Method to transiently detect samples in atomic force microscopes

1. A method to detect transient cantilever movement in an atomic force microscope, the cantilever having a tip, the method comprising the steps of:

receiving an input signal and tip deflection data corresponding to the input signal;

generating an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude with additive white Gaussian noise; and

computing maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test.

2. The method of claim 1 wherein the step of computing maximum likelihood estimates includes the step of selecting a threshold ε such that a false alarm rate is below one percent.

3. The method of claim 1 further comprising the step of updating the estimated dynamic state with the maximum likelihood estimates of the arrival time and the magnitude.

4. The method of claim 1 wherein the dynamic profile is defined by the equation

Y i;θ =H [Φ( i ,θ)− FX ( i− 1;θ)]

where

X ( i ;θ)= K i Y i;θ +FX ( i −1;θ)

and where Φ(i,θ)=Π j=θ i F, X(i;i)=K i H, K i is a Kalman filter gain, F is a function of the first resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever tip position and cantilever tip velocity.

5. The method of claim 4 wherein step of generating an estimated dynamic state includes the step of generating a maximum likelihood estimate of arrival time, the maximum likelihood estimate of arrival time being given by a value θ≦n such that {circumflex over (θ)} n =arg max 1≦θ≦n l(n;θ) where l(n;θ)=d T (i;θ)C −1 (n;θ)d(n;θ),

C

⁡

(

n

;

θ

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=

∑

i

=

θ

n

⁢

Y

i

;

θ

T

⁢

V

-

1

⁢

Y

i

;

θ

,

d

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(

n

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=

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=

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n

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Y

i

;

θ

T

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V

-

1

⁢

Y

i

,

V

=

HPH

T

+

R

,

H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

6. The method of claim 5 wherein the step of generating the maximum likelihood estimate of arrival time includes the step of generating the maximum likelihood estimate of arrival time in a data window of finite length.

7. The method of claim 4 wherein the maximum likelihood estimate of magnitude is given by a value {circumflex over (ν)} n (θ)=C −1 (n;θ)d(n;θ) where

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n

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,

V=HPH T +R, H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

8. The method of claim 7 wherein the step of generating the maximum likelihood estimate of magnitude includes the step of generating the maximum likelihood estimate of magnitude in a data window of finite length.

9. A method of estimating the transient deflection of a cantilever operating in dynamic mode in an atomic force microscope system, the method comprising the steps of:

receiving, by a Kalman observer, an input signal and tip deflection data corresponding to the input signal;

generating, by the Kalman observer, an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude;

computing, by the Kalman observer, maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and

estimating the transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude.

10. The method of claim 9 wherein the step of generating an estimated dynamic state includes the step of generating a measurement residual, the measurement residual equal to the dynamic profile plus an offset, the offset being a non-jump measurement residual.

11. The method of claim 10 wherein the non-jump measurement residual comprises white Gaussian noise.

12. The method of claim 9 wherein the dynamic profile is defined by the equation

Y i;θ =H [Φ( i ,θ)− FX ( i −1;θ)]

where

X ( i ;θ)= K i Y i,θ +FX ( i −1;θ)

and where Φ(i,θ)=Π j=θ i F, X(i;i)=K i H, K i is a Kalman filter gain, F is a function of the first resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever tip position and cantilever tip velocity.

13. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value θ≦n such that {circumflex over (θ)} n =arg max 1≦θ≦n l(n;θ) where l(n;θ)=d T (i;θ)C −1 (n;θ)d(n;θ),

C

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n

;

θ

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;

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Y

i

,

V=HPH T +R , H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

14. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value {circumflex over (ν)} n (θ)=C −1 (n;θ)d(n;θ) where

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V

=

HPH

T

+

R

,

H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

15. A control system adapted to perform the method of claim 9 comprising:

a first mode approximation model of the cantilever;

a tip-sample interaction model in communication with the first mode approximation model; and

a Kalman observer designed to estimate the dynamic state of the cantilever movement, the Kalman observer in communication with the first mode approximation model of the cantilever.

16. The control system of claim 15 wherein the tip-sample interaction model is modeled as an impulsive force applied to the cantilever.

17. The control system of claim 16 wherein the impulsive force comprises a repulsive part of the tip-sample interaction.

18. The control system of claim 15 wherein the first mode approximation model of the cantilever and the Kalman observer are adapted to receive an input signal of the cantilever.

19. The control system of claim 15 wherein the Kalman observer is adapted to:

receive an input signal and tip deflection data corresponding to the input signal;

generate an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude;

compute maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and

estimate a transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude.