# Compensation of Hysteresis on Piezoelectric Actuators Based on Tripartite PI Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Causes of Hysteresis

#### 2.1. Micromechanism

#### 2.2. Analysis of the Causes of Hysteresis

## 3. Piezoelectric Ceramic Deformation Speed Law

#### 3.1. Derivation of Deformation Speed Law

#### 3.2. Analysis of Deformation Speed Law

_{1}, n

_{2}represent the numbers of nucleations per unit time per unit area, δ is the activation of the electric field, and k

_{1}, k

_{2}are constants.

_{c}, the total number of domains contained in the piezoelectric ceramic is

_{0}, E

_{1}, E

_{2}, E

_{3}…, E

_{n}

_{-1}, E

_{n}as the field strength points at equal intervals on the axis of coordinates, while the distance between the two adjacent pressure points is defined as h

_{i}= E

_{i}− E

_{i}

_{−1}. Assuming that E

_{c}is an inflection point electric field (can be considered as a constant), and using the geometric meaning of definite integral, we can get

_{c}is a constant, M can also be considered as a constant. Similarly,

_{1}and α

_{2}are electric field functions. However, the exponential function grows much faster than the power function. Therefore, the growth rate of α

_{1}is obviously greater than the growth rate of α

_{2}, which means that the deformation rate of the piezoelectric ceramics in the range 0~E

_{c}is greater than the deformation rate of piezoelectric ceramics in the range E

_{c}~2E

_{c}. In the piezoelectric deformation curve, the deformation speed of the voltage rise phase first increases and then decreases, and there is an inflection point of deformation rate. Combining this with the applied voltage period, we can determine the piezoelectric ceramic hysteresis curve inflection point voltage.

## 4. Hysteresis Modeling

#### 4.1. Play Operator and Prandtle–Ishlinskii Model

_{0}is 0.

_{i}is the weight of each hysteresis operator in the mathematical sense, n is the number of operators, Y(k) is the output of the model at the moment k, and r

_{i}is the threshold of the hysteresis operator. The vector form of Equation (11) is

_{1},···, w

_{i},···, w

_{n})

^{T}, the state vector of the operator at the moment k is y(k) = (y

_{1}(k),···, y

_{i}(k),···, y

_{n}(k))

^{T}, and the state vector of the operator at initial time is y(0) = (y

_{1}(0),···, y

_{i}(0),···, y

_{n}(0))

^{T}.

_{max}, the operator output is u(k) − r. When the input voltage drops from peak u

_{max}to u

_{max}− 2r, the operator output is u

_{max}− r. After this, the operator output y(k) is u(k) + r, until the voltage drops to zero. The output of the operator whose threshold r ≥ 0.5u

_{max}does not have an output of u(k) + r.

#### 4.2. Traditional PI Modeling and Inverse Model

#### 4.3. Tripartite PI Model Based on the Deformation Rate of Piezoelectric Ceramics

- (1)
- The selection of operators is based on the principles of concave-convex consistency, which means that in the hysteresis curve, the concave and convex parts of the curve correspond to the boost part and the depressurization part of the play operator, respectively.
- (2)
- The rising curve rises from zero voltage to the inflection point voltage u
_{if}(u_{if}refers to the voltage indicated by the arrows in Figure 6 and Figure 7), i.e., when the deformation speed rises from 0 to the maximum. The relationship between the voltage and displacement is described by a single lateral play operator as shown in Figure 13 (the dotted portion). - (3)
- The rising curve rises from the inflection voltage u
_{if}to maximum voltage u_{max}(u_{max}refers to the maximum point voltage applied to the piezoelectric ceramic during the whole rising cycle. It is 150 V here). Voltage–position relation in this part is described by a single lateral play operator as shown in Figure 13 (the solid line). One side play operators and hysteresis curves have a counter clock directivity. The reducing portion and rising process in the second part manifest the epirelief characteristic. The reducing portion of play operators point to the origin of coordinates while the second rising hysteresis curve deviates from it. Therefore, we need to model in reverse when we use play operators in the reducing part to describe the second rising process of the hysteresis curve. - (4)
- The retraced curve’s relation that reduces from the maximum to zero voltage is described by a single lateral play operator as shown in Figure 13 (the solid line).

## 5. Experiment Results and Discussion

**.**The driving power communicated with the host computer through the standard parallel port (SPP) parallel communication port. Here, we refer to the Preisach model parameters used in Song et al.’s study [16] and we used the experimental data in this study to establish the PCAs Preisach model. The desired displacement was taken as the input of the PI inverse model, the Preisach inverse model, and the tripartite PI inverse model. Thus, we got three sets of control voltages. These three sets of voltages were used to control the piezoelectric ceramic via the driving power. The output displacement was collected and recorded in real time by the laser interferometer. The control block diagram is shown in Figure 18. After the experiment, the data was processed and compared.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Tuma, T.; Lygeros, J.; Kartik, V.; Sebastian, A.; Pantazi, A. High-speed multiresolution scanning probe microscopy based on Lissajous scan trajectories. Nanotechnology
**2012**, 23, 185501. [Google Scholar] [CrossRef] [PubMed] - Braunsmann, C.; Schäffer, T.E. High-speed atomic force microscopy for large scan sizes using small cantilevers. Nanotechnology
**2010**, 21, 225705. [Google Scholar] [CrossRef] [PubMed] - Bhagat, U.; Shirinzadeh, B.; Tian, Y.; Zhang, D. Experimental analysis of laser interferometry-based robust motion tracking control of a flexure-based mechanism. IEEE Trans. Autom. Sci. Eng.
**2013**, 10, 267–275. [Google Scholar] [CrossRef] - Hansma, P.K.; Schitter, G.; Fantner, G.E.; Prater, C. High-speed atomic force microscopy. Appl. Phys.
**2006**, 314, 601–602. [Google Scholar] - Tuma, T.; Sebastian, A.; Lygeros, J.; Pantazi, A. The four pillars of nanopositioning for scanning probe microscopy: The position sensor, the scanning device, the feedback controller, and the reference trajectory. Control Syst. IEEE
**2013**, 33, 68–85. [Google Scholar] [CrossRef] - Clayton, G.M.; Tien, S.; Leang, K.K.; Zou, Q.; Devasia, S. A review of feedforward control approaches in nanopositioning for high-speed SPM. J. Dyn. Syst. Meas. Control
**2009**, 131, 636–650. [Google Scholar] [CrossRef] - Park, G.; Bement, M.T.; Hartman, D.A.; Smith, R.E.; Farrar, C.R. The use of active materials for machining processes: A review. Int. J. Mach. Tools. Manuf.
**2007**, 47, 2189–2206. [Google Scholar] [CrossRef] - Gozen, B.A.; Ozdoganlar, O.B. Design and evaluation of a mechanical nanomanufacturing system for nanomilling. Precis. Eng.
**2012**, 36, 19–30. [Google Scholar] [CrossRef] - Huang, S.; Tan, K.K.; Tong, H.L. Adaptive sliding-mode control of piezoelectric actuators. IEEE Trans. Ind. Electron.
**2009**, 56, 3514–3522. [Google Scholar] [CrossRef] - Zhu, W.; Rui, X. Hysteresis modeling and displacement control of piezoelectric actuators with the frequency-dependent behavior using a generalized Bouc–Wen model. Precis. Eng.
**2016**, 43, 299–307. [Google Scholar] [CrossRef] - Janaideh, A.; Farhan, M. Generalized Prandtl-Ishlinskii hysteresis model and its analytical inverse for compensation of hysteresis in smart actuators. Mech. Ind. Eng.
**2009**, 9, 307–312. [Google Scholar] - Li, P.; Li, P.; Sui, Y. Adaptive fuzzy hysteresis internal model tracking control of piezoelectric actuators with nanoscale application. IEEE Trans. Fuzzy Syst.
**2016**, 24, 1246–1254. [Google Scholar] [CrossRef] - Gu, G.Y.; Li, C.X.; Zhu, L.M.; Su, C.Y. Modeling and identification of piezoelectric-actuated stages cascading hysteresis nonlinearity with linear dynamics. IEEE/ASME Trans. Mechatron.
**2016**, 21, 1792–1797. [Google Scholar] [CrossRef] - Mokaberi, B.; Requicha, A.A.G. Compensation of Scanner Creep and Hysteresis for AFM Nanomanipulation. IEEE Trans. Autom. Sci. Eng.
**2008**, 5, 197–206. [Google Scholar] [CrossRef] - Gu, G.Y.; Zhu, L.M.; Su, C.Y. Integral resonant damping for high-bandwidth control of piezoceramic stack actuators with asymmetric hysteresis nonlinearity. Mechatronics
**2014**, 24, 367–375. [Google Scholar] [CrossRef] - Song, X.; Duggen, L.; Lassen, B.; Mangeot, C. Modeling and identification of hysteresis with modified preisach model in piezoelectric actuator. In Proceedings of the IEEE International Conference on Advanced Intelligent Mechatronics, Munich, Germany, 3–7 July 2017; pp. 1538–1543. [Google Scholar]
- Cao, Y.; Chen, X.B. A survey of modeling and control issues for piezo-electric actuators. J. Dyn. Syst. Meas. Control
**2015**, 137, 14001. [Google Scholar] [CrossRef] - Jiles, D.C.; Atherton, D.L. Theory of ferromagnetic hysteresis (invited). J. Magn. Mag.Mater.
**1986**, 61, 48–60. [Google Scholar] [CrossRef] - Carrera, Y.; Avila-de La Rosa, G.; Vernon-Carter, E.J.; Alvarez-Ramirez, J. A fractional-order Maxwell model for non-Newtonian fluids. Phys. A Stat. Mech. Its Appl.
**2017**, 482, 276–285. [Google Scholar] [CrossRef] - Malczyk, R.; Izydorczyk, J. The frequency-dependent Jiles–Atherton hysteresis model. Phys. B Condens. Matter
**2015**, 463, 68–75. [Google Scholar] [CrossRef] - Liu, Y.; Liu, H.; Wu, H.; Zou, D. Modelling and compensation of hysteresis in piezoelectric actuators based on Maxwell approach. Electron. Lett.
**2015**, 52, 188–190. [Google Scholar] [CrossRef] - Liu, L.; Tan, K.K.; Chen, S.L.; Huang, S.; Lee, T.H. SVD-based Preisach hysteresis identification and composite control of piezo actuators. ISA Trans.
**2012**, 51, 430–438. [Google Scholar] [CrossRef] [PubMed] - Hassani, V.; Tjahjowidodo, T.; Do, T.N. A survey on hysteresis modeling, identification and control. Mech. Syst. Signal Process.
**2014**, 49, 209–233. [Google Scholar] [CrossRef] - Chen, H.; Tan, Y.; Zhou, X.; Dong, R.; Zhang, Y. Identification of dynamic hysteresis based on duhem model. In Proceedings of the International Conference on Intelligent Computation Technology and Automation, Shenzhen, China, 28–29 March 2011; pp. 810–814. [Google Scholar]
- Lin, C.J.; Lin, P.T. Tracking control of a biaxial piezo-actuated positioning stage using generalized Duhem model. Comput. Math. Appl.
**2012**, 64, 766–787. [Google Scholar] [CrossRef] - Wang, G.; Chen, G.; Bai, F. Modeling and identification of asymmetric Bouc–Wen hysteresis for piezoelectric actuator via a novel differential evolution algorithm. Sens. Actuators A Phys.
**2015**, 235, 105–118. [Google Scholar] [CrossRef] - Huang, X.; Zeng, J.; Ruan, X.; Zheng, L.; Li, G. Structure, electrical and thermal expansion properties of PZnTe-PZT ternary system piezoelectric ceramics. J. Am. Ceram. Soc.
**2017**, 101, 274–282. [Google Scholar] [CrossRef] - Zhong, W. Physics of Ferroelectrics; Science Press: Beijing, China, 1996; pp. 294–297, 391–394. [Google Scholar]
- Bridger, K.; Jones, L.; Poppe, F.; Brown, S.A.; Winzer, S.R. High-force cofired multilayer actuators. Proc. SPIE
**1996**, 2721, 341–352. [Google Scholar] - Lancée, C.T.; Souquet, J.; Ohigashi, H.; Bom, N. Transducers in medical ultrasound: Part One. Ferro-electric ceramics versus polymer piezoelectric materials. Ultrasonics
**1985**, 23, 138. [Google Scholar] [CrossRef] - Rabe, K.M.; Ahn, C.H.; Triscone, J.M. Physics of Ferroelectrics; Springer: Berlin/Heidelberg, Germany, 2007; pp. 203–234. [Google Scholar]
- Merz, W.J. Domain Formation and Domain Wall Motions in Ferroelectric BaTiO
_{3}Single Crystals. Phys. Rev.**1954**, 95, 690–698. [Google Scholar] [CrossRef] - Merz, W.J. Switching Time in Ferroelectric BaTiO
_{3}and Its Dependence on Crystal Thickness. J. Appl. Phys.**1956**, 27, 938–943. [Google Scholar] [CrossRef]

**Figure 2.**Piezoelectric effect diagram (Red dashed lines indicate after deformation): (

**a**) Direct piezoelectric effect diagram; (

**b**) Inverse piezoelectric effect diagram. The black rectangle represents the original shape of the piezoelectric ceramic block, and the red dashed rectangle represents the deformed shape.

**Figure 4.**Schematic diagram of the spontaneous polarization alignment: (

**a**) before; (

**b**) during; (

**c**) after presence of an electric field.

**Figure 6.**Deformation rate of piezoelectric actuators at an applied triangle wave voltage of 150 V and a frequency of (

**a**) 0.2 Hz; (

**b**) 0.4 Hz; (

**c**) 1 Hz. Below the timeline, the voltage is loaded from the minimum voltage (0 V) to the maximum voltage (150 V). Above the timeline, the voltage drops from the maximum voltage (150 V) to the minimum voltage (0 V).

**Figure 7.**Deformation rate of piezoelectric actuators for an applied voltage of 150 V with frequency 1 Hz in (

**a**) triangular wave form; (

**b**) sign-wave form, u = 150(sinπt/5) positive half cycle; (

**c**) Manually added, 0 V–150 V–0 V, at steps of 15 V. Below the timeline, the voltage is loaded from the minimum voltage (0 V) to the maximum voltage (150 V). Above the timeline, the voltage drops from the maximum voltage (150 V) to the minimum voltage (0 V).

**Figure 15.**The tripartite PI model and its inverse model: (

**a**) experimental and tripartite PI model hysteresis curves of a piezoelectric ceramic actuator; (

**b**) experimental and tripartite PI inverse model hysteresis curves of a piezoelectric ceramic actuator.

**Figure 19.**Positioning accuracies of three kinds of inverse models: (

**a**) PI inverse model; (

**b**) Preisach inverse model; (

**c**) Tripartite PI inverse model.

**Table 1.**Parameters of the Prandtl–Ishlinskii (PI) model. (i is the number of sampling point, r

_{i}is the threshold of the play operator, w

_{i}is the weight of each hysteresis operator in the mathematical sense).

i | r_{i} | w_{i} |
---|---|---|

1 | 0 | 0.0493 |

2 | 15 | 0.0298 |

3 | 30 | 0.0120 |

4 | 45 | 0.0090 |

5 | 60 | 0 |

6 | 75 | 0 |

7 | 90 | 0 |

8 | 105 | 0 |

9 | 120 | 0 |

10 | 135 | 0 |

**Table 2.**Parameters of tripartite PI model. (i is the number of sampling point, r

_{1}is the threshold of the first stage play operator, w

_{1}is the weight of the first stage hysteresis operator in the mathematical sense; r

_{2}is the threshold of the second stage play operator, w

_{2}is the weight of the second stage hysteresis operator in the mathematical sense; r

_{3}is the threshold of the first stage play operator, w

_{3}is the weight of the first stage hysteresis operator in the mathematical sense).

i | r_{1} | w_{1} | r_{2} | w_{2} | r_{3} | w_{3} |
---|---|---|---|---|---|---|

1 | 0 | 0.0415 | 0 | 0.0529 | 0 | 0.0322 |

2 | 6.42 | 0.0097 | 15 | 0.0027 | 15 | 0.0081 |

3 | 12.84 | 0.0082 | 30 | 0.0067 | 30 | 0.0054 |

4 | 19.26 | 0.0067 | 45 | 0.0022 | 45 | 0.0044 |

5 | 25.68 | 0.0051 | 60 | 0 | 60 | 0.0065 |

6 | 32.10 | 0.0043 | 75 | 0 | 75 | 0.0039 |

7 | 38.52 | 0.0026 | 90 | 0 | 90 | 0.0087 |

8 | 44.94 | 0.0016 | 105 | 0 | 105 | 0.0016 |

9 | 51.36 | 0 | 120 | 0 | 120 | 0.0009 |

10 | 57.78 | 0 | 135 | 0 | 135 | 0.0008 |

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**MDPI and ACS Style**

An, D.; Li, H.; Xu, Y.; Zhang, L.
Compensation of Hysteresis on Piezoelectric Actuators Based on Tripartite PI Model. *Micromachines* **2018**, *9*, 44.
https://doi.org/10.3390/mi9020044

**AMA Style**

An D, Li H, Xu Y, Zhang L.
Compensation of Hysteresis on Piezoelectric Actuators Based on Tripartite PI Model. *Micromachines*. 2018; 9(2):44.
https://doi.org/10.3390/mi9020044

**Chicago/Turabian Style**

An, Dong, Haodong Li, Ying Xu, and Lixiu Zhang.
2018. "Compensation of Hysteresis on Piezoelectric Actuators Based on Tripartite PI Model" *Micromachines* 9, no. 2: 44.
https://doi.org/10.3390/mi9020044