Abstract

Uncertainty in the interaction matrix between sensors and actuators can lead to performance degradation or instability in control of segmented mirrors (typically the telescope primary). The interaction matrix is ill conditioned, and thus the position estimate required for control can be highly sensitive to small errors in knowledge of the matrix, due to uncertainty or temporal variations. The robustness to different types of uncertainty is bounded here using the small gain theorem and structured singular values. The control is quite robust to moderate uncertainty in actuator gain, sensor gain, or the ratio of sensor dihedral and height sensitivity. However, the control is extremely sensitive to small errors in geometry, with the maximum error that can be tolerated scaling inversely with the number of segments. The same tools can be applied to adaptive optics; however, the interaction matrix here is better conditioned and so uncertainty is less of an issue, with the tolerable error scaling inversely with the square root of the number of actuators.

© 2009 Optical Society of America

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  1. J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
    [CrossRef]
  2. R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” 7012, 701219 (2008).
  3. G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation,” Appl. Opt. 43, 1223-1232(2004).
    [CrossRef] [PubMed]
  4. R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
    [CrossRef]
  5. J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences (Oxford University Press, 1998).
  6. T. J. Brennan, “Stability margin loss due to wavefront sensor misregistration: amelioration with spatial filtering techniques,” Proc. SPIE 4376, 88-98 (2001).
    [CrossRef]
  7. K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press, 2008).
  8. J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (MacMillan, 1992).
  9. A. Packard and J. Doyle, “The complex structured singular value,” Automatica 29, 71-109 (1993).
    [CrossRef]
  10. D. G. MacMartin and G. Chanan, “Measurement accuracy in control of segmented-mirror telescopes,” Appl. Opt. 43, 608-615 (2004).
    [CrossRef] [PubMed]
  11. D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Control of many coupled oscillators and application to segmented-mirror telescopes,” presented at the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii, 18-21 August 2008.

2008

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

2004

2001

T. J. Brennan, “Stability margin loss due to wavefront sensor misregistration: amelioration with spatial filtering techniques,” Proc. SPIE 4376, 88-98 (2001).
[CrossRef]

1993

A. Packard and J. Doyle, “The complex structured singular value,” Automatica 29, 71-109 (1993).
[CrossRef]

1990

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Andreae, S.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Arthur, A. A.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Åström, K. J.

K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press, 2008).

Biocca, A.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Brennan, T. J.

T. J. Brennan, “Stability margin loss due to wavefront sensor misregistration: amelioration with spatial filtering techniques,” Proc. SPIE 4376, 88-98 (2001).
[CrossRef]

Chanan, G.

Cohen, R. W.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Doyle, J.

A. Packard and J. Doyle, “The complex structured singular value,” Automatica 29, 71-109 (1993).
[CrossRef]

Doyle, J. C.

J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (MacMillan, 1992).

Francis, B. A.

J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (MacMillan, 1992).

Franck, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Fuertes, J. M.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Gabor, G.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Gilmozzi, R.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” 7012, 701219 (2008).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences (Oxford University Press, 1998).

Jared, R. C.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Llacer, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

MacMartin, D. G.

MacMynowski, D. G.

D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Control of many coupled oscillators and application to segmented-mirror telescopes,” presented at the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii, 18-21 August 2008.

Mast, T.

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation,” Appl. Opt. 43, 1223-1232(2004).
[CrossRef] [PubMed]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Meng, J.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Merrick, T.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Minor, R.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Murray, R. M.

K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press, 2008).

Nelson, J.

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

G. Chanan, D. G. MacMartin, J. Nelson, and T. Mast, “Control and alignment of segmented-mirror telescopes: matrices, modes, and error propagation,” Appl. Opt. 43, 1223-1232(2004).
[CrossRef] [PubMed]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Orayani, M.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Packard, A.

A. Packard and J. Doyle, “The complex structured singular value,” Automatica 29, 71-109 (1993).
[CrossRef]

Salz, P.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Sanders, G. H.

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

Schaefer, B.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Sirota, M. J.

D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Control of many coupled oscillators and application to segmented-mirror telescopes,” presented at the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii, 18-21 August 2008.

Spyromilio, J.

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” 7012, 701219 (2008).

Tannenbaum, A. R.

J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (MacMillan, 1992).

Thompson, P. M.

D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Control of many coupled oscillators and application to segmented-mirror telescopes,” presented at the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii, 18-21 August 2008.

Witebsky, C.

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

Appl. Opt.

Automatica

A. Packard and J. Doyle, “The complex structured singular value,” Automatica 29, 71-109 (1993).
[CrossRef]

Proc. SPIE

J. Nelson and G. H. Sanders, “The status of the Thirty Meter Telescope project,” Proc. SPIE 7012, 70121A (2008).
[CrossRef]

R. C. Jared, A. A. Arthur, S. Andreae, A. Biocca, R. W. Cohen, J. M. Fuertes, J. Franck, G. Gabor, J. Llacer, T. Mast, J. Meng, T. Merrick, R. Minor, J. Nelson, M. Orayani, P. Salz, B. Schaefer, and C. Witebsky, “The W. M. Keck Telescope segmented primary mirror active control system,” Proc. SPIE 1236, 996-1008 (1990).
[CrossRef]

T. J. Brennan, “Stability margin loss due to wavefront sensor misregistration: amelioration with spatial filtering techniques,” Proc. SPIE 4376, 88-98 (2001).
[CrossRef]

Other

K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University Press, 2008).

J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory (MacMillan, 1992).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series on Optical and Imaging Sciences (Oxford University Press, 1998).

R. Gilmozzi and J. Spyromilio, “The 42 m European ELT: status,” 7012, 701219 (2008).

D. G. MacMynowski, P. M. Thompson, and M. J. Sirota, “Control of many coupled oscillators and application to segmented-mirror telescopes,” presented at the AIAA Guidance, Navigation and Control Conference, Honolulu, Hawaii, 18-21 August 2008.

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Figures (8)

Fig. 1
Fig. 1

Segmentation layout for a 492-segment mirror.

Fig. 2
Fig. 2

Detail on three segments illustrating locations of actuator “+” and intersegment sensor “•”. Parameters used in examples are segment radius a = 0.715 m , actuator radius t = 0.742 a , and distance from sensor to vertex f = 0.14 a .

Fig. 3
Fig. 3

Definition of sensor moment arm L eff ; a sensor measuring relative height a distance L eff from the segment edge will give output z = h + L eff θ . At Keck, the dihedral sensitivity resulted from an actual moment arm; other sensor geometries give rise to dihedral sensitivity from other mechanisms but can still be characterized in terms of L eff .

Fig. 4
Fig. 4

Schematic of uncertainty Δ in interaction matrix A with estimator A + and control K ( s ) : (a) uncertainty in actuator gain, (b) in sensor gain, (c) in dihedral gain, and (d) in matrix entries. In each case, the block diagram can be transformed into the form on the right, with T ( s ) M given in the text.

Fig. 5
Fig. 5

Inverse stability bound μ as a function of number of segments in the mirror, for constant L eff / a and for L eff = 0 . In the latter case, focus mode is not controlled; in the former, μ scales inversely with L eff / a . Only the upper bound on μ is shown; the lower bound is roughly 15% lower. Values for t / a and f / a as in Fig. 2.

Fig. 6
Fig. 6

Worst-case (nonzero) eigenvalue of A + A evaluated as a function of uncertainty magnitude, for a 492-segment primary mirror with L eff / a = 0.014 . For geometry errors, both the analytical bound and the results of Monte Carlo analysis to estimate the 95% confidence bounds are shown. Instability will occur for any nonzero control bandwidth if the minimum eigenvalue drops below zero.

Fig. 7
Fig. 7

Representative actuator/sensor geometry for AO simulations (with the central obscuration ignored for simplicity) where “•” represent actuators and “+” a two-axis tip/tilt measurement.

Fig. 8
Fig. 8

Inverse stability bound μ as a function of number of deformable mirror actuators for single-conjugate AO.

Equations (24)

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

y = A x .
A = A h + A θ .
A = P A ̲ R ,
P = [ I m I m I m ] ,
A + = lim ϵ 0 ( A T A + ϵ I ) 1 A T .
A = U Σ V T A + = V Σ 1 U T ,
A = [ U h U f U g U np ] [ Σ h 0 0 0 Σ f 0 0 0 Σ g 0 0 0 ] [ V h T V f T V g T ] ,
= U h Σ h V h T + U f Σ f V f ,
A + = V h Σ h 1 U h T + V f Σ f 1 U f T .
A i j = A i j × ( 1 + δ j ) ,
A i j = A i j × ( 1 + δ i ) ,
A i j = ( A h ) i j + ( A θ ) i j × ( 1 + δ i ) ,
A i j = A i j + δ i j .
T ( s ) = k ( s ) 1 + k ( s ) .
Δ < M T 1 ,
A θ A + U f U f T ,
det [ I T ( s ) A + ( I + Δ 1 ) A ( I + Δ 2 ) ] = 0
Δ = P Δ ̲ R .
max Q Q ρ ( Q M ) = μ σ max ( M ) ,
Δ = [ d 1 I 1 0 0 d 2 I 2 ] ,
1.2 × 10 4 μ 1.4 × 10 4 .
204 μ 245.
13 μ 25.
50 μ 98.

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