Abstract

This paper presents a linear-quadratic-Gaussian (LQG) design based on the equivalent discrete-time model of an adaptive optics (AO) system. The design model incorporates deformable mirror dynamics, an asynchronous wavefront sensor and zero-order hold operation, and a continuous-time model of the incident wavefront. Using the structure of the discrete-time model, the dimensions of the Riccati equations to be solved are reduced. The LQG controller is shown to improve AO system performance under several conditions.

© 2008 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).
  2. M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).
  3. D. P. Looze, “Discrete-time model of an adaptive optics system,” J. Opt. Soc. Am. A 24, 2850-2863 (2007).
    [CrossRef]
  4. E. Gendron and P. Léna, “Astronomical adaptive optics. I: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).
  5. B. L. Ellerbroek, C. van Loan, N. P. Pitsianis, and R. J. Plemmons, “Optimizing closed-loop adaptive-optics performance with use of multiple control bandwidths,” J. Opt. Soc. Am. A 11, 2871-1886 (1994).
    [CrossRef]
  6. C. Dessenne, P.-Y. Madec, and G. Rousset, “Optimization of a predictive controller for closed-loop adaptive optics,” Appl. Opt. 37, 4623-4633 (1998).
    [CrossRef]
  7. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier-transform wavefront control,” J. Opt. Soc. Am. A 22, 1515-1526 (2005).
    [CrossRef]
  8. R. N. Paschall and D. J. Anderson, “Linear quadratic Gaussian control of a deformable mirror adaptive optics system with time-delayed measurements,” Appl. Opt. 32, 6347-6358 (1993).
    [CrossRef] [PubMed]
  9. D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
    [CrossRef]
  10. B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
    [CrossRef]
  11. D. M. Wiberg, C. E. Max, and D. T. Gavel, “A spatial non-dynamic LQG controller. Part II, theory,” in Procedings of 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.
  12. D. P. Looze, “Minimum variance control structure for adaptive optics systems,” J. Opt. Soc. Am. A 23, 603-612 (2006).
    [CrossRef]
  13. C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
    [CrossRef]
  14. C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
    [CrossRef]
  15. K. Hinnen, M. Verhaegen, and N. Doelman, “Exploiting the spatiotemporal correlation in adaptive optics using data-driven H2--optimal control,” J. Opt. Soc. Am. A 24, 1714-1725 (2007).
    [CrossRef]
  16. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
    [CrossRef] [PubMed]
  17. H. S. Witsenhausen, “Separation of estimation and control for discrete time systems,” Proc. IEEE 59, 1557-1566 (1971).
    [CrossRef]
  18. E. I. Jury, Sampled-Data Control Systems (Wiley, 1958).
  19. K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).
  20. K. J. Åström and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1997).
  21. P. Dorato and A. H. Levis, “Optimal linear regulators: the discrete-time case,” IEEE Trans. Autom. Control AC-16, 613-620 (1971).
  22. T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).
  23. R. Köhler, “CHEOPS performance simulations,” MPIA Doc. no. CHEOPS-TRE-MPI-00033 (Max Planck Institüt fur Astronomie, 2004).
  24. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

2007

2006

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

D. P. Looze, “Minimum variance control structure for adaptive optics systems,” J. Opt. Soc. Am. A 23, 603-612 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

2005

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier-transform wavefront control,” J. Opt. Soc. Am. A 22, 1515-1526 (2005).
[CrossRef]

2004

2003

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

1998

1994

1993

1971

H. S. Witsenhausen, “Separation of estimation and control for discrete time systems,” Proc. IEEE 59, 1557-1566 (1971).
[CrossRef]

P. Dorato and A. H. Levis, “Optimal linear regulators: the discrete-time case,” IEEE Trans. Autom. Control AC-16, 613-620 (1971).

Anderson, D. J.

Åström, K. J.

K. J. Åström and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1997).

Beker, O.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

Chen, T.

T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

Conan, J.-M.

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

de Lesegno, P. V.

Dessenne, C.

Doelman, N.

Dorato, P.

P. Dorato and A. H. Levis, “Optimal linear regulators: the discrete-time case,” IEEE Trans. Autom. Control AC-16, 613-620 (1971).

Ellerbroek, B. L.

Francis, B. A.

T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

Fusco, T.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Gavel, D. T.

D. M. Wiberg, C. E. Max, and D. T. Gavel, “A spatial non-dynamic LQG controller. Part II, theory,” in Procedings of 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

Gendron, E.

E. Gendron and P. Léna, “Astronomical adaptive optics. I: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

Hinnen, K.

Hippler, S.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

Jury, E. I.

E. I. Jury, Sampled-Data Control Systems (Wiley, 1958).

Kasper, M.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

Köhler, R.

R. Köhler, “CHEOPS performance simulations,” MPIA Doc. no. CHEOPS-TRE-MPI-00033 (Max Planck Institüt fur Astronomie, 2004).

Kulcsár, C.

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Le Roux, B.

Léna, P.

E. Gendron and P. Léna, “Astronomical adaptive optics. I: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Levis, A. H.

P. Dorato and A. H. Levis, “Optimal linear regulators: the discrete-time case,” IEEE Trans. Autom. Control AC-16, 613-620 (1971).

Looze, D. P.

Madec, P.-Y.

Max, C. E.

D. M. Wiberg, C. E. Max, and D. T. Gavel, “A spatial non-dynamic LQG controller. Part II, theory,” in Procedings of 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

Montri, J.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

Mugnier, L. M.

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Ogata, K.

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).

Paschall, R. N.

Petit, C.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

Pitsianis, N. P.

Plemmons, R. J.

Poyneer, L. A.

Rabaud, D.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

Raynaud, H.-F.

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers, and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261-1276 (2004).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Rousset, G.

van Loan, C.

Verhaegen, M.

Weiss, R.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Wiberg, D. M.

D. M. Wiberg, C. E. Max, and D. T. Gavel, “A spatial non-dynamic LQG controller. Part II, theory,” in Procedings of 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

Witsenhausen, H. S.

H. S. Witsenhausen, “Separation of estimation and control for discrete time systems,” Proc. IEEE 59, 1557-1566 (1971).
[CrossRef]

Wittenmark, B.

K. J. Åström and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1997).

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

Appl. Opt.

Astron. Astrophys.

E. Gendron and P. Léna, “Astronomical adaptive optics. I: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

C. R. Phys.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “Optimal control for multi-conjugate adaptive optics,” C. R. Phys. 6, 1059-1069 (2005).
[CrossRef]

Exp. Astron.

D. P. Looze, M. Kasper, S. Hippler, O. Beker, and R. Weiss, “Optimal compensation and implementation for adaptive optics systems,” Exp. Astron. 15, 67-88 (2003).
[CrossRef]

IEEE Trans. Autom. Control

P. Dorato and A. H. Levis, “Optimal linear regulators: the discrete-time case,” IEEE Trans. Autom. Control AC-16, 613-620 (1971).

J. Opt. Soc. Am. A

Opt. Express

Proc. IEEE

H. S. Witsenhausen, “Separation of estimation and control for discrete time systems,” Proc. IEEE 59, 1557-1566 (1971).
[CrossRef]

Proc. SPIE

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, J. Montri, and D. Rabaud, “First laboratory demonstration of closed-loop Kalman based optimal control for vibration filtering and simplified MCAO,” Proc. SPIE , 6272, 62721T-1-62721T-12 (2006).
[CrossRef]

Other

E. I. Jury, Sampled-Data Control Systems (Wiley, 1958).

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).

K. J. Åström and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1997).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

M. C. Roggemann and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

D. M. Wiberg, C. E. Max, and D. T. Gavel, “A spatial non-dynamic LQG controller. Part II, theory,” in Procedings of 2004 IEEE Conference on Decision and Control (IEEE, 2004), pp. 3333-3338.

T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

R. Köhler, “CHEOPS performance simulations,” MPIA Doc. no. CHEOPS-TRE-MPI-00033 (Max Planck Institüt fur Astronomie, 2004).

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, 1981).

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Figures (7)

Fig. 1
Fig. 1

Model of a classic AO system. Continuous-time signals (solid line) in the feedback loop are the incident phase ϕ tur , the mirror surface s m , the residual phase ϕ res , and the actuator inputs a. Discrete-time signals (dashed line, tilde over signal name) include the measurement y ̃ , the measurement noise θ ̃ , and the commanded actuator inputs a ̃ .

Fig. 2
Fig. 2

Solid curve is the estimated power spectrum of data obtained from a simulated von Karman phase screen that is representative of the Paranal Observatory site, Chile. The dashed line is the power spectral density (PSD) of the IWF model G a . The dashed–doted curve is the PSD of a discrete-time (DT) IWF model used by the LQG controllers other than the hybrid LQG controller.

Fig. 3
Fig. 3

Residual RMS as a function of the computational loop delay for an AO system using LQG controllers based on different models: the hybrid LQG (solid curve), the discrete-time (DT) LQG controller (dashed curve), and the discrete-time IWF LQG controller (dashed–dotted curve).

Fig. 4
Fig. 4

Zoomed view of the residual RMS of an AO system with the hybrid LQG controller (solid line), the discrete-time LQG controller (dashed curve), and the discrete-time IWF LQG controller (dashed–doted).

Fig. 5
Fig. 5

Residual RMS as a function of the computational loop delay for an AO system using the hybrid LQG controller (solid curve) and the gain-optimized integral controller (dashed curve). Three measurement noise levels are shown.

Fig. 6
Fig. 6

Residual RMS as a function of actuator time constant for an AO system using the hybrid LQG controller (solid curve), the gain-optimized integral controller (dashed curve), and the hybrid LQG controller designed using a stiff DM model (dashed–doted curve).

Fig. 7
Fig. 7

(a) Loop bandwidth and (b) controller phase at the loop bandwidth as a function of actuator time constant for an AO system using the hybrid LQG controller (solid curve) and the gain-optimized integral controller (dashed curve) x 2 .

Equations (53)

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

x ̇ m ( t ) = A m x m ( t ) + B m a ( t ) , x m ( t ) R n m , a ( t ) R m m ,
s m ( t ) = C m x m ( t ) + D m a ( t ) , s m ( t ) R q ,
x ̇ a ( t ) = A a x a ( t ) + B a ζ ( t ) , x a ( t ) R n a , ζ ( t ) R m a ,
ϕ tur ( t ) = C a x a ( t ) , ϕ tur ( t ) R q ,
ϕ res ( t ) = ϕ tur ( t ) s m ( t ) R q ,
θ ̃ k D θ θ ̃ n k , D θ = ϴ R p × p .
y ̃ k = g ( k T f ) + θ ̃ k , g ( t ) = 1 T f H t T f τ c t τ c ϕ res ( σ ) d σ .
x ̃ k + 1 = A K x ̃ k + B K y ̃ k , x ̃ k R n K ,
a ̃ k = C K x ̃ k + D K y ̃ k , a ̃ k R m .
a ( t ) = a ̃ k for k T f t < ( k + 1 ) T f .
V = lim W 1 W 0 W E { ϕ res ( σ ) T ϕ res ( σ ) } d σ = lim W 1 W 0 W tr [ P res ( σ ) ] d σ ,
x ̃ k + 1 = A x ̃ k + B 1 d ζ ̃ k + B 2 d a ̃ k , x ̃ k R n aug ,
e ̃ k = C 1 d x ̃ k + D d a ̃ k , e ̃ k R p e ,
y ̃ k = C 2 d x ̃ k + D θ θ ̃ n k , ζ ̃ k R m ζ ,
V d = lim N 1 N k = 0 N E { e ̃ k T e ̃ k } .
x ̃ T = [ x ̃ m T x ̃ m 2 T x ̃ m 1 T x ̃ a T x ̃ a 2 T x ̃ a 1 T ] ,
x ̃ m R n m ; x ̃ a R n a ; x ̃ m 1 , x ̃ m 2 , x ̃ a 1 , x ̃ a 2 R q .
A = [ A d m 0 0 A d a ] = [ A m d 0 0 0 0 0 A m 1 0 0 0 0 0 A m 2 I 0 0 0 0 0 0 0 A a d 0 0 0 0 0 A a 1 0 0 0 0 0 A a 2 I 0 ] .
B 1 d = [ 0 B a 1 d ] = [ 0 0 0 B a d B 11 B 12 ] , B 2 d = [ B m 2 d 0 ] = [ B m d B 21 B 22 0 0 0 ] ,
C 1 d = [ C m 1 d 0 0 C a 1 d 0 0 ] ,
C 2 d = [ 0 0 ( 1 T f ) H 0 0 ( 1 T f ) H ] .
V = 1 T f ( V 0 + V d ) ,
V 0 = tr { C a 0 T f 0 σ e A a γ B a B a T e A a T γ d γ d σ C a T } .
V a = lim W 1 N k = 0 N E { a ̃ k T D a T D a a ̃ k } ,
V = V d + V a .
Q = C 1 d T C 1 d , Q x u = C 1 d T D d , R = D d T D d + D a T D a ,
Ξ = B 1 d B 1 d T , Θ = D θ D θ T .
x ̂ k + 1 = ( A L C 2 d ) x ̂ k + B 2 d a ̃ k + L y ̃ k ,
a ̃ k = K ( x ̂ k + L f ( y ̃ k C 2 d x ̂ k ) ) .
S = A T S A + Q ( A S B 2 d + Q x u ) ( B 2 d T S B 2 d + R ) 1 ( A S B 2 d + Q x u ) T ,
K = ( B 2 d T S B 2 d + R ) 1 ( B 2 d T S A T + Q x u T ) .
Σ = A Σ A T + Ξ A Σ C 2 T ( C 2 Σ C 2 T + Θ ) 1 C 2 Σ A T ,
L f = Σ C 2 T ( C 2 Σ C 2 T + Θ ) 1 , L = A L f .
x o ¯ T = [ 0 x ̃ m 2 T x ̃ m 1 T 0 x ̃ a 2 T x ̃ a 1 T ] ,
x ̃ m 1 , x ̃ m 2 , x ̃ a 1 , x ̃ a 2 R q .
A c = [ A m 0 0 A a ] , B c = [ B m d 0 ] , S c = [ S 11 S 14 S 14 T S 44 ] ,
Q x u c = [ C m 1 d T D d C a 1 d T D d ] , R = D d T D d + R a ,
Q c = [ C m 1 d T C m 1 d C m 1 d T C a 1 d C a 1 d T C m 1 d C a 1 d T C a 1 d ] .
S c = A c T S c A c + Q c ( A c S c B c + Q x u c ) ( B c T S B c + R ) 1 ( A c S c B c + Q x u c ) T .
x u T = [ 0 0 0 x ̃ a T x ̃ a 2 T x ̃ a 1 T ] ,
x ̃ a R n a ; x ̃ a 1 , x ̃ a 2 R p .
C a = [ 0 0 ( 1 T f ) H ] , Ξ a = B a d 1 B a d 1 T ,
Σ a = [ Σ 44 Σ 45 Σ 46 Σ 45 T Σ 55 Σ 56 Σ 46 T Σ 56 T Σ 66 ] .
Σ a = A d a Σ a A d a T + Ξ a A d a Σ a C a T ( C a Σ a C a T + Θ ) 1 C a Σ a A d a T .
L f a = Σ a C a T ( C a Σ a C a T + Θ ) 1 R ( n a + 2 p ) × p ,
L f = [ 0 L f a ] , L = [ 0 A d a L f a ] .
G m ( s ) = 1 , G a ( s ) = 5000 s + 1 .
H = 1 , G WFS ( s ) = 1 e s T f s T f , T f = 0.001 s .
G DM ( s ) = 1 .
G m ( z ) = k D M z α z 2 , α = τ c T f τ c .
G K ( z ) = K I 1 z 1 .
x ̇ m ( t ) = 1 τ m x m + 1 τ m a ( t ) ,
s m ( t ) = x m ( t ) .

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