Damping and Tracking Control Schemes For Nanopositioning
Damping and Tracking Control Schemes For Nanopositioning
Damping and Tracking Control Schemes For Nanopositioning
I. INTRODUCTION
ANOPOSITIONING devices are used for high-resolution
positioning, including, but not limited to, scanning probe
microscopy and its many applications for imaging and manipulation. Some tasks require periodic reference trajectory tracking,
typically for imaging, while other tasks require arbitrary reference trajectory tracking, such as for manipulation, fabrication,
and lithography. To improve throughput in such applications,
high-bandwidth control is required [1][3].
Most high-speed nanopositioning devices use piezoelectric
actuators, as they produce large forces and provide friction-less
motion. As such, they are ideal for high-speed, high-resolution
positioning. A positioning device applying piezoelectric actua-
Manuscript received June 12, 2012; revised October 11, 2012; accepted
December 8, 2012. Date of publication February 14, 2013; date of current
version February 20, 2014. Recommended by Technical Editor Y. Sun. This
work was supported by the Norwegian University of Science and Technology
and the Norwegian Research Council. Part of this work was carried out during
the tenure of an ERCIM Alain Bensoussan Fellowship Programme. This Programme is supported by the Marie Curie Co-Funding of Regional, National and
International Programmes of the European Commission.
The authors are with the Department of Engineering Cybernetics, Norwegian
University of Science and Technology, N-7491 Trondheim, Norway (e-mail:
eielsen@itk.ntnu.no; Marialena.Vagia@itk.ntnu.no; Tommy.Gravdahl@itk.
ntnu.no; Kristin.Y.Pettersen@itk.ntnu.no).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TMECH.2013.2242482
tors typically exhibit lightly damped resonances. This is a disadvantage, as it limits the usable bandwidth because reference
signals with high-frequency components will excite the vibration modes, prohibiting accurate positioning. It also makes the
device susceptible to environmental disturbances, such as sound
and floor vibrations. The hysteresis and creep nonlinearities in
piezoelectric actuators is an additional challenge. These are loss
phenomena that prevent the system from having a linear response, introducing bounded input disturbances dependent on
the driving voltage signal [3]. There also exist several sources
of dynamic uncertainty. Hysteresis, in addition to introducing
an input disturbance, change the effective gain of the actuator depending on the amplitude and frequency of the driving
voltage signal. Actuator gain is also dependent on temperature
and reduces over time due to depolarization [4], [5, Ch. 4]. In
addition, users typically need to position payloads of various
masses; thus, resonant frequencies and the effective gain of the
mechanical structure can change as a result.
Tracking control for nanopositioning devices can be achieved
using feed-forward and feedback control techniques. Although
feed-forward techniques can provide very good results [2], [6],
feedback control may be necessary in order to reduce sensitivity to uncertainty and disturbances. Combining feed-forward
and feedback control can improve overall tracking performance
[7], [8]. In order to control lightly damped vibrational modes in
active structures, several control schemes that introduce damping have been developed. These include fixed-structure, loworder control laws, such as positive position feedback (PPF)
[9], integral force feedback (IFF) [10], passive shunt-damping
(PSD) [11], resonant control [12], and integral resonant control
(IRC) [13]. By coupling such schemes with an integral control law, significantly better reference tracking performance can
be achieved. With the exception of PSD, this has been experimentally demonstrated in [14][16]. The main reason for the
increased performance is that a reduction of the dominant resonant peak of the system leads to an increased gain margin,
enabling much higher gain to be used for the reference tracking
integral control law [16].
General model-based control laws can also be used, such as
H -control law synthesis [17], the linear-quadratic-Gaussian
regulator [18], and output feedback control laws such as pole
placement and model reference control (MRC) [18], [19]. H control has seen widespread application on nanopositioning systems [20][23]. Nonlinear control approaches, such as slidingmode control, have also been applied [7], [24].
The advantage of using fixed-structure, low-order control
laws is mostly practical, as such control laws are simple
to implement and have low computational complexity. This
1083-4435 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
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allows for the highest possible sampling frequency when implementing using digital signal processing equipment, which
will reduce the noise floor due to quantization in the analog-todigital converter [25]. The simplicity also makes them feasible
for implementation using analog circuit elements. This can be
beneficial, as it avoids noise due to sampling and quantization
altogether. The disadvantage of using fixed-structure, low-order
control laws, is a lack of methods for optimal tuning, and this
is a long standing and challenging control engineering problem [26], [27].
A. Contributions
Six different damping and tracking control schemes are presented and applied to a nanopositioning system for experimental comparison. All the schemes combine integral action with a
control law that introduces damping of the dominant vibration
mode of the system. The damping control schemes considered
are PPF, IRC, IFF, and PSD.
A control scheme based on the work in [28] is also presented,
and the tuning methodology therein is generalized and also
applied to the presented control schemes based on PPF, IRC,
IFF, and PSD. IRC has been applied in [15], and IFF in [16]. PPF
and polynomial-based control (also known as pole placement)
are applied in [14]. Furthermore, in this paper, pole placement
in the form of MRC is applied, also incorporating integral action
and filtering, in order to reduce sensitivity to disturbances and
uncertainty, and to reduce quantization noise.
II. SYSTEM DESCRIPTION AND MODELING
A. Description of the Experimental System
The experimental setup consists of a dSPACE DS1103
hardware-in-the-loop (HIL) system, an ADE 6810 capacitive
gauge, an ADE 6501 capacitive probe from ADE Technologies, a Piezodrive PDX200 voltage amplifier, two SIM 965
programmable filters from Stanford Research Systems, and
the custom-made long-range serial-kinematic nanopositioner
shown in Fig. 1. The nanopositioner is fitted with a Noliac
SCMAP07-H10 actuator, where one of the stack elements is
used as a force transducer. The transducer current is measured
433
Fig. 3.
System diagram.
434
(1)
where kp (N V1 = C N1 ) is the effective gain of the piezoelectric actuator from voltage to force, and ua (V) is the applied
voltage. Piezoelectric actuators introduce hysteresis and creep
when driven by an external voltage signal. These effects occur
in the electrical domain [31], and it is a reasonable assumption
to consider this behavior as a bounded disturbance added to the
input, represented by the term w (V) [32].
The dynamics due to an applied voltage ua or disturbance
w of a point d (m) on the flexible structure, as observed by
a colocated sensor, is adequately described by the following
lumped parameter, truncated linear model [33]:
d
d
i
(s)
+ Dr
2 + 2 s + 2
fa
s
i i
i
i=1
Gd (s) = kp
(2)
Fig. 4. Response and uncertainty for the displacement. (a) Frequency response
for the displacement model G d (s). (b) Multiplicative uncertainty weight d (s).
C. Charge
q = kp d + Cp ua = Cp (ua + d)
where Cp (F) is the capacitance of the piezoelectric stack actuator, and (Vm1 ) is a constant determining the amount of
charge generated by the direct piezoelectric effect due to the displacement d of the mechanical structure. The transfer function
from applied voltage ua to induced charge q is therefore
Gq (s) =
q
(s) = Cp (1 + Gd (s)).
ua
(3)
D. Force Transducer
The IFF scheme utilizes a colocated piezoelectric force transducer. The force transducer generates a charge, depending on
the applied force. The current or charge produced by the force
transducer is typically converted to a voltage signal using a simple op-amp circuit with a high input impedance. The output
voltage from such a sensor when measuring the charge can be
435
TABLE I
IDENTIFIED MODEL PARAMETERS
Fig. 7.
Fig. 5. Response and uncertainty for the charge. (a) Frequency response for
the charge model G q (s). (b) Multiplicative uncertainty weight q (s).
Fig. 6. Response and uncertainty for the force. (a) Frequency response for the
force model G f (s). (b) Multiplicative uncertainty weight f (s).
single-input multiple-output (SIMO), the uncertainty description of the models from the scalar input up to the output vector
yp has the form [36]
yp i = Gi (s)(1 + i (s)i (s))up ; i (s) 1
up = C(s)(r F (s)yp )
(6)
(5)
where i denotes the index into the output vector yp , such that
Gi (s) corresponds to the transfer function from the input up
to the output yp i , and i (s) is the corresponding frequency
dependent uncertainty weight. The uncertainty weights {i (s)}
(7)
436
(8)
(9)
(10)
(11)
(12)
ny
i=1
C. Tuning
Control design for fixed-structure, low-order control laws using output feedback is a long-standing and challenging problem
in control engineering. A common approach to output feedback
problems is to use H -synthesis. If the control law is allowed to
have any order and every matrix of the control law is freely tunable, H -synthesis guarantees a solution to the control design
problem by convex optimization.
For a control law with a fixed structure and with a lower order
than the plant, this approach cannot be applied. There exist some
results for fixed low-order control problems, solved with the use
of linear matrix inequalities, but these methods do not allow for
the use of unstructured uncertainty, do not guarantee global,
and in many cases not even local, convergence, and might not
accommodate for control laws where the structure is fixed in
addition to the order [26], [27]. In other words, there does not
exist any general control design method for output feedback
using fixed-structure, low-order control laws.
In this paper, a practical optimization procedure is, therefore,
proposed in order to obtain good tracking performance.
Control design is often a tradeoff between conflicting goals.
For nanopositioning systems, it is desirable to have a high bandwidth for E(s) in order to have good reference tracking. Also,
the system is required to be well damped in order to avoid excessive vibrations. This translates to an absence of peaks in T (s).
To counter hysteresis and creep, as well as environmental disturbances, D(s) must provide a high degree of attenuation. In
addition, the amplification of sensor noise should be as small
as possible, meaning that N (s) should have the smallest bandwidth possible. Due to the restriction imposed by the Bode
sensitivity integral [18], it is impossible to meet these criteria
simultaneously.
As the purpose of damping control is to reduce peaks in the
closed-loop response due to lightly damped vibration modes,
and since ideal tracking performance is achieved when
d = r T (s) = 1
it appears that a good overall performance criterion is the flatness
of |T (j)|. Let c be the vector of control law parameters. It is
here proposed that the flatness criterion can be expressed using
the cost function
J(c ) = 1 |T (c ; j)|2
(14)
(15)
where {i } is the set of eigenvalues for the closed-loop system. The optimization problem can be solved either by using
Fig. 8.
437
(16)
where kd > 0 is the control law gain, d is the damping coefficient, and d is the cutoff frequency. The tracking control law
is an integral control law with a negative gain, which will be
inverted by the negative gain of the filter (16)
kt
.
(17)
s
Here, kt > 0 is the gain of the integral term.
To analyze the nominal performance of the control scheme,
the control structure in Fig. 8 can be put on the equivalent
formulation adhering to the control structure in Fig. 7. The
feed-forward filter is found as
Ct (s) =
(18)
(19)
kd
.
s kd Df
(20)
(22)
(23)
and
438
TABLE III
OPTIMAL PARAMETERS FOR (20) AND (21)
TABLE IV
OPTIMAL PARAMETER FOR (24)
(28)
ki
s
(24)
(29)
f
s + f
(25)
(26)
(31)
(30)
Ci (s)Gd (s)
1 Ci (s)Gf (s) + Wlp (s)Ci (s) [Gd (s) + Gf (s)]
Ci (s)Gd (s)
1 Ci (s)Gf (s)
(32)
(27)
(33)
439
Fig. 10.
ki
s
(34)
(35)
(36)
(37)
(38)
ki
s
(39)
2
c
.
s2 + 2c s + c 2
(40)
(41)
(42)
440
Fig. 12.
Nominal frequency responses. (a) PPF. (b) IRC. (c) IFF. (d) PSD. (e) DI. (f) MRC.
TABLE VI
OPTIMAL PARAMETERS FOR (39) AND (40)
stage in [40]. The synthesis equations for the MRC scheme are
summarized in the Appendix.
For the MRC design, the displacement model of the system
Gd (s) is truncated to only include the dominant piston mode at
1660 Hz, the second mode of the positioning stage,
2
d (s) = Gd (0)
G
.
2 /2 2 s2 + 22 2 s + 2 2
(43)
(44)
p (s),
and is of seventh order. In addition to the plant model G
there are two additional design choices with regards to the
TABLE VII
CONTROL LAW PARAMETERS FOR MRC
TABLE VIII
BANDWIDTH OF E(s), N (s)2 FROM n d TO d, D(s) FROM w TO d, 1/,
RMSE, AND ME OBTAINED FOR THE CONTROL SCHEMES
control law: the reference model Wm (s) and the output filter
1/(s). The main limiting factor in determining these filters is
the uncertainty of the plant model, which for the system at hand
is mostly due to nonmodeled high-frequency dynamics.
For simplicity, the reference model Wm (s) is chosen to be a
seventh-order Butterworth filter with cutoff frequency m . Since
p (s) does not have any zeros, (s) should be a
the plant model G
polynomial of sixth order. The zeros of (s) were chosen to have
a Butterworth pattern with radius l . The reconstruction and
antialiasing filters have a user-programmable cutoff frequency
c , which can be tuned, given that c is below the Nyquist
frequency.
The design problem is as such reduced to three tunable control
law parameters
c = [c , m , l ]T
the cutoff frequencies c , m , l . The reference model is a
Butterworth filter, which is maximally flat by definition. The
optimality criterion described in (14) is, therefore, satisfied for
all m , l , and c that render a stable closed-loop system. The
cutoff frequencies were, therefore, tuned manually, attempting
to obtain the highest bandwidth for E(s) while still having a
robustly stable closed-loop system. The control law parameters
used are found in Table VII.
The actual implementation of the scheme in the Appendix is
done by augmenting the filter (49) by an integrator, i.e.,
1
C(s)
= C(s)
s
(45)
and using the filter F (s) in (50) as it is. The parameters for the
filters C(s)
and F (s) are found using (51) and (52), as given in
the Appendix.
With regard to the general control structure in Fig. 7, the analysis in terms of the sensitivity (7), complementary sensitivity
(8), noise attenuation (9), error attenuation (10), and disturbance
rejection (12) is done using
Gp (s) = Gd (s)
441
(46)
C(s) = C(s)W
r (s)
(47)
(48)
and
mance when using a triangle wave reference signal with a fundamental frequency of 80 Hz and an amplitude of 1 m was
recorded for each scheme. The fundamental frequency of the
reference signal was chosen in order for the 21st harmonic of
the signal to be close to the dominant vibration mode. The
displacement for all the schemes was measured on a separate
channel using an antialiasing filter with a 35-kHz cutoff frequency. The generated current from the force transducer was
measured, and integrated numerically. The cutoff frequency for
the antialiasing filter for this measurement was always 20 kHz
for all the experiments.
A. Results and Discussion
Nominal frequency responses for the various schemes are
found in Fig. 12(a)(f). The measures from Section III are summarized in Table VIII. Note that the values for 1/ are not
directly comparable between SISO and SIMO systems.
The results when tracking a triangle-wave reference signal
are presented in Fig. 13. The maximum error (ME) ranges from
15% to 24%, and the root-mean-square error (RMSE) ranges
from 0.11 to 0.20 m. The error values are also summarized
in Table VIII. Note that tracking performance can be increased
by adding feed-forward, but this is not done in order for the
error signals to be significantly larger than the noise in the
measured displacement signal to avoid obfuscating the actual
results achieved due to feedback.
The best performing control schemes in terms of the error
are the scheme using IFF and the MRC scheme. The worst performance is obtained when using PPF and the damping integral
control (DI) scheme, while when using IRC and PSD, errors in
the middle of the range are obtained.
The error figures in terms of ME and RMSE can be changed
by the control law tuning, but a reduction in RMSE typically
leads to an increase in ME, due to a more oscillatory response.
The MRC scheme is the most complex scheme. It requires
the implementation of a sixth-order and a seventh-order filter,
a total of 13 integrators. By comparison, the IFF-based scheme
only requires three integrators, but with the disadvantage of
reduced range due to the force transducer, and it requires more
instrumentation and good calibration. On the other hand, the
noise performance is superior, due to the extremely low noise
density of the force transducer, although this benefit it lost for
a digital implementation, due to quantization noise and DAC
artifacts, as discussed below.
442
Fig. 14. Time derivative of force measurement, when having a filter with high
cutoff frequency (PPF, IRC, and IFF) versus low cutoff frequency (PSD, DI,
and MRC) in the signal chain before the voltage is applied to the piezoelectric
actuator.
The simplest control schemes to implement on a digital platform are the PSD based and DI schemes, as they only require
one integrator. The DI scheme is the simplest with regard to extra instrumentation, as it is not necessary to add a shunt circuit,
although for the PSD-based control law, the antialiasing filter is
not necessary and can be omitted. For an analog implementation,
the DI scheme and the schemes based on PPF, IRC, and PSD
are almost equivalent in terms of complexity. The MRC scheme
is likely too complex for an efficient analog implementation.
As quantization noise is the dominant noise source in the
experimental system, it is not possible to obtain reliable closedloop noise measurements. However, due to the low noise and
high sensitivity of the force transducer, the effect of quantization noise and DAC artifacts can be measured. An example of
this is shown in Fig. 14, where the time derivative of the force
measurement is shown when using the IFF-based scheme and
the MRC scheme. The MRC scheme, as well as the PSD and
DI scheme, has a low-pass filter with a low cutoff frequency
before the voltage is applied to the piezoelectric actuator, and
so the noise and disturbances coming from the DAC are effectively attenuated. For the PPF-, IRC-, and IFF-based schemes,
the reconstruction filter has a cutoff frequency of 20 kHz, and
F (s)yp )
up = C(s)(r
where C(s)
is a feed-forward filter given as
c0 (s)
(s) 1 T (s)
C(s)
=
(49)
(50)
Here,
(s) = [sn p 2 , sn p 3 , . . . , s, 1]T ,
for np 2
(s) = 0,
for np = 1
and
(s) = 0 (s)Zm (s)
is a monic and Hurwitz polynomial of degree np 1. Thus,
0 (s) is a monic and Hurwitz polynomial of degree n0 = np
1 qm . The control law parameter vector is
T
c = [1 T , 2 T , 3 , c0 ]
(51)
(52)
443
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444
Marialena Vagia received the B.Sc. and Ph.D. degrees from the University of Patras, Greece, in 2005
and 2009, respectively.
She is currently a Postdoctoral Researcher at the
Norwegian University of Science and Technology
(NTNU), Trondheim, Norway, working in the area
of robotics. Previously, she was awarded with postdoctoral scholarships such as the one funded by the
ERCIM Marie Curie Network completed at NTNU,
the postdoctoral scholarship from the Greek State
Scholarships Foundation (IKY) completed at the
University of Patras, and the Post Doctoral Researchers Scholarships funded
by the EU and General Secretariat Research and Technology, Greece. She has
also served as an Adjunct Lecturer at the University of Patras. Her research
interests include robust and nonlinear control, switching control, electrostatic
microactuators, microcantilever beams, nanostructures, micropiezo actuators,
nanopositioners, and robotic manipulators. She has published ten journal papers and 20 conference papers. She has also served as a reviewer for many
international conferences and journals.
Dr. Vagia received the Best Student Paper Award at the IEEE Mediterranean
Conference on Control and Automation in 2007.