Analysis of closed-loop systems involving hysteresis is important to both the understanding of these systems and the synthesis of control schemes. However, such analysis is challenging due to the nonsmooth nature of hysteresis nonlinearities. In this paper, singular perturbation techniques are employed to derive an analytical approximation to the tracking error for a system consisting of fast linear dynamics preceded by a piecewise linear hysteresis nonlinearity, which is motivated by applications such as piezo-actuated nanopositioning. The control architecture considered combines hysteresis inversion and proportional-integral feedback, with and without a constant feedforward control. The analysis incorporates the effect of uncertainty in the hysteresis model, and offers insight into how the tracking performance depends on the system parameters and the references, thereby offering guidance in the controller design. Simulation and experimental results on a piezo-actuated nanopositioning system are presented to support the analysis. In particular, the control scheme incorporating the feedforward element consistently outperforms the classical PI controller in tracking a variety of references.

References

References
1.
Devasia
,
S.
,
Eleftheriou
,
E.
, and
Moheimani
,
S. O.
,
2007
, “
A Survey of Control Issues in Nanopositioning
,”
IEEE Trans. Control Syst. Technol.
,
15
(
5
), pp.
802
823
.10.1109/TCST.2007.903345
2.
Yong
,
Y. K.
,
Aphale
,
S. S.
, and
Moheimani
,
S. R.
,
2009
, “
Design, Identification, and Control of a Flexure-Based XY Stage for Fast Nanoscale Positioning
,”
IEEE Trans. Nanotechnol.
,
8
(
1
), pp.
46
54
.10.1109/TNANO.2008.2005829
3.
Kuhnen
,
K.
, and
Janocha
,
H.
,
2001
, “
Inverse Feedforward Controller for Complex Hysteretic Nonlinearities in Smart-Material Systems
,”
Control Intell. Syst.
,
29
(
3
), pp.
74
83
.
4.
Janaideh
,
M. A.
,
Rakheja
,
S.
, and
Su
,
C.-Y.
,
2011
, “
An Analytical Generalized Prandtlishlinskii Model Inversion for Hysteresis Compensation in Micropositioning Control
,”
IEEE/ASME Trans. Mechatronics
,
16
(
4
), pp.
734
744
.10.1109/TMECH.2010.2052366
5.
Mayergoyz
,
I.
,
1986
, “
Mathematical Models of Hysteresis
,”
IEEE Trans. Magn.
,
22
(
5
), pp.
603
608
.10.1109/TMAG.1986.1064347
6.
Kuhnen
,
K.
,
2003
, “
Modeling, Identification, and Compensation of Complex Hysteretic Nonlinearities—A Modified Prandtl-Ishlinskii Approach
,”
Eur. J. Control
,
9
(
4
), pp.
407
418
.10.3166/ejc.9.407-418
7.
Tao
,
G.
, and
Kokotovic
,
P.
,
1995
, “
Adaptive Control of Plants With Unknown Hysteresis
,”
IEEE Trans. Autom. Control
,
40
(
2
), pp.
200
212
.10.1109/9.341778
8.
Croft
,
D.
,
Shedd
,
G.
, and
Devasia
,
S.
,
2000
, “
Creep, Hysteresis, and Vibration Compensation for Piezo Actuators: Atomic Force Microscopy Application
,”
Proceedings of the 2000 American Control Conference
, pp.
2123
2128
.
9.
Iyer
,
R.
, and
Tan
,
X.
,
2009
, “
Control of Hysteretic Systems Through Inverse Compensation: Inversion Algorithms, Adaptation, and Embedded Implementation
,”
IEEE Control Syst Mag.
,
29
(
1
), pp.
83
99
.10.1109/MCS.2008.930924
10.
Chen
,
X.
,
Hisayama
,
T.
, and
Su
,
C.-Y.
,
2008
, “
Pseudo-Inverse-Based Adaptive Control for Uncertain Discrete Time Systems Preceded by Hysteresis
,”
Automatica
,
45
, pp.
469
476
.10.1016/j.automatica.2008.08.004
11.
Tan
,
X.
, and
Baras
,
J. S.
,
2004
, “
Modeling and Control of Hysteresis in Magnetostrictive Actuators
,”
Automatica
,
40
(
9
), pp.
1469
1480
.10.1016/j.automatica.2004.04.006
12.
Tan
,
X.
, and
Baras
,
J.
,
2005
, “
Adaptive Identification and Control of Hysteresis in Smart Materials
,”
IEEE Trans. Autom. Control
,
50
(
6
), pp.
827
839
.10.1109/TAC.2005.849215
13.
Tan
,
X.
, and
Khalil
,
H. K.
,
2009
, “
Two-Time-Scale Averaging of Systems Involving Operators and Its Application to Adaptive Control of Hysteretic Systems
,”
Proceedings of the 2009 American Control Conference
, pp.
4476
4481
.
14.
Esbrook
,
A.
,
Tan
,
X.
, and
Khalil
,
H. K.
,
2013
, “
Control of Systems With Hysteresis Via Servocompensation and Its Application to Nanopositioning
,”
IEEE Trans. Control Syst. Technol.
,
21
(
3
), pp.
725
738
.10.1109/TCST.2012.2192734
15.
Kuhnen
,
K.
, and
Janocha
,
K.
,
1999
, “
Adaptive Inverse Control of Piezoelectric Actuators With Hysteresis Operator
,”
Proceedings of the European Control Conference
, Germany, Paper F 0291.
16.
Shen
,
J.
,
Jywea
,
W.
,
Chiang
,
H.
, and
Shub
,
Y.
,
2008
, “
Precision Tracking Control of a Piezoelectric-Actuated System
,”
Precis. Eng.
,
32
, pp.
71
78
.10.1016/j.precisioneng.2007.04.002
17.
Bashash
,
S.
, and
Jalili
,
N.
,
2007
, “
Robust Multiple Frequency Trajectory Tracking Control of Piezoelectrically Driven Micro/Nanopositioning Systems
,”
IEEE Trans. Control Syst. Technol.
,
15
(
5
), pp.
867
878
.10.1109/TCST.2007.902949
18.
Su
,
C. Y.
,
Stepanenko
,
Y.
,
Svoboda
,
J.
, and
Leung
,
T. P.
,
2000
, “
Robust Adaptive Control of a Class of Nonlinear Systems With Unknown Backlash-Like Hysteresis
,”
IEEE Trans. Autom. Control
,
45
, pp.
2427
2432
.10.1109/9.895588
19.
Bashash
,
S.
, and
Jalili
,
N.
,
2009
, “
Robust Adaptive Control of Coupled Parallel Piezo-Flexural Nanopositioning Stages
,”
IEEE/ASME Trans. Mechatronics
,
14
, pp.
11
20
.10.1109/TMECH.2008.2006501
20.
Zhong
,
J.
, and
Yao
,
B.
,
2008
, “
Adaptive Robust Precision Motion Control of a Piezoelectric Positioning Stage
,”
IEEE Trans. Control Syst. Technol.
,
16
, pp.
1039
1046
.10.1109/TCST.2007.916319
21.
Valadkhan
,
S.
,
Morris
,
K.
, and
Khajepour
,
A.
,
2008
, “
Robust PI Control of Hysteretic Systems
,”
Proceedings of the 47th IEEE Conference on Decision and Control
, pp.
3787
3792
.
22.
Wu
,
Y.
, and
Zou
,
Q.
,
2007
, “
Iterative Control Approach to Compensate for Both the Hysteresis and the Dynamics Effects of Piezo Actuators
,”
IEEE Trans. Control Syst. Technol.
,
15
(
5
), pp.
936
944
.10.1109/TCST.2007.899722
23.
Ge
,
P.
, and
Jouaneh
,
M.
,
1996
, “
Tracking Control of a Piezoceramic Actuator
,”
IEEE Trans. Control Syst. Technol.
,
4
(
3
), pp.
209
216
.10.1109/87.491195
24.
Prempain
,
E.
, and
Postlethwaite
,
I.
,
2001
, “
Feedforward Control: A Full-Information Approach
,”
Automatica
,
37
(
1
), pp.
17
28
.10.1016/S0005-1098(00)00118-7
25.
Youla
,
D.
, and
Bongiorno
,
J.
,
1985
, “
A Feedback Theory of Two-Degree-of-Freedom Optimal Wiener-Hopf Design
,”
IEEE Trans. Autom. Control
,
30
(
7
), pp.
652
665
.10.1109/TAC.1985.1104023
26.
Howze
,
J.
, and
Bhattacharyya
,
S.
,
1997
, “
Robust Tracking, Error Feedback, and Two-Degree-of-Freedom Controllers
,”
IEEE Trans. Autom. Control
,
42
(
7
), pp.
990
983
.10.1109/9.599977
27.
Hara
,
S.
, and
Sugie
,
T.
,
1988
, “
Independent Parameterization of Two-Degree-of-Freedom Compensators in General Robust Tracking Systems
,”
IEEE Trans. Autom. Control
,
33
(
1
), pp.
59
67
.10.1109/9.361
28.
Edardar
,
M.
,
Tan
,
X.
, and
Khalil
,
H. K.
,
2012
, “
Tracking Error Analysis for Singularly Perturbed Systems Preceded by Piecewise Linear Hysteresis
,”
Proceedings of the 51th IEEE Conference on Decision and Control
, pp.
3139
3144
.
29.
Zhou
,
K.
, and
Doyk
,
J.
,
1998
,
Essentials of Robust Control
,
Tom Robbins
, Upper Saddle River, NJ.
30.
Khalil
,
H.
,
2002
,
Nonlinear Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
31.
Kokotovic
,
P.
,
Khalil
,
H. K.
, and
O'reilly
,
J.
,
1999
, “
Singular Perturbation Methods in Control Analysis
,”
SIAM
, Philadelphia, PA.
32.
Edardar
,
M.
,
2013
, “
Robust Control of Systems With Piecewise Linear Hysteresis
,” Ph.D. thesis, Michigan State University, East Lansing, MI.
33.
Wei
,
Q.
,
Hu
,
C.
, and
Zhang
,
D.
,
2010
, “
Precision Control of Piezoelectric Actuated Mechanism in Scanning Tunneling Microscope
,”
Proceedings of the 8th World Congress on Intelligent Control and Automation (WCICA)
, pp.
1262
1267
.
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