Abstract

Lyapunov stability criteria are derived for a first-order closed-loop adaptive-optics servo system, resulting in a linear matrix equation that includes the system geometry, servo parameters, and wave-front reconstruction matrix. It is demonstrated that instability zones depend on the choice of matrix estimator and the servo-loop gain. Divergence of the error propagator gives results that are consistent with the Lyapunov equation. The significance of these results is discussed.

© 1998 Optical Society of America

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References

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  1. W. J. Wild, E. J. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1985).
    [CrossRef]
  2. W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
    [CrossRef]
  3. Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control (Academic, San Diego, Calif., 1995).
  4. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
    [CrossRef]
  5. R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
    [CrossRef]
  6. S. Lefschetz, Differential Equations:?Geometric Theory (Dover, New York, 1977).
  7. W. J. Wild, Proc. SPIE 3126, 278 (1997).
    [CrossRef]
  8. W. J. Wild, Opt. Lett. 21, 1433 (1996).
    [CrossRef] [PubMed]
  9. W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
    [CrossRef]
  10. J. Herrmann, J. Opt. Soc. Am. 70, 28 (1980).
    [CrossRef]
  11. W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).
  12. A. S. Willsky, Proc. IEEE 66, 996 (1978).
    [CrossRef]
  13. D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
  14. S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
    [CrossRef]

1997 (1)

W. J. Wild, Proc. SPIE 3126, 278 (1997).
[CrossRef]

1996 (1)

1995 (1)

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

1994 (1)

W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

1985 (1)

1980 (1)

1978 (1)

A. S. Willsky, Proc. IEEE 66, 996 (1978).
[CrossRef]

Balakrishnan, V.

S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
[CrossRef]

Boyd, S.

S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
[CrossRef]

Cochran, G. M.

W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).

El Ghaoui, L.

S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
[CrossRef]

Farmiga, N.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

Faron, E.

S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
[CrossRef]

Gajic, Z.

Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control (Academic, San Diego, Calif., 1995).

Herrmann, J.

Horn, R. A.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

Kibblewhite, E.

W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Kibblewhite, E. J.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1985).
[CrossRef]

Lefschetz, S.

S. Lefschetz, Differential Equations:?Geometric Theory (Dover, New York, 1977).

Leunberger, D. G.

D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

Moretti, W.

W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).

Qureshi, M. T. J.

Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control (Academic, San Diego, Calif., 1995).

Scor, V.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Shi, F.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

Steinhoff, K. E.

W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).

Tyler, G. A.

W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).

Vuilleumier, R.

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1985).
[CrossRef]

Wild, W. J.

W. J. Wild, Proc. SPIE 3126, 278 (1997).
[CrossRef]

W. J. Wild, Opt. Lett. 21, 1433 (1996).
[CrossRef] [PubMed]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, and R. Vuilleumier, Opt. Lett. 20, 955 (1985).
[CrossRef]

Willsky, A. S.

A. S. Willsky, Proc. IEEE 66, 996 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Proc. IEEE (1)

A. S. Willsky, Proc. IEEE 66, 996 (1978).
[CrossRef]

Proc. SPIE (3)

W. J. Wild, Proc. SPIE 3126, 278 (1997).
[CrossRef]

W. J. Wild, E. J. Kibblewhite, R. Vuilleumier, V. Scor, F. Shi, and N. Farmiga, Proc. SPIE 2534, 194 (1995).
[CrossRef]

W. J. Wild, E. Kibblewhite, and V. Scor, Proc. SPIE 2201, 726 (1994).
[CrossRef]

Other (7)

Z. Gajic and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control (Academic, San Diego, Calif., 1995).

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, Cambridge, 1991).
[CrossRef]

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge U. Press, Cambridge, 1985).
[CrossRef]

S. Lefschetz, Differential Equations:?Geometric Theory (Dover, New York, 1977).

W. Moretti, G. M. Cochran, K. E. Steinhoff, and G. A. Tyler, (Optical Sciences Company, Anaheim, Calif., 1988).

D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

S. Boyd, L. El Ghaoui, E. Faron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1994).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

9×9 actuator (discrete phase point)–subaperture geometry used in the simulations, with the local subaperture tilts indicated by the arrows. This configuration is commonly referred to as the Fried geometry. The active controlled region is the circular pupil inscribed within the 9×9 region.

Fig. 2
Fig. 2

Family of curves for four estimators, showing gn versus servo-loop gain k. For the solutions of the Lyapunov matrix equation the minimum unity eigenvalue of P specifies when P is positive definite and hence when the system will exhibit asymptotic stability. The horizontal lines, which are shown separated by a constant, show the region in which P is positive definite. The lines, from top to bottom, describe η=1 and v=0 POE, the optimal estimator, the least-squares estimator, and the successive symmetric overrelaxation estimators. The corresponding error propagator curves diverge when the line for positive definite P ends. Note that the POE has a value of gn that is approximately an order of magnitude lower than the least-squares matrix for all k.

Equations (8)

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

ϕti+1=a0ϕti+kMsti+1-Aϕti=kMsti+1+Fϕti,
ϕti+1=Fϕti,
ΔVϕi=ϕi+1TPϕi+1-ϕiTPϕi,
FTPF-P=-L,
vecP=II-FTF-1vecL,
gn=k2NatrLimΩj=0Ωj=0ΩFΩ-jMMT FTΩ-j,
kFTFkkFTkFk
gn=k2NatrI-FTF-1MMT.

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