Abstract

This paper presents a theoretical and experimental investigation of the Brownian motion control of an optically trapped probe. The Langevin equation is employed to describe the motion of the probe experiencing random thermal force and optical trapping force. Since active feedback control is applied to suppress the probe’s Brownian motion, actuator dynamics and measurement delay are included in the equation. The equation of motion is simplified to a first-order linear differential equation and transformed to a discrete model for the purpose of controller design and data analysis. The derived model is experimentally verified by comparing the model prediction to the measured response of a 1.87μm trapped probe subject to proportional control. It is then employed to design the optimal controller that minimizes the variance of the probe’s Brownian motion. Theoretical analysis is derived to evaluate the control performance of a specific optical trap. Both experiment and simulation are used to validate the design as well as theoretical analysis, and to illustrate the performance envelope of the active control. Moreover, adaptive minimum variance control is implemented to maintain the optimal performance in the case in which the system is time varying when operating the actively controlled optical trap in a complex environment.

© 2009 Optical Society of America

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2009 (2)

Y. Huang, J. Wan, M.- C. Cheng, Z. Zhang, S. M. Jhiang, and C.- H. Menq, “Three-axis rapid steering of optically propelled micro/nano particles,” Rev. Sci. Instrum. 80, 063107 (2009).
[CrossRef] [PubMed]

J. Wan, Y. Huang, S. M. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832-4841 (2009).
[CrossRef] [PubMed]

2008 (5)

A. Patete, K. Furuta, and M. Tomizuka, “Self-tuning control based on generalized minimum variance criterion for auto-regressive models,” Automatica 44, 1970-1975(2008).
[CrossRef]

A. E. Wallin, H. Ojala, E. Haggstrom, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[CrossRef]

A. E. Cohen and W. E. Moerner, “Controlling Brownian motion of single protein molecules and single fluorophores in aqueous buffer,” Opt. Express 16, 6941-6956 (2008).
[CrossRef] [PubMed]

M.- T. Wei, A. Zaorski, H. C. Yalcin, J. Wang, M. Hallow, S. N. Ghadiali, A. Chiou, and H. D. Ou-Yang, “A comparative study of living cell micromechanical properties by oscillatory optical tweezers,” Opt. Express 16, 8594-8603 (2008).
[CrossRef] [PubMed]

M. B. Rasmussen, L. B. Oddershede, and H. Siegumfeldt, “Optical tweezers cause physiological damage to Escherichia coli and Listeria bacteria,” Appl. Environ. Microbiol. 74, 2441-2446 (2008).
[CrossRef] [PubMed]

2007 (3)

2006 (1)

S. Ayano, Y. Wakamoto, S. Yamashita, and K. Yasuda, “Quantitative measurement of damage caused by 1064 nm wavelength optical trapping of Escherichia coli cells using on-chip single cell cultivation system,” Biochem. Biophys. Res. Commun. 350, 678-684 (2006).
[CrossRef] [PubMed]

2005 (3)

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

A. Ranaweera and B. Bamieh, “Modelling, identification, and control of a spherical particle trapped in an optical tweezer,” Int. J. Robust Nonlinear Control 15, 747-768 (2005).
[CrossRef]

2004 (3)

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594-612(2004).
[CrossRef]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

H. Huang, R. D. Kamm, and R. T. Lee, “Cell mechanics and mechanotransduction: pathways, probes, and physiology,” Am. J. Physiol., Cell Physiol. 287, C1-C11 (2004).
[CrossRef] [PubMed]

2003 (2)

C. Bustamante, Z. Bryant, and S. B. Smith, “Ten years of tension: single-molecule DNA mechanics,” Nature 421, 423-427(2003).
[CrossRef] [PubMed]

J. Zlatanova and S. H. Leuba, “Magnetic tweezers: a sensitive tool to study DNA and chromatin at the single-molecule level,” Biochem. Cell Biol. 81, 151-159 (2003).
[CrossRef] [PubMed]

2002 (3)

C. Gosse and V. Croquette, “Magnetic tweezers: micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

Q. Zhang, “Adaptive observer for multiple-input-multiple-output (MIMO) linear time-varying systems,” IEEE Trans. Autom. Control 47, 525-529 (2002).
[CrossRef]

2000 (1)

C. Bustamante, J. C. Macosko, and G. J. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell. Biol. 1, 130-136 (2000).
[CrossRef]

1999 (1)

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]

1998 (2)

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282, 902-907 (1998).
[CrossRef] [PubMed]

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Eur. Biophys. J. 27, 75-81(1998).
[CrossRef]

1996 (2)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529-541 (1996).
[CrossRef]

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813-1822 (1996).
[CrossRef] [PubMed]

1990 (1)

P. R. Kumar, “Convergence of adaptive control schemes using least-squares parameter estimates,” IEEE Trans. Autom. Control. 35, 416-424 (1990).
[CrossRef]

Alchenberger, D.

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Astrom, K. J.

K. J. Astrom, Introduction to Stochastic Control Theory (Academic, 1970).

Ayano, S.

S. Ayano, Y. Wakamoto, S. Yamashita, and K. Yasuda, “Quantitative measurement of damage caused by 1064 nm wavelength optical trapping of Escherichia coli cells using on-chip single cell cultivation system,” Biochem. Biophys. Res. Commun. 350, 678-684 (2006).
[CrossRef] [PubMed]

Bamieh, B.

A. Ranaweera and B. Bamieh, “Modelling, identification, and control of a spherical particle trapped in an optical tweezer,” Int. J. Robust Nonlinear Control 15, 747-768 (2005).
[CrossRef]

Berg-Sorensen, K.

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594-612(2004).
[CrossRef]

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

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, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Box, G.

G. Box, G. M. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, 1994).

Bryant, Z.

C. Bustamante, Z. Bryant, and S. B. Smith, “Ten years of tension: single-molecule DNA mechanics,” Nature 421, 423-427(2003).
[CrossRef] [PubMed]

Bustamante, C.

C. Bustamante, Z. Bryant, and S. B. Smith, “Ten years of tension: single-molecule DNA mechanics,” Nature 421, 423-427(2003).
[CrossRef] [PubMed]

C. Bustamante, J. C. Macosko, and G. J. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell. Biol. 1, 130-136 (2000).
[CrossRef]

Carter, A. R.

Cheng, M.- C.

Y. Huang, J. Wan, M.- C. Cheng, Z. Zhang, S. M. Jhiang, and C.- H. Menq, “Three-axis rapid steering of optically propelled micro/nano particles,” Rev. Sci. Instrum. 80, 063107 (2009).
[CrossRef] [PubMed]

Chiou, A.

Chu, S.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813-1822 (1996).
[CrossRef] [PubMed]

Clark, R. L.

Cohen, A. E.

A. E. Cohen and W. E. Moerner, “Controlling Brownian motion of single protein molecules and single fluorophores in aqueous buffer,” Opt. Express 16, 6941-6956 (2008).
[CrossRef] [PubMed]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

Cole, D. G.

Croquette, V.

C. Gosse and V. Croquette, “Magnetic tweezers: micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

Fabry, B.

P. Kollmannsberger and B. Fabry, “High-force magnetic tweezers with force feedback for biological applications,” Rev. Sci. Instrum. 78, 114301 (2007).
[CrossRef] [PubMed]

Finer, J. T.

R. M. Simmons, J. T. Finer, S. Chu, and J. A. Spudich, “Quantitative measurements of force and displacement using an optical trap,” Biophys. J. 70, 1813-1822 (1996).
[CrossRef] [PubMed]

Flyvbjerg, H.

K. Berg-Sorensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594-612(2004).
[CrossRef]

Franklin, G. F.

G. F. Franklin, J. D. Powell, and M. Workman, Digital Control of Dynamic Systems, 3rd ed. (Addison-Wesley, 1998).

Furuta, K.

A. Patete, K. Furuta, and M. Tomizuka, “Self-tuning control based on generalized minimum variance criterion for auto-regressive models,” Automatica 44, 1970-1975(2008).
[CrossRef]

Gelles, J.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Ghadiali, S. N.

Gittes, F.

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Eur. Biophys. J. 27, 75-81(1998).
[CrossRef]

Gosse, C.

C. Gosse and V. Croquette, “Magnetic tweezers: micromanipulation and force measurement at the molecular level,” Biophys. J. 82, 3314-3329 (2002).
[CrossRef] [PubMed]

Haggstrom, E.

A. E. Wallin, H. Ojala, E. Haggstrom, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[CrossRef]

Hallow, M.

Halsey, W.

Hamid, S.

A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).

Hansen, P. M.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529-541 (1996).
[CrossRef]

Harrit, N.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Huang, H.

H. Huang, R. D. Kamm, and R. T. Lee, “Cell mechanics and mechanotransduction: pathways, probes, and physiology,” Am. J. Physiol., Cell Physiol. 287, C1-C11 (2004).
[CrossRef] [PubMed]

Huang, Y.

Y. Huang, J. Wan, M.- C. Cheng, Z. Zhang, S. M. Jhiang, and C.- H. Menq, “Three-axis rapid steering of optically propelled micro/nano particles,” Rev. Sci. Instrum. 80, 063107 (2009).
[CrossRef] [PubMed]

J. Wan, Y. Huang, S. M. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832-4841 (2009).
[CrossRef] [PubMed]

Jenkins, G. M.

G. Box, G. M. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, 1994).

Jhiang, S. M.

J. Wan, Y. Huang, S. M. Jhiang, and C.-H. Menq, “Real-time in situ calibration of an optically trapped probing system,” Appl. Opt. 48, 4832-4841 (2009).
[CrossRef] [PubMed]

Y. Huang, J. Wan, M.- C. Cheng, Z. Zhang, S. M. Jhiang, and C.- H. Menq, “Three-axis rapid steering of optically propelled micro/nano particles,” Rev. Sci. Instrum. 80, 063107 (2009).
[CrossRef] [PubMed]

Kamm, R. D.

H. Huang, R. D. Kamm, and R. T. Lee, “Cell mechanics and mechanotransduction: pathways, probes, and physiology,” Am. J. Physiol., Cell Physiol. 287, C1-C11 (2004).
[CrossRef] [PubMed]

King, G. M.

Kollmannsberger, P.

P. Kollmannsberger and B. Fabry, “High-force magnetic tweezers with force feedback for biological applications,” Rev. Sci. Instrum. 78, 114301 (2007).
[CrossRef] [PubMed]

Kumar, P. R.

P. R. Kumar, “Convergence of adaptive control schemes using least-squares parameter estimates,” IEEE Trans. Autom. Control. 35, 416-424 (1990).
[CrossRef]

Landick, R.

M. D. Wang, M. J. Schnitzer, H. Yin, R. Landick, J. Gelles, and S. M. Block, “Force and velocity measured for single molecules of RNA polymerase,” Science 282, 902-907 (1998).
[CrossRef] [PubMed]

Lee, R. T.

H. Huang, R. D. Kamm, and R. T. Lee, “Cell mechanics and mechanotransduction: pathways, probes, and physiology,” Am. J. Physiol., Cell Physiol. 287, C1-C11 (2004).
[CrossRef] [PubMed]

Leuba, S. H.

J. Zlatanova and S. H. Leuba, “Magnetic tweezers: a sensitive tool to study DNA and chromatin at the single-molecule level,” Biochem. Cell Biol. 81, 151-159 (2003).
[CrossRef] [PubMed]

Macosko, J. C.

C. Bustamante, J. C. Macosko, and G. J. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell. Biol. 1, 130-136 (2000).
[CrossRef]

Mazo, R. M.

R. M. Mazo, Brownian Motion: Fluctuations, Dynamics, and Applications (Oxford University Press, 2002).

Mendel, J. M.

J. M. Mendel, Lessons in Etimation Theory for Signal Processing, Communications, and Control, 2nd ed. (Prentice-Hall, 1995).

Menq, C.- H.

Y. Huang, J. Wan, M.- C. Cheng, Z. Zhang, S. M. Jhiang, and C.- H. Menq, “Three-axis rapid steering of optically propelled micro/nano particles,” Rev. Sci. Instrum. 80, 063107 (2009).
[CrossRef] [PubMed]

Menq, C.-H.

Moerner, W. E.

A. E. Cohen and W. E. Moerner, “Controlling Brownian motion of single protein molecules and single fluorophores in aqueous buffer,” Opt. Express 16, 6941-6956 (2008).
[CrossRef] [PubMed]

A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. 86, 093109 (2005).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Oddershede, L.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Oddershede, L. B.

M. B. Rasmussen, L. B. Oddershede, and H. Siegumfeldt, “Optical tweezers cause physiological damage to Escherichia coli and Listeria bacteria,” Appl. Environ. Microbiol. 74, 2441-2446 (2008).
[CrossRef] [PubMed]

Ojala, H.

A. E. Wallin, H. Ojala, E. Haggstrom, and R. Tuma, “Stiffer optical tweezers through real-time feedback control,” Appl. Phys. Lett. 92, 224104 (2008).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. Hamid, Signals and Systems, 2nd ed. (Prentice-Hall, 1996).

Ou-Yang, H. D.

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Patete, A.

A. Patete, K. Furuta, and M. Tomizuka, “Self-tuning control based on generalized minimum variance criterion for auto-regressive models,” Automatica 44, 1970-1975(2008).
[CrossRef]

Perkins, T. T.

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Powell, J. D.

G. F. Franklin, J. D. Powell, and M. Workman, Digital Control of Dynamic Systems, 3rd ed. (Addison-Wesley, 1998).

Ranaweera, A.

A. Ranaweera and B. Bamieh, “Modelling, identification, and control of a spherical particle trapped in an optical tweezer,” Int. J. Robust Nonlinear Control 15, 747-768 (2005).
[CrossRef]

Rasmussen, M. B.

M. B. Rasmussen, L. B. Oddershede, and H. Siegumfeldt, “Optical tweezers cause physiological damage to Escherichia coli and Listeria bacteria,” Appl. Environ. Microbiol. 74, 2441-2446 (2008).
[CrossRef] [PubMed]

Reinsel, G.

G. Box, G. M. Jenkins, and G. Reinsel, Time Series Analysis: Forecasting and Control, 3rd ed. (Prentice-Hall, 1994).

Rohrbach, A.

A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

Schmidt, C. F.

F. Gittes and C. F. Schmidt, “Thermal noise limitations on micromechanical experiments,” Eur. Biophys. J. 27, 75-81(1998).
[CrossRef]

Schnitzer, M. J.

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Figures (6)

Fig. 1
Fig. 1

Brownian motion control of an optically trapped probe.

Fig. 2
Fig. 2

Optical trapping force exerted on a 1.87 μm probe along the x axis. The laser power is 80 mW at focus. The force curve is calculated using the probe’s averaged step response of 200 measurements.

Fig. 3
Fig. 3

Comparison of experimental PSDs and theoretical predictions of proportionally controlled Brownian motion of a 1.87 μm probe, wherein a number of control gains are employed. The experimental PSDs are calculated using the FFT algorithm and averaged in frequency domain from 100 sets of data to enhance the result. They are overlaid with their corresponding theoretical predictions (black solid lines) based on Eq. (12) and open-loop system calibration.

Fig. 4
Fig. 4

Comparison of analytical and experimental PSDs of the controlled Brownian motion of a 1.87 μm probe between the best P control and the MVC.

Fig. 5
Fig. 5

(a) Measured motion of a 1.87 μm probe and (b) standard deviation of the motion with and without adaptive control. Each data point in (b) is statistically determined from observing 0.3 s long data of the corresponding time in (a).

Fig. 6
Fig. 6

Estimated φ in adaptive MVC experiment. Inset, sample stage’s trajectory.

Equations (23)

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m x ¨ ( t ) 0 = γ x ˙ ( t ) viscous drag force + F L ( t ) random thermal force + F OT ( x u ) optical trapping force + F E ( t ) external force .
γ x ˙ + k OT x = k OT u + F L ( t ) .
x m ( t ) = x ( t τ m ) ,
u c ( t ) = u c [ x m ( t k 1 τ c ) ] , t [ t k 1 , t k ) | k = 1 , 2 , .
u ( t ) = u c ( t τ a ) .
γ x ˙ m ( t ) + k OT x m ( t ) = k OT u c [ x m ( t k 1 τ T ) ] + F L ( t ) , t [ t k 1 , t k ) | k = 1 , 2 , ,
γ x ˙ m d ( t ) + k OT x m d ( t ) = k OT u c [ x m ( t k 1 τ T ) ] , t [ t k 1 , t k ) , γ v ˙ ( t ) + k OT v ( t ) = F L ( t ) U ( t t k 1 ) , t [ t k 1 , t k ) .
x m [ k ] = φ x m [ k 1 ] + ( 1 φ ) u c { x m [ k 1 Δ ] } + v [ k ] ,
σ v 2 = k B T k OT ( 1 φ 2 ) 2 k B T γ τ s ,
R v [ i , j ] = k B T k OT ( 1 φ 2 ) · δ [ i , j ] = { k B T k OT ( 1 φ 2 ) , i = j 0 , i j ,
S v ( z ) = k B T k OT ( 1 φ 2 ) ,
H ( z ) = X m ( z ) V ( z ) = 1 1 φ z 1 + ( 1 φ ) C ( z ) z ( Δ + 1 ) .
S x c ( f ) = τ s · S x [ z | z = exp ( j 2 π f τ s ) ] , f ( 1 2 τ s , 1 2 τ s ] = k B T ( 1 φ 2 ) τ s k OT | 1 φ e j 2 π f τ s + ( 1 φ ) C ( e j 2 π f τ s ) e j ( Δ + 1 ) 2 π f τ s | 2 .
C M ( z ) = φ Δ + 1 / 1 φ 1 + φ z 1 + + φ Δ z Δ .
H M ( z ) = X m ( z ) V ( z ) = 1 + φ z 1 + + φ Δ z Δ .
σ x 2 = E ( { v [ k ] + φ v [ k 1 ] + + φ Δ v [ k Δ ] } 2 ) = k B T k OT [ 1 φ 2 ( Δ + 1 ) ] 2 k B T γ ( τ T + τ s ) .
S M ( f ) = k B T ( 1 φ 2 ) τ s k OT | 1 + φ e j 2 π f τ s + + φ Δ e j 2 π Δ f τ s | 2 .
σ F OT 2 = E ( k OT 2 { v [ k ] + φ v [ k 1 ] + + ( φ Δ + φ Δ + 1 1 φ ) v [ k Δ ] } 2 ) = k B T k OT [ 1 φ 2 ( Δ + 1 ) + 2 φ 2 Δ + 1 ( 1 + φ ) + φ 2 ( Δ + 1 ) ( 1 + φ ) 1 φ ] 2 k B T γ τ s .
A ( z 1 ) y [ n + k ] = B ( z 1 ) u [ n ] + D ( z 1 ) e [ n + k ] ,
D ( z 1 ) = A ( z 1 ) F ( z 1 ) + z k G ( z 1 ) ,
y [ n + k ] = F ( z 1 ) e [ n + k ] + G ( z 1 ) D ( z 1 ) y [ n ] + B ( z 1 ) F ( z 1 ) D ( z 1 ) u [ n ] .
E { y [ n + k ] } 2 = E { F ( z 1 ) e [ n + k ] } 2 Controller-independent + E { G ( z 1 ) D ( z 1 ) y [ n ] + B ( z 1 ) F ( z 1 ) D ( z 1 ) u [ n ] } 2 Controller-dependent σ e 2 j = 0 k 1 f j 2 ,
C M ( z ) = U ( z ) Y ( z ) = G ( z 1 ) B ( z 1 ) F ( z 1 ) .

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