Abstract

Frequency scanning interferometry (FSI) with a single external cavity diode laser (ECDL) and time-invariant Kalman filtering is an effective technique for measuring the distance of a dynamic target. However, due to the hysteresis of the piezoelectric ceramic transducer (PZT) actuator in the ECDL, the optical frequency sweeps of the ECDL exhibit different behaviors, depending on whether the frequency is increasing or decreasing. Consequently, the model parameters of Kalman filter appear time varying in each iteration, which produces state estimation errors with time-invariant filtering. To address this, in this paper, a time-varying Kalman filter is proposed to model the instantaneous movement of a target relative to the different optical frequency tuning durations of the ECDL. The combination of the FSI method with the time-varying Kalman filter was theoretically analyzed, and the simulation and experimental results show the proposed method greatly improves the performance of dynamic FSI measurements.

© 2017 Optical Society of America

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References

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  1. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
    [PubMed]
  2. B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).
  3. G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).
  4. A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).
  5. P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).
  6. H. J. Yang, T. Chen, and K. Riles, “Frequency scanned interferometry for ILC tracker alignment,” https://arxiv.org/abs/1109.2582 (2011).
  7. H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).
  8. P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).
  9. J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).
  10. A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).
  11. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014).
    [PubMed]
  12. M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
    [PubMed]
  13. R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).
  14. B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).
  15. Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).
  16. Z. Liu, Z. Liu, Z. Deng, and L. Tao, “Interference signal frequency tracking for extracting phase in frequency scanning interferometry using an extended Kalman filter,” Appl. Opt. 55(11), 2985–2992 (2016).
    [PubMed]
  17. A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
  18. B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).
  19. Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

2017 (2)

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

2016 (1)

2015 (2)

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

2014 (3)

2007 (1)

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

2005 (2)

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

2004 (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

2003 (1)

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

1999 (1)

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

1981 (1)

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

1960 (1)

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

Anandarajah, P. M.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Arulampalam, S.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Bhattacharya, N.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Braat, J. J. M.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Bu, M.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Cabral, A.

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

Campbell, M. A.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Coe, P. A.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Copner, N.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Dale, J.

Deng, Z.

Deng, Z. W.

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Fox-Murphy, A. F.

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Gibson, S. M.

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Gordon, N.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Hong, J.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Howell, D. F.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Hughes, B.

Hughes, E. B.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Jain, A.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Kalman, R. E.

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

Krishnamurthy, P. K.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Lancaster, A. J.

Landais, P.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

Lewis, A. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[PubMed]

Liu, Z.

Liu, Z. G.

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Martinez, J. J.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

Meng, X. S.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Mitra, A.

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Mo, Y. L.

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

Moore, J. B.

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Nickerson, R. B.

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

Nyberg, S.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Rebordao, J.

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

Reichold, A. J. H.

Riles, K.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Ristic, B.

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Shi, G.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Sinopoli, B.

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

Swinkels, B. L.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Tao, L.

Warden, M. S.

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014).
[PubMed]

Weidberg, A. R.

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Wielders, A. A.

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

Xu, X. H.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Yang, H. J.

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

Zhang, F. M.

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Zhang, M.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Zhang, W.

Zhou, Y.

Zhu, Y.

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

Appl. Opt. (1)

Bol. soc. mat. mexicana (1)

R. E. Kalman, “Contribution to the theory of optimal control,” Bol. soc. mat. mexicana 5(63), 102–119 (1960).

IEEE Photonics J. (1)

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).

IEEE Photonics Technol. Lett. (1)

J. J. Martinez, M. A. Campbell, M. S. Warden, E. B. Hughes, N. Copner, and A. J. Lewis, “Dual-sweep frequency scanning interferometry using four wave mixing,” IEEE Photonics Technol. Lett. 27(7), 733–736 (2015).

IEEE Trans. Aerosp. Electron. Syst. (1)

B. Ristic, S. Arulampalam, and N. Gordon, “Beyond the Kalman filter: Particle filters for tracking applications,” IEEE Trans. Aerosp. Electron. Syst. 19(7), 37–38 (2003).

Meas. Sci. Technol. (1)

P. A. Coe, D. F. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004).

Nucl. Instrum. Methods Phys. Res. (2)

H. J. Yang, S. Nyberg, and K. Riles, “High-precision absolute distance measurement using dual-laser frequency scanned interferometry under realistic conditions,” Nucl. Instrum. Methods Phys. Res. 575(3), 395–401 (2007).

A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R. Weidberg, “Frequency scanned interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry,” Nucl. Instrum. Methods Phys. Res. 383(1), 229–237 (1999).

Opt. Eng. (1)

G. Shi, F. M. Zhang, X. H. Xu, and X. S. Meng, “High-resolution frequency-modulated continuous-wave laser ranging for precision distance metrology applications,” Opt. Eng. 53(12), 122402 (2014).

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

Z. W. Deng and Z. G. Liu, “Precision improvement in frequency-scanning interferometry based on suppressing nonlinear optical frequency sweeping,” Opt. Rev. 22(5), 724–730 (2015).

Proc. SPIE (2)

B. L. Swinkels, N. Bhattacharya, A. A. Wielders, and J. J. M. Braat, “Absolute distance metrology for space interferometers,” Proc. SPIE 554, 559–561 (2005).

A. Cabral and J. Rebordao, “Absolute distance metrology with frequency sweeping interferometry,” Proc. SPIE 5879, 195–204 (2005).

Rev. Sci. Instrum. (1)

M. Zhang, Z. Liu, Y. Zhu, M. Bu, and J. Hong, “Integral force feedback control with input shaping: Application to piezo-based scanning systems in ECDLs,” Rev. Sci. Instrum. 88(7), 075006 (2017).
[PubMed]

SIAM J. Control Optim. (1)

B. D. O. Anderson and J. B. Moore, “Detectability and stabilizability of time-varying discrete-time linear systems,” SIAM J. Control Optim. 19(1), 20–32 (1981).

Other (3)

Y. L. Mo and B. Sinopoli, “Kalman filtering with intermittent observations: critical value for second order system,” 2003 Proc. IEEE Conf. Decision Control, 1453–1464 (2004).

P. A. Coe, A. Mitra, S. M. Gibson, D. F. Howell, and R. B. Nickerson, ” Frequency scanning interferometry - a versatile, high precision, multiple distance measurement technique,” Seventh International Workshop on Accelerator Alignment (2002).

H. J. Yang, T. Chen, and K. Riles, “Frequency scanned interferometry for ILC tracker alignment,” https://arxiv.org/abs/1109.2582 (2011).

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Figures (12)

Fig. 1
Fig. 1 Schematic illustration of the principles of FSI.
Fig. 2
Fig. 2 The FSI system. In the laboratory environment, the setup was constructed on a floating optical table to reduce interference caused by external noise.
Fig. 3
Fig. 3 The state estimation for dynamic FSI.
Fig. 4
Fig. 4 Statistical distribution of T and t. Part a (blue) represents the sampling period T of the Kalman filter, and Part b (red) represents the FSI period t. These plots are based on 6,000 FSI measurement samples.
Fig. 5
Fig. 5 Length errors between the time-invariant and time-varying Kalman filters.
Fig. 6
Fig. 6 Velocity errors between the time-invariant and time-varying Kalman filters.
Fig. 7
Fig. 7 The simulation results of vibration tests.
Fig. 8
Fig. 8 The RMSE of the vibration in variable amplitude.
Fig. 9
Fig. 9 The simulations of vibration test in variable frequencies.
Fig. 10
Fig. 10 Conditions of vibration measurement.
Fig. 11
Fig. 11 Results of the 1D dynamic velocity measurement. a, the measured absolute length of the moving target. b, Estimated velocity using the FSI-Kalman filter. c, Estimated acceleration of the target. d, Deviation between the estimated velocity and the velocity of the moving stage.
Fig. 12
Fig. 12 The results of the vibration measurement. a, measured length results of standard FSI (black) and the FSI-Kalman filter (red); b, the measured velocity (deep blue). c, the estimated acceleration of FSI-Kalman filter; d, the measurements results of vibration amplitude by using eddy3010 and FSI-Kalman filter.

Tables (1)

Tables Icon

Table 1 Simulation Parameters

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

I= I 1 + I 2 +2 I 1 I 2 cos( ϕ 1 ϕ 2 ),
L= 1 2n Δϕ 2π c Δv ,
Δ ϕ ' = 4nπ c [ ( L+Δε ) v 2 L v 1 ].
L M = cΔ ϕ ' 4nπΔv = L true + v 2 v 2 ν 1 Δε= L true +ΩΔε ,
L k+1 = L k +sT+ 1 2 a T 2 ,
L kM = L k + Ω k s k t k + 1 2 Ω k a k t k 2 ,
x k+1 = Φ k x k + w k ,
y k = H k x k + υ k ,
Q k = σ w 2 [ T k 6 36 T k 5 12 T k 4 6 T k 5 12 T k 4 4 T k 3 2 T k 4 6 T k 3 2 T k 2 ], R k =[ σ υ 2 ],
T k ={ N( T up , σ Tup ) N( T down , σ Tdown ) , t k ={ N( t up , σ tup ) N( t down , σ tdown ) ,
Φ k =[ 1 T k T k 2 /2 0 1 T k 0 0 1 ]={ Φ up Φ down , H k =[ 1 Ω k t k Ω k t k 2 2 ]={ H up H down .
Ω k ={ Ω up = v 2 v 2 v 1 Ω down = v 1 v 1 v 2 .
x ^ k|k = x ^ k|k1 + K k ( y k H k x ^ k|k1 ),
P k|k = P k| k1 K k H k P k| k1 ,
x ^ k+1|k = Φ k+1 x ^ k|k , P k+1|k = Φ k+1 P k|k Φ k+1 T + Q k ,
K k = P k| k1 H k1 T ( H k P k| k1 H k T +R ) 1 , x ^ 0| 1 = x 0 , P 0| 1 = P 0 ,
P 0 , s.t. sup E P k =0p p c
sup k Φ k 2 1.01799, sup k H k 2 67.45363
0< ξ 1 I i=kN+1 k Φ k,i Q i,i1 Q i,i1 T Φ k,i T 0< ξ 2 I j=kN+1 k Φ j,k T H j T H j Φ j,k .
i=kN+1 k Φ k,i Q i,i1 Q i,i1 T Φ k,i T = σ w 2 [ f 11 ( N,T ) f 12 ( N,T ) f 13 ( N,T ) f 21 ( N,T ) f 22 ( N,T ) f 23 ( N,T ) f 31 ( N,T ) f 32 ( N,T ) f 33 ( N,T ) ]3.85× 10 11 σ w 2 I>0 .
j=kN+1 k Φ j,k T H j T H j Φ j,k = [ N f 12 ( Ω,N,T,t ) f 13 ( Ω,N,T,t ) f 21 ( Ω,N,T,t ) f 22 ( Ω,N,T,t ) f 23 ( Ω,N,T,t ) f 31 ( Ω,N,T,t ) f 32 ( Ω,N,T,t ) f 33 ( Ω,N,T,t ) ]1.15× 10 9 I>0
t k ={ t up ,k( 2n1 ) t down ,k( 2n ) , T k ={ T up ,k( 2n1 ) T down ,k( 2n ) ,n N + .
Ω k ={ Ω up = v 2 v 2 v 1 ,k( 2n1 ) Ω down = v 1 v 1 v 2 ,k( 2n ) ,n N + .

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