REVIEW OF SCIENTIFIC INSTRUMENTS 77, 083704 共2006兲

Dynamic proportional-integral-differential controller for high-speed
atomic force microscopy
Noriyuki Kodera and Mitsuru Sakashita
Department of Physics, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan

Toshio Ando
Department of Physics, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japanand CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan

共Received 4 April 2006; accepted 10 July 2006; published online 31 August 2006兲
In tapping mode atomic force microscopy, the cantilever tip intermittently taps the sample as the tip
scans over the surface. This mode is suitable for imaging fragile samples such as biologicalmacromolecules, because vertical oscillation of the cantilever reduces lateral forces between the tip
and sample. However, the tapping force 共vertical force兲 is not necessarily weak enough for delicate
samples, particularly for biomolecular systems containing weak inter- or intramolecular interactions.
Light tapping requires an amplitude set point 共i.e., a constant cantilever amplitude to be maintained
during scanning兲 to be set very close to its free oscillation amplitude. However, this requirement
does not reconcile with fast scans, because, with such a set point, the tip may easily be removed
from the surface completely. This article presents two devices to overcome this difficulty; a new
feedback controller 共named as “dynamic proportional-integral-differential controller”兲 and acompensator for drift in the cantilever-excitation efficiency. Together with other devices optimized
for fast scan, these devices enable high-speed imaging of fragile samples. © 2006 American
Institute of Physics. 关DOI: 10.1063/1.2336113兴

I. INTRODUCTION

The atomic force microscope 共AFM兲 has become an indispensable tool in imaging biological samples at high spatial resolutions 共see reviews1,2兲. Amongst the various operating modes, tapping mode3 has been most used when imaging
biological samples in aqueous solutions because the oscillating cantilever tip exerts little lateral force on the sample. In
order to gain the capability of tracing a protein in action athigh temporal resolution with AFM, various efforts have recently been carried out.4–11 The following devices and techniques have been developed, focusing mainly on enhancing
the scan speed; small cantilevers4–6,12 with a high resonant
frequency and a small spring constant, an optical deflection
detection system compatible with small cantilevers,4–6 a fastrms-to-dc converter to quickly measure the oscillation amplitude of a cantilever,4 a high-speed scanner with minimal
structural resonance,4,5 an active damping technique to eliminate resonant vibrations of the piezoactuators,13 and a feedforward controller14–16 capable of lightening the task of the
feedback loop that maintains a constant tapping amplitude. A
combination of some of these efforts has produced a highspeed AFM that can capture moving protein molecules onvideo at 80 ms/ frame.4,5,17,18 However, efforts to minimize
the tapping force have not extensively been carried out. For
example, forces involved in a highly dynamic protein-protein
interaction are very weak. In order to image molecular processes that contain such weak interactions, the oscillating
cantilever tip should barely come in contact with the sample.
First of all, the spring constant of small cantilevers has
to be minimized to reduce the tapping force. However, it is0034-6748/2006/77共8兲/083704/7/$23.00

almost determined by the balance with the resonant frequency required for fast imaging. Although a large quality
factor Q can reduce the tapping force,19,20 it slows the cantilever response, which is incompatible with fast imaging.
Reduction of the cantilever’s oscillation amplitude is one
possibility but increases lateral forces between the tip andsample. In order for the oscillating cantilever tip to touch thesample surface with minimal force, the amplitude set point
As 共the peak-to-peak oscillation amplitude to be maintained
during scanning兲 should be set very close to the free oscillation peak-to-peak amplitude 2A0. This too is incompatible
with fast imaging because in this situation the cantilever tip
tends to be detached completely from the sample surface,especially at steep downhill locations of the sample. Once
detached, the tip will not quickly land again on the surface
共parachuting兲, because of feedback saturation 关the error signal is saturated at small values of 共2A0 − As兲, irrespective of
how far the tip is separated from the surface at the end of its
bottom swing兴. At faster scan speeds, the parachuting effectwould be increased. In addition, a small drift lowering thecantilever-excitation efficiency significantly affects the small
difference, 共2A0 − As兲, which may make 2A0 less than As andconsequently lead to complete detachment between the tip
and sample. Thus, it is very difficult to make the fast scanand “light touching” compatible with each other. In this article we overcome this difficulty by developing a new feedback controller 关“dynamic proportional-integral-differential
共PID兲 controller”兴 and a compensator for drift in the
cantilever-excitation efficiency. The dynamic PID controller
can avoid feedback saturation and give much less dependence of the feedback bandwidth on As. The compensator

77, 083704-1© 2006 American Institute of Physics

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Rev. Sci. Instrum. 77, 083704 共2006兲

Kodera, Sakashita, and Ando

FIG. 1. The residual topography to be sensed by a cantilever tip under
feedback control. When the maximum height of the residual topography islarger than the difference 共2A0 − As兲, the tip completely detaches from thesurface. The untouched areas are shown in gray. The average tip-surface
separation 具d典 at the end of cantilever’s bottom swing is given by 具d典
t0= 共1 / 2t0兲兰−t
关−2A0共1 − r兲 + h0 sin共␾ / 2兲cos共2␲ ft兲兴dt, where t0 = ␤ / 2␲ f 共see0
the text兲. This integral results in 具d典 = 2A0共1 − r兲关tan ␤ / ␤ − 1兴.

stabilizes the cantilever’s free oscillation amplitude. With theuse of these controls together, high-speed and stable successive imaging becomes possible even for fragile samples
which would either be destroyed or not be imaged quickly
with a conventional PID controller. Brief descriptions of preliminary work on the dynamic PID control were previously
presented.17,18
II. THEORETICAL CONSIDERATIONS

h0关sin共2␲ ft兲 − sin共2␲ ft − ␾兲兴2= h0 sin冉冊␾␾
cos 2␲ ft −
,
2
2

共1兲

where h0 is the maximum height of the sample. The maximum height of the residual topography h0 sin共␾ / 2兲 should
be smaller than 共2A0 − As兲, otherwise the cantilever tip would
occasionally detach itself completely from the sample surface. This condition restricts the maximum value of r
⬅ As / 2A0 according to Eq. 共2兲 given below, as a function of
h0 / 2A0 and ␾ 共see Fig. 2兲,

␾
h0
sin .
r⬍1−
2A0
2

and resonant frequency of the cantilever and Qs and f s are
the quality factor and resonant frequency of the z scanner,respectively. These delays and other delays 共the total ␦兲 givethe following relationship:f=Here we analyze the dependence of feedback bandwidth
on the set point As, the free oscillation amplitude A0, thesample height h0, and other factors. Similar analyses werepreviously presented by Sulchek et al. in a qualitative
manner10 and by us in a semiquantitative manner.21 Here, we
present analytical expressions. Suppose that a sample on asubstrate has a periodicity of ␭ and that the sample stage is
moved horizontally with a velocity Vs, the spatial frequency
1 / ␭ is converted to a temporal frequency f = Vs / ␭. This is the
feedback frequency at which the sample stage is moved in
the z direction. When the phase of the feedback signal isdelayed by ␾, a cantilever tip senses the “residual sample
topography” ⌬S共t兲 as a function of time 共see Fig. 1兲. ⌬S共t兲 isexpressed as⌬S共t兲 =

FIG. 2. The maximum amplitude set point rmax allowed for the cantilever tip
to trace the sample surface without complete detachment from the surface,
and its dependence on the ratio of the sample height h0 to the free oscillation
peak-to-peak amplitude 2A0 of the cantilever. The number attached to eachline indicates the phase delay of the feedback operation.

共2兲

The phase delay ␾ is given by 2␲ f⌬␶, where ⌬␶ is the timedelay of the feedback control. The time delay is caused byvarious factors; the main delays are in the time of reading thecantilever’s oscillation amplitude 共it takes at least 1 / 2f c兲, thecantilever’s response time 共Qc / ␲ f c兲, and the z-scanner’s response time 共Qs / ␲ f s兲, where Qc and f c are the quality factor

␾fc
2␲

冒冉

冊

1 Qc Qs f c+++ f c␦ .
2 ␲
␲fs

共3兲

Here, we have to note that the delays can be compensated forto some extent by a differential operation with the PID feedback controller. When the cantilever tip is completely detached from the sample surface at the end of its bottom
swing, it takes a time to touch the surface again. This adds anadditional delay ⌬␶d. For the first approximation, Eq. 共1兲 isassumed to hold even in this case. The average separation
during detachment is given by 2A0共1 − r兲关共tan ␤兲 / ␤ − 1兴,
where ␤ is cos−1关2A0共1 − r兲 / h0 sin共␾ / 2兲兴 共see Fig. 1兲. The
feedback gain is usually set to a level at which the separation
distance of 2A0共1 − r兲 diminishes roughly in a single periodof the cantilever oscillation. Therefore, the additional timedelay ⌬␶d is roughly given by 关共tan ␤兲 / ␤ − 1兴 / f c. By introducing this additional delay into Eq. 共3兲, we can obtain the
feedback bandwidth as a function of various parameters 关Eq.
共4兲 and Fig. 3, where the phase delay is set at ␲ / 4兴,
f=

fc
8

冒冉

冊

tan ␤ 1Qc Qs f c+
+ f c␦ +
−
.
2
␲
␲fs
␤

共4兲

When r is smaller than 关1 − 共h0 / 2A0兲sin共␲ / 8兲兴, ␤ becomeszero and thus the feedback bandwidth is independent of r. As
seen in Fig. 3, the feedback bandwidth decreases with increasing r and rapidly approaches zero at r ⬎ 0.9.
III. MATERIALS AND METHODS
A. High-speed AFM apparatus

A high-speed atomic force microscope developed in ourlaboratory and nearly the same as that described previously4,5
was used. The z-piezoactuators were replaced with thosehaving a higher resonance frequency of 400 kHz 共custom
made, NEC-Tokin, Japan兲. In addition, vibrations of thez-piezoactuators are actively damped using an active damping technique,13 resulting in no resonant vibrations. The
bandwidth of this z scanner is about 150 kHz. The PID feed-

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Dynamic PID for fast AFM

FIG. 3. Theoretically derived feedback bandwidth as a function of the ratio
r of the amplitude set point to the free oscillation peak-to-peak amplitude of
the cantilever. The number attached to each curve indicates the ratio 2A0 / h0.
The feedback bandwidths were obtained under the following conditions: thecantilever’s resonant frequency, 1.2 MHz; quality factor of the cantilever
oscillation, 3; the resonant frequency of the z scanner, 150 kHz; and quality
factor of the z scanner, 0.5.

FIG. 5. Schematic showing the principle of the dynamic PID control. Solidline: an amplitude-distance curve; gray line: an error signal used in theconventional PID control; and broken line: an error signal used in the dynamic PID control.

C. Sample preparation and imaging

back circuit was replaced with the dynamic PID controller
whose design and characteristics are presented in this article.
However, which of these two controllers is active can beselected by a switch. Small cantilevers used here were supplied by Olympus.12 They have a resonant frequency of⬃1.2 MHz in water and a spring constant of ⬃200 pN/ nm.
The tips are grown by electron-beam deposition. The tip
length was adjusted to approximately 1.0 ␮m. The second
harmonic amplitude of the oscillating cantilevers was detected by using a lock-in amplifier 共SR844-RF, Stanford Research Systems, Sunnyvale兲.

B. Mock AFM circuit

To execute quick and precise tests of the feedback loop
performance, we developed a mock AFM circuit 共Fig. 4兲.
This circuit consists of two sets of second-order low-passfilters and a threshold circuit that can simulate a decrease in
the cantilever’s oscillation amplitude caused by tip-sample
interaction. One of the low-pass filters has resonant properties 共and hence a transfer function兲 very similar to those of
the z scanner 共resonant frequency: 150 kHz, Q: 18兲, and theother low-pass filter has resonant properties very similar to
those of the small cantilevers 共resonant frequency:⬃1.2 MHz, Q: 2–3兲. In addition, the mock sample topography is produced by a wave-function generator. The mock z
scanner is actively damped with a controller.13

FIG. 4. Circuit diagram of a mock AFM system. Disturbance signals fed
into the input 2 simulate sample topography. The output simulates the oscillation of a cantilever tip interacting with a sample surface. The amplitudechange caused by the interaction is given by the diode.

Myosin V was extracted from chick brains and purified
as previously described.22 Myosin V was stored at 0 ° C inbuffer A 关25 mM KCl, 25 mM imidazole 共pH 7.6兲, 2 mM
MgCl2, 1 mM ethylene glycol tetraacetic acid 共EGTA兲, and2 mM dithiothreitol兴. Actin was prepared from rabbit skeletal muscles as previously described.23 The purified actin
共⬃50 ␮M兲 was stored as F-actin in buffer B 关100 mM KCl,
2 mM MgCl2, 0.2 mM CaCl2, tris-HCl 共pH 8.0兲, and
0.2 mM adenosine triphoshate 共ATP兲兴 on ice. Just before use,an aliquot of the F-actin solution was centrifuged 共150 000 g,
1 h兲 to remove ATP and unpolymerized actin, and the pellet
was suspended in buffer C 关100 mM KCl, 2 mM MgCl2,
1 mM EGTA, 0.1 mM NaN3, and 25 mM imidazole 共pH
7.6兲兴. For imaging myosin V attached to actin filaments, a
few nanomolar myosin V was mixed with actin filaments
共2 ␮M兲 in buffer C. One drop 共⬃1.5 ␮l兲 of the sample wasplaced on freshly cleaved mica 共1 mm ␾兲 for 3 min, rinsed
with buffer C, and imaged in buffer C.
IV. RESULTS
A. Dynamic PID feedback controller

With a conventional PID feedback circuit, the gain parameters cannot be automatically altered during scanning
based on the topographic features of the sample. When an
oscillating cantilever tip is completely detached from thesample surface, the error signal becomes saturated at 共2A0
− As兲 共Fig. 5兲. When As is very close to 2A0, the saturated
error signal is very small, and thus the detached tip will not
quickly land again on the surface 共parachuting兲. The PID
gains could be increased to shorten the parachuting period.
However, this increase promotes overshooting at uphill regions of the sample, especially near the peak of a local maximum on the sample, and consequently introduces instability
in the feedback operation. This problem can be solved if the
PID gains are regulated based on the cantilever’s peak-topeak oscillation amplitude Ap-p relative to As. We devised
such a feedback controller 共dynamic PID controller兲 by inserting a circuit 共termed “dynamic operator”兲 between theerror signal output and the input of a conventional PID cir-Downloaded 04 Sep 2006 to 133.28.47.30. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp

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Kodera, Sakashita, and Ando

FIG. 6. Diagram of feedback loop with dynamic PID operator.

cuit 共Fig. 6兲. This dynamic operator functions as follows 共seeFig. 5兲. A threshold level Aupper is set between As and 2A0.
When the cantilever’s peak-to-peak oscillation amplitude
Ap-p exceeds Aupper, the differential signal 共Ap-p − Aupper兲 isamplified and added to the error signal. The error signal that
contains an extra signal is fed to the conventional PID. The“false error signal” which is larger than the “true error signal” produces a quicker feedback response, and therefore
Ap-p quickly becomes smaller than Aupper and the feedback
operation automatically returns to the normal mode. Thus,
even with As very close to 2A0, the parachuting period isshortened drastically, and hence no feedback saturation occurs. A similar manipulation of the error signal can be madeas well when Ap-p is smaller than As. In this case a new
threshold level Alower is set lower than As to an appropriate
extent. When Ap-p becomes lower than Alower, the differential
signal 共Ap-p − Alower兲 is amplified and then added to the error
signal 共Ap-p − As兲. This manipulation can keep the cantilever
tip from pushing into the sample too strongly, especially at
steep uphill regions of the sample.
These manipulative operations are implemented by a circuit shown in Fig. 7. The circuit has three branches in thehorizontal direction. The true error signal passes through the
middle branch. A dc signal corresponding to 共Aupper − As兲 is

FIG. 7. Circuit diagram of dynamic operator. For details, see the text.

Rev. Sci. Instrum. 77, 083704 共2006兲

FIG. 8. Pseudo-AFM images of a sample with rectangles with two different
heights. The images were obtained using a conventional PID controller 共a兲
or using the dynamic PID controller 共c兲. Lower panels 共b兲 and 共d兲 show line
profiles of images 共a兲 and 共c兲, respectively. These simulations with the mock
AFM system were made under the same condition shown in Fig. 3 captions.
The line scan speed: 1 mm/ s; the line scan frequency: 1 kHz; and the frame
rate: 100 ms/ frame.

fed to the upper branch input terminal and passes though asubtracter to produce 共Ap-p − Aupper兲. The output is amplified
共gain, Gupper兲, passed through a precision diode circuit, andfinally summed with the true error signal. The precision diode circuit compensates for an offset inherent in the diodechip. The final output, L关Ap-p − Aupper兴 ⫻ Gupper + 共Ap-p − As兲, is
fed to the conventional PID input terminal. Here, L is theoperator that acts as L关x兴 = x 共if x ⬎ 0兲 or L关x兴 = 0 共if x 艋 0兲.
The signal 共Ap-p − Aupper兲 could be constructed directly from
Ap-p and Aupper. However, in this case, Aupper has to be tuned
depending on As, because Aupper has to be larger than As. On
the other hand, using the method we employed above, thepositive dc signal, corresponding to 共Aupper − As兲, is tuned independently from As. The false error signal, −L关Alower− Ap-p兴 ⫻ Glower + 共Ap-p − As兲, is similarly constructed for the
lower branch.
B. Performance test of dynamic PID using a mock
AFM

Feedback performance of the dynamic PID controller
was compared with that of a conventional PID controller
using a mock AFM. Herein, a mock cantilever with Q = 3
oscillating at its resonant frequency of 1.2 MHz is scannedover a mock sample surface 共rectangular shapes with two
different heights兲 from left to right at scan speed of 1 mm/ s
共frame rate of 100 ms/ frame兲. Here, 2A0 is the same as thetaller sample height, and As is set at 0.9⫻ 2A0. With theconventional PID controller, the topographic image became
blunt 关Fig. 8共a兲兴. As seen in the line profile 关Fig. 8共b兲兴, parachuting occurred significantly at steep downhill regions. On
the other hand, use of the dynamic PID controller produces aclear image 关Fig. 8共c兲兴 and almost no parachuting occurred关Fig. 8共d兲兴.
Next, we examined performance of the dynamic PID bymeasuring feedback bandwidth as a function of h0 / 2A0 and
r共=As / 2A0兲, using the mock AFM, and compared it with thatof the conventional PID. Here, we did not use the dynamic
operator of the lower branch 共i.e, Glower = 0, see Fig. 7兲. The

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Dynamic PID for fast AFM

C. Performance test of dynamic PID using a highspeed AFM

FIG. 9. Feedback bandwidth as a function of the ratio r = As / 2A0, measured
using the mock AFM system or the real AFM system. Solid-line curves:
feedback bandwidths measured using the mock AFM system with a conventional PID controller; dotted-line curves: feedback bandwidths measured
using the mock AFM system with the dynamic PID controller; closed marks:
feedback bandwidths with 2A0 / h0 = 0.5共⽧兲, 1共䉲兲, 2共䉱兲, and 5共䊏兲 measured
using the real AFM system with the conventional PID controller; and open
marks: feedback bandwidths with 2A0 / h0 = 0.5共〫兲, 1共䉮兲, 2共䉭兲, and 5共䊐兲
measured using the real AFM system with the dynamic PID controller. The
solid-line curves and the dotted-line curves are aligned from top to bottom
according to the ratio 2A0 / h0 = 5, 2, 1, and 0.5.

feedback bandwidth was determined as a frequency of disturbance signals 共sinusoidal waves that simulate sample topography兲 that gave 45° phase delay to the feedback controller output. Although the disturbance should be applied to themock cantilever, it was applied to the mock z scanner for areason mentioned in the next section. First, the gain parameters of the conventional PID controller were adjusted toprovide the best performance 共a minimal error signal rms兲for each r and for h0 / 2A0 = 1. The gain parameters were notaltered for various ratios of h0 / 2A0. After measuring various
bandwidths, the gain parameters of the dynamic PID controller were adjusted 共the P, I, and D gains were slightly attenuated兲 for the same r, and then the gain parameter 共Gupper, seeFig. 7兲 of the dynamic operator was adjusted. The thresholdlevel Aupper was set to be slightly higher than As. The adjustment of the PID gains and Gupper was performed so as toproduce no shift of the dc level of the mock cantilever deflection from As and a minimal rms value of the true error
signals. The results of these measurements are summarized
in Fig. 9. The maximum feedback bandwidth obtained wasabout 70 kHz, which was higher than the frequency expectedfrom Eq. 共3兲 with ␦ = 0 共about 55 kHz兲, due to a compensation effect by the D operator of the PID feedback control.
With the conventional PID control, feedback bandwidth decreased with increasing r and h0 / 2A0. These behaviors werevery similar to the results obtained from theoretical analysis
共Fig. 3兲. On the other hand, feedback bandwidth was nearly
even over the set point range examined 共0.6⬍ r ⬍ 0.95兲 using
the dynamic PID control. In addition, the maximum feedback bandwidth observed at each h0 / 2A0 was always higher
than the corresponding bandwidth obtained with the conventional PID control. When Aupper was varied between As and
2A0, the resulting feedback bandwidth did not change as longas optimum adjustment of Gupper and the PID gain parameters was made.

We examined whether the excellent performance of dynamic PID witnessed during testing using the mock AFM
was also true with a real AFM system. In this examination,
A0 and As were fixed at 5.5 nm and 0.9⫻ 2A0 共9.9 nm兲, respectively, to measure feedback bandwidth. Because it wasdifficult to prepare test samples with sinusoidal wave topographies, mica 共immersed in water兲 on the sample stage was
moved in the z direction at various frequencies and withvarious amplitudes. This is the reason why in the aforementioned experiment, perturbation was applied to the mock z
scanner. A sinusoidal signal for moving the sample stage wasadded to the output of the conventional PID circuit or the
dynamic PID circuit and their sum was fed into az-piezoactuator driving amplifier. As indicated in Fig. 9, thefeedback bandwidths obtained here were very similar to thecorresponding values obtained using the mock AFM, except
for the case 2A0 / h0 = 5. The disagreement observed with
2A0 / h0 = 5 arises from a relatively low signal to noise ratio of
the sensor signal when the z scanner is perturbed by sinusoidal waves with a small amplitude 共h0 = 2.2 nm兲. The gain
parameters of the dynamic PID controller could not be increased much due to the sensor noise.
We have to note that the feedback bandwidth measured
using the mock or real AFM systems is underestimated, because the perturbation was applied to the 共mock兲 z scanner.
When the perturbation was applied to the mock cantilever
共which is more close to the actual situation in imaging兲, theobserved feedback bandwidths were always 30%–40%
higher than the corresponding values given in Fig. 9. This isdue to the slower response speed of the 共mock兲 z scanner,compared with the small cantilevers.
D. Drift compensation

Compared with contact mode imaging, tapping mode
imaging is not significantly affected by drifts in various components of AFM. This is because in tapping mode, the tipsample interaction is reflected only on the ac component of
the signal from the cantilever-deflection detector and thus
drift in its dc component hardly affects the imaging performance. Drifts, however, are still problematic even in tapping
mode imaging. Some efforts to compensate for drifts in thedeflection sensor signal were previously carried out. Kindt etal.24 controlled the deflection set point or amplitude set point
using a cross correlation of the feature richness between two
traces 共forward and backward traces兲 at slightly different set
points. Although this method works well for both contact andtapping modes, calculations for the cross correlation must be
made for every line scan, and therefore additional efforts are
required to make it compatible with fast imaging. In tapping
mode, drift in the cantilever-excitation efficiency is the mostproblematic one, particularly when As is set very close to 2A0
to minimize the tapping force. The AFM apparatus misunderstands this drift-caused change in the cantilever oscillation amplitude and interprets the change as a result of tipsample interaction. For example, when the excitation
efficiency is lowered, the AFM apparatus interprets this as

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Kodera, Sakashita, and Ando

image was obtained because of complete detachment between the tip and sample.
V. DISCUSSION

FIG. 10. Successive imaging of myosin V attached to actin filaments using
a compensator for drift in the cantilever-excitation efficiency. The imaging
was successively made for 3 min at frame rate of 10 frames/ s and with
A0 = 2.5 nm and r共As / 2A0兲 = 0.92. Only five images obtained at times indicated by arrows are shown. Black line: the output from the drift compensator; gray line: the second harmonic amplitude. At 3 min, the compensator
was switched off.

the tip interacting with the sample too strongly. Therefore,
the feedback responds by withdrawing the sample stage fromthe tip, which is an incorrect direction. Without stability inthe excitation efficiency or A0, successive imaging under asmall tapping force cannot be realized. The drift may becaused by an increase in the temperature of the oscillating
piezoactuator 共“excitation actuator” for driving cantilever oscillation兲, by a change in an area of the fluid cell in contact
with a buffer solution, and by a change in the cantileverresonance frequency. It is difficult to eliminate these causes.
In addition, we cannot detect the free amplitude A0 while
imaging. This problem was previously challenged by
Schiener et al.25 They used the second harmonic amplitude
of cantilever oscillation to detect drifts. The second harmonic
amplitude is sensitive to tip-sample interaction, and therefore, drift in A0 is reflected in the amplitude averaged over a
period longer than the image-acquisition time. Instead ofcontrolling A0, they controlled As in order to maintain theconstant difference 共2A0 − As兲. However, this control varies
the tapping force and feedback bandwidth because h0 / 2A0changes. To compensate for drift in the cantilever-excitation
efficiency here, we also used the second harmonic amplitude
of cantilever oscillation, but instead of controlling As we
controlled the output gain of a wave generator 共WF-1946A,
NF Corp., Osaka, Japan兲 connected to the excitation piezoactuator. We used only a type I controller whose time constant was adjusted to 1 – 2 s 共about ten times longer than theimage-acquisition time兲. The performance of this drift compensation is shown in Fig. 10. A sample of myosin V boundto actin filaments in solution was imaged successively for
3 min at 100 ms/ frame. Very stable imaging was achieved,
even with the small difference 共2A0 − As兲 = 0.4 nm. In addition, the fragile actin filaments were never disassembled during imaging. The output signal from the type I controller wasincreasing with time, indicating that the cantilever-excitation
efficiency was declining with time. On the other hand, thesecond harmonic amplitude was kept constant. When the thetype I controller output was disconnected after 3 min, no

Feedback bandwidth is the most important factor inhigh-speed AFM imaging. In tapping mode, feedback bandwidth 共defined by 45° phase delay兲 cannot exceed one-fourthof the cantilever’s resonant frequency 关see Eq. 共4兲兴. This isbecause at least a half period of the cantilever oscillation isrequired for reading its amplitude. In addition, the cantilever’s Q factor slows its response to the tip-sample interaction
and hence lowers the feedback bandwidth. Thus, high-speed
tapping mode AFM requires cantilevers having a small Q
and a high resonant frequency that is at least five to six times
greater than the required feedback bandwidth. However,there is a practical limit in reconciling the high resonancefrequency with a small spring constant. Accordingly, fast imaging is difficult to achieve under a nondestructive imaging
condition.
One of the main purposes of fast imaging of biologicalsamples is to observe their dynamic behaviors as they function in solution. Hence, minimization of the tip-sample interaction force is essential. It requires a wide feedback bandwidth as well as a small tapping force. In a conventional PID
circuit, gains are the same for both regimes of Ap-p ⬎ As andAp-p ⬍ As. In order to shorten the parachuting period 共where
Ap-p ⬎ As兲, the gains must be large. However, to avoid overshoots at uphill regions of the sample 共where Ap-p ⬍ As兲, thegains have to be attenuated. Thus, the gains have to be determined via a balance between these two aspects. In the
dynamic PID control, appropriate gains are separately adjusted for the two regimes. This is the key for its excellent
performance. The dynamic PID control improves both feedback bandwidth and tapping force simultaneously and will
therefore become indispensable in high-speed AFM imaging
of delicate samples.
Its implementation can be made with a simple analog
circuit 共the dynamic operator and a conventional PID controller兲 and should be able to be implemented in a digital
signal processor system. We inserted the dynamic operator
between the error signal output and the input to a conventional PID controller. However, there may be some variations. For example, the outputs from the upper and lower
branches 共see Fig. 7兲 can be added directly to the conventional PID controller output. A similar function can be obtained with a different design; a voltage-controlled variable
gain amplifier is placed on each output of the two PI or three
PID components, where the error signal is used for controlling the gains. Better performance may be obtained using
one of these methods or possibly other variations.
The dynamic PID control for regime Ap-p ⬍ As is not as
effective as it is for regime Ap-p ⬎ As, particularly when As isset very close to A0, because in the regime Ap-p ⬍ As the error
signal hardly saturates. In addition, this control for the regime Ap-p ⬍ As may produce ill effects on the feedback operation when the threshold level Alower is set close to As. It
promotes overshoot of the cantilever tip and hence induces
its parachuting. However, this control is still useful when the

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083704-7

threshold level Alower is set far from As and is close to zero. It
can avoid tip-sample contact that is too strong and thus prevents damage of both the sample and tip. A similar control
system for avoiding overly strong tip-sample contact can beconstructed and used for the preimaging operation of thesample stage approaching the cantilever tip.
Compared with the first generation of high-speed
AFM,4,5 faster, more stable, and nondestructive imaging has
become possible, owing to the dynamic PID controller andthe drift compensator developed here. From this ability andimprovements that will come forth, high-speed AFM is anticipated to play an active role in biological sciences in thenear future.
ACKNOWLEDGMENTS

This work was supported by CREST/JST, Special Coordination Funds for Promoting Science and Technology 共Effective Promotion of Joint Research with Industry, Academia, and Government兲 from JST, and a Grant-in-Aid for
Basic Research 共S兲 from the Ministry of Education, Culture,
Sports, Science, and Technology of Japan.
1Rev. Sci. Instrum. 77, 083704 共2006兲

Dynamic PID for fast AFM

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