Abstract

To correct wavefront aberrations, commonly employing proportional-integral control in adaptive optics (AO) systems, the control process depends strictly on the response matrix of the deformable mirror. The alignment error between the Hartmann–Shack wavefront sensor and the deformable mirror is caused by various factors in AO systems. In the conventional control method, the response matrix can be recalibrated to reduce the impact of alignment error, but the impact cannot be eliminated. This paper proposes a control method based on a deep learning control model (DLCM) to compensate for wavefront aberrations, eliminating the dependence on the deformable mirror response matrix. Based on the wavefront slope data, the cost functions of the model network and the actor network are defined, and the gradient optimization algorithm improves the efficiency of the network training. The model network guarantees the stability and convergence speed, while the actor network improves the control accuracy, realizing an online identification and self-adaptive control of the system. A parameter-sharing mechanism is adopted between the model network and the actor network to control the system gain. Simulation results show that the DLCM has good adaptability and stability. Through self-learning, it improves the convergence accuracy and iterations, as well as the adjustment tolerance of the system.

© 2019 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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2015 (4)

J. Tesch, T. Truong, R. Burruss, and S. Gibson, “On-sky demonstration of optimal control for adaptive optics at Palomar Observatory,” Opt. Lett. 40, 1575–1578 (2015).
[Crossref]

H. Xiaochuan, P. Jiaqi, and Z. Bin, “Thermal distortion of deformable mirror and its influence on beam quality,” Chin. J. Lasers 42, 45–53 (2015).

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

2014 (2)

Y. M. Guo, X. Y. Ma, and C. H. Rao, “Optimal closed-loop bandwidth of tip-tilt correction loop in adaptive optics system,” Acta Phys. Sinca 63, 069502 (2014).
[Crossref]

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

2013 (3)

Z. Y. Zheng, C. W. Li, B. M. Li, and S. J. Zhang, “Analysis and demonstration of PID algorithm based on arranging the transient process for adaptive optics,” Chin. Opt. Lett. 11, 110101 (2013).
[Crossref]

Z. J. Yan, X. Y. Li, and C. H. Rao, “Multi channel adaptive control algorithm for closed-loop adaptive optics system,” Acta Opt. Sin. 33, 0301002 (2013).
[Crossref]

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

2011 (1)

J. Duchi, E. Hazan, and Y. Singer, “Adaptive subgradient methods for online learning and stochastic optimization,” J. Mach. Learn. Res. 12, 2121–2159 (2011).

2010 (1)

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

2009 (2)

2007 (2)

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

2006 (1)

2003 (1)

H. Jing, J. Wenhan, and L. Ning, “The misalignment errors of Hartmann-Shack wavefront sensors and deformable mirror in the two kinds of adaptive optics systems,” J. Opt. 23, 750–755 (2003).

1988 (2)

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

W. Jang, S. Huang, and B. Xu, “Hill-climbing adaptive optics wavefront correction system,” Chin. Phys. Lasers 15, 27–31 (1988).

1983 (1)

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k 2),” Sov. Math. Dokl. 27, 372–376(1983).

1977 (1)

Abbeel, P.

A. Punjani and P. Abbeel, “Deep learning helicopter dynamics models,” in IEEE International Conference on Robotics and Automation (ICRA) (IEEE, 2015), pp. 3223–3230.

Adler, J.

Anderson, C. W.

C. W. Anderson, M. Lee, and D. L. Elliott, “Faster reinforcement learning after pretraining deep networks to predict state dynamics,” in International Joint Conference on Neural Networks (IJCNN) (IEEE, 2015), pp. 1–7.

Ao, M.

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

Ba, H.

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Beresnev, L.

Bin, Z.

H. Xiaochuan, P. Jiaqi, and Z. Bin, “Thermal distortion of deformable mirror and its influence on beam quality,” Chin. J. Lasers 42, 45–53 (2015).

Burruss, R.

Cao, F.

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

Carhart, G.

Chen, L. Z.

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

Cheon, K.

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

Dong, L.

L. Dong, P. Yang, and B. Xu, “Adaptive aberration correction based on ant colony algorithm for solid-state lasers: numerical simulations,” Appl. Phys. B 96, 527–533 (2009).
[Crossref]

Duchi, J.

J. Duchi, E. Hazan, and Y. Singer, “Adaptive subgradient methods for online learning and stochastic optimization,” J. Mach. Learn. Res. 12, 2121–2159 (2011).

Elliott, D. L.

C. W. Anderson, M. Lee, and D. L. Elliott, “Faster reinforcement learning after pretraining deep networks to predict state dynamics,” in International Joint Conference on Neural Networks (IJCNN) (IEEE, 2015), pp. 1–7.

Gibson, S.

Gu, N. T.

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

Gudimetla, V. S. R.

Guo, Y. M.

Y. M. Guo, X. Y. Ma, and C. H. Rao, “Optimal closed-loop bandwidth of tip-tilt correction loop in adaptive optics system,” Acta Phys. Sinca 63, 069502 (2014).
[Crossref]

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Hamadache, M.

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

Hazan, E.

J. Duchi, E. Hazan, and Y. Singer, “Adaptive subgradient methods for online learning and stochastic optimization,” J. Mach. Learn. Res. 12, 2121–2159 (2011).

Hinton, G.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Hu, S.

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

Huang, L. H.

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

Huang, S.

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

W. Jang, S. Huang, and B. Xu, “Hill-climbing adaptive optics wavefront correction system,” Chin. Phys. Lasers 15, 27–31 (1988).

Jang, W.

W. Jang, S. Huang, and B. Xu, “Hill-climbing adaptive optics wavefront correction system,” Chin. Phys. Lasers 15, 27–31 (1988).

Jiang, W.

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

Jiaqi, P.

H. Xiaochuan, P. Jiaqi, and Z. Bin, “Thermal distortion of deformable mirror and its influence on beam quality,” Chin. J. Lasers 42, 45–53 (2015).

Jing, H.

H. Jing, J. Wenhan, and L. Ning, “The misalignment errors of Hartmann-Shack wavefront sensors and deformable mirror in the two kinds of adaptive optics systems,” J. Opt. 23, 750–755 (2003).

Kim, J.

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

Knepper, R.

I. Lenz, R. Knepper, and A. Saxena, “Deep MPC: learning deep latent features for model predictive control,” in Robotics: Science and Systems (RSS), Rome, Italy (2015).

LeCun, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Lee, D.

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

Lee, M.

C. W. Anderson, M. Lee, and D. L. Elliott, “Faster reinforcement learning after pretraining deep networks to predict state dynamics,” in International Joint Conference on Neural Networks (IJCNN) (IEEE, 2015), pp. 1–7.

Lenz, I.

I. Lenz, R. Knepper, and A. Saxena, “Deep MPC: learning deep latent features for model predictive control,” in Robotics: Science and Systems (RSS), Rome, Italy (2015).

Levine, S.

S. Levine, “Exploring deep and recurrent architectures for optimal control,” arXiv: 1311.1761 (2013).

Li, B. M.

Li, C. W.

Li, X. Y.

Z. J. Yan, X. Y. Li, and C. H. Rao, “Multi channel adaptive control algorithm for closed-loop adaptive optics system,” Acta Opt. Sin. 33, 0301002 (2013).
[Crossref]

Li, Y.

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

Ling, N.

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

Lipson, S. G.

Liu, Y.

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

Long, C.

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

Ma, X. Y.

Y. M. Guo, X. Y. Ma, and C. H. Rao, “Optimal closed-loop bandwidth of tip-tilt correction loop in adaptive optics system,” Acta Phys. Sinca 63, 069502 (2014).
[Crossref]

Nesterov, Y.

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k 2),” Sov. Math. Dokl. 27, 372–376(1983).

Ning, L.

H. Jing, J. Wenhan, and L. Ning, “The misalignment errors of Hartmann-Shack wavefront sensors and deformable mirror in the two kinds of adaptive optics systems,” J. Opt. 23, 750–755 (2003).

O’Meara, T. R.

Punjani, A.

A. Punjani and P. Abbeel, “Deep learning helicopter dynamics models,” in IEEE International Conference on Robotics and Automation (ICRA) (IEEE, 2015), pp. 3223–3230.

Rao, C. H.

Y. M. Guo, X. Y. Ma, and C. H. Rao, “Optimal closed-loop bandwidth of tip-tilt correction loop in adaptive optics system,” Acta Phys. Sinca 63, 069502 (2014).
[Crossref]

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Z. J. Yan, X. Y. Li, and C. H. Rao, “Multi channel adaptive control algorithm for closed-loop adaptive optics system,” Acta Opt. Sin. 33, 0301002 (2013).
[Crossref]

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

Ribak, E. N.

Riker, J.

Roberts, L. C.

Saxena, A.

I. Lenz, R. Knepper, and A. Saxena, “Deep MPC: learning deep latent features for model predictive control,” in Robotics: Science and Systems (RSS), Rome, Italy (2015).

Singer, Y.

J. Duchi, E. Hazan, and Y. Singer, “Adaptive subgradient methods for online learning and stochastic optimization,” J. Mach. Learn. Res. 12, 2121–2159 (2011).

Tesch, J.

Truong, T.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

Voronstov, M.

Wei, K.

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Wenhan, J.

H. Jing, J. Wenhan, and L. Ning, “The misalignment errors of Hartmann-Shack wavefront sensors and deformable mirror in the two kinds of adaptive optics systems,” J. Opt. 23, 750–755 (2003).

Weyrauch, T.

Wu, X.

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

Xiaochuan, H.

H. Xiaochuan, P. Jiaqi, and Z. Bin, “Thermal distortion of deformable mirror and its influence on beam quality,” Chin. J. Lasers 42, 45–53 (2015).

Xu, B.

L. Dong, P. Yang, and B. Xu, “Adaptive aberration correction based on ant colony algorithm for solid-state lasers: numerical simulations,” Appl. Phys. B 96, 527–533 (2009).
[Crossref]

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

W. Jang, S. Huang, and B. Xu, “Hill-climbing adaptive optics wavefront correction system,” Chin. Phys. Lasers 15, 27–31 (1988).

Yan, Z. J.

Z. J. Yan, X. Y. Li, and C. H. Rao, “Multi channel adaptive control algorithm for closed-loop adaptive optics system,” Acta Opt. Sin. 33, 0301002 (2013).
[Crossref]

Yang, P.

L. Dong, P. Yang, and B. Xu, “Adaptive aberration correction based on ant colony algorithm for solid-state lasers: numerical simulations,” Appl. Phys. B 96, 527–533 (2009).
[Crossref]

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

P. Yang, M. Ao, Y. Liu, B. Xu, and W. Jiang, “Intracavity transverse modes control by an genetic algorithm based on Zernike mode coefficients,” Opt. Express 15, 17051–17062 (2007).
[Crossref]

Yang, Z. P.

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

Zhang, A.

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Zhang, S. J.

Zheng, Z. Y.

Zommer, S.

Acta Opt. Sin. (1)

Z. J. Yan, X. Y. Li, and C. H. Rao, “Multi channel adaptive control algorithm for closed-loop adaptive optics system,” Acta Opt. Sin. 33, 0301002 (2013).
[Crossref]

Acta Phys. Sinca (2)

Y. M. Guo, X. Y. Ma, and C. H. Rao, “Optimal closed-loop bandwidth of tip-tilt correction loop in adaptive optics system,” Acta Phys. Sinca 63, 069502 (2014).
[Crossref]

Y. M. Guo, C. H. Rao, H. Ba, A. Zhang, and K. Wei, “Direct computation of the interaction matrix of adaptive optical system,” Acta Phys. Sinca 63, 149501 (2014).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

L. Dong, P. Yang, and B. Xu, “Adaptive aberration correction based on ant colony algorithm for solid-state lasers: numerical simulations,” Appl. Phys. B 96, 527–533 (2009).
[Crossref]

Chin. J. Lasers (1)

H. Xiaochuan, P. Jiaqi, and Z. Bin, “Thermal distortion of deformable mirror and its influence on beam quality,” Chin. J. Lasers 42, 45–53 (2015).

Chin. Opt. Lett. (1)

Chin. Phys. Lasers (1)

W. Jang, S. Huang, and B. Xu, “Hill-climbing adaptive optics wavefront correction system,” Chin. Phys. Lasers 15, 27–31 (1988).

J. Automation Control Eng. (1)

K. Cheon, J. Kim, M. Hamadache, and D. Lee, “On replacing PID controller with deep learning controller for DC motor system,” J. Automation Control Eng. 3, 452–456 (2015).

J. Mach. Learn. Res. (1)

J. Duchi, E. Hazan, and Y. Singer, “Adaptive subgradient methods for online learning and stochastic optimization,” J. Mach. Learn. Res. 12, 2121–2159 (2011).

J. Opt. (2)

H. Jing, J. Wenhan, and L. Ning, “The misalignment errors of Hartmann-Shack wavefront sensors and deformable mirror in the two kinds of adaptive optics systems,” J. Opt. 23, 750–755 (2003).

N. T. Gu, Z. P. Yang, L. H. Huang, and C. H. Rao, “Measurement method of alignment error between Hartmann-Shack sensor and deformable mirror in adaptive optics system,” J. Opt. 12, 095504 (2010).
[Crossref]

J. Opt. Soc. Am. (1)

Mod. Appl. Phys. (1)

F. Cao, L. Z. Chen, C. Long, and Y. Li, “Analysis of the influence of spatial mismatch between deformable mirror and wavefront sensor on fitting accuracy,” Mod. Appl. Phys. 4, 5–8 (2013).

Nature (1)

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).
[Crossref]

Opt. Commun. (1)

P. Yang, Y. Liu, M. Ao, S. Hu, and B. Xu, “A wavefront sensor-less adaptive optical system for a solid-state laser,” Opt. Commun. 278, 377–381 (2007).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

W. Jiang, S. Huang, N. Ling, and X. Wu, “Hill-climb–ing wavefront correction system for large laser engineering,” Proc. SPIE 965, 266–272 (1988).
[Crossref]

Sov. Math. Dokl. (1)

Y. Nesterov, “A method of solving a convex programming problem with convergence rate O(1/k 2),” Sov. Math. Dokl. 27, 372–376(1983).

Other (6)

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

S. Levine, “Exploring deep and recurrent architectures for optimal control,” arXiv: 1311.1761 (2013).

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https://github.com/rconan/OOMAO .

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Figures (12)

Fig. 1.
Fig. 1. Controller for an AO system. (a) PI; (b) DLCM.
Fig. 2.
Fig. 2. Spatial relationship of deformable mirror and HS sensor.
Fig. 3.
Fig. 3. Network structure of the DNN. (a) Model network; (b) actor network.
Fig. 4.
Fig. 4. Correction capability of PI and DLCM under various rotations α. (a)–(e) System convergence process; (f) SR value after system convergence.
Fig. 5.
Fig. 5. Correction capability of PI and DLCM under various translations Δx. (a)–(e) System convergence process; (f) SR value after system convergence. When Δx=0, the SR of PI is 0.971. The SR of the DLCM is 0.985; under other Δx conditions, the SR value of PI is obviously less than that of the DLCM.
Fig. 6.
Fig. 6. Correction capability of PI and DLCM under various translations Δy. (a)–(e) System convergence process; (f) SR value after system convergence. When Δy=0, the SR of PI is 0.971. The SR of the DLCM is 0.985.
Fig. 7.
Fig. 7. Mean of the SR after 100 aberrations are corrected. (a)–(c) SR mean after system convergence.
Fig. 8.
Fig. 8. Correction capability of PI and DLCM under various rotations α. (a)–(e) System convergence process; (f) SR value after system convergence.
Fig. 9.
Fig. 9. Correction capability of PI and DLCM under various translations Δy. (a)–(e) System convergence process; (f) SR value after system convergence.
Fig. 10.
Fig. 10. Correction capability of PI and DLCM under various translations Δx. (a)–(e) System convergence process; (f) SR value after system convergence.
Fig. 11.
Fig. 11. Mean of the SR after 100 aberrations are corrected.
Fig. 12.
Fig. 12. Comparison result of convergence speed of AGD and GD for deep network.

Tables (3)

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Algorithm 1. Computing AGD update at time t

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Table 1. Comparison Result of the Control Model (Δx=0, Δy=0, α=0)a

Equations (25)

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

Sn=BV,
V=B+Sn.
Ve=fn((f2(f1(Sθm1)θm2))θmn).
J(θm)=12j(VjVej)2.
S=BV.
ΔS+S=B(ΔV+V).
S=BΔV.
{Xc=XiIiIi=λf2πSAϕ(x,y)xdxdy=λf2πGXYc=YiIiIi=λf2πSAϕ(x,y)xdxdy=λf2πGY,
J0(GX,GY)=i=1N(GXi2+GYi2)N,
θaJ0=VJ0θaV(θa).
θa=θaαθaJ0,
θm=θmβ1θmJ(θm).
θmJ0=VJ0θmVe(θm).
θm=θmβ2θmJ0.
vt=γvt1+αθJ(θγvt1),
θ=θvt,
E[g2]t=γE[g2]t1+(1γ)gt2,
Δθt=αE[g2]t+gt.
Δθt=αRMS[g]tgt.
E[Δθ2]t=γE[Δθ2]t1+(1γ)Δθt2.
RMS[Δθ]t=E[Δθ2]t.
Δθt=RMS[Δθ]t1RMS[g]tgt,
θ=θ+Δθt.
θaτθm+(1τ)θa.
SR|max[I(i)]||i=1N[I(i)]2|,