Based Optimal Inversion for Output Tracking: Application to Scanning Tunneling Microscopy
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
375
Preview-Based Optimal Inversion for
Output Tracking: Application to Scanning
Tunneling Microscopy
Qingze Zou, Member, IEEE, and Santosh Devasia, Member, IEEE
AbstractOptimal inversion of system dynamics can be used to
design inputs that achieve precision output tracking. However, a
challenge in implementing the optimal-inversion approach is that
the resulting inverse input tends to be noncausal. The noncausality
of the optimal inverse implies that the desired output trajectory
must be pre-specified and cannot be changed online. Therefore, the
optimal inverse can only be used in trajectory-planning applica-
tions (where the desired output is known in advance for all future
time). The main contribution of this article is the development of a
technique to compute the noncausal optimal inverse when the de-
sired output trajectory is known in advance for only a finite time
interval. This future time interval, during which the desired output
trajectory is specified, is referred to as the preview time. Addition-
ally, this article develops a time-domain implementation of the op-
timal inverse and quantifies the required preview time in terms
of the specified accuracy in output tracking, the system dynamics,
and the cost function used to develop the optimal inverse. The pro-
posed approach is applied to precision (subnanoscale) positioning
of a scanning tunneling microscope (STM), which is a key enabling
tool in emerging nanotechnologies. Experimental results are pre-
sented which show that finite preview of the desired output trajec-
tory is sufficient to operate the STM at high speeds.
Index TermsNanotechnology, output tracking, scanning, scan-
ning tunneling microscope (STM), system inversion.
I. I
NTRODUCTION
M
ODEL-BASED inversion of system dynamics [1][3],
can be used to find inputs that achieve high-precision
output tracking; this input is referred to as the inverse input.
The inversion technique has been applied to a number of output-
tracking applications; for example, in the precision control of
flexible manipulators [4], [5], aircraft control [6], and high-pre-
cision positioning of piezo probes for nanoscale imaging using
scanning-probe microscopy [7]. However, the model-based in-
version approach suffers from two problems: 1) the inverse input
will be erroneous if the modeling uncertainty is large and 2)
the inverse input will be unacceptable if it violates input energy
or bandwidth limitations. These two problems have been ad-
dressed by the development of the optimal-inversion technique
in [8]. In particular, the optimalinversion technique can be used
Manuscript received May 15, 2002; revised December 24, 2002. Manuscript
received in final form June 17, 2003. Recommended by Associate Editor
A. Kelkar. This work was supported by NASA Ames Research Center Grant
NAG 2-1450 and NSF Grants CMS 0196214 and CMS 0301787.
The authors are with the Mechanical Engineering Department, University
of Washington, Seattle, WA 98195-2600 USA (e-mail: qzouatuw@u.wash-
ington.edu; devasia@u.washington.edu).
Digital Object Identifier 10.1109/TCST.2004.824797
to account for modeling errors by only inverting the system
model in frequency regions where the modeling uncertainty is
sufficiently small [9]. Additionally, actuator constraints such as
input energy and bandwidth limitations can be accounted for
by trading off the precision needed in output tracking. Thus,
the optimal-inversion technique extends the standard-inversion
theory to design inverse inputs in the presence of modeling un-
certainties and actuator limitations. However, a challenge in im-
plementing the optimal-inversion approach is that the resulting
inverse input tends to be noncausal [9]. (Even the exact inverse
input is noncausal when the system is nonminimum phase [2].)
The noncausality of the optimal inverse implies that the desired
output trajectory must be pre-specified for all time and cannot be
changed online. Therefore, the optimal inverse can only be used
in trajectory-planning applications (where the desired output is
known in advance for all future time).
The main contribution of this article is the development of a
technique to compute the noncausal optimal inverse when the
desired output trajectory is known, in advance, for only a fi-
nite-time interval. This future time interval, during which the de-
sired output trajectory is specified, is referred to as the preview
time. We note that such a preview-based implementation enables
the online specification of the desired output trajectory (see, e.g.,
[10]). Additionally, this article develops a time-domain imple-
mentation of the preview-based optimal inverse as opposed to
the frequency-domain computation in [8]. The time-domain rep-
resentation enables the quantification of the required preview
time in terms of the specified accuracy in output tracking, the
system dynamics and the cost function used to develop the op-
timal inverse.
The finite preview based optimal-inversion technique is
illustrated by using it for precision positioning of a probe
during high-speed surface imaging with a scanning tunneling
microscope (STM). It is noted that the STM is a key enabling
tool in emerging technologies such as nanofabrication [11].
However, in spite of its atomic-level resolution, the oper-
ating-speed (throughput) limitation prevents the STM-based
nanofabrication from competing (at least at present) with more
established techniques like electron beam (EB) and X-ray
lithography [11]. A limitation to increasing STMs throughput
is the positioning error caused by movement-induced vibrations
in the piezo-based positioning system (i.e., piezo scanner). As
the STMs scanning frequency (also called scan rate) is in-
creased relative to the smallest resonant-vibrational frequency
of the piezo scanner, the vibrational modes of the piezo scanner
are excited. These movement-induced vibrations increase with
1063-6536/04$20.00 © 2004 IEEE 376
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
scan rate and result in errors in the STM probes positioning,
thereby limiting the maximum operating speed of the STM [7].
In practice, the achievable scan rate is substantially smaller
(around 100 times smaller) than the smallest resonant-vibra-
tional frequency of the STM scanner. This inability to operate
the STM at high speed currently hinders the investigation and
manipulation of ultrafast processes at the nanoscale. Therefore,
there is a need to develop high-speed precision-positioning
techniques for STM.
In general, the tracking performance of piezo-based posi-
tioning systems can be improved by using feedback control,
for example, to reduce positioning errors due to creep and
hysteresis (see, e.g., [12][18]). However, a problem with using
feedback-based approach is the low-gain margin, of piezo-based
positioners, that limits the achievable improvements because
high-gain feedback tends to destabilize piezo-based STM
scanners [19]. (The low gain margin is due to low structural
damping in piezo-actuators that results in high-quality factor
, i.e., a sharp-resonant peak accompanied by a rapid-phase
drop in the frequency response.) In practice, a compromise is
sought between performance and instability; feedback gains
are adjusted to improve performance without instability. Thus,
the tendency to become unstable at high gains (due to low-gain
margins) has limited the success of typical feedback-based
techniques to achieve high-speed positioning in STM applica-
tions. Furthermore, conventional sensors have relatively low
resolution during high-speed operation (because the sensor
noise tends to increase with operating speed); therefore, cur-
rently available sensors cannot be used to implement feedback
controllers for subnanometer scale positioning when operating
STMs at high speeds (under normal room temperatures).
Feedforward control approaches have been successful in in-
creasing the operating speed of STM [20]. However, when the
inverse input is noncausal, the computation of the inverse input
requires prespecification of the entire output trajectory. This
prevents the use of the inversion-based technique in online-STM
applications such as nanofabrication, where the desired output
trajectories may not be completely pre-specified and may have
to be changed online. This problem is addressed by the devel-
opment of the preview-based approach, which enables the on-
line implementation of the optimal inverse. The preview-based
optimal inversion technique is applied to a STM system and ex-
perimental results are presented to show that the STM can be
operated at high speeds when only a finite preview of the de-
sired output trajectory is available.
The paper is organized in the following format. The optimal-
inversion approach and its finite-preview-based implementation
is presented in Section II. In Section III, the preview-based ap-
proach is applied to a STM system and results (simulation and
experimental) are presented and discussed in Section IV. Our
conclusions are in Section V.
II. P
REVIEW
-B
ASED
O
PTIMAL
-I
NVERSION
The optimal-inversion problem is presented as the minimiza-
tion of a quadratic-cost function and the optimal inverse is ob-
tained as a filter
(as developed in [8]). Properties of
the optimal-inverse filter
are analyzed and a state-
space representation of the optimal inverse is developed for a