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Research Papers

Nonlinear Proportional Plus Integral Control of Optical Traps for Exogenous Force Estimation

[+] Author and Article Information
D. G. Cole1

 Department of Mechanical Engineering and Materials Science, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA 15261dgcole@pitt.edu

J. G. Pickel

 Department of Mechanical Engineering and Materials Science, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA 15261jgp2@pitt.edu

Here high stiffness implies that the trap’s cut-off frequency k/γ is greater than the bandwidth of the signal of interest, so that k>>γωb.

1

Corresponding author.

J. Dyn. Sys., Meas., Control 134(1), 011020 (Dec 06, 2011) (7 pages) doi:10.1115/1.4004774 History: Received August 26, 2010; Revised June 06, 2011; Published December 06, 2011; Online December 06, 2011

This article explores nonlinear proportional plus integral (PI) feedback for controlling the position of an object held in an optical trap. In general, nonlinearities in the spatial dependence of the optical force complicate feedback control for optical traps. Nonlinear PI control has been shown to provide all of the benefits of integral control: disturbance rejection, servo tracking, and force estimation. The controller also linearizes the closed-loop system. More importantly, the nonlinear controller is shown to be equivalent to an estimator of the exogenous force. The ability of nonlinear PI control to lower the measurement SNR is evaluated and compared to the variational open-loop case. A simulation demonstrating the performance of the nonlinear PI control is presented.

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Copyright © 2012 by American Society of Mechanical Engineers
Topics: Force , Feedback
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Figures

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Figure 2

This schematic illustrates a typical single-molecule experiment. In this two-beam setup, microspheres are held in each trap. A long chain polymer, e.g., DNA, is tethered between the two spheres, and the force on the molecule can be determined by measuring the position of each microsphere in its trap. Feedback control can be used to control the molecule’s elongation or the force applied to it.

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Figure 3

(a) A plot of the normalized optical trapping force, ft /(kw0 ). (b) A plot of the normalized local stiffness, ∇ft/k. The optical force acting on a trapped object is a nonlinear function of the displacement of the object within the trap. Two parameters characterize the trap: the trap stiffness k and the peak trapping force ft,max  = 0.61kw0 , where w0 is a characteristic dimension of the trap.

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Figure 4

The controller can also be viewed as an estimator that estimates the exogenous force over some bandwidth. This is done in a feedback scheme to make the estimation error small. The control signal contains three feedback loops: Loop 1 cancels the nonlinear optical force; Loop 2 stabilizes the system; and Loop 3 uses integral control for improved tracking, disturbance rejection, and force estimation.

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Figure 5

(a) The relative displacement z for a move-and-settle motion with a rise segment (at a constant velocity) to a final setpoint. The final setpoint is chosen to be 817 nm (0.95w0 ); that is, we require the bead to nearly reach the boundaries of the optical trap. Note: the reference signal zsp has been displaced up 200 nm for clarity. (b) The integral state z∧ during the move-and-settle motion. Note that in the presence of an exogenous force, 〈z∧〉 will be different from 〈z〉 as shown in Eq. 18.

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Figure 6

Shown is the exogenous force and estimated force for a 70 fN variation about the final setpoint of the move-and-settle motion. The SNR for this force estimate is 2 (6 dB).

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Figure 1

Optical traps are typically built around a microscope, which provides a high NA objective and a means for imaging trapped objects. A collimated beam is introduced to the microscope at the front focal plane of the objective, making a tight focus at the specimen plane. Diffraction of the light at the focus by dielectric objects results in the optical forces that hold the object. Light from a laser can be steered using various means, and the position of the trapped object within the trap can be sensed using a quadrant photo-diode positioned at the objective’s back focal plane.

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