Proceedings of the American Control Conference Anchorage, AK May 810,2002
Design, Identification and Control of a Fast Nanopositioning Device
S.
Salapakalt
,
A.
Sebastian2 ,
.P.
Cleveland3§
and
M.V.
Salapaka4$
'salpax@engineering,ucsb.edu,
*abuseb@liastate.edu,
jason@asylumresearch.com
nd
4murti@iastate.edu
t
Department
of
Mechanical and
Environmental
Engineering,
UCSB, Santa Barbara, CA 93106
t
Department
of
Electrical and Computer Engineering, Iowa State
University,
Ames,
IA 50014
§Asylum
Research, 601C Pine Avenue, Goleta, CA 93117
Abstract
This paper presents the design, identification and con trol
of
a
nanopositioning device. The device is actu ated by
a
piezoelectric stack and its motion is sensed by
a
Linear Variable Differential Transformer
LVDT).
A
fourth order single input single output model has been identified
to
describe its dynamics.
It
is
demon strated that
PI
control law does not meet the band width requirements for positioning. This motivated the design and implementation of an
'H
controller which demonstrates substantial improvements in the positioning speed and precision besides eliminating the undesirable nonlinear effects
of
the actuator. The char acterization
of
the device in terms
of
bandwidth,
reso
lution, and repeatability is also shown.
Introduction
The advent of new techniques to explore properties
of
near atomicscale structures has led to the development of the new field of nanotechnology. In the past decade, it has be come evident that nanotechnology will make fundamental contributions to science and technology. Inevitably, most schemes of nanotechnology impose severe specifications on positioning. This demand of ultrahigh positioning preci sion forms a pivotal requirement in many applications of nanotechnology.
For
example micro/nanopositioning sys tems are essential in auto focus systems
[l]
n optics; disk spin stands and vibration cancellation in disk drives
[2];
wafer and mask positioning in microelectronics
[3];
piezo hammers
[4]
in precision mechanics; and cell penetration and micro dispensing devices
[5]
in medicine and biology. Besides this requirement of high precision, there is also an increasing need for high bandwidth positioning systems. For example in the field of cell biology, there are attrac tive proposals to employ nanoprobes to track events in the cell. These events often have time scale in microsecond or nanosecond regimes. This necessitates high bandwidth po sitioning systems. To meet the dual goal of high precision positioning
at
high bandwidth, novel sensors and actuators have been studied and developed. Most of the current high precision position ing devices utilize piezoelectric materials for actuation. The crystal lattices of these materials deform on the application
of
an electric field and these deformations are used for po sitioning with high accuracy. However, the precision in po sitioning is significantly reduced due to nonlinear hysteresis effects especially when the piezoactuators are used in rela
U.S.
Government work not protected
by U.S.
Copyright
1966
tively long range positioning applications.
For
instance, the maximum error due to hysteresis can be
as
much
as
1015
of the path covered. Another cause for the loss in precision is the drift due to creep effects. These effects become notice able when piezoactuation is required over extended periods of time, i.e. during slow operation modes of the positioning device. Most of the commercially available devices circum vent these nonlinear effects
at
the cost of their performance by restricting the devices to low drive applications, where the behaviour is approximately linear and/or limiting their motions to some specific trajectories, for which nonlinear effects have been accounted for and appropriately compen sated. In the recent past, some efforts have been made to address these nonlinear effects. In
[6),
charge control (in contrast to voltage control) has been proposed to diminish the hys teresis effects. However these techniques lead to increased drift and saturation problems and lead to further reduc tion of the travel range and the positioning bandwidth of the piezoactuator. In
[7],
postcorrection techniques for re moving creep and hysteresis effects from SPMimages have been presented. However, these postprocessing methods cease to be useful for applications (such
as
cell biological studies) in which real time compensation is needed. In
[8],
the design of a feedback controller using an optical sensor attachment to a commercial
AFM
to enhance its perfor mance has been described. The feedback laws have been demonstrated to eliminate these nonlinear effects and con siderably increase the positioning bandwidths. In
[9],
the problem of nonlinear effects has been addressed by a care ful identification/modeling of these nonlinearities and then using a model based inversion approach to compensate for their adverse effects. However, the efficacy of this approach depends on the extent
of
accuracy
of
modeling
of
the non linearities. But, this design methodology can be used in conjunction with the feedback design to achieve better per formance and also robustly account for modeling uncertain ties. In this paper, we present
a
nanopositioning device with
a
piezoelectric actuator and
a
Linear Variable Differential Transformer (LVDT) sensor. The device described here is an independent unit, and is not a modification or an en hancement of an already existing device. The piezoactua tor used here is a stackpiezo, in contrast to piezoelectric tube in
[8]
and
[9],
which has better hysteresis and creep properties but is much more expensive. We also show that the traditional and commonly
used
ProportionalIntegra1
(PI)
control laws in the scanning probe community have
a
highly unsatisfactory performance. This motivated us to
carry out an
'H
control design, which gave significantly better results. For instance, we were able to obtain band widths
as
high
as
130
Hz
by this design in contrast to less than 3
Hz
with the
PI
control law.
1 Device Description
A schematic of the device is shown in Figure
1.
It consists of
a
flexure stage with
a
sample holder, an evaluation stage, an actuation system,
a
detection system and a control system.
+
Actuation Flexure Detection System Stage system
Control
Svstem
1
Figure
1:
A
schematic block diagram
of
the device.
slot
for
LVDT
base plde sample holder
slot
orpiezo
stack
a)
b)
Figure
2:
(a) The base plate of the flexure stage. (b) The exploded view of the flexure and evaluation stages. The flexure stage consists of two components: the base plate and the top plate. The base plate (see Figure 2(a)) is 20
cm
x
20
cm
x
5
cm
and is made of steel. From its centre protrudes a cylindrical block (the sample holder) which has a provision for the sample to be placed on it. This part of the base plate that seats the sample holder executes mo tion relative to its periphery. This motion is obtained by the serpentine spring design (see Figure 2(c)) where design grooves (of about 150pm wide) are cut in the base plate that make it possible for the central block to move rela tive to the frame. The top plate sits on the base plate and is of similar dimensions
as
the base plate. It provides the sample an access to the evaluation stage (see Figure 2(b)). The evaluation stage consists of
a
modified Atomic Force Microscope head where the features on the sample cause the cantilever to deflect in vertically
as
the sample moves under its tip. The resulting deflection signal
is
processed to infer the topography of the surface. The actuation system consists of
a
voltage amplifier and
a
piezostack arrangement. This arrangement sits in the slot in the flexure stage (see Figure 2(a)). The piezo deforma tion imparts the motion to the central block
of
the flexure stage. The input to the amplifier (which has gain of 15) is restricted to be negative and to be less than 10
V
in magni tude since the piezostack saturates beyond this limit. The piezo stacks are
at
an angle
a
x
7.5 degrees
(see
Figure 2) because this arrangement, besides providing the sufficient force to the central block, also achieves a mechanical gain of 1/sin(a). The detection system consists of a LVDT and associated demodulation circuit. It has a resolution in the order of
a
few angstroms. The LVDT has been adjusted
so
that it outputs
0
V
when an input of 5
V
is given to the ac tuation system. The corresponding position of the central block is called the null position. The LVDT output signal is proportional to the motion of the central block. The design of the of the control laws have been presented in Section 3. These laws were implemented on a Texas Instruments
C44
digital signal processor.
2
Identification
of
the system
The modeling
of
the device
was
done
using
the
blackbox
identification technique where we chose a specific point in the operating range of the device (where
its
behaviour is approximately linear) and obtain a model of the device at this point by studying its frequency response over
a
prespecified bandwidth. For this purpose, we used a HP signal analyzer, which gave a series of sinusoidal inputs,
U
=
5
+
Asin
wt
(V
,
with frequencies spanning over a bandwidth of
2
d.z.
T)he amplitude of the signals,
A
were chosen small enough
(5
50
mV)
o be in the linear regime of the device. The frequency response of the plant
G
at
the null point was then obtained by recording steady state val ues of the output
y
for each
U
(see Figure 3). Accordingly, a fourth order non minimum phase transfer function: 9.7
x
1oqs

7.2
f
.44
x
103)
G(s)
=
(s
+
(1.9 4.5i)
x
103)(s
+
(0.12
*
5.2i)
x
102)
was
fit
to this data. Figure
3
shows that there is a good match between this frequency response data and the one simulated from the model,
G(s).
3
Control Design
The model inferred for the device at the null position was employed to design the feedback laws. The schematic of the closed loop system is shown in Figure
4.
Here
e
=
Sr

n
is the error signal where
S
=
1
+
GK)
is the sensitivity function and
T
=
(1
+
GK) GK
is the complementary sensitivity function. The primary objective of the control design is to achieve good tracking with high bandwidths. Also, the feedback laws were constrained to provide control signals that were negative and within actuator saturation limits (10
V
to
0
V).
Besides these implementation con straints, the presence of RHP zeros imposes fundamental
1967
Figure
3:
A
comparison of experimentally obtained and simulated frequency responses of the plant.
Figure
4:
A
schematic block diagram of the closed loop system. constraints.
For
example, the classical root locus analysis predicts high gain instability of the system. This rules out high gain feedback laws. They also impose a fundamental limit on the achievable bandwidth of the closed loop system. It has been shown in
[lo,
111,
that
a
complex pair of
RHP
ze
roes,
z1,2
=
zfiy
(as
n this case
z1 2
=
(1.72f7.364
x
lo3), the ideal controller leads to the following sensitivity func tion,
42s
S=
(s
+
z
+
y) s
+
z

y)
By
the above criteria the achievable bandwidth (the
fre
quency at which
(S(jw)/
crosses 3dB from below) is ap proximately 415
Hz
for. the system. This controller is ideal in the sense that it may not be realizable and which, for a unit step reference
~ t),
enerates an input
u t)
which minimizes the integral square tracking error: In industry, it is
a
common practice to design proportional
P)
or proportionalintegral
PI)
controllers. In the next section, we show
that
these controllers do not perform well and it becomes inevitable to look for more sophisticated designs. In particular, we present
Ha
control design which shows
a
great improvement over
P
and
PI
controllers.
P
and
PI
controller designs:
From the analysis of the rootlocus plot of the open loop system, it is seen that the closed loop system is unstable for feedback gains greater than
0.1674
which rules out the use of proportional control. In the design of
PI
control law
k,
+
ki/s),
since we know the structure of the controller, we can determine the regions in
k,ki
plane which guarantee closed loop stability. Figure 5(a) shows this plot.
It
should be noted that the region (in the
k,k,)
that gives high bandwidth
(see
plot
(d))
is
the region with low gain margin
see
plot (c)), i.e. there is
a
tradeoff between robustness and performance.
So
we chose
k,
=
0.01 and
ki
=
75 which guarantees a gain margin of 1.57 and
a
phase margin of
90
and the corresponding bandwidth of the closed loop transfer function is 2.12 Hz. It
0
.2571
0.2
1968
Figure
5:
(a) The region in
kik,
plane that guarantees closed loop stability. The contour plots show ing (b)the phase margins (c) the gain margins (d) the bandwidths of the closed loop systems corresponding to different points in the
kik,
plane. should be noted that the bandwidth
(<
3Hz) attained here is much less than the achievable bandwidth
(x
415
Hz)
as
described in the previous section.
Ha
controller design:
The main advantage of
us
ing this design is that
it
includes the performance objec tives in the problem formulation itself. In this setting, the
'H
optimal control problem amounts to finding the control feedback law,
K,
such that
11
P,,
11
is infimized, where
P.,
is the transfer function from
w
to
z.
The first step towards
pz
G
+
Figure
6:
(a) The generalized plant framework. (b) The closed loop system with regulated outputs.
Ha
control design is to form the generalized plant,
P.
In the system (see Figure 6(b)), the exogenous input
w
is the reference signal
T,
the control input
is
'U.
and the measured output
v
is the error signal
e.
In
order to reflect the per formance objectives and physical constraints, the regulated outputs were chosen to be the weighted transfer function,
21
=
Wle,
the weighted system output,
zz
=
W2y
and the weighted control input,
23
=
W3u.
This choice of out puts implies
P,,
=
[WlS
W2T
W3KSlT.
The weighting functions
Wi,
=
1,2
and
3
are used to scale these closed loop transfer functions to specify the frequency information of the performance objectives and system limitations. The
transfer function,
Wl
,
s chosen such that its inverse (an ap proximate upper bound on sensitivity function) has a gain of
0.1
at
low frequencies
(<
1
Hz)
and a gain of
x
5
around
200
Ht.
It was chosen to be a first order transfer function, given by
0.1667s
+
2827
wl s)=
f2.827
.
This weighting function puts a lower bound on the band width of the closed loop system but does not specify the roll of the open loop system to prevent high frequency noise am plification and limiting the bandwidth to be below Nyquist frequency. To do this, we scale the complementary sensi tivity function,
T,
by
+
235.6
w

0.01s
+
1414
which has low gains
at
high frequencies and vice versa. The transfer function,
KS
was scaled by a constant weighting
W3
=
0.1
to restrict the magnitude of the input signals such that they are within the saturation limits. This weighting constant gives control signals that are at most six times the reference signals. In practice it is not usually necessary to obtain an optimal controller and often it is computationally simpler to design a suboptimal one (i.e. one close to optimal one in the sense of
31
norm). In particular, for any
y
>
yopt
>
0 we can find a controller transfer function
K
such that
IIPz,
[Im
<
y where
yopt
is the optimal value. The controller is designed (using the function,
hinfsyn
in MATLAB) for
y
=
2.415
and the weighting functions described above. The following sixth order controller transfer function,
K
was obtained with
a
DC gain of
2.2599
x
lo3,
its poles at
1.14959
x
lo7,
1.4137
x
lo5,
5.6432
x
lo3,
2.8274
and
(1.5676f 5.84382)
x
lo3,
and its zeroes at
1.4137
x
lo5,
1.8647f 4.49582)
x
lo3
and
1.1713
x
lo
1.52051
x
lo3.
The corresponding controller and the sensitivity transfer functions are shown in Figure 7. The bandwidth of the sensitivity transfer function is found to be
138
Hz
t should be noted that this is an enormous improvement over the PI controller.
Also,
this controller provides a gain margin of
2.57
and a phase margin of
62.3'
as
opposed to the values of
1.57
and
89'
in the PI controller. In this figure, we have compared this sensitivity function with the ideal one introduced earlier in the section. Figure
8
shows the performance of this controller
for
a
25,
50
and
100
Hz
triangular and
100
Hz
reference sig nals. It can be seen that the closed loop system tracks well the
25
Hz
and
50
Hz
signals (see plots (a) and (b)). The mismatch in the case of the
100
Hz
(in (c)) signal is due to the accentuation of higher modes of the triangular wave. In contrast, the
100
Hz
sinusoidal signal shown in (d) does not have higher harmonics and the closed loop system shows much improvement in its tracking.
4
Characterization
of
the device
In this section, we characterize the closed loop and the open loop device in terms of its resolution, bandwidth, and the range. First, the device was calibrated using a calibration sample which had
180
nm
high grooves every
5
pm.
This grid was placed on the sample holder and probed by the evaluation stage.
A
triangular input of amplitude
2
V
was
controller ransfer unction
Figure
7:
he controller and the sensitivity transfer func tions
CJ
5
U
5
o
O.OJ
o,m
a03
u.ru
U
o.01
a112
aw
u.w
Figure
8:
Tracking of (a)
25
Hz
(b)
50
Hz
(c)
100
Hz
tri angular waves and
(d)100
Hz
sinusoidal wave using the
'H
controller. given and the resulting LVDT output showed the presence of
7.26
grooves. This implies the device has a static sensi tivity of
18.15
pm/V.
As
already mentioned, piezoactuators do not have any back lash
or
friction and therefore have
very
fine resolutions. The resolution of the device, therefore, depends on the experi mental environment and it is limited by thermal and elec tronic noise. Since the effect of the noise increases with the bandwidth, finer resolutions are expected at lower band widths. To determine the resolution at different band widths,
Wb,
we gave a constant input of
5
o the ac tuation system and recorded the resulting noise signal from the LVDT (the detection system). The resolution was de fined
as
three times the standard deviation
(T
of this
LVDT
output signal. The standard deviations at different band widths were obtained by calculating the area under the Power Spectral Density
S W))
urve up to the correspond ing bandwidths, therefore, resolution
=
3
(s,
s(~) )
.
The experiments to determine resolution were done both in open and closed loop configurations. The advantages of closing the loop is that, in this case the creep and drift effects are compensated and thus better resolutions are
1969
obtained
(see
Figure
9).
As
seen in the previous section (see Figure
7 ,
the
Hw
controller achieved an approximate bandwidth of
138
Hz,
however, for slower (low bandwidth) application, one can decrease the bandwidth appropriately and achieve high resolutions.
It
was
seen that the input voltage of approximately
4
V
can be given without reach ing the limits of the actuator. This guarantees a travel range of
70pm.
10.5
9
7.5
E
.$
+
4.5
P
1.5
.
.


O
20
40
60
80
IO0
bandwidth
Hz)
Figure
9:
The comparison of resolution vs bandwidth plots between open loop and closed loop with PI and
Hw
controllers.
Elimination
of
nonlinear effects
The positioning precision of the piezoactuators is signif icantly reduced due to nonlinear effects such
as
hystere sis, drift and creep. Hysteresis effects are significant when the sensors are used for relatively long ranges. Therefore piezoactuators are typically operated in linear ranges to avoid positioning effects. However, with the feedback con trol laws, these nonlinear effects are compensated and thus
the
closed
loop
device
does not
show
any hysteresis.
To
study this, we first plotted hysteresis curves by operating the device in open loop and then compared them with the curves obtained by repeating the experiment in the closed loop. Input signals (less than
1
Hz
triangular pulses) of increasing amplitudes (1
V
to
4
V)
were given in open loop and the corresponding output signals recorded (see Figure
10).
It is observed that the hysteresis effects are dominant in higher amplitudes (longer travels) and become smaller
as
the travel lengths are reduced. The hysteresis is quantified numerically in terms of maximum input
(or
output) hystere
sis
usually given
as
a percentage of the full scale. Figure 10 shows that the feedback control laws virtually eliminate all hysteretic effects, and the output and the reference signals match well. Creep is another undesirable nonlinear effect common with piezoelectric actuators. It is related to the effect of the applied voltage on the remnant polarization of the piezo ceramics. If the operating voltage of a piezoactuator (open loop) is increased (decreased), the remnant polarization (piezo gain) continues to increase (decrease), manifesting itself in a slow creep (positive or negative) after the voltage change is complete. This effect is approximately described by the equation, Y(t)
=
YO l+
7log(t/to)), where
to
is the time at which the creep effect
is
discernible,
yo
is the value of the signal at
to
and
y
is a constant, called
I

0
1.5
4
2
o
2
4
1.5
inpd
(VJ
Reference
(VJ
open loop closed loop
62.3
nm
0.11
)
mar.
ow.
hysl.
mas
np.
hyst.
mar.
ow.
hyrr.:
I
0.74
pm
(
7.2 ) OS4 V
(5.8 )
..
2 2.09 pm(9.3 J 0.36V(7.5 )
mas.inp.hgrr.:
3
3.46
~(9.8 )
0.56V(7.7 ) 2 mV (0.07
)
4 4.93
pm
(10.0
)
0.73 V(7.6 J
Figure
10:
Hysteresis in (a) the open loop configuration, (b) its elimination in the closed loop configu ration. the creep factor, that characterizes this nonlinear effect. To measure this effect, we studied
a
step response of the device in open and closed loop configurations. We see that the output in the open loop case responds to the reference signal but instead of reaching a steady state value it continues to decrease at a very slow rate. The response
y t)
was found to approximately satisfy the creep law with
a
creep factor of
0.55
The same experiment conducted with the closed loop shows that the feedback laws virtually eliminate this effect and the system tracks the reference signal exactly.
4.2
open loop
I.
p 0
=
0.4303(J+0.5538/og(2t))
Figure
11:
(a) responses to the step input of open loop and closed loop systems (b) The elimination of creep in the closed loop system and its ap proximation by a creep law.
A
significant adverse effect of the nonlinearities in the open loop is that of non repeatability. This was seen clearly in the calibration experiment described in the previous section. In the open loop case, the grooves that were observed when traveling in one direction were not concomitant with those in the other direction. This
has
been shown in Figure 12(a). The corresponding hysteresis curve is shown in (c). These effects were removed with feedback control and in (b), we see that the grooves in the forward and reverse direction fall exactly on top of each other. The corresponding hysteresis plot is shown in (d). The mismatch in the open loop is more clearly seen in Figure
13(b)
where the image obtained in one direction is kept behind the one got in the other direction
1970