This paper addresses the optimal tracking of piezo-positioners, which are fundamental tools to achieve high precision positioning for nanotechnology applications. This is particularly important as piezo-positioners are increasingly used in high-speed positioning for applications like nano-storage of information and nanofabrication. The optimal tracking approach can be used to (a) optimize the positioning trajectory, and (b) restrict trajectory to the available band-width and saturation-limits of the amplifiers used to apply voltage to the piezo-positioners. The approach is applied to the output tracking of a piezo-probe actuator and experimental results are presented.

1 Introduction

Piezo-positioners are used in scanning probe-based microscopy (SPM), which are key enabling tools for nanotechnology. In addition to their use as surface analysis tools, their potential for surface modification is now being exploited for nano-lithography. For example, SPM's have been used in both (a) additive techniques like localized chemical vapor deposition and (b) material removal techniques like localized etching of the surface with a gas or electrolyte (see for example, Gentili et al., 1993; Weck et al., 1997; Wiesendanger, 1994; Xie et al., 1997; and the references therein). A key advantage of such piezo-based nanofabrication is that it is the least expensive tool able to produce structures with dimensions below 100nm. Piezo-positioners are also used in the storage of information (Hosaka et al., 1997) and in the study of material and biological properties at the nano-scale (for a list of SPM applications see for example, Marrian, 1997; Regis, 1995; Vettiger et al., 1996; Wiesendanger, 1994). Thus, piezo-positioner based SPMs are poised to play a significant role in future technological progress, especially, when manufacturing processes in electronics, optics, and precision machining begin to require nanometer-scale control over features. However, a critical problem of current SPM-based (piezo-based) techniques is that they suffer from throughput-limitations, which are present in all serial processing techniques -- the piezo-probe must visit each point where the surface analysis or modification is needed. This can take a prohibitive amount of time unless the positioning-speeds of piezo-based systems can be increased. The issue of high-speed optimal-positioning for piezo-based systems is studied in this paper.

The positioning speed of piezo-based systems is limited by the lowest structural vibrational frequency, because structural vibrations become substantial at frequencies close to the resonant frequency, and can cause significant positioning errors (Pohl, 1986). For example, in scanning operations, turnaround transients can substantially limit the maximum scan-rates achieved (Barrett and Quate, 1991). These transients are proportional to the velocity changes (as in turnarounds) in the scan-path, and depend on the product of scan size and scan frequency -- thus any attempt to increase scan velocities increases the transients and thus decreases the useful part of the scan. Thus, the only choice with present approaches is to use piezo-positioners with fast dynamic response or to modify the dynamic response through feedback. Faster dynamic response can be achieved by using shorter piezo-tubes or piezo plate-scanners (Koops and Sawatzky, 1992), however, they tend to have limited positioning-ranges. An alternative approach is to improve the dynamic response by using feedback. For example, adding derivative feedback tends to suppress residual errors due to vibrations and thus allows faster scanning speeds. However, there are limits to the improvements achievable through feedback schemes because high feedback-gains tend to destabilize the system (Barrett and Quate, 1991). Further, feedback is not always available in piezo-based nano-positioning systems. To conclude, the system-dynamics must be considered when determining the input voltages applied to piezo-positioners for achieving high-bandwidth positioning.

Recent results have clearly demonstrated that significant improvements in positioning speed can be achieved by inverting the piezo-dynamics to find inputs that compensate for the structural vibrations (Croft et al., 1998b). Such inversion-based approaches have also been developed to account for hysteresis nonlinearities of the piezo-positioners when large range displacements are needed (Croft and Devasia, 1998a). The approach is to linearize the system dynamics by inverting the hysteresis nonlinearity, which is modeled as an input-nonlinearity. The inversion-based approach is then applied to the linearized piezo-dynamics. Although inversion-based approaches achieve exact tracking of the desired position-trajectory, a key issue is the design of the output position trajectory. For example, some output trajectories might require very large inputs to achieve exact-tracking -- such inputs might accelerate the depoling of the piezo-positioners. Further, the exact-tracking inputs found from inversion might exceed bandwidth limitations of the available piezo-amplifiers that apply the input voltages to the piezo-positioners. However, the exact-tracking inputs found from inversion are unique and therefore, the only recourse is to give up some of the exact-tracking requirement. Hence an optimal input is sought that trades the exact tracking requirement to achieve other goals like reduction of input-bandwidth and input-amplitudes. In this paper, a recently developed theory for optimal inversion (Dewey et al., 1998) is used to find optimal input voltages to piezo-positioners and experimental results are also presented.

We begin with a formulation and solution of the general optimal-inversion
problem in Section 2. In Section 3, we apply this technique to a piezo-probe
and experimental results are presented. Our Conclusions are in Section
4.

2 Problem Formulation and Solution

In this section the output-tracking problem for a general piezo-positioner
is posed and solved in the frequency domain.
(MATLAB code to find the optimal inverse can be obtained by email to
santosh@eng.utah.edu)

2.1 System Inversion for Exact Tracking

Let the dynamics of the piezo-positioner be described by the following
transfer function relationship in the frequency domain

(1)

where u is the input and y is the output. Given a desired
output trajectory, y_{d} , the inverse input trajectory,
u_{d} ,that achieves exact tracking can be obtained by inverting
the system's dynamics (Equation 1) to obtain (Bayo, 1987)

(2)

2.2 The Performance Criterion

The exact-output tracking input, found from Equation 2, is unique (Bayo, 1987; Devasia et al., 1996; Ledesma et al., 1994). For a generic desired output trajectory, this exact tracking input might be large and therefore depolarize the piezo-positioner or saturate the piezo-amplifiers. The input might also excite unwanted resonances in the piezo-positioner and in the structure that houses the piezo-positioner. The input-bandwidth needed to achieve this positioning trajectory might also not be available through the piezo-amplifier. Therefore, the goal of the optimal inverse is to generate a modified output trajectory as well as to find the input needed to achieve exact tracking of this new trajectory.

Note that desired output trajectories are typically chosen in an ad-hoc manner. For example, the output is typically chosen as triangular to-and-fro scans, in scanning probe microscopy. It has been shown that these triangular scans cannot be tracked exactly, because the velocities are discontinuous at the corners leading to infinite accelerations. Therefore, the corners have to be smoothed to achieve exact tracking (Tamer and Dahleh, 1994). The optimal redesign of the output can be achieved by the following approach, in which an optimal input to the piezo-positioner is also found.

We formulate this optimal inversion problem as the minimization (over
all possible inputs, u) of a quadratic cost functional of the type

(3)

where the superscript * denotes complex conjugate transpose.

In this optimization criterion, R(jw ) and Q(jw ) are non-negative frequency-dependent scalars that represent the weights on control-input, and the output-tracking-error, respectively, and y_{d} is a desired output-trajectory which has an inverse (for requirements on y_{d} , see Bayo, 1987; Devasia et al., 1996). The weights, R(jw ) and Q(jw ), should not be both simultaneously zero at any frequency, w .

2.3 Optimal Input

Our main result is given by the following lemma, which finds the optimal
input trajectory. A generic result for linear multi-input multi-output
systems can be found in Dewey et al., 1998.

Lemma The optimal inverse input trajectory u_{opt},
is given by

(4)

The optimal modified output trajectory is given by

(5)

Proof:

The cost functional (Equation 3) can be rewritten as (see Lemma in reference
Dewey et al., 1998 for a similar argument)

(6)

where the explicit dependence on jwis not written for ease in notation and the superscript * implies complex
conjugate. Note that the only term with u inside the integral is
quadratic in u and hence the Lemma follows by setting this quadratic
term to zero.

Remark 1

We point out two extreme cases. First case: if the weight on the tracking
error is zero, Q=0, but R is positive definite then the input
is u_{opt}=0, i.e., the best strategy is not to track the
desired trajectory at all. Second case: if the weight on the inputs are
zero, i.e., R=0 but with Q positive definite then y_{opt}=
y_{d}. This implies that exact output-tracking is optimal,
and the cost is again zero. The resulting optimal inverse is the exact-inverse
input that achieves exact tracking of the prescribed output, y_{d}
without any modification.

Remark 2

Frequency dependent weights (R and Q) on the input and output
error can be used, for example, to account for actuator-bandwidth-limitations
(Gupta, 1980). This is illustrated with experimental results in the following
section.

3 Experimental Verification

In this section the experimental system is described and the response of
the experimental system is presented and discussed.

3.1 Modeling of the Experimental System

The experimental piezo-positioning system (see Figure 1) consisted of a
sectored piezo-actuator that was used for lateral displacement and an inductive
sensor that measured this lateral displacement. The lateral motion of the
sectored piezo-actuator was achieved by applying voltages across opposite
electrodes of the sectored piezo-actuator. The complete experimental system
also included amplifiers for the piezo-actuator and the electronics needed
to condition the sensor outputs.

Figure 1: Experimental Piezo-positioning System

Figure 2: Determination of the Structural Dynamics of the Experimental
Piezo-positioning system

Figure 3:Bode plots of experimental system. The model captures
the system behavior up to the first fundamental vibrational frequency of
600 Hz. (Solid Line: Experimental Dashed
Line: Model)

The piezo-system was modeled as a cascade of two submodels. The model was
decoupled into (a) a hysteresis nonlinearity and (b) the structural dynamics
of the piezo-system. The hysteresis nonlinearity was first determined by
applying a triangular shaped voltage profile to the system at low frequency
(1 Hz) and measuring the input-output response. Note that at low input
frequencies relative the first resonant frequency of the piezoactuator
system, the structural vibrations have little effect and can be neglected.
An inverse hysteresis model for the resulting input-output hysteresis response
was then modeled as separate third order polynomials for an increasing
or decreasing voltage profile (see Croft and Devasia, 1998b). The linear
structural model was then determined using a Dynamic Signal Analyzer (HP3650A).
As shown in Figure 2, the input command from the signal analyzer was passed
through the inverse hysteresis model (implemented using a personal computer)
and the resulting signal was then sent to the experimental piezo-system.
The measured signal from the inductive sensor was then returned to the
signal analyzer. The resulting input-output response measured from the
signal analyzer was then model as a linear system. The Bode plot, found
experimentally from the signal analyzer, is given in Figure 3.

The resulting transfer function (model) from the signal analyzer can
be written as

(7)

where k = .454; a = -1.93x10^{4}; b= -1.61x10^{8}; c =
-2.75x10^{12}; d = -3.87x10^{16}; e = 5.62x10^{3};
f = 4.28x10^{8}; g^{ }= 2.08x10^{12}; h = 1.80x10^{16};
i = 3.13x10^{19}; and j = 1.64x10^{23} for the experimental
piezo-positioning system used in our experiments. This linear model then
can be inverted using equation (2) to find the inverse input trajectory
or using equation (4) to find the optimal inverse input trajectory.

3.2 Results

Three sets of experiments were performed: (1) output tracking without using
information about the system's dynamics, i.e. applying y_{d}
divided by the DC gain of the system only (2) using the inverse input trajectory
without frequency weighting (i.e. R=0 for all frequencies) and (3)
using the inverse input trajectory with optimal frequency weighting with
Q=1 for all frequencies and the weighting on the control input was
chosen to be R = [0 0 .02 .02] at frequencies w
= [0 550 650 100500] Hz, (where the values of R at other frequencies
were obtained by linearly interpolated between the specified values). Plots
for the response of the piezo-positioning system for these three cases
are given in Figures 4-7. The applied voltage profiles for select scanning
frequencies for cases (2) and (3) are given in Figure 8.

The desired output trajectory (displacement trajectory) was chosen as
a back and forth scanning (triangular shaped) of the piezo-positioning
system (see, for example, Croft and Devasia, 1998a).

Figure 4:Scan path tracking for y_{d} divided
by the DC gain of the system only. Scan path tracking is achieved for
a 25 Hz scan rate but structural vibrations cause loss of tracking for
higher scan rates such as the 75 Hz scan rate.

Figure 5: Scan path tracking for the inverse input trajectory without
optimal frequency weighting. The inverse input trajectory substantially
improves the 75 Hz scan rate (compare with Figure 4), but fails to sufficiently
track at higher frequencies (such as 150 Hz) due to the unmodeled frequency
content in the input signal.

Figure 6:Scan path tracking for inverse input trajectory with
optimal frequency weighting. At lower scanning frequencies, the addition
of optimal frequency weighting to the inverse input trajectory achieves
scan path tracking comparable to the scan path tracking achieved by the
inverse input trajectory without optimal frequency weighting (compare the
results in Figure 6 with those in Figure 5).

Figure 7: Scan path tracking for vibration compensation with optimal
frequency weighting. At higher scanning frequencies, the addition of
optimal frequency weighting to the inverse input trajectory substantially
improves scan path tracking at the 150 Hz scan rate (compare with Figure
5). Higher scan rates (350 Hz) are also achieved with some "smoothing"
of the corners of the scan path, caused by the frequency weighting of the
high frequency components of the input signal.

Figure 8: Applied voltages to piezo-positioning system. At low scan
frequencies (75 Hz), the applied voltages for the inverse input and the
optimal inverse input are similar. At higher scan frequencies, the high
frequency components have been removed from the optimal inverse input (compare
the inverse input and the optimal inverse input for the 150 Hz scan rate).
At even higher scan rates (350 Hz), the applied voltage for the inverse
input becomes large compared to the applied voltage for the optimal inverse
input (at 350 Hz). Note that these inputs are also trying to force tracking
at bandwidths where the modeling is relatively poor. In contrast, the optimal
inverse input is smaller and achieves better tracking.

3.3 Discussion of results

As shown in Figure 4, scan path tracking without considering the system
dynamics (i.e., using inputs that only use the DC gain factor) can only
achieve good tracking at low scan rates, such as 25 Hz. However as
the scan rate increases, structural vibrations of the piezo-positioning
system adversely affect the tracking performance of the piezo-positioning
system. By the 75 Hz scan rate, these vibrations have become substantial
and scan path tracking is poor.

In contrast, Figure 5, shows that the tracking of the system for the inverse input without optimal frequency weighting (R = 0), improves at the 75 Hz scan rate (compare with Figure 4). However, as the scan rate increases further, the tracking of the system degrades due to the unmodeled frequency content in the feed-forward applied voltages required to achieve the scan path. For example, Figure 5 shows that the 150 Hz scan path tracking is poor due to this unmodeled frequency content in the feed-forward applied voltages.

With the addition of the optimal frequency weighting to the inverse
input trajectory, scan path tracking is improved remarkably at the 150
Hz scan rate and is extended to higher scan rates, such as 350 Hz (see
Figures 6 and 7). This improved tracking is due to the removal of the higher
frequency content in the feed-forward input. It is also noted that the
high-frequency content in the input (seen in Figure 8) corresponds to the
zero-frequency in the system dynamics close to 2000 Hz (see the dip in
the Bode Plot in Figure 3). These high-amplitude inputs at near the zero-frequencies,
have relatively low impact on the desired output and are removed in the
optimal inversion process. However, when the higher frequency content of
the scan signal is filtered out, the resulting scan path is "smoothed"
at the corners due to the loss of the higher frequency components of the
feed-forward voltage. The effects of this can be seen in the 150 Hz and
350 Hz scan rate of Figure 7.

The applied feed-forward voltages for the inverse input trajectory without
optimal frequency weighting and the inverse input trajectory with optimal
frequency are compared in Figure 8. At the lower scan rate of 75 Hz, there
is little difference between the applied feed-forward voltages. At the
higher scan rate of 150 Hz, the applied feed-forward voltages are significantly
different. The poorly modeled high frequency content of the inverse input
voltage without optimal frequency weighting has been removed from the inverse
input trajectory with frequency weighting, which leads to improved tracking
for the optimal inverse case.

4 Conclusions

The optimal inversion-based approach is used to successfully exploit
the known piezo-dynamics and optimally modify the desired position trajectory.
The approach provides a systematic method to an optimal tradeoff between
tracking the desired trajectory and other goals like vibration reduction
or reduction of required inputs. The approach demonstrated a substantial
increase in the effective bandwidth of the experimental piezo-positioning
system (more than an order of magnitude increase, from 25 Hz for the output
tracking without using information about the system's dynamics, to 350
Hz for the optimal inverse input).

5 References

Barrett, R. C. and Quate, C. F. (1991) Rev. Sci. Instrument., 62 No. 6,
June, pages 1393-1399, Optical scan-correction system applied to atomic
force microscopy

Bayo, E. (1987) J. of Robotic Systems, 4 No.1, pages 63-75, A finite-element
approach to control the end-point of a single-link flexible robot

Croft, D. and Devasia, S. (1998a) AIAA Journal of Guidance, Control, and
Dynamics, In Press, Hysteresis and vibration compensation for piezoactuators

Croft, D.; McAllister, D.; and Devasia, S. (1998b) ASME Journal of Manufacturing
Science and Engineering, In Press, Scheduled to Appear August, High-speed
scanning of piezo-actuators for nanofabrication

Devasia, S.; Chen, D.; and Paden, B. (1996) IEEE Transactions on Automatic
Control, 41 No. 7, pages 930-943, Nonlinear inversion-based output tracking

Dewey, J. S.; Leang, K.; and Devasia, S. (1998) ASME Journal of Dynamic
Systems, Measurement and Control, Accepted, Experimental and theoretical
results in output trajectory redesign for flexible structures

Gentili, M.; Giovannella, C.; and Selci, S., editors (1993) Nanolithography:
A Boderland between STM, EB, IB, and X-ray Lithographies, NATO ASI
Series E: Applied Science, 264, Kluwer Academic Publishers

Gupta, N. K. (1980) Journal of Guidance and Control, 3 No. 6, November,
pages 529-535, Frequency shaped cost functions: Extension of linear-quadratic-gaussian
design methods

Hosaka, S.; Kikikawa, A.; Koyanagi, H.; Shintani, T.; Miyamoto, M.; Nakamura,
K.; and Etoh, E. (1997) Nanotechnology: Special issue on Scanning Probe-based
Nanoelectronics, 8 No. 3A, September, pages A58-A62

Koops, R. and Sawatzky, G. A. (1992) Rev. Sci. Instrument., 63 No. 8, August,
pages 4008-4009, New scanning device for scanning tunneling microscope
applications

Ledesma, R.; Devasia, S.; and Bayo, E. (1994) Journal of Robotic Systems,
11 No. 4, pages 327-338, Inverse dynamics of spatial open-chain flexible
manipulators with lumped and distributed actuators

Marrian, C. R. K., editor (1997) Special Issue on Nanometer-scale Science
and Technology in the Proceedings of the IEEE, 85 No. 4, April

Pohl, D. W. (1986) IBM J. Res. Dev., 30 No. 4, July, pages 417-427, Some
Design Criteria in Scanning Tunneling microscopy

Regis, E., editor (1995) Nano, Little, Brown and Company

Tamer, N. and Dahleh, M. A. (1994) Proceedings of Control and Decision
Conference, Lake Buena Vista, Florida, pages 1826-1831, Feedback control
of piezoelectric tube scanners

Vettiger, P.; Staufer, U.; and Kern, D.P., editors (1996) Special Issue
on Nanotechnology in Journal of Microelectronic Engineering, 32, Elsevier,
New York

Weck, M.; Fischer, S.; and Vos, M. (1997) Nanotechnology , 8 No. 3, September,
pages 145-148. Fabrication of microcomponents using ultraprecision machine
tools

Wiesendanger, R., editor (1994) Scanning Probe Microscopy and Spectroscopy,
Cambridge University Press

Xie, W.; Dai, X.; Xu, L. S.; Allee, D. A.; and Spector, J. (1997) Nanotechnology,
8 No. 2, June, pages 88-93, Fabrication of cr nanostructures with scanning
tunnelling microscope

^{*}Corresponding Address:
Santosh Devasia
50 S. Campus Dr. Rm. 2202, Mechanical Engineering Dept.
University of Utah
Salt Lake City, UT 84112-9208
Phone: 801-581-4613 , Fax: 801-585-9826
E-mail: santosh@eng.utah.edu, Web: http://www.mech.utah.edu/~santosh