A Decomposition Approach to Distributed Control of Dynamic Deformable Mirrors

Deformable mirrors with spatially invariant dynamic response can be considered as part of the class of decomposable systems. Such systems can be thought of as the interconnection of a number of identical subsystems, and they can be used to model certain classes of large scale systems. We show in this article that the technique allows the design of a distributed ℋ2 controller for a deformable mirror of any size, with a computational cost that does not increase with the size of the mirror. The method shows a performance close to that of the centralized ℋ2 optimal controller in a simulation example, which is about two times better (in terms of ℋ2 norm) than the performance of a decentralized PI controller, with additional notches to suppress the high resonance frequencies of the deformable mirror.

[1]  Ye.A. Gorin,et al.  Fourier analysis on groups: Rudin, W., New York and London, 1962☆ , 1963 .

[2]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

[3]  Albrecht Böttcher,et al.  Spectral properties of banded Toeplitz matrices , 1987 .

[4]  David W. Miller,et al.  Robust control of the Multiple Mirror Telescope adaptive secondary mirror , 1999 .

[5]  Torben Andersen,et al.  Novel concept for large deformable mirrors , 2006 .

[6]  Michel Verhaegen,et al.  Distributed control of vehicle formations: A decomposition approach , 2008, 2008 47th IEEE Conference on Decision and Control.

[7]  W. G. Bickley,et al.  Relaxation Methods in Theoretical Physics , 1947 .

[8]  Karen M. Hampson,et al.  Adaptive optics and vision , 2008 .

[9]  Richard M. Murray,et al.  Information flow and cooperative control of vehicle formations , 2004, IEEE Transactions on Automatic Control.

[10]  M. Kasper,et al.  Adaptive Optics for Astronomy , 2012, 1201.5741.

[11]  M Maarten Steinbuch,et al.  Large adaptive deformable mirror: design and first prototypes , 2005, SPIE Optics + Photonics.

[12]  Dimitry M. Gorinevsky,et al.  Feedback controller design for a spatially distributed system: the paper machine problem , 2003, IEEE Trans. Control. Syst. Technol..

[13]  Bernhard Brandl,et al.  Application of distributed control techniques to the adaptive secondary mirror of Cornell's Large Atacama Telescope , 2003, SPIE Astronomical Telescopes + Instrumentation.

[14]  G. TEMPLE,et al.  Relaxation Methods in Theoretical Physics , 1946, Nature.

[15]  W. Rudin,et al.  Fourier Analysis on Groups. , 1965 .

[16]  Michel Verhaegen,et al.  Adaptive optics application of distributed control design for decomposable systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[17]  S. BRODETSKY,et al.  Theory of Plates and Shells , 1941, Nature.

[18]  Michel Verhaegen,et al.  Distributed Control for Identical Dynamically Coupled Systems: A Decomposition Approach , 2009, IEEE Transactions on Automatic Control.