Hysteresis modeling and compensation of piezo actuator with sparse regression

doi:10.23919/JSEE.2023.000172

Journal of Systems Engineering and Electronics ›› 2025, Vol. 36 ›› Issue (1): 48-61.doi: 10.23919/JSEE.2023.000172

• ELECTRONICS TECHNOLOGY • Previous Articles

Hysteresis modeling and compensation of piezo actuator with sparse regression

Yu JIN¹(), Xucheng WANG¹(), Yunlang XU²(), Jianbo YU²(), Qiaodan LU³(), Xiaofeng YANG¹^,*()

¹ Academy for Engineering & Technology, Fudan University, Shanghai 200433, China
² School of Microelectronics, Fudan University, Shanghai 200433, China
³ Shanghai Yinguan Semiconductor Technology Co., Ltd, Shanghai 201206, China

Received:2023-06-27 Accepted:2023-11-02 Online:2025-02-18 Published:2025-03-18
Contact: Xiaofeng YANG E-mail:21110860033@m.fudan.edu.cn;21110860046@m.fudan.edu.cn;xuyunlang@fudan.edu.cn;jb_yu@fudan.edu.cn;luqd@yg-st.com;xf_yang@fudan.edu.cn
About author:
JIN Yu was born in 1987. He received his M.S. degree in control engineering from Shanghai Jiao Tong University, Shanghai, China, in 2017. He is currently a Ph.D. student at the Academy for Engineering & Technology, Fudan University. His research interests include model identification and mechanical vibration. E-mail: 21110860033@m.fudan.edu.cn

WANG Xucheng was born in 1993. He received his M.S. degree in automation from Northwestern Polytechnical University, Xi’an, Shaanxi, China, in 2020. He is currently a Ph.D. student at the Academy for Engineering & Technology, Fudan University. His research interests include predictive control and robot. E-mail: 21110860046@m.fudan.edu.cn

XU Yunlang was born in 1993. He received his B.S. degree in mechanical and electrical engineering from Central South University, Changsha, China, in 2015, and Ph.D. degree in mechanical and electrical engineering from Huazhong University of Science and Technology, Wuhan, China, in 2021. He is currently a postdoctoral research fellow with Fudan University, Shanghai, China. His research interests include global optimization algorithm, adaptive control, sliding mode control, maglev systems, and hysteresis modeling. E-mail: xuyunlang@fudan.edu.cn

YU Jianbo was born in 1994. He received his B.E. degree in automation and Ph.D. degree in control science and engineering from East China University of Science and Technology, Shanghai, China, in 2016 and 2022, respectively. He is currently engaded in postdoctoral research with the School of Microelectronics, Fudan University. His research interests include explainable machine learning, deep learning models, process modeling and anomaly detection control, maglev systems, and hysteresis modeling. E-mail: jb_yu@fudan.edu.cn

LU Qiaodan was born in 1994. He received his B.S. degree in electrical engineering and automation from East China University of Science and Technology in 2017. Currently he is working as a simulation engineer in Shanghai YinGuan Semiconductor Technology Company. His research interests include precision motion control, modeling and simulation of flexible nano-motion stage. E-mail: luqd@yg-st.com

YANG Xiaofeng was born in 1964. He received his M.S. degree in automation control from Northeastern University, Shenyang, China, in 1989, and Ph.D. degree in mechanical engineering from Sophia University, Tokyo, Japan, in 1997. He is a full professor of control engineering with the School of Microelectronics, Fudan University, Shanghai, China. His research interests include advanced control method of precision motion control system, special actuator and sensor technology of nanometer precision multi-degree of freedom motion stage. E-mail: xf_yang@fudan.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (62203118)

Abstract

Abstract:

Piezo actuators are widely used in ultra-precision fields because of their high response and nano-scale step length. However, their hysteresis characteristics seriously affect the accuracy and stability of piezo actuators. Existing methods for fitting hysteresis loops include operator class, differential equation class, and machine learning class. The modeling cost of operator class and differential equation class methods is high, the model complexity is high, and the process of machine learning, such as neural network calculation, is opaque. The physical model framework cannot be directly extracted. Therefore, the sparse identification of nonlinear dynamics (SINDy) algorithm is proposed to fit hysteresis loops. Furthermore, the SINDy algorithm is improved. While the SINDy algorithm builds an orthogonal candidate database for modeling, the sparse regression model is simplified, and the Relay operator is introduced for piecewise fitting to solve the distortion problem of the SINDy algorithm fitting singularities. The Relay-SINDy algorithm proposed in this paper is applied to fitting hysteresis loops. Good performance is obtained with the experimental results of open and closed loops. Compared with the existing methods, the modeling cost and model complexity are reduced, and the modeling accuracy of the hysteresis loop is improved.

Key words: sparse identification of nonlinear dynamics (SINDy), hysteresis loop, relay operator, sparse regression, piezo actuator

Yu JIN, Xucheng WANG, Yunlang XU, Jianbo YU, Qiaodan LU, Xiaofeng YANG. Hysteresis modeling and compensation of piezo actuator with sparse regression[J]. Journal of Systems Engineering and Electronics, 2025, 36(1): 48-61.

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References 27

1	YU Z Q, ZHANG Y M, JIANG B PID-type fault-tolerant prescribed performance control of fixed-wing UAV. Journal of Systems Engineering and Electronics, 2021, 32 (5): 1053- 1061. doi: 10.23919/JSEE.2021.000090
2	LI Y, JIANG K, ZENG T, et al Belief reliability modeling and analysis for planetary reducer considering multi-source uncertainties and wear. Journal of Systems Engineering and Electronics, 2021, 32 (5): 1246- 1262. doi: 10.23919/JSEE.2021.000106
3	LI Z, LIU S N, SU C Y. A novel analytical inverse compensation approach for preisach model. Intelligent Robotics and Applications, Part II, 2013, 8103: 656−665.
4	YUE H Y, LU Y, DANG C, et al Transfer matrix model and experimental validation for the integrated piezo longitudinal actuators. Journal of Intelligent Material Systems and Structures, 2023, 34 (3): 352- 363. doi: 10.1177/1045389X221109252
5	LI Z, SHAN J J Inverse compensation based synchronization control of the piezo-actuated fabry-perot spectrometer. IEEE Trans. on Industrial Electronics, 2017, 64 (11): 8588- 8597. doi: 10.1109/TIE.2017.2711511
6	KREBS S. Modeling of a clamping-based piezo actuator in triangular configuration. Proc. of the IEEE 17th International Conference on Advanced Motion Control, 2022: 150−156.
7	PREISACH F About the magnetic aftereffect. Zeitschrift Fur Physik, 1935, 94 (5/6): 277- 302. doi: 10.1007/BF01349418
8	KRASNOSELSKII M A, POKROVSKII A V, CHERNORUTSKII V V, et al Dynamics of controlled systems described by parabolic equations with hysteresis nonlinearities. Automation and Remote Control, 1992, 53 (11): 1705- 1711.
9	LI Z, SHAN J J, GABBERT U Inverse compensation of hysteresis using Krasnoselskii-Pokrovskii model. IEEE/ASME Trans. on Mechatronics, 2018, 23 (2): 966- 971. doi: 10.1109/TMECH.2018.2805761
10	MANG H W, XU Y, AN D, et al Compensation of hysteresis on piezo actuators based on tripartite PI model. Nanotechnology and Precision Engineering, 2017, 15, 53- 60.
11	WEN Y K Method for random vibration of hysteretic systems. Journal of the Engineering Mechanics Division, 1976, 102 (2): 249- 263. doi: 10.1061/JMCEA3.0002106
12	COLEMAN B D, HODGDON M L On a class of constitutive relations for ferromagnetic hysteresis. Archive for Rational Mechanics and Analysis, 1987, 99, 375- 396. doi: 10.1007/BF00282052
13	LI Z, SHAN J J Modeling and inverse compensation for coupled hysteresis in piezo-actuated fabry-perot spectrometer. IEEE/ASME Trans. on Mechatronics, 2017, 22 (4): 1903- 1913. doi: 10.1109/TMECH.2017.2703167
14	LI Z, SU C Y, CHAI T Y Compensation of hysteresis nonlinearity in magnetostrictive actuators with inverse multiplicative structure for Preisach model. IEEE Trans. on Automation Science and Engineering, 2014, 11 (2): 613- 619. doi: 10.1109/TASE.2013.2284437
15	XU Y L, SHU F, SU X Y, et al A composite neural network-based adaptive sliding mode control method for reluctance actuator maglev system. Neural Computing & Applications, 2023, 35 (21): 15877- 15890.
16	CHENG L, LIU W, HOU Z G, et al Neural-network-based nonlinear model predictive control for piezoelectric actuators. IEEE Trans. on Industrial Electronics, 2015, 62 (12): 7717- 7727.
17	KOGA K, TAKEMOTO K Simple black-box universal adversarial attacks on deep neural networks for medical image classification. Algorithms, 2022, 15 (5): 144. doi: 10.3390/a15050144
18	WANG Y F, ZHOU M L Data driven adaptive control with hysteresis input for a piezo-actuated stage. Proc. of the IEEE 10th Data Driven Control and Learning Systems Conference, 2021, 218- 223.
19	LI C T, TAN Y H Adaptive control of system with hysteresis using neural networks. Journal of Systems Engineering and Electronics, 2006, 17 (1): 163- 167. doi: 10.1016/S1004-4132(06)60028-5
20	MANGAN N M, BRUNTON S L, PROCTOR J L, et al Inferring biological networks by sparse identification of nonlinear dynamics. IEEE Trans. on Molecular, Biological and Multi-Scale Communications, 2016, 2 (1): 52- 63. doi: 10.1109/TMBMC.2016.2633265
21	SU W J J. When is the first spurious variable selected by sequential regression procedures? Biometrika, 2018, 105(3): 517−527.
22	MANGAN N M, ASKHAM T, BRUNTON S L, et al Model selection for hybrid dynamical systems via sparse regression. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2019, 475 (2223): 20180534. doi: 10.1098/rspa.2018.0534
23	CARPENTIER A, COLLIER O, COMMINGES L, et al Estimation of the L(2)-norm and testing in sparse linear regression with unknown variance. Bernoulli, 2022, 28 (4): 2744- 2787.
24	FASEL U, KUTZ J N, NROUNTON B W, et al. Ensemble-SINDy: robust sparse model discovery in the low-data, high-noise limit, with active learning and control. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2022, 478(2260): 20210904.
28	KAHEMAN K, KUTZ J N, BRUNTON S L SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proceedings of the Royal Society A, 2020, 476 (2242): 20200279. doi: 10.1098/rspa.2020.0279
29	HU L, YI G X, HUANG C. A sparse algorithm for adaptive pruning least square support vector regression machine based on global representative point ranking. Journal of Systems Engineering and Electronics, 2021, 32(1): 151−162.
30	LIN W, SHI P X, FENG R, et al Variable selection in regression with compositional covariates. Biometrika, 2014, 101 (4): 785- 797. doi: 10.1093/biomet/asu031
31	ZHANG X Y, XU H Z, LI Z, et al Adaptive neural piecewise implicit inverse controller design for a class of nonlinear systems considering butterfly hysteresis. IEEE Trans. on Systems Man Cybernetics-Systems, 2023, 53 (6): 3695- 3706. doi: 10.1109/TSMC.2022.3231261
32	PEREV K L Rational function approximation of the relay with hysteresis nonlinear element. IFAC Papersonline, 2021, 54 (14): 19- 24. doi: 10.1016/j.ifacol.2021.10.322
33	LI Z, SHAN J J, GABBERT U A direct inverse model for hysteresis compensation. IEEE Trans. on Industrial Electronics, 2020, 68 (5): 4173- 4181.
34	ZHANG X Y, LIU Y H, CHEN X K, et al. Adaptive pseudoinverse control for constrained hysteretic nonlinear systems and its application on dielectric elastomer actuator. IEEE/ASME Trans. on Mechatronics, 2023, 28(4): 2142−2154.
35	LIU L, SHAN L, DAI Y W, et al Improved quantum bacterial foraging algorithm for tuning parameters of fractional-order PID controller. Journal of Systems Engineering and Electronics, 2018, 29 (1): 166- 175. doi: 10.21629/JSEE.2018.01.17
25	KIM S H, BOUKOUVALA F. Machine learning-based surrogate modeling for data-driven optimization: a comparison of subset selection for regression techniques. Optimization Letters, 2020, 14(4): 989−1010.
26	MOJAHEDI H, SANGAR AB, MASDARI M. Towards tax evasion detection using improved particle swarm optimization algorithm. Mathematical Problems in Engineering, 2022, 2022: 1027518.
27	MESSENGER D A, BORTZ D M Weak SINDy: Galerkin-based data-driven model selection. Multiscale Modeling & Simulation, 2021, 19 (3): 1474- 1497.

Model	P-I	Relay-SINDy
RMS/μm	0.3339	0.2468
Relative error/%	1.0644	0.7867

Signal input	Frequency/Hz	Relay-SINDy/(μm/%)	P-I/(μm/%)
Sine signal	1	1.558/7.79	2.118/10.59
	5	1.565/7.826	2.182/10.910
	10	1.553/7.76	2.300/11.504
Complex signal	1	1.399/6.996	1.915/9.575
	5	1.371/6.985	1.902/9.514
	10	1.404/7.022	1.939/9.695

Signal input	Frequency/Hz	Relay-SINDy/(μm/%)	P-I/(μm/%)
Sine signal	1	2.828/14.140	3.689/18.45
	5	2.715/13.575	3.944/19.72
	10	2.849/14.245	3.915/19.575
Complex signal	1	2.782/13.910	3.846/19.23
	5	2.790/13.950	3.684/18.420
	10	2.794/13.970	3.712/18.560

Signal input	Frequency/Hz	Relay-SINDy/(nm/%)	SINDy/(nm/%)	P-I/(nm/%)
Sine signal	1	1.63/0.01	74.64/0.37	5.261/0.03
	5	16.28/0.08	130.43/0.65	15.51/0.08
	10	34.23/0.17	234.54/1.17	45.28/0.23
Complex signal	1	0.31/0.001	18.99/0.09	2.32/0.01
	5	3.24/0.02	49.97/0.25	4.89/0.03
	10	7.56/0.04	66.91/0.33	8.59/0.04

Signal input	Frequency/Hz	Relay-SINDy/(nm/%)	SINDy/(nm/%)	P-I/(nm/%)
Sine signal	1	2.57/0.01	114.89/0.57	26.94/0.13
	5	24.19/0.12	286.81/1.43	37.02/0.19
	10	50.90/0.25	493.94/2.47	94.04/0.47
Complex signal	1	0.67/0.003	38.64/0.19	27.48/0.14
	5	7.27/0.04	105.48/0.53	25.34/0.13
	10	16.53/0.08	165.79/0.83	31.29/0.16

Hysteresis modeling and compensation of piezo actuator with sparse regression

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Signal input	Frequency/Hz	Relay-SINDy/(nm/%)	SINDy/(nm/%)	P-I/(nm/%)
Sine signal	1	38.98/0.19	91.96/0.45	54.26/ 0.27
	5	266.92/1.33	590.44/2.95	328.63/1.64
	10	492.99/2.47	1032.25/5.16	662.34/3.31
Complex signal	1	7.33/0.04	59.43/0.29	49.21/0.24
	5	67.49/0.34	145.23/ 0.72	82.65/0.41
	10	120.16/0.60	229.58/1.14	130.34/0.65

Signal input	Frequency/Hz	Relay-SINDy/(nm/%)	SINDy/(nm/%)	P-I/(nm/%)
Sine signal	1	86.64/0.43	230.89/1.15	158.54/0.79
	5	486.39/2.43	1063.52/5.31	601.15/3.00
	10	1076.43/5.38	1662.83/8.31	1082.23/5.41
Complex signal	1	26.12/0.13	190.43/0.95	165.80/0.82
	5	183.51/0.92	303.04/1.51	177.80/0.88
	10	286.15/1.34	531.91/2.65	259.93/1.29