Abstract

Using the subspace identification technique, we identify a finite dimensional, dynamical model of a recently developed prototype of a thermally actuated deformable mirror (TADM). The main advantage of the identified model over the models described by partial differential equations is its low complexity and low dimensionality. Consequently, the identified model can be easily used for high-performance feedback or feed-forward control. The experimental results show good agreement between the dynamical response predicted by the model and the measured response of the TADM.

© 2013 Optical Society of America

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  1. O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, Opt. Lett. 25, 52 (2000).
  2. S. Piehler, B. Weichelt, A. Voss, M. Abdou Ahmed, and T. Graf, Opt. Lett. 37, 5033 (2012).
    [CrossRef]
  3. S. K. Ravensbergen, “Adaptive optics for extreme ultraviolet lithography: actuator design and validation for deformable mirror concepts,” Ph.D. thesis (Technische Universiteit Eindhoven, 2012).
  4. M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, Appl. Opt. 52, 2909 (2013).
    [CrossRef]
  5. R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, Opt. Lett. 29, 2635 (2004).
  6. B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).
  7. G. Vdovin and M. Loktev, Opt. Lett. 27, 677 (2002).
    [CrossRef]
  8. S. K. Ravensbergen, P.C.J. N. Rosielle, and M. Steinbuch, Precis. Eng. 37, 353 (2013).
    [CrossRef]
  9. S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).
  10. A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
    [CrossRef]
  11. K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University, 2009).
  12. H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).
  13. A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).
  14. A. Haber, A. Polo, C. S. Smith, S. F. Pereira, H. P. Urbach, and M. Verhaegen, Appl. Opt. 52, 2363 (2013).
    [CrossRef]
  15. W. Nowacki, Dynamic Problems of Thermoelasticity (Springer, 1975).
  16. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).
  17. M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).
  18. M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).
  19. A. Chiuso, Automatica 43(6), 1034 (2007).
  20. D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).

2013 (3)

2012 (4)

M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

S. Piehler, B. Weichelt, A. Voss, M. Abdou Ahmed, and T. Graf, Opt. Lett. 37, 5033 (2012).
[CrossRef]

2011 (1)

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

2010 (1)

A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
[CrossRef]

2007 (1)

A. Chiuso, Automatica 43(6), 1034 (2007).

2004 (2)

R. Lawrence, D. Ottaway, M. Zucker, and P. Fritschel, Opt. Lett. 29, 2635 (2004).

S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).

2002 (1)

2000 (1)

Abdou Ahmed, M.

Albert, O.

Astrom, K. J.

K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University, 2009).

Canuel, B.

M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, Appl. Opt. 52, 2909 (2013).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Cavalier, F.

Chiuso, A.

A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
[CrossRef]

A. Chiuso, Automatica 43(6), 1034 (2007).

Chung, J. H.

S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).

Day, R.

M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, Appl. Opt. 52, 2909 (2013).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Ferrari, M.

M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).

Fraanje, R.

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

Fritschel, P.

Genin, E.

M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, Appl. Opt. 52, 2909 (2013).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Graf, T.

Haber, A.

Hugot, E.

M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).

Kasprzack, M.

Laslandes, M.

M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).

Lawrence, R.

Loktev, M.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).

Marchetti, E.

A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
[CrossRef]

Marque, J.

M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, Appl. Opt. 52, 2909 (2013).
[CrossRef]

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Mourou, G.

Muradore, R.

A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
[CrossRef]

Murray, R. M.

K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University, 2009).

Norris, T. B.

Nowacki, W.

W. Nowacki, Dynamic Problems of Thermoelasticity (Springer, 1975).

Ottaway, D.

Penna, P. La

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Pereira, S. F.

Piehler, S.

Polo, A.

Ravensbergen, S. K.

S. K. Ravensbergen, P.C.J. N. Rosielle, and M. Steinbuch, Precis. Eng. 37, 353 (2013).
[CrossRef]

S. K. Ravensbergen, “Adaptive optics for extreme ultraviolet lithography: actuator design and validation for deformable mirror concepts,” Ph.D. thesis (Technische Universiteit Eindhoven, 2012).

Rosielle, P.C.J. N.

S. K. Ravensbergen, P.C.J. N. Rosielle, and M. Steinbuch, Precis. Eng. 37, 353 (2013).
[CrossRef]

Schitter, G.

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

Sentenac, D.

Sherman, L.

Smith, C. S.

Song, H.

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

Steinbuch, M.

S. K. Ravensbergen, P.C.J. N. Rosielle, and M. Steinbuch, Precis. Eng. 37, 353 (2013).
[CrossRef]

Timoshenko, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

Urbach, H. P.

Vajente, G.

Vdovin, G.

Velinsky, S. A.

S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Verhaegen, M.

A. Haber, A. Polo, C. S. Smith, S. F. Pereira, H. P. Urbach, and M. Verhaegen, Appl. Opt. 52, 2363 (2013).
[CrossRef]

A. Polo, A. Haber, S. F. Pereira, M. Verhaegen, and H. P. Urbach, Opt. Express 20, 27922 (2012).

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

Voss, A.

Weichelt, B.

Winters, S. E.

S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).

Woinowsky-Krieger, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

Zucker, M.

Appl. Opt. (2)

Automatica (1)

A. Chiuso, Automatica 43(6), 1034 (2007).

Class. Quantum Grav. (1)

B. Canuel, R. Day, E. Genin, P. La Penna, and J. Marque, Class. Quantum Grav. 29, 085012 (2012).

Eur. J. Control (1)

H. Song, R. Fraanje, G. Schitter, G. Vdovin, and M. Verhaegen, Eur. J. Control 17, 290 (2011).

IEEE Trans. Control Syst. Technol. (1)

A. Chiuso, R. Muradore, and E. Marchetti, IEEE Trans. Control Syst. Technol. 18, 705 (2010).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

M. Laslandes, E. Hugot, and M. Ferrari, J. Eur. Opt. Soc. Rapid Pub. 7, 12036 (2012).

Mech. Based Des. Struc. Mach. (1)

S. E. Winters, J. H. Chung, and S. A. Velinsky, Mech. Based Des. Struc. Mach. 32, 195 (2004).

Opt. Express (1)

Opt. Lett. (4)

Precis. Eng. (1)

S. K. Ravensbergen, P.C.J. N. Rosielle, and M. Steinbuch, Precis. Eng. 37, 353 (2013).
[CrossRef]

Other (6)

K. J. Astrom and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers (Princeton University, 2009).

S. K. Ravensbergen, “Adaptive optics for extreme ultraviolet lithography: actuator design and validation for deformable mirror concepts,” Ph.D. thesis (Technische Universiteit Eindhoven, 2012).

W. Nowacki, Dynamic Problems of Thermoelasticity (Springer, 1975).

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

M. Verhaegen and V. Verdult, Filtering and System Identification: A Least Squares Approach (Cambridge University, 2007).

D. Malacara, Optical Shop Testing (Wiley-Interscience, 2007).

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Figures (6)

Fig. 1.
Fig. 1.

Sketch of the thermally actuated DM.

Fig. 2.
Fig. 2.

Step response of the TADM when the step functions are applied to the third actuator: (a) the RMS y(k)2. (b) Steady-state value of the RMS y()2.

Fig. 3.
Fig. 3.

Test of the superposition principle: (a) 5th Zernike coefficient and (b) the RMS of the total measured wavefront.

Fig. 4.
Fig. 4.

(a) Singular values of the data matrix used in the identification. (b) The VAF values for different order n of the state-space model (1,2).

Fig. 5.
Fig. 5.

Identification results when only actuator 3 is active: (a) the input used for identification and the rms of the output. (b) Bode plot of the transfer function between v3 and α3. (c) and (d) Prediction and measurements of Zernike coefficients.

Fig. 6.
Fig. 6.

Identification results when all 19 actuators are active: (a) singular values of the data matrix used for identification; (b) VAF for different model orders n; (c) and (d) prediction performance of the identified model.

Equations (6)

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

x(k+1)=Ax(k)+Bu(k)
y(k)=Cx(k)+e(k),
y̲2=O^x(0)+D^u̲2+e̲2,
y̲2=[y2(0)y2(1)y2(M)],u̲2=[u2(0)u2(1)u2(M)],e̲2=[e2(0)e2(1)e2(M)],O^=[C^C^A^C^A^M],D^=[0000C^B^000C^A^B^C^B^00C^A^M1B^C^A^M2B^C^B^0]
minx(0)z̲O^x(0)22,
VAF=max{0,(1var(y2y^2)var(y2))×100%}.

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