## An adaptive system identification approach to optical trap calibration

Optics Express, Vol. 16, Issue 7, pp. 4420-4425 (2008)

http://dx.doi.org/10.1364/OE.16.004420

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### Abstract

A method of adaptive system identification for the modeling of an optical trap’s system dynamics is presented. The system dynamics can be used to locate the corner frequency for trapping stiffness calibration using the power spectral method. The method is based on an adaptive least-mean-square (LMS) algorithm, which adjusts weights of a tapped delay line filter using a gradient descent method. The identified model is the inverse of the high order finite impulse response (FIR) filter. The model order is reduced using balanced model reduction, giving the corner frequency which can be used to calibrate the trapping stiffness. This method has an advantage over other techniques in that it is quick, does not explicitly require operator interaction, and can be acquired in real time. It is also a necessary step for an adaptive controller that can automatically update the controller for changes in the trap (*e.g.*, power fluctuations) and for particles of different sizes and refractive indices.

© 2008 Optical Society of America

## 1. Introduction

1. A. Ashkin and J.M. Dziedzic, “Optical Trapping andManipulation of Viruses and Bacteria,” Science **235**, 1517–1520 (1987). [CrossRef] [PubMed]

3. K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature **400**, 184–189 (1999). [CrossRef] [PubMed]

4. M. D Wang, H. Yin, R. Landick, J. Gelles, and S.M. Block, “Stretching DNA with optical tweezers,” Biophys. J. **72**, 1335–1346 (1997). [CrossRef] [PubMed]

*F*=

*kx*), where

*F*is the force,

*k*is the trapping stiffness, and

*x*is the displacement from the center of the trap. Many methods exist for measuring the particle’s displacement [5

5. M. J. Lang and S. M. Block, “Resource letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. **71**, 201–215 (2003). [CrossRef]

*mẍ*+

*γẋ*+

*kx*=

*kd̃*, where

*m*is the object’s mass,

*x*is the position of the object with respect to the trapping center, and d̃ is the fluctuating Brownian disturbance, which, in this case, is white, zero mean, with a variance of

*γẋ*+

*kx*=

*kd̃*. A continuous-time frequency-domain representation of this system can be made by Laplace transforming this equation (assuming zero initial conditions). A frequency-domain transfer function is the ratio of the output of the system to the input. The transfer function for this system is:

*G*(

*s*)=

*X*(

*s*)/

*D̃*(

*s*)=

*k*/(

*γs*+

*k*), where

*s*is the Laplace variable. The auto-spectrum of this system is

*ω*is the frequency in radians/second. The corner frequency, Ω, dictates the speed of which a trapped particle can be manipulated.

*e.g.*, power fluctuations) and for particles of different sizes and refractive indices.

## 2. Adaptive least-mean-square algorithm

*z*-domain where

*z*

^{-1}represents a single time step delay while

*z*

^{-m}represents an

*m*th time step delay. For an

*m*th order filter, the output of the filter,

*y*, at a discrete time step

*n*is defined as:

*y*(

*n*)=

**w**′(

*n*)

**u**(

*n*), where

**w**(

*n*)=[

*w*

_{0}(

*n*),

*w*

_{1}(

*n*), …,

*w*(

_{m}*n*)]′ is the vector of tap weights, and

**u**(

*n*)=[

*u*(

*n*),

*u*(

*n*-1), …,

*u*(

*n*-

*m*)]′ is the vector of tap inputs.

*e*, which will be defined later. The LMS algorithm has a mean-squared error cost function:

*ℰ*{·} denotes the statistical expectation operator. The cost function is quadratic in

**w**, creating a surface on which a global minimum can be found using a gradient descent method. To minimize the cost function, the tap weights are adjusted in the negative gradient direction of the cost function. The minimum is achieved when the gradient is zero, ∇

*J*=-

*ℰ*{

**u**(

*n*)

*e*(

*n*)}=0; that is, when the filter error and tap input are orthogonal. Since determining the expected value of the gradient is difficult, an instantaneous estimate of the gradient is used in the LMS algorithm. The resultant weight update equation is

*µ*is the step size.

*u*(

*n*)=

*x*(

*n*). The resulting FIR filter is an inverse of the system’s transfer function. In the

*z*-domain, the resulting transfer function model of the system dynamics is

*Ĝ*(

*z*) is stable since the zeros of the FIR filter could be anywhere in the

*z*-plane; but, in this situation this is not important since the objective is to match the frequency response of the system dynamics with the identified model, and ultimately determine the stiffness. It is important to note that this configuration only finds the corner frequency of the system and not the magnitude of the response. The magnitude is fit using the equipartition theorem.

*w*

_{0}=1, and the remaining

*m*weights are allowed to adapt. This is done by defining the error function as:

*e*(

*n*)=

*y*(

*n*)+

*x*(

*n*). Intuitively, this is can be thought of as comparing a Laplace transform to a

*z*transform, which for the system presented here is:

*z*transform is unity and the second value is dependent on the corner frequency,

*a*, and the sample period,

*T*. By allowing the rest of the weights to adapt, the discrete time transfer function mimics the actual continuous time system.

## 3. Experimental methods and results

11. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. **23**, 7–9 (1998). [CrossRef]

*m*=3). Multiple delays, from 1 to 10 were tested, with 3 being the smallest size that accurately captured the system dynamics. The step-size,

*µ*, was set to 0.08. The adaptive system identification program was then executed and allowed 1 min for the weights to converge. Particles with higher corner frequencies converged in less time, but 1 min was used for consistency. The movement of the particle was tracked on the QPD and recorded, as well as the final value of the weight vector. The resulting identified plant transfer function was then reduced to a first order system and plotted along with the auto-spectrum of the data. The magnitude of the reduced order model was fit using the equipartition theorem. As can be seen in Fig. 3, the modeled system accurately tracks the corner frequency as well as the magnitude. The corner frequencies vary from 17 Hz for the 5 µm to 4213 Hz for the 0.5 µm microspheres.

*k*=Ω

*γ*. The results for various trapping conditions are presented in Table 1. These results are comparable to those achieved previously [12

12. L. P. Ghislain, N. A. Switz, and W.W. Webb, “Measurement of Small Forces Using an Optical Trap,” Rev. Sci. Instrum. **65**, 2762–2768 (1994). [CrossRef]

## 4. Conclusions

## References and links

1. | A. Ashkin and J.M. Dziedzic, “Optical Trapping andManipulation of Viruses and Bacteria,” Science |

2. | J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, “Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers,” Biophys. J. |

3. | K. Visscher, M. J. Schnitzer, and S. M. Block, “Single kinesin molecules studied with a molecular force clamp,” Nature |

4. | M. D Wang, H. Yin, R. Landick, J. Gelles, and S.M. Block, “Stretching DNA with optical tweezers,” Biophys. J. |

5. | M. J. Lang and S. M. Block, “Resource letter: LBOT-1: Laser-based optical tweezers,” Am. J. Phys. |

6. | M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum. |

7. | K. D. Wulff, D. G. Cole, and R. L. Clark, “Servo control of an optical trap,” Appl. Opt. |

8. | B. Widrow and S. D. Stearns, |

9. | S. S. Haykin, |

10. | M. Green and D. J. N. Limebeer, |

11. | F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. |

12. | L. P. Ghislain, N. A. Switz, and W.W. Webb, “Measurement of Small Forces Using an Optical Trap,” Rev. Sci. Instrum. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(180.0180) Microscopy : Microscopy

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: July 18, 2007

Revised Manuscript: December 11, 2007

Manuscript Accepted: December 14, 2007

Published: March 17, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Kurt D. Wulff, Daniel G. Cole, and Robert L. Clark, "An adaptive system identification
approach to optical trap calibration," Opt. Express **16**, 4420-4425 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4420

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### References

- A. Ashkin and J. M. Dziedzic, "Optical Trapping andManipulation of Viruses and Bacteria," Science 235, 1517-1520 (1987). [CrossRef] [PubMed]
- J. E. Molloy, J. E. Burns, J. C. Sparrow, R. T. Tregear, J. Kendrickjones, and D. C. S. White, "Single-Molecule Mechanics of Heavy-Meromyosin and S1 Interacting with Rabbit or Drosophila Actins Using Optical Tweezers," Biophys. J. 68, S298-S305 (1995).
- K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999). [CrossRef] [PubMed]
- M. D. Wang, H. Yin, R. Landick, J. Gelles, and S.M. Block, "Stretching DNA with optical tweezers," Biophys. J. 72, 1335-1346 (1997). [CrossRef] [PubMed]
- M. J. Lang and S. M. Block, "Resource letter: LBOT-1: Laser-based optical tweezers," Am. J. Phys. 71, 201-215 (2003). [CrossRef]
- M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, "Calibration of optical tweezers with differential interference contrast signals," Rev. Sci. Instrum. 73, 1687-1696 (2002). [CrossRef]
- K. D. Wulff, D. G. Cole, and R. L. Clark, "Servo control of an optical trap," Appl. Opt. 46, 4923-4931 (2007). [CrossRef] [PubMed]
- B. Widrow and S. D. Stearns, Adaptive signal processing, Prentice-Hall signal processing series (Prentice-Hall, Englewood Cliffs, N.J., 1985).
- S. S. Haykin, Adaptive filter theory, Prentice Hall information and system sciences series, 2nd ed. (Prentice Hall, Englewood Cliffs, NJ, 1991).
- M. Green and D. J. N. Limebeer, Linear robust control (Prentice Hall, Englewood Cliffs, N.J., 1995).
- F. Gittes and C. F. Schmidt, "Interference model for back-focal-plane displacement detection in optical tweezers," Opt. Lett. 23, 7-9 (1998). [CrossRef]
- L. P. Ghislain, N. A. Switz, and W. W. Webb, "Measurement of Small Forces Using an Optical Trap," Rev. Sci. Instrum. 65, 2762-2768 (1994). [CrossRef]

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