Abstract

Hysteresis is a nonlinear characteristic ubiquitously exhibited by smart material sensors and actuators, such as piezoelectric actuators and shape memory alloys. The Prandtl–Ishlinskii (PI) operator is widely used to describe hysteresis of smart material systems due to its simple structure and the existence of analytical inverse. A PI operator consists of a weighted superposition of play (backlash) operators. While adaptive estimation of the weights for PI operators has been reported in the literature, rigorous analysis of parameter convergence is lacking. In this article, we establish persistent excitation and thus parameter convergence for adaptive weight estimation under a rather modest condition on the input to the PI operator. The analysis is further supported via simulation, where a recursive least square (RLS) method is adopted for parameter estimation.

References

References
1.
Tian
,
Y.
,
Shirinzadeh
,
B.
, and
Zhang
,
D.
,
2009
, “
A Flexure-Based Five-Bar Mechanism for Micro/Nano Manipulation
,”
Sens. Actuators., A.
,
152
(
1
), pp.
96
104
. 10.1016/j.sna.2009.04.022
2.
Stefanski
,
F.
,
Minorowicz
,
B.
,
Persson
,
J.
,
Plummer
,
A.
, and
Bowen
,
C.
,
2017
, “
Non-Linear Control of a Hydraulic Piezo-Valve Using a Generalised Prandtl-Ishlinskii Hysteresis Model
,”
Mech. Syst. Signal Process.
,
82
, pp.
412
431
. 10.1016/j.ymssp.2016.05.032
3.
Minorowicz
,
B.
,
Leonetti
,
G.
,
Stefanski
,
F.
,
Binetti
,
G.
, and
Naso
,
D.
,
2016
, “
Design, Modelling and Control of a Micro-Positioning Actuator Based on Magnetic Shape Memory Alloys
,”
Smart Mater. Struct.
,
25
(
7
), p.
075005
. 10.1088/0964-1726/25/7/075005
4.
Tan
,
X.
, and
Baras
,
J.
,
2004
, “
Modeling and Control of Hysteresis in Magnetostrictive Actuators
,”
Automatica
,
40
(
9
), pp.
1469
1480
. 10.1016/j.automatica.2004.04.006
5.
Zhu
,
Z.
,
To
,
S.
,
Li
,
Y.
,
Zhu
,
W.
, and
Bian
,
L.
,
2018
, “
External Force Estimation of a Piezo-Actuated Compliant Mechanism Based on a Fractional Order Hysteresis Model
,”
Mech. Syst. Signal Process.
,
110
, pp.
296
306
. 10.1016/j.ymssp.2018.03.012
6.
Gu
,
G. Y.
,
Li
,
C. X.
,
Zhu
,
L. M.
, and
Su
,
C. -Y.
,
2015
, “
Modeling and Identification of Piezoelectric-Actuated Stages Cascading Hysteresis Nonlinearity With Linear Dynamics
,”
IEEE/ASME Trans. Mech.
,
21
(
3
), pp.
1792
1797
. 10.1109/TMECH.2015.2465868
7.
Li
,
Y.
,
Tong
,
S.
, and
Li
,
T.
,
2012
, “
Adaptive Fuzzy Output Feedback Control of Uncertain Nonlinear Systems With Unknown Backlash-Like Hysteresis
,”
Inform. Sci.
,
198
(
7
), pp.
130
146
. 10.1016/j.ins.2012.02.050
8.
Oh
,
J.
, and
Bernstein
,
D. S.
,
2005
, “
Semilinear Duhem Model for Rate-Independent and Rate-Dependent Hysteresis
,”
IEEE Trans. Auto. Control
,
50
(
5
), pp.
631
645
. 10.1109/TAC.2005.847035
9.
Zhou
,
M.
,
Yang
,
P.
,
Wang
,
J.
, and
Gao
,
W.
,
2016
, “
Adaptive Sliding Mode Control Based on Duhem Model for Piezoelectric Actuators
,”
IETE Tech. Rev.
,
33
(
5
), pp.
557
568
. 10.1080/02564602.2015.1126202
10.
Wen
,
Y.-K.
,
1976
, “
Method for Random Vibration of Hysteretic Systems
,”
J. Eng. Mech. Div.
,
102
(
2
), pp.
249
263
. 10.1061/JMCEA3.0002106
11.
Xu
,
R.
,
Zhang
,
X.
,
Guo
,
H.
, and
Zhou
,
M.
,
2018
, “
Sliding Mode Tracking Control With Perturbation Estimation for Hysteresis Nonlinearity of Piezo-Actuated Stages
,”
IETE Access
,
6
, pp.
30617
30629
. 10.1109/ACCESS.2018.2840538
12.
Mayergoyz
,
I. D.
,
2003
,
Mathematical Models of Hysteresis and Their Applications
,
Academic Press
.
13.
Li
,
Z.
,
Su
,
C.-Y.
, and
Chai
,
T.
,
2014
, “
Compensation of Hysteresis Nonlinearity in Magnetostrictive Actuators With Inverse Multiplicative Structure for Preisach Model
,”
IEEE Trans. Auto. Sci. Eng.
,
11
(
2
), pp.
613
619
. 10.1109/TASE.2013.2284437
14.
Krasnosel’skiĭ
,
M. A.
, and
Pokrovskiĭ
,
A. V.
,
1989
,
Systems With Hysteresis
,
Springer
,
New York
.
15.
Xu
,
R.
, and
Zhou
,
M.
,
2017
, “
Elman Neural Network-Based Identification of Krasnosel’skii-Pokrovskii Model for Magnetic Shape Memory Alloys Actuator
,”
IEEE. Trans. Magn.
,
53
(
11
), pp.
1
4
. 10.1109/TMAG.2018.2792846
16.
Brokate
,
M.
, and
Sprekels
,
J.
,
1996
,
Hysteresis and Phase Transitions
, Vol.
121
,
Springer Science & Business Media
,
New York
.
17.
Al Janaideh
,
M.
, and
Tan
,
X.
,
2019
, “
Adaptive Estimation of Threshold Parameters for a Prandtl-Ishlinskii Hysteresis Operator
,”
Proceedings of the 2019 American Control Conference
,
Philadelphia, PA
,
July 10–12
, pp.
3770
3775
.
18.
Zhang
,
J.
,
Merced
,
E.
,
Sepulveda
,
N.
, and
Tan
,
X.
,
2015
, “
Optimal Compression of Generalized Prandtl-Ishlinskii Hysteresis Models
,”
Automatica
,
57
(
7
), pp.
170
179
. 10.1016/j.automatica.2015.04.012
19.
Li
,
Y.
,
Feng
,
Y.
,
Feng
,
J.
, and
Liu
,
Y.
,
2019
, “
Parameter Identification Based on PSO Algorithm for Piezoelectric Actuating System With Rate-Dependent Prandtl-Ishlinskii Hysteresis Modeling Method
,”
Proceedings of the 2019 IEEE 4th International Conference on Advanced Robotics and Mechatronics (ICARM)
,
Osaka, Japan
,
July 3–5
, pp.
36
41
.
20.
Xie
,
S.
,
Mei
,
J.
,
Liu
,
H.
, and
Wang
,
Y.
,
2018
, “
Hysteresis Modeling and Trajectory Tracking Control of the Pneumatic Muscle Actuator Using Modified Prandtl-Ishlinskii Model
,”
Mech. Mach. Theory.
,
120
, pp.
213
224
. 10.1016/j.mechmachtheory.2017.07.016
21.
Su
,
Q.
,
Wang
,
C-Y.
,
Chen
,
X.
, and
Rakheja
,
S.
,
2005
, “
Adaptive Variable Structure Control of a Class of Nonlinear Systems With Unknown Prandtl-Ishlinskii Hysteresis
,”
IEEE Trans. Auto. Control
,
50
(
12
), pp.
2069
2074
. 10.1109/TAC.2005.860260
22.
Kuhnen
,
K.
, and
Janocha
,
H.
,
1999
, “
Adaptive Inverse Control of Piezoelectric Actuators With Hysteresis Operators
,”
Proceedings of the 1999 European Control Conference (ECC)
,
Karlsruhe, Germany
,
Aug. 31–Sept. 3
, pp.
791
796
.
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