Adaptive linearization of hysteresis for

International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
Adaptive Linearization of Hysteresis for Enhanced
Positional Accuracy of Robotic Arm
1
Saikat Kumar Shome, 2Mangal Prakash
1,4
Electronics & Instrumentation Group, CSIR-CMERI
Durgapur, India
1
[email protected]
2
[email protected]
Abstract—Precision control of multi-axis piezo actuated
micro / nanopositioning stages suffers not only from the inherent
nonlinearities but also from parametric variations and
uncertainties. In order to effectively implement any real-time
control theory to mitigate the effects of nonlinearity, an adaptive
mechanism to update the control system is a prerequisite. This
paper investigates the effects of hysteretic nonlinearity in a
second order Dahl model based piezoelectric manipulator. The
plant performance is known to improve with injection of a
suitable dose of noise called “dither”. The concept of dither based
control is interwoven with the conventional PID controller to
introduce a novel adaptive dither control strategy. A double axis
piezo stage is considered for controller design and validation. The
controller performance is ascertained using three specific tests- a)
Tracking Error Test, b) Multi-frequency, multi-amplitude
tracking test and c) Two-axis Circular Contour Tracking Test.
Results show an enhanced positioning accuracy of the
manipulator in the presence of proposed control scheme than
without it. The proposed control transpires as an effective
control to improve tracking accuracy and resist external
disturbances.
Keywords—hysteresis; piezoelectric actuator; adaptive control;
dither; Stochastic Resonance (SR); auto-tuning; nanopositioning.
I.
INTRODUCTION
For past few decades, piezoelectric type micro/nano
positioners have revolutionized nano scale trajectory tracking
in industrial applications like Scanning Probe Microscopy [1],
semiconductor fabrication [2], micro robotics [3] and adaptive
optics[4-5]amongst many others. The inherent high resolution
of displacement, high stiffness, large blocking force and quick
response characteristics have enabled piezo based
manipulators supplant the traditional positioning systems.
However, the piezo-driven stages have always been plagued
with hysteresis which manifests itself as the major obstruction
in achieving the desired positioning accuracy. Hysteresis
introduces severe open loop positioning error which may even
be as large as 15% of the entire travel range. Hysteresis can
also destabilize the closed loop operation of piezoelectric
actuator and limits its applicability [6-7].
To counter these shortfalls, extensive research has been done
on piezoelectric ceramic hysteretic modeling and its control.
The main idea behind every modeling has been the
978-1-4799-5202-1/14/$31.00 ©2014 IEEE
3
Sourav Pradhan, 4Arpita Mukherjee
2,3
Dept. of Electrical Engg., National Institute of Technology
Durgapur, India
3
[email protected]
4
[email protected]
development of a mathematical formulation that can faithfully
and effectively capture the characteristics of hysteresis under
widely varying operating conditions. In fact, the level of
precision obtainable from a piezo actuated stage is heavily
dependent on the accuracy with which it is modeled. Various
modeling frameworks have been designed over years to
successfully encapsulate the dynamic characteristics of
hysteresis. Some standout frameworks include PrandtlIshlinskii model, Duhem model, Preisach model, Maxwell
model, Dahl model, Bouc Wen model and other memory
based frameworks [8-13]. Most of these models have a
convoluted structure to account for the multiple-loop behavior
of hysteresis. A charge-driven circuit has been proposed
in [14] to get around the complications involved in modeling
hysteresis, but at the expense of costly instruments, reduction
in system responsiveness and the amplification of
measurement noise.
The design of effective control systems is equally important
in order to achieve the desired level of trajectory tracking of
time varying signals. The control spectrum of piezoelectric
actuators can be broadly divided into two categories:
feedforward and feedback control. Feedforward control
structures essentially rely upon the inverse hysteresis models
to generate a compensated voltage and hence, reduce
hysteresis [15-16]. Such a control is known to operate with
near perfection provided no uncertainties and external
disturbances exist in the system. But practical applications are
inadvertently dynamic in nature and the possibility of
disturbances cannot be rejected completely. Feedback control
covers these drawbacks of the feedforward controller and
provides improved levels of positioning [10], [15]. Yet, no
methodic stability and robustness are given for the feedback
systems [17].
The general perception of noise as a foe in modern systems
and applications has undergone a radical transformation in
recent decades. In fact, noise has been used as a powerful tool
in the fields of control systems and signal processing to realize
better levels of performance. Recent studies have investigated
noise induced performance boost in the domain of
piezoelectric actuators [18-21]. The noise which is used to
corrupt the otherwise pure system for augmenting the system
capabilities is generally called dither. Dithering is presently
International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
creating a buzz in physics, chemistry, engineering, bio
sciences and several others fields. In the context of
piezoelectric actuators, it is found to be inducing another wellknown phenomenon called Stochastic Resonance (SR). As a
consequence of this actuality, the system tracking error
undergoes a minimum with a variation of injected noise
intensity.
The presence of parameter uncertainties and external
disturbances are arguably the greatest challenges in
implementing any real-time control scheme. This elucidates
the need of adaptive and robust control techniques when highperformance applications are demanded. Researchers have
devised adaptive controllers to meet this end for
nanopositioning applications via piezo stages. Most of these
schemes employ a representative hysteresis model [22-23].
Asymptotic robust schemes like Sliding mode control [24]
have enjoyed considerable success but their practicality is
limited by chattering phenomenon. The control system design
assumes a greater challenge when the system is subjected to
other sources of uncertainties like cross-coupling effect in
piezo stages.
This article employs a second order Dahl model to envisage
the hysteresis behavior of a dual axis parallel piezoelectric
nanopositioning stage. As a general control strategy,
feedforward topology realized by inverse Dahl model is
deployed here. Dithering in tandem with SR is used to support
the effort in developing a dither based control scheme. The
benefits of adaptive control are incorporated in the system by
adaptively tuning the dither amplitude with the help of a
classical PID controller. Results indicate effective dual axis
tracking and robustness to sudden variations in input. The
computationally simple algorithm and the ease of
implementation advocate the potential of the proposed novel
control structure.
II.
DYNAMIC MODEL OF THE SYSTEM
The piezoelectric actuator used for experimentations is a
P-841 series translator manufactured by Physik Instrumente
which can provide sub nanometer resolution as well as sub
millisecond response. It is equipped with highly reliable
multilayer piezo ceramic stacks protected by a non-magnetic
stainless steel case with internal spring preload. P-841.40
preloaded piezoactuator with integrated feedback strain gauge
sensor has been used. The strain gauge sensor provides high
resolution for closed loop operation. The absence of polymer
insulation and the high Curie temperature ensure optimal
ultra-high vacuum compatibility, i.e., minimum outgassing.
The allowable operating temperature for this setup lies within
-20° to +80° C. A data acquisition system (cRIO-9073) is used
to interface the equipment with LABVIEW real-time
workshop. FPGA mode of cRIO-9073 is used which consists
of an 8-slot integrated 266 MHz real-time controller. cRIO is
equipped with NI-9205 32 channel, + 10 V, 250 kS/s, 16-bit
analog input module, NI 9219 4 channel-channel isolated, 24bit, + 60 V, 100 kS/s, universal analog input module and NI
9263 4 channel, 16 bit, + 10 V, 100 kS/s/ch, analog output
Fig. 1: Experimental Setup of the piezoelectric actuator
module. A schematic representation of the experimental setup
is shown in Fig. 1.
The hysteretic modeling of piezoelectric actuators presents
two major challenges: a) the non-local memory of the
hysteresis loops and b) the asymmetry of hysteresis loops
between ascending and descending paths. A popular hysteretic
model that captures hysteretic characteristics appreciably well
and describes the dynamics of piezo-positioning mechanism is
second order Dahl model. It has been known to portray
hysteresis loops with near exactness and has lesser number of
parameters compared to the higher order Dahl models [25].
The second order Dahl model in the presence of hysteresis is
analyzed below.
For two dimensional motion tracking (along x and y axes),
two separate and decoupled micromanipulators can be
assumed. The mathematical treatment of manipulator along x
axis is described here. A similar approach is valid for the
analysis of micro-manipulator along y axis. Considering the
mechanical parts of micromanipulator system to be linear and
of second order, the dynamic model of the system in presence
of nonlinear hysteresis is given by (1).
Mx + Dx + Kx = Tu − Fh
(1)
where the parameters x, K, D and M represent the x-axis
displacement, stiffness, damping coefficient and equivalent
mass of the XY micromanipulator respectively. T denotes the
piezoelectric coefficient, U stands for the input voltage and Fh
indicates the hysteretic effect in terms of force. Equation (2) is
the state space representation of the second order Dahl model
based entire manipulator system incorporating hysteretic force
term [25].
V = AhVx + Bh u p x
(2)
Fh = ChV
(3)
where, up is a constant (set to 30), V = [q1 q2]T is the
intermediate state vector and
Ah =
⎡ 0
⎢ −a
⎣ 2
⎤
− sgn( x ) a1 ⎥⎦
Bh =
1
⎡0⎤
⎢1 ⎥
⎣ ⎦
(4)
(5)
International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
Ch = [ b1
sgn( x )b0 ]
(6)
TABLE I: IDENTIFIED DYNAMIC MODEL PARAMETERS OF THE MICRO
MANIPULATOR WITH DAHL HYSTERESIS
Parameter
Value
Unit
M
0.1828
kg
D
2.5973 x 103
N s/m
K
2.6065 x104
N/m
T
0.0468
C/m
a1
121.9874
-
a2
1.1773 x 106
-
b1
1.8485 x 106
-
b0
0
-
(a)
(b)
Fig. 2: a) System floor plan of the plant using Dahl Model, b) Dahl model for
calculating the Hysteretic term Fh
The identified Dahl model parameters for the XY
micromanipulator system is given in Table I. LABVIEW
implementation of the identified dynamic model with Dahl
hysteresis is shown in Fig. 2(a) and Fig. 2(b).
III.
CONTROLLER DESIGN
A properly designed controller is needed as a remedy for
the nonlinear hysteresis in order to achieve a high performance
tracking control. Once the trajectory to be tracked is decided,
the controller design centers on the input voltage necessary to
track the desired trajectory.
A. Pure Feedforward Control (PFC)
Feedforward control depicted in Fig. 4 is one of most
fundamental control strategies which has been known to
effectively perform required control actions in open loop. As
an anticipative control structure, it guarantees a perfect control
action provided the model dynamics are known with certainty.
In this control algorithm, an inverse Dahl model is used in the
feedforward path, Fig. 3. The controller efficacy depends upon
the accuracy with which the hysteresis is modeled. To
implement an open loop control scheme, the inverse Dahl
model maps the compensated voltage corresponding to the
desired position to be tracked. The output of the inverse Dahl
model is fed as an input to the Dahl model in order to force the
system to track the desired signal. The inverse Dahl model is
directly implemented for online identification of hysteresis.
This is essential as hysteresis if estimated off-line will not
match the actuator nonlinearities and there exists a possibility
of an error between the desired and the actual compensated
outputs.
B. Unadaptive Dithered Nonlinear Control Paradigm
As explained earlier, dither inputs are high frequency noise
signals which maximize the information content of an
otherwise pure system or enhance its performance. Dither
inputs have been used in control systems and signal processing
applications to alleviate the effects of nonlinearity, hysteresis,
quantization, gear backlash, etc. Studies have shown that
dithering holds amazing capabilities when it comes to
linearizing the effects of hysteretic nonlinearity in piezoelectric
actuators [18-21]. Injection of a suitable dose of noise
stimulates the manifestation of the popular phenomenon of SR.
Since its emergence as an explanation for the periodic
recurrence of Earth’s ice ages [26], it has traversed many
disciplines ranging from electronic systems [27] to sensing
neurons [28] and from bidirectional ring lasers [29] to super
conducting quantum loops [30]. The three basic ingredients
necessary for demonstration of SR are: a nonlinear system with
a threshold, a sub-threshold periodic signal and noise.
It has been shown in [31] that given these conditions, the
system potential can be modeled as a symmetric/asymmetric
bistable potential well and the system will oscillate between its
two stable states with frequency equal to the frequency of
periodic forcing causing a peak in Power Spectral Density. In
context of piezoelectric actuator, the second order based Dahl
model acts as the nonlinear system whereas the trajectory to
be tracked (low frequency sinusoidal signal in this work)
represents the periodic forcing. Noise is modeled by the
externally added dither signal. The piezo potential operating
inside the sample surface potential can be described by (7)
which represents an unsymmetrical bistable potential.
Δ6θ
K 2
θ
+
x −
(7)
2M
( x + Z ) 210( x + Z ) 7
where Δ and θ are parameters depending upon the material
of piezo tip and sample on which piezo is operating and Z is
the distance. In piezo, SR is manifested by a dip in tracking
error or by a peak in tracking when a suitable dose of noise is
injected with sinusoiadal periodic forcing.
V ( x) =
International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
amplitude A for the desired dither. The value of A given
by (10) should be in the range of 1 to 10. If A goes past the
range 1 to 10, then the effective dither amplitude will change
by a factor of 10. This will result in a greater tracking error as
the effective dither amplitude is not equal to the best dither
amplitude.
d = et − c
(8)
Fig. 3: Inverse Dahl Model for Compensated Voltage Calculation
Fig. 4: Pure Feedforward Control Strategy
C. Displacement Dither Based PID Adaptive Loop Control
Precision tracking control of the two-axis nano positioning
stage presents certain non-idealities like parametric
uncertainties, external disturbances and the unmodeled
dynamics including hysteresis modeling uncertainties and the
effects of cross-coupling. In order to tackle all these
roadblocks, an efficient adaptive controller is essential. In this
section, an adaptive dither based control strategy is proposed.
The schematic diagramic of this control is demonstrated in
Fig. 5.
In this control, a classical PID controller is utilized to tune
the dither amplitude adaptively in accordance with a change in
the tracking error. For successful implementation of this
control strategy, a few pertinent issues like what is the
criterion of reduction of nonlinearity? and what is a suitable
dither amplitude?, need to addressed. The limiting condition
for nonlinearity is taken as the minimum condition which if
present can reduce the system tracking error below a prespecified value (assumed to be 10% of the tracking error
obtained for undithered system in this case). Since the
proposed adaptive control strategy exploits the dithering
capabilities in displacement dither mode (dither is injected as
displacement), the best displacement dither obtained for
unadaptive displacement dithering has been used as the
suitable dither amplitude in this scheme. The algorithm for
adaptively varying the dither amplitude is described below.
The tracking error et in this case is compared with a
constant c, (its value being 10% of the tracking error obtained
for unadaptively dithered system). This assumption is valid as
it effectively aims at keeping the tracking error within a range
of 10% of that obtained with an unadaptive dithered system.
The difference d thus obtained is worked upon by a PID
Controller to generate Ad. The PID controller output (Ad) is
compared with a constant A1 to generate the adaptive
(
)
Ad = K p d + K i ∫ ddt + K d d * K
A = A1 − Ad
(9)
(10)
where A1 is the initial best estimate of suitable dither
amplitude and Ad is the compensated adaptively generated
dither amplitude.
The selection of control parameters is equally important in
this case as well. The sine dither used has the same value as
that found out for best displacement dither in unadaptive
displacement dither control paradigm. A1 is selected such that
‫ ܣ‬ൌ ‫ܣ‬ଵ െ ‫ܣ‬ௗ is constrained between 1 to 10 for better
adaptivity. Constant c is computed to be 37.3e (-10), A1= 6
and PID parameters of controller are P = 100, I = -0.1, D = 0.
IV.
RESULTS AND DISCUSSIONS
The effectiveness of the proposed control strategies are
tested rigorously using a sinusoidal input having amplitude 110
μm and frequency 1 Hz. The input covers the entire range of
the workspace.
Commencing with arbitrary dither amplitude, the optimum
dither intensity in unadaptive displacement dithering mode is
found out by RMS tracking error test. The dither amplitude
which ensures the least RMS tracking error is established as the
best dither. SR also manifests itself during the search for the
best dither. When the dither amplitude is gradually changed, a
resonance like behavior appears. The RMS tracking error
attains a minima for a particular value of dither amplitude (10e
(-11)). For dither amplitudes other than this optimum value, the
system performance degrades. SR is evident in Fig. 6 where
tracking error is plotted against varying dither amplitudes in
displacement dithering mode. As evident from Fig. 7 and Table
II, unadaptive displacement dithering mode achieves a
reduction of hysteretic curve area by 15.788 % as compared to
the undithered system. The best displacement dither found
herein is used for further simulations in adaptive control
paradigm. PID Adaptive Loop Control further enhances
tracking performance with a reduced tracking error of 29.546
nm. The results of tracking error test with the proposed control
paradigms are presented in Fig. 8.
To ascertain the robustness and adaptive capabilities of the
proposed controller, multi-amplitude, multi-frequency test is
undertaken as well. The system is subjected to input amplitudes
of 90 μm, 110 μm and 130 μm and frequency variation
includes 0.25 Hz, 0.5 Hz and 1 Hz respectively. From the plots
obtained for actual position versus desired position in Fig. 9, it
is observed that the control action is robust and successfully
mitigates the impact of sudden input variation.
International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
Fig. 5: Pure Feedforward cum PID Adaptive Control Strategy
Fig. 9: Input vs. Output plot for Multi-Frequency Multi-Amplitude Tracking.
Fig. 6: Stochastic Resonance Plots for various systems.
Fig. 10: Two Axis Tracking for x-y axes inputs differing by angle 90° for
various systems.
Fig. 7: Hysteresis plots
TABLE II. SYSTEM PERFORMANCE WITH VARIOUS CONTROL STRATEGIES
Type
Control
Fig. 8: Tracking Error Plots for Various Systems
The two-dimensional motion tracking of the manipulator
needs to be tested as well apart from the single axis tracking
undertaken above. Contouring error is defined as the error
between the actual position and desired position along an
orthogonal direction to the trajectory. Contouring has been
done for input signals having magnitudes of 110 µm and
of
RMS
Error (in
nm)
%
Error
Reduction
RMS Contouring
Error (in nm)
Undithered
System
37.22
0
85.483
Unadaptive
Best
Displacement
Dithered
31.42
15.788
79.319
PID
Adaptive
Scheme
29.546
20.617
70.964
frequencies 1 Hz along both x and y axes. Fig. 10 depicts the
contour tracking performance of the PID Adaptive Loop
controller with two axis inputs differing by a phase angle of
90ο.
International Conference on Magnetics, Machines & Drives (AICERA-2014 iCMMD)
V.
CONCLUSION
An adaptive displacement dither based control paradigm
with good performance characteristics is designed for efficient
dual-axis tracking of micro/nano positioning stage. The
controller imparts 20.617 % effective hysteresis compensation
as compared to the undithered system. The reduced tracking
error of 29.546 nm validates the results. Additionally, the
controller is shown to be robust in presence of sudden input
variations. Multi-amplitude, multi-frequency trajectory was
tracked with high accuracy by the PID Adaptive Loop
controlled nanopositioning platform. The cross-coupling
effects of the micro/nano manipulator are also mitigated
considerably with a contour tracking error of only 70.964 nm
obtained for the two axes inputs differing in phase by 90ο. The
tracking performance improvement over both unadaptive and
undithered systems is noteworthy, making this strategy
suitable for real-time industrial applications.
ACKNOWLEDGMENT
The authors acknowledge the continuous support of
Director, CSIR-CMERI for carrying out this research.
REFERENCES
[1]
S. Gonda, T. Doi, T. Kurosawa, Y. Tanimura, N. Hisata,
T. Yamagishi, H. Fujimoto and H. Yukawa, “Accurate topographic
images using a measuring atomic force microscope,” Appl. Surf. Sci.,
vol. 144/145, pp. 505–509, 1999.
[2] K. Kajiwara, M. Hayatu, S. Imaoka and T. Fujita, “Application of large
scale active microvibration control system using piezoelectric actuators
to semiconductor manufacturing equipment,” in Proc. SPIE, Bellingham,
WA, 1997, vol. 3044, pp. 258–269.
[3] J. Hesselbach, R. Ritter, R. Thoben, C. Reich and G. Pokar, “Visual
control and calibration of parallel robots for micro assembly,” in Proc.
SPIE, Boston, MA, Nov. 1998, vol. 3519, pp. 50–61.
[4] A. Henke, M.A. K¨ummel and J. Wallaschek, “A piezoelectrically
driven wire feeding system for high performance wedge wedge-bonding
machines,” Mechatronics, vol. 9, pp. 757–767, 1999.
[5] S. Aoshima, N. Yoshizawa and T. Yabuta, “Compact mass axis
alignment device with piezo elements for optical fibers,” IEEE Photon.
Technol. Lett., vol. 4, no. 5, pp. 462–464, May 1992.
[6] Z. X- Long and T.Y- Hong, “Intelligent Modeling for Hysteresis
Nonlinearity in Piezoceramic Actuator, ” Journal of System Simulation,
vol. 18, no. 1, Jan. 2006.
[7] F-J. Lin, H-J. Shieh, P-K. Huang and L-T. Teng.“Adaptive Control with
Hysteresis Estimation and Compensation Using RFNN for PiezoActuator,” IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, vol. 53, no. 9, September 2006.
[8] K. Kuhnen and H. Janocha, “Complex hysteresis modeling of a
broadcclass of hysteretic nonlinearities,” in Proc. 8th Int. Conf. New
Actuators, Bremen, Germany, Jun. 2002, pp. 688–691.
[9] Y. Stepanenko and C.Y. Su, “Intelligent control of piezoelectric
actuators,” in Proc. 37th IEEE Conf. Decision Control, 1998, vol. 4,
pp. 4234–4239.
[10] G. Ping and J. Musa, “Generalized Preisach model for hysteresis
nonlinearityof piezoceramic actuators,” Precision Eng., vol. 20,
pp. 99–111, 1997.
[11] H. Tang and Y. Li, “Design, Analysis and Test of a Novel 2-DOF
Nanopositioning System Driven by Dual-Mode, ” IEEE Transactions on
Robotics, February 2013.
[12] A. I. Cahyadi and Y. Yamamoto, “Modelling a Micro Manipulation
System With Flexure Hinge,” Proceedings of the IEEE International
Conference on Robotics, Automation and Mechatronics, 2006, pp. 1–5.
[13] S. Bashash and N. Jalili, “Intelligent rules of hysteresis in
feedforwardtrajectory control of piezoelectrically-driven nanostagers,”
J. Micromech.Microeng., vol. 17, pp. 342–349, 2007.
[14] K. Furutani, M. Urushibata and N. Mohri, “Displacement control
ofpiezoelectric element by feedback of induced charge,”
Nanotechnology, vol. 9, pp. 93–98, 1998.
[15] R. Changhai and S. Lining, “Improving positioning accuracy of
piezoelectricactuators by feedforward hysteresis compensation based on
a new mathematical model,” Rev. Sci. Instrum., vol. 76,
pp. 095111-1–095111-8, 2005.
[16] S. Lining, “Tracking control of piezoelectric actuator based on a
newmathematical model,” J. Micromech. Microeng., vol. 14,
pp. 1439–1444, 2004.
[17] S. Bashash and N. Jalili, “Robust Adaptive Control of Coupled Parallel
Piezo Flexural Nanopositioning Stages,” IEEE Trans. on Mechatronics,
vol. 14, no. 1, pp. 11-20, 2009.
[18] S.K. Shome, M. Prakash, A. Mukherjee and U. Datta, “Dither Control
for Dahl Model Based Hysteresis Compensation,” IEEE International
Conference on Signal Processing, Computing and Control”, 2013,
pp. 1-6.
[19] S. Pradhan, S. K. Shome, M. Prakash and M. K. Patel, “Performance
Evaluation of Dither Distributions of Piezo Electric Actuators for NanoPositioning,” IEEE International Conference on Control, Automation,
Robotics and Embedded System, 2013, pp. 1-6.
[20] M. Prakash, S. K. Shome, S. Pradhan and M. K. Patel “A Comparison of
Dithers for Hysteresis Alleviation in Second Order Dahl Model Based
Piezoelectric Actuator,” IEEE International Conference on Control,
Automation, Robotics and Embedded System, 2013, pp. 1-6.
[21] S.K. Shome, S. Pradhan, A. Mukherjee and U. Datta, “Dither Based
Precise Position Control of Piezo Actuated Micro-Nano Manipulator,”
39th Annual Conference of the IEEE Industrial Electronics Society,
Austria, 2013, pp. 3486-3491.
[22] S. Bashash and N. Jalili, “Robust multiple-frequency trajectory
trackingcontrol of piezoelectrically-driven micro/nano-positioning
systems,” IEEE Trans. Control Syst. Technol., vol. 15, no. 5,
pp. 867–878, Sep. 2007.
[23] C.L. Hwang, Y.M. Chen and C. Jan, “Trajectory tracking of large
displacementpiezoelectric actuators using a nonlinear observer-based
variable structure control,” IEEE Trans. Control Syst. Technol., vol. 13,
no. 1, pp. 56–66, Jan. 2005.
[24] J.J. Slotine and S.S. Sastry, “Tracking control of nonlinear systems
usingsliding surface with application to robot manipulators,” Int. J.
Control, vol. 38, pp. 465–492, 1983.
[25] Q. Xu and Y. Li, “Dahl Model-Based Hysteresis Compensation and
PrecisePositioning Control of an XY Parallel Micromanipulator With
Piezoelectric Actuation,” Journal of Dynamic System, Measurement and
Control, ASME, pp. 041011-1-041011-12, 2010.
[26] R. Benzi, G. Parisi, A. Sutera and A. Vulpiani, “Stochastic resonance in
climatic change,” Tellus, 1982.
[27] V.I. Malnikov, “Schmitt trigger: a solvable model of stochastic
resonance,” Phys. Rev. E, vol. 48, no. 4, pp. 2481–2489, 1993.
[28] J.K. Douglas, L. Wilkens, E. Pantazelou and F. Moss, “Noise
enhancement of information transfer in crayfish mechanoreceptors by
stochastic resonance,” Nature, vol. 365, pp.337–339, 1993.
[29] B. McNamara, K. Wiesenfeld, R. Roy, “Observation of
stochasticresonance in a ring laser,” Phys. Rev. Lett., vol. 60,
pp. 2626–2629, 1988.
[30] K. Wiesenfeld and F. Moss, “Stochastic resonance and the benefits
ofnoise: from ice ages to crayfish and SQUIDs,” Nature, vol. 373,
pp. 33–36, 1995.
[31] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, “Stochastic
Resonance,” Review of Modern Physics, vol. 70, no. 1, 1998.