the Nie-Tan method [36]. It can reduce the computational burden without much loss of performance compared with the iterative Karnik-Mendal method [37]. Besides, it is feasible to derive the analytic gradient of the output function in (10) of the modelling parameters ${c}_{l},{\sigma}_{l1},{\sigma}_{l2},{q}_{l1},{q}_{l2},$ and ${q}_{l3}$, which gives much convenience of using gradient based optimization method. Moreover, due to the computational simplicity, the proposed IT2 fuzzy system can be practically applied to the open-loop feedforward controller for compensating the hysteresis effect.
- There are 2 variables, ${y}_{k-1}$ and ${x}_{k}$, in the consequent part of the fuzzy rule whilst only 1 variable ${y}_{k-1}$ in the antecedent part. This design is vital for obtaining the analytic inverse of the fuzzy system without ${x}_{k}$ in the antecedent part of the fuzzy rule. In fact, the proposed fuzzy rule in (7) is the same as the IT2 fuzzy rule:${\tilde{R}}^{l}:\phantom{\rule{3.33333pt}{0ex}}IF\phantom{\rule{3.33333pt}{0ex}}y(k-1)\phantom{\rule{3.33333pt}{0ex}}\mathrm{is}\phantom{\rule{3.33333pt}{0ex}}{\tilde{A}}^{l}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}x\left(k\right)\phantom{\rule{3.33333pt}{0ex}}\mathrm{is}\phantom{\rule{3.33333pt}{0ex}}{\tilde{A}}_{x}^{l},\phantom{\rule{3.33333pt}{0ex}}\mathrm{THEN}\phantom{\rule{3.33333pt}{0ex}}y\left(k\right)={q}_{l1}y(k-1)+{q}_{l2}x\left(k\right)+{q}_{l3}$ where ${\tilde{A}}_{x}^{l}$ is a IT2 fuzzy set whose LMF and UMF are constantly equal to 1, i.e., ${\underline{\mu}}_{{\tilde{A}}_{x}^{l}}^{l}\left({x}_{k}\right)={\overline{\mu}}_{{\tilde{A}}_{x}^{l}}^{l}\left({x}_{k}\right)=1$. This design also simplifies the identification of the modelling parameters and the computation of their partial derivative.

#### 2.3. Optimization

## 3. Experiments

#### 3.1. Experimental Platform

#### 3.2. Model Identification

#### 3.3. Model Validation

#### 3.4. Feedforward Control

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IT2 | Interval type-2 |

MF | Membership function |

FOU | Footprint of uncertainty |

LMF | Lower membership function |

UMF | Upper membership function |

T-S | Takagi-Sugeno |

SGS | Strain gauge sensor |

RTCP | Real-time control platform |

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**Figure 1.**Experimental platform: (

**top**) main hardware including piezoelectric actuator, power amplifier, strain gauge sensor signal conditioner, and real-time control platform AD5436A, (

**bottom**) signal flow.

**Figure 2.**Model Identification: (

**top**) input voltage signal, (

**bottom**) identification result and error.

**Figure 4.**Validation results of the identified hysteresis model under different input signals: (

**top**) 20 Hz sinusoidal, (

**center**) 40 Hz sinusoidal, and (

**bottom**) 25 Hz triangular.

**Figure 5.**Validation results of the identified hysteresis model under the signal which is the sum of 2 different sinusoidal profiles with 100 Hz and 50 Hz frequencies.

**Figure 7.**Tracking performance of feedforward controller: (

**top**) tracking results, (

**bottom**) hysteresis compensation result.

Rule | Antecedent Parameters | Consequent Parameters |
---|---|---|

1 | (4.3427, 17.8879, 0.0892) | (0.8943, 0.0196, 1.7504) |

2 | (2.5647, 11.5798, 8.6067) | (0.9140, 0.0129, 0.8773) |

3 | (2.1349, 19.8162, 18.7381) | (1.1882, 0.0148, −6.2312) |

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**MDPI and ACS Style**

Li, P.-Z.; Zhang, D.-F.; Hu, J.-Y.; Lennox, B.; Arvin, F. Hysteresis Modelling and Feedforward Control of Piezoelectric Actuator Based on Simplified Interval Type-2 Fuzzy System. *Sensors* **2020**, *20*, 2587.
https://doi.org/10.3390/s20092587

**AMA Style**

Li P-Z, Zhang D-F, Hu J-Y, Lennox B, Arvin F. Hysteresis Modelling and Feedforward Control of Piezoelectric Actuator Based on Simplified Interval Type-2 Fuzzy System. *Sensors*. 2020; 20(9):2587.
https://doi.org/10.3390/s20092587

**Chicago/Turabian Style**

Li, Peng-Zhi, De-Fu Zhang, Jun-Yan Hu, Barry Lennox, and Farshad Arvin. 2020. "Hysteresis Modelling and Feedforward Control of Piezoelectric Actuator Based on Simplified Interval Type-2 Fuzzy System" *Sensors* 20, no. 9: 2587.
https://doi.org/10.3390/s20092587