An observer based Q control method for a cantilever in an atomic force microscopy is provided that provides for a “dual” Q behavior such that a particular effective Q is achieved when a sample is present and another effective Q when a sample is absent. In the control method, the transfer function from dither input to photo-diode output is independent of the observer so that the cantilever effectively behaves like a spring-mass-damper system. The effective quality factor and stiffness of the cantilever can be changed by appropriately choosing the state feedback gain. The method provides sample-imaging using transient atomic force microscopy.
1. A method to perform high-speed high resolution imaging of a sample from detecting transient cantilever movement in an atomic force microscope, the cantilever having a tip, the method comprising the steps of:
receiving an input dither signal and tip deflection data corresponding to the input dither signal;
generating an innovation signal based upon the input dither signal and tip deflection data, the innovation signal having a dynamic profile of unknown arrival time and unknown magnitude with additive white noise;
computing maximum likelihood estimates of the arrival time and magnitude using a likelihood ratio test;
deriving an image of the sample using the maximum likelihood estimates of the arrival time and magnitude; and
wherein the likelihood ratio test is in accordance with
r
(
M
)
=
e
_
T
e
_
T
V
where V is a covariance matrix, ē=[e 1 , e 2 , . . . , e M ] T , and e is an error in an estimate of the output of an observer.
2. The method of claim 1 wherein the step of generating the innovation signal comprises the step of generating the innovation signal according to
e
(
s
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=
η
(
s
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+
(
s
2
+
ω
0
Q
s
+
ω
0
2
)
υ
(
s
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+
(
s
+
ω
0
Q
)
v
1
+
v
2
s
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(
ω
0
Q
+
l
1
)
s
+
(
ω
0
2
+
l
2
+
ω
0
Q
l
1
)
where L=[l 1 l 2 ] T is an observer gain such that
(
ω
0
Q
+
l
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>
0
and
(
ω
0
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+
ω
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Q
l
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>
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,
ω 0 is the resonant frequency, ν is measurement noise, η is thermal noise, Q is the quality factor and v=[v 1 , v 2 ] is a state-jump that occurs when the cantilever tip hits the sample surface.
3. The method of claim 1 further comprising the step of comparing a likelihood ratio from the likelihood ratio test to a threshold value to arrive at a decision whether the dynamic profile is present.
4. The method of claim 3 wherein the threshold value is chosen such that a false alarm rate is below a predefined level.
5. The method of claim 3 wherein the threshold value is chosen such that a detection probability is above a predefined level.
6. A method to perform transient imaging of a sample from detecting transient cantilever movement in an atomic force microscope using Q control, the cantilever having a tip, the method comprising the steps of:
receiving an input dither signal and tip deflection data corresponding to the input dither signal;
generating an estimated position and an estimate velocity of the tip;
generating an innovation signal based upon the input dither signal, the tip deflection data, the estimated position and the estimated velocity, the innovation signal having a dynamic profile of unknown arrival time and unknown magnitude with additive white noise;
computing maximum likelihood estimates of the arrival time and magnitude using a likelihood ratio test;
deriving an image of the sample using the maximum likelihood estimates of the arrival time and magnitude; and
further comprising the step of adding the estimated position and the estimate velocity in a feedback loop to add to the input dither signal, thereby actively damping a quality factor (“Q”) of the cantilever.
7. The method of claim 6 further comprising the step of comparing a likelihood ratio from the likelihood ratio test to a threshold value to arrive at a decision whether the dynamic profile is present.
8. The method of claim 7 wherein the threshold value is chosen such that a false alarm rate is below a predefined level.
9. The method of claim 7 wherein the threshold value is chosen such that a detection probability is above a predefined level.
10. A method to perform transient imaging of a sample from detecting transient cantilever movement in an atomic force microscope using Q control, the cantilever having a tip, the method comprising the steps of:
receiving an input dither signal and tip deflection data corresponding to the input dither signal;
generating an estimated position and an estimate velocity of the tip;
generating an innovation signal based upon the input dither signal, the tip deflection data, the estimated position and the estimated velocity, the innovation signal having a dynamic profile of unknown arrival time and unknown magnitude with additive white noise;
computing maximum likelihood estimates of the arrival time and magnitude using a likelihood ratio test;
deriving an image of the sample using the maximum likelihood estimates of the arrival time and magnitude; and
further comprising the step of adding the estimated position and the estimate velocity in a feedback loop to add to the input dither signal, thereby actively enhancing a quality factor (“Q”) of the cantilever.
11. A method to perform transient imaging of a sample from detecting transient cantilever movement in an atomic force microscope using Q control, the cantilever having a tip, the method comprising the steps of:
receiving an input dither signal and tip deflection data corresponding to the input signal;
generating an estimated position and an estimate velocity of the tip;
generating an innovation signal based upon the input dither signal, the tip deflection data, the estimated position and the estimated velocity, the innovation signal having a dynamic profile of unknown arrival time and unknown magnitude with additive white noise;
computing maximum likelihood estimates of the arrival time and magnitude using a likelihood ratio test;
deriving an image of the sample using the maximum likelihood estimates of the arrival time and magnitude;
wherein the step of generating an estimated position and an estimate velocity comprises the step of generating an estimated position and an estimate velocity using an observer; and
wherein dynamics of the observer are given in accordance with:
{circumflex over ({dot over (x)})}=A{circumflex over (x)}+Bu+L ( y−C{circumflex over (x)} ); {circumflex over (x)} (0)= {circumflex over (x)} 0
u= ( g+F{circumflex over (x)} )
{circumflex over (p)}=C{circumflex over (x)}
{circumflex over (ν)}=D{circumflex over (x)}
where {circumflex over ({dot over (x)})} is the estimate dynamic state of the cantilever, L is the observer gain, F is the state feedback gain, g is the input dither signal, {circumflex over (p)} is the estimate position, A, B, C, and D are state matrices realized from the frequency response from a dither-piezo input to a photo-diode output, u is the dither-piezo input, y is the photo-diode output, and {circumflex over (ν)} is the estimated velocity.
12. The method of claim 11 wherein a transfer function from dither input to photo-diode output is independent of the observer.
13. The method of claim 11 wherein the state feedback gain F does not affect the effective resonant frequency of the cantilever.