This paper presents an approach employing disturbance observers to enhance the performance of inverse-based hysteresis compensation based on the generalized Prandtl–Ishlinskii model in feedback control reference-tracking applications. It is first shown that the error resulting from inexact hysteresis compensation is an L-bounded signal. Hence, a disturbance observer (DOB) is designed to cancel its effect and improve the closed loop robust tracking performance in the presence of plant dynamics uncertainty. The design of the DOB makes use of an equivalent internal model-based estimation of exogenous disturbances, where the internal model dynamics is designed to have at least an eigenvalue at the origin. The synthesis is then formulated as an H weighted-sensitivity optimization for static output feedback (SOF) gain of a Luenberger observer. A linearization heuristic is then implemented to solve the bilinear-matrix-inequality (BMI) constrained semidefinite program (SDP) for a (sub)optimal static gain. Simulation results indicate that tracking performance is indeed improved using the combined inversion-based compensation and the DOB.

References

References
1.
Smith
,
R. C.
,
2005
,
Smart Material Systems: Model Development
,
Society for Industrial and Applied Mathematics
,
Philidelphia, PA
.
2.
Al Janaideh
,
M.
, and
Krejčí
,
P.
,
2012
, “
Inverse Rate-Dependent Prandtl-Ishlinskii Model for Feedforward Compensation of Hysteresis in a Piezomicropositioning Actuator
,”
IEEE/ASME Trans. Mechatron.
,
PP
(
99
), pp
1
10
.10.1109/TMECH.2012.2205265
3.
Esbrook
,
A.
,
Guibord
,
M.
,
Tan
,
X.
, and
Khalil
,
H.
,
2010
, “
Control of Systems With Hysteresis via Servocompensation and its Application to Nanopositioning
,”
Proceedings of the American Control Conference
,
Baltimore, MD
, pp.
6531
6536
.
4.
Krejčí
,
P.
,
Al Janaideh
,
M.
, and
Deasy
,
F.
,
2012
, “
Inversion of Hysteresis and Creep Operators
,”
Phys. B
,
407
(
9
), pp.
1354
1356
.10.1016/j.physb.2011.06.020
5.
Al Janaideh
,
M.
, and
Krejčí
,
P.
,
2011
, “
An Inversion Formula for a Prandtl-Ishlinskii Operator with Time-Dependent Thresholds
,”
Phys. B
,
406
(
8
), pp.
1528
1532
.10.1016/j.physb.2011.01.062
6.
Tan
,
X.
, and
Khalil
,
H.
,
2007
, “
Control Unknown Dynamic Hysteretic Systems Using Slow Adaption: Preliminary Results
,”
Proceedings of the American Control Conference
,
New York
, pp.
3294
3299
.
7.
Krejčí
,
P.
, and
Kuhnen
,
K.
,
2001
, “
Inverse Control of Systems With Hysteresis and Creep
,”
IEEE Proc. Control Theory Appl.
,
148
(
3
), pp.
185
192
.10.1049/ip-cta:20010375
8.
Doyle
,
J.
,
Francis
,
B.
, and
Tannenbaum
,
A.
,
1990
,
Feedback Control Theory
,
Macmillan Publishing Co.
,
New York
.
9.
Al Janaideh
,
M.
, and
Krejčí
,
P.
,
2012
, “
Prandtl-Ishlinskii Hysteresis Models for Complex Time Dependent Hysteresis Nonlinearities
,”
Phys. B
,
407
(
9
), pp.
1365
1367
.10.1016/j.physb.2011.09.041
10.
Davino
,
D.
,
Natale
,
C.
,
Pirozzi
,
S.
, and
Visone
,
C.
,
2004
, “
A Phenomenological Dynamic Model of a Magnetostrictive Actuator
,”
Phys. B
,
343
(
1–4
), pp.
112
116
.10.1016/j.physb.2003.08.080
11.
Al Janaideh
,
M.
,
Yan
,
J.
,
D'Amato
,
A.
, and
Bernstein
,
D.
,
2012
, “
Retrospective-Cost Adaptive Control of Uncertain Hammerstein-Wiener Systems with Memoryless and Hysteretic Nonlinearities
,”
Proceedings of AIAA Guidance, Navigation, and Control Conference
,
Minneapolis, MN
, Paper No. AIAA-2012-4449-671, pp.
1
26
.
12.
Al Janaideh
,
M.
,
Sumer
,
E.
,
Yan
,
J.
,
Anthony
,
D.
,
Drincic
,
B.
,
Aljanaideh
,
K.
, and
Bernstein
,
D.
,
2012
, “
Adaptive Control of Uncertain Linear Systems With Uncertain Hysteretic Input Nonlinearities
,”
Proceedings of the 5th Annual Dynamic Systems and Control Conference
, Ft.
Lauderdale, FL
, pp.
1
10
.
13.
Nealis
,
J.
, and
Smith
,
R. C.
,
2007
, “
Model-Based Robust Control Design for Magnetostrictive Transducers Operating in Hysteretic and Nonlinear Regimes
,”
IEEE Trans. Control Syst. Technol.
,
15
(
1
), pp.
22
39
.10.1109/TCST.2006.883235
14.
Al Janaideh
,
M.
,
Feng
,
Y.
,
Rakheja
,
S.
,
Tan
,
Y.
, and
Su
,
C.-Y.
,
2009
, “
Generalized Prandtl-Ishlinskii Hysteresis: Modeling and Robust Control
,”
Proceedings of the IEEE Conference on Decision and Control
,
Shanghai, China
, pp.
7279
7284
.
15.
Al Janaideh
,
M.
,
Rakheja
,
S.
, and
Su
,
C.-Y.
,
2009
, “
Experimental Characterization and Modeling of Rate-Dependent Hysteresis of a Piezoceramic Actuator
,”
Mechatronics
,
19
(
5
), pp.
656
670
.10.1016/j.mechatronics.2009.02.008
16.
Visone
,
C.
, and
Sjöström
,
M.
,
2004
, “
Exact Invertible Hysteresis Models Based on Play Operators
,”
Phys. B
,
343
, pp.
148
152
.10.1016/j.physb.2003.08.087
17.
Al Janaideh
,
M.
,
Rakheja
,
S.
, and
Su
,
C.-Y.
,
2011
, “
An Analytical Generalized Prandtl-Ishlinskii Model Inversion for Hysteresis Compensation in Micro Positioning Control
,”
IEEE/ASME Trans. Mechatron.
,
16
(
4
), pp.
734
744
.10.1109/TMECH.2010.2052366
18.
Kuhnen
,
K.
,
2003
, “
Modeling, Identification and Compensation of Complex Hysteretic Nonlinearities-A Modified Plandtl-Ishlinskii Approach
,”
Eur. J. Control
,
9
(
4
), pp.
407
418
.10.3166/ejc.9.407-418
19.
Bickel
,
R.
, and
Tomizuka
,
M.
,
1999
, “
Passivity-Based Versus Disturbance Observer Based Robot Control: Equivalence and Stability
,”
J. Dyn. Syst., Meas. Control
,
121
(
1
), pp.
41
47
.10.1115/1.2802440
20.
El-Shaer
,
A. H.
,
Zhang
,
T.
, and
Al Janaideh
,
M.
,
2011
, “
On Robust Distrubance Observer Design Using Semi-Definite Programming
,”
Proceedings of the Dynamic Systems Control Conference
,
Arlington, VA
, pp.
1
7
.
21.
Schrijver
,
E.
, and
Dijk
,
J. V.
,
2002
, “
Disturbance Observers for Rigid Mechanical Systems: Equivalence
,”
Stab. Des., J. Dyn. Syst., Meas. Control
,
124
(
4
), pp.
539
548
.10.1115/1.1513570
22.
Shahruz
,
S. M.
,
2000
, “
Performance Enhancement of a Class of Nonlinear Systems by Disturbance Observers
,”
IEEE/ASME Trans. Mechatron.
,
5
(
3
), pp.
319
323
.10.1109/3516.868924
23.
Wang
,
C. C.
, and
Tomizuka
,
M.
,
2004
, “
Design of Robustly Stable Disturbance Observers Based on Closed Loop Consideration Using H∞ Optimization and its Applications to Motion Control
,”
Proceedings of the American Control Conference
,
Boston, MA
, pp.
3764
3769
.
24.
Johnson
,
C. D.
,
1971
, “
Accommodation of External Disturbances in Linear Regulator and Servomechanism Problems
,”
IEEE Trans. Autom. Control
,
16
(
6
), pp.
635
644
.10.1109/TAC.1971.1099830
25.
Lofberg
,
J.
,
2010
, “
YALMIP
,” http://control.ee.ethz.ch/joloef/yalmip.php
26.
Mita
,
T.
,
Hirata
,
M.
,
Murata
,
K.
, and
Zhang
,
H.
,
1998
, “
H∞ Control Versus Disturbance-Observer-Based Control
,”
IEEE Trans. Ind. Electron.
,
45
(
3
), pp.
488
495
.10.1109/41.679007
27.
Dullerud
,
G. E.
, and
Paganini
,
F.
,
2000
,
A Course in Robust Control Theory
,
Springer-Verlag
,
New York
.
28.
Söffker
,
D.
,
Yu
,
T.
, and
Müller
,
P. C.
,
1995
, “
State Estimation of Dynamical Systems With Nonlinearities by Using Proportional-Integral Observer
,”
Int. J. Syst. Sci.
,
26
(
9
),
1571
1582
.10.1080/00207729508929120
29.
Al
Janaideh
,
M.
,
2009
, “
Generalized Prandtl-Ishlinskii Hysteresis Model and its Analytical Inverse for Compensation of Hysteresis in Smart Actuators
,” Ph.D. thesis, Concordia University, Montreal, Canada.
30.
Cao
,
C.
, and
Hovakimyan
,
N.
,
2006
, “
Design and Analysis of a Novel L1 Adaptive Control Architecture with Guaranteed Transient Performance
,”
Proceedings of the American Control Conference
,
Minneapolis, MN
, pp.
3397
3402
.
31.
Fan
,
X.
, and
Smith
,
R.
,
2008
, “
Model-Based L1 Adaptive Control of Hysteresis in Smart Materials
,”
Proceedings of the IEEE Conference on Decision and Control
,
Cancun, Mexico
, pp.
3251
3256
.
32.
Umeno
,
T.
,
Kaneko
,
T.
, and
Hori
,
Y.
,
1993
, “
Robust Servosystem Design with Two Degrees of Freedom and its Application to Novel Motion Control of Robot Manipulators
,”
IEEE Trans. Ind. Electron.
,
40
(
5
), pp.
473
485
.10.1109/41.238016
33.
Kim
,
B. K.
, and
Chung
,
W. K.
,
2003
, “
Advanced Disturbance Observer Design for Mechanical Positioning Systems
,”
IEEE Trans. Ind. Electron.
,
50
(
6
), pp.
1207
1216
.10.1109/TIE.2003.819695
34.
Gahinet
,
P.
, and
Apkarian
,
P.
,
1994
, “
A Linear Matrix Inequality Approach to H∞ Control
,”
Int. J. Robust Nonlinear Control
,
4
(
4
), pp.
421
448
.10.1002/rnc.4590040403
35.
Leibfritz
,
F.
,
2001
, “
An LMI-Base Algorithm for Designing Suboptimal Static H2/H∞ Output Feedback Controllers
,”
SIAM J. Control Optim.
,
39
(
6
), pp.
1711
1735
.10.1137/S0363012999349553
36.
L.
El Ghaoui
,
F.
Oustry
, and
M.
AitRami
, 1997, “
A Cone Complementarity Linearization Algorithm for Static Output-Feedback and Related Problems,” IEEE Transactions on Automatic Control
,
42
(8)
, pp.
1171
1176
.
37.
Al Janaideh
,
M.
,
Feng
,
Y.
,
Rakheja
,
S.
,
Su
,
C.-Y.
, and
Rabbath
,
C.
,
2009
, “
Hysteresis Compensation for Smart Actuators Using Inverse Generalized Prandtl-Ishlinskii Model
,”
Proceedings of the American Control Conference
,
St. Louis, MO
, pp.
307
312
.
38.
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
You do not currently have access to this content.