Abstract

In many scenarios, an adaptive optics (AO) control system operates in the presence of temporally non-white noise. We use a Kalman filter with a state space formulation that allows suppression of this colored noise, hence improving residual error over the case where the noise is assumed to be white. We demonstrate the effectiveness of this new filter in the case of the estimated Gemini Planet Imager tip–tilt environment, where there are both common-path and non-common-path vibrations. We discuss how this same framework can also be used to suppress spatial aliasing during predictive wavefront control assuming frozen flow in a low-order AO system without a spatially filtered wavefront sensor, and present experimental measurements from Altair that clearly reveal these aliased components.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2010

2009

D. P. Looze, “Linear-quadratic-Gaussian control for adaptive optics systems using a hybrid model,” J. Opt. Soc. Am. A 26, 1–9 (2009).
[CrossRef]

J.-P. Véran and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in 1st AO4ELT Conference—Adaptative Optics for Extremely Large Telescopes, Y.Clénet, J.-M.Conan, T.Fusco, and G.Rousset, eds. (EDP Sciences, 2009), 05002.

L. A. Poyneer, M. A. van Dam, and J.-P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. 26, 833–846 (2009).
[CrossRef]

L. A. Poyneer, A. Norton, and D. Dillon, “Open-loop shaping of a 4k mems with Fourier-domain pre-compensation,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC3.

C. Petit, J.-M. Conan, C. Kulcsár, and H.-F. Raynaud, “Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis,” J. Opt. Soc. Am. A 26, 1307–1325 (2009).
[CrossRef]

2008

2007

2006

D. G. MacMynowski, K. Vogiatzis, G. Z. Angeli, J. Fitzsimmons, and J. E. Nelson, “Wind loads on ground-based telescopes,” Appl. Opt. 45, 7912–7923 (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, “Étude de la commande optimale en OA et OAMC, validation numérique et expérimentale,” Ph.D. thesis (L’Université Paris 13, 2006).

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A 23, 382–394 (2006).
[CrossRef]

2005

2004

2002

1998

F. J. Rigaut, J.-P. Véran, and O. Lai, “Analytical model for Shack-Hartmann-based adaptive optics systems,” Proc. SPIE 3353, 1038–1048 (1998).
[CrossRef]

1997

1995

Angeli, G. Z.

Bauman, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Beuzit, J.-L.

Brase, J. M.

Candy, J.

J. Candy, Model-Based Signal Processing (Wiley, 2005).
[CrossRef]

Conan, J. M.

Conan, J.-M.

Conan, R.

Correia, C.

Dillon, D.

L. A. Poyneer, A. Norton, and D. Dillon, “Open-loop shaping of a 4k mems with Fourier-domain pre-compensation,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC3.

Doyon, R.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Erikson, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Evans, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Fitzsimmons, J.

Fusco, T.

Gavel, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Gavel, D. T.

Graham, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Jolissaint, L.

Kulcsár, C.

Lai, O.

F. J. Rigaut, J.-P. Véran, and O. Lai, “Analytical model for Shack-Hartmann-based adaptive optics systems,” Proc. SPIE 3353, 1038–1048 (1998).
[CrossRef]

Larkin, J.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Le Roux, B.

Looze, D. P.

Macintosh, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
[CrossRef]

Macintosh, B. A.

MacMynowski, D. G.

Madec, P.-Y.

Maitre, H.

Morzinski, K.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Mugnier, L. M.

Nawab, S. H.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Nelson, J. E.

Norton, A.

L. A. Poyneer, A. Norton, and D. Dillon, “Open-loop shaping of a 4k mems with Fourier-domain pre-compensation,” in Adaptive Optics: Methods, Analysis and Applications, OSA Technical Digest (CD) (Optical Society of America, 2009), paper JTuC3.

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Oppenheimer, B.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Palmer, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Petit, C.

Phillion, D.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Poyneer, L.

J.-P. Véran and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in 1st AO4ELT Conference—Adaptative Optics for Extremely Large Telescopes, Y.Clénet, J.-M.Conan, T.Fusco, and G.Rousset, eds. (EDP Sciences, 2009), 05002.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Poyneer, L. A.

Pueyo, L.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

Raynaud, H. F.

Raynaud, H.-F.

Rigaut, F.

Rigaut, F. J.

F. J. Rigaut, J.-P. Véran, and O. Lai, “Analytical model for Shack-Hartmann-based adaptive optics systems,” Proc. SPIE 3353, 1038–1048 (1998).
[CrossRef]

Rouan, D.

Rousset, G.

Saddlemyer, L.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Shao, M.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

Sivaramakrishnan, A.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Soummer, R.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Thibault, S.

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

van Dam, M. A.

L. A. Poyneer, M. A. van Dam, and J.-P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. 26, 833–846 (2009).
[CrossRef]

Véran, J.-P.

L. A. Poyneer, M. A. van Dam, and J.-P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. 26, 833–846 (2009).
[CrossRef]

J.-P. Véran and L. Poyneer, “Evaluation of the T/T conditions at Gemini South using NICI AO telemetry data,” in 1st AO4ELT Conference—Adaptative Optics for Extremely Large Telescopes, Y.Clénet, J.-M.Conan, T.Fusco, and G.Rousset, eds. (EDP Sciences, 2009), 05002.

L. A. Poyneer and J.-P. Véran, “Predictive wavefront control for adaptive optics with arbitrary control loop delays,” J. Opt. Soc. Am. A 25, 1486–1496 (2008).
[CrossRef]

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A 23, 382–394 (2006).
[CrossRef]

F. J. Rigaut, J.-P. Véran, and O. Lai, “Analytical model for Shack-Hartmann-based adaptive optics systems,” Proc. SPIE 3353, 1038–1048 (1998).
[CrossRef]

J.-P. Véran, F. Rigaut, H. Maitre, and D. Rouan, “Estimation of the adaptive optics long-exposure point-spread function using control loop data,” J. Opt. Soc. Am. A 14, 3057–3069 (1997).
[CrossRef]

Vogiatzis, K.

Wallace, J. K.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

B. Macintosh, J. Graham, D. Palmer, R. Doyon, D. Gavel, J. Larkin, B. Oppenheimer, L. Saddlemyer, J. K. Wallace, B. Bauman, J. Evans, D. Erikson, K. Morzinski, D. Phillion, L. Poyneer, A. Sivaramakrishnan, R. Soummer, S. Thibault, and J.-P. Véran, “The Gemini Planet Imager,” Proc. SPIE 6272, 62720L (2006).
[CrossRef]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Systems, 2nd ed. (Prentice Hall, 1997).

Appl. Opt.

J. Astrophys. Astron.

A. Sivaramakrishnan, R. Soummer, L. Pueyo, J. K. Wallace, and M. Shao, “Sensing phase aberrations behind Lyot coronagraphs,” J. Astrophys. Astron. 688, 701–708 (2008).
[CrossRef]

J. Opt. Soc. Am.

L. A. Poyneer, M. A. van Dam, and J.-P. Véran, “Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry,” J. Opt. Soc. Am. 26, 833–846 (2009).
[CrossRef]

J. Opt. Soc. Am. A

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002).
[CrossRef]

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004).
[CrossRef]

J.-M. Conan, G. Rousset, and P.-Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. A 12, 1559–1570 (1995).
[CrossRef]

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

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
[CrossRef]

D. P. Looze, “Linear-quadratic-Gaussian control for adaptive optics systems using a hybrid model,” J. Opt. Soc. Am. A 26, 1–9 (2009).
[CrossRef]

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “On the optimal reconstruction and control of adaptive optical systems with mirror dynamics,” J. Opt. Soc. Am. A 27, 333–349 (2010).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Error [top] and noise [bottom] transfer functions for the GPI tilt simulation. Use of a higher-order model for amtospheric tilt plus common-path vibrations results in an error transfer function with better low-frequency rejection and notches for the vibrations, as compared to an optimized-gain integrator. Additionally modeling non-common-path vibration produces a noise transfer function that notches out spurious signals. (Shallow secondary notches are a result of the common-path vibration prediction.)

Fig. 2
Fig. 2

Maps of found peaks and aliases, Altair April 4, 2008, data set. Each square panel is a f x , f y mapping of the spatial frequencies with dashed lines as the axes. The color of each point is the temporal frequency of a peak found in that Fourier mode’s temporal PSD. In “True layer peaks” each peak corresponds to a layer, which is identified with 90% likelihood. In "Aliased layer peaks" each identified peak matches with an alias of that layer. This shows the controllable spatial frequencies that are aliased into. Note the opposite pattern of temporal frequency increase from the true layer peaks. At bottom in “Peaks repositioned,” the aliases are returned to their parent high spatial frequencies, forming a layer map over a larger spatial frequency range. The solid square box outlines the controllable spatial frequencies of the AO system.

Fig. 3
Fig. 3

Temporal power spectrum of Fourier mode 2,6 from a 1 -minute -long Altair AO closed-loop run April 4, 2008. The single layer of wind identified for this observation causes the strong peak at 4.8 Hz . The secondary peak at 12.8 Hz is caused by an alias from Fourier mode k = 2 , l = 10 . Nearby mode 4,7 has similar behavior and is also shown.

Fig. 4
Fig. 4

Block diagram of anti-alias predictive controller for a single Fourier mode. First the layers are predicted in parallel with layer integrators. The result is summed and sent through a second filter. This filter predicts the aliases and subtracts them to produce the DM command. The area shaded in gray is the new portion due to incorporation of alias removal. The regular PFC filter has the exact same structure as the white region.

Tables (2)

Tables Icon

Table 1 Tilt Error (mas RMS), 32 Trials, for GPI Tilt Simulation with Different Controllers a

Tables Icon

Table 2 Information on Aliased Components That Might Appear for Controllable Mode k = 2 , l = 6 in Altair Data

Equations (54)

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x [ t + 1 ] = A x [ t ] + B w [ t ] ,
y [ t ] = C x [ t ] + D u [ t ] + v [ t ] ,
x c [ t + 1 ] = A c x c [ t ] + B c w c [ t ] .
y [ t ] = C c x c [ t ] + D u [ t ] + z [ t ] .
n [ t + 1 ] = A n n [ t ] + B n r [ t ] ,
z [ t ] = C n n [ t ] + v [ t ] .
x [ t ] = ( x c T [ t ] , n T [ t ] ) T ,
w [ t ] = ( w c T [ t ] , r T [ t ] ) T .
A = ( A c 0 0 A n ) ,
B = ( B c 0 0 B n ) ,
C = ( C c C n ) .
C mod = ( C c 0 )
( a r [ t + 1 ] a i [ t + 1 ] ) = ( α r α i α i α r ) ( a r [ t ] a i [ t ] ) + ( w r [ t ] w i [ t ] ) .
g [ t + 1 ] = G g [ t ] + B g w g [ t ] ,
x c = [ g T , a T , φ [ t 1 ] ] T .
A c = ( G 0 0 0 0 0 0 T α r , 1 α i , 1 0 0 0 0 T α i , 1 α r , 1 0 0 0 0 T 0 0 α r , N a α i , N a 0 0 T 0 0 α i , N a α r , N a 0 1 , 0 , 0 1 0 1 0 0 ) .
B c = ( B g 0 0 I 2 N a 0 0 T ) .
A n = ( β r , 1 β i , 1 0 0 0 β i , 1 β r , 1 0 0 0 0 0 β r , N b β i , N b 0 0 0 β i , N b β r , N b 0 1 0 1 0 0 ) .
B n = ( I 2 N b 0 T ) .
x ̂ [ t | t ] = ( I K s C ) A x ̂ [ t 1 | t 1 ] + K s ( y [ t ] D u [ t ] ) .
K s = P s C H ( C P s C H + P v ) 1 .
P s = A P s A H + B P w B H A P s C H ( C P s C H + P v ) 1 C P s A H .
φ ̂ [ t + 1 | t ] = C mod A A x ̂ [ t | t ] .
A ̃ = ( ( 0 0 1 0 ) 0 0 A ) ,
B ̃ = [ 1 , 0 ] T ,
C aug = [ 0 , 1 ] ,
C ̃ = [ C aug , C ] ,
W ̃ = ( 0 B ) .
Q = ( C aug T Q p C aug C aug T Q p C mod C mod T Q p C aug C mod T Q p C mod ) .
G = ( r exp ( j θ ) 1 0 r exp ( j θ ) ) .
f t = f x v x + f y v y = k v x + l v y N d .
f t = ( k + N n k ) v x + ( l + N n l ) v y N d .
a [ t ] = α a [ t 1 ] + w [ t ] .
a [ t ] = ( a 0 [ t ] , a 1 [ t ] , , a N a [ t ] )
A α = Diag ( α 0 , α 1 , , α N a ) .
P w = Diag ( σ a 0 2 , σ a 1 2 , , σ a N a 2 ) .
x c [ t ] = ( a [ t ] , φ [ t 1 ] ) T .
A c = ( A α 0 1 T 0 ) ,
B c = ( I 0 T ) .
C c = ( 0 T , 1 ) ,
n [ t ] = ( b 1 [ t ] , , b N b [ t ] , ρ [ t 1 ] ) .
A β = Diag ( β 1 , , β N b ) .
A n = ( A β 0 1 T 0 ) .
B n = ( I 0 T ) .
P r = Diag ( σ b 1 2 , , σ b N b 2 ) .
C ( z ) = ( k = 0 N a K k α k 1 α k z 1 ) ( 1 + z 1 D 1 ) 1 ,
K k = Q 1 p N a + 1 , k ,
D 1 = Q 1 k = 0 N a p N a + 1 , k ,
Q = p N a + 1 + σ v 2 .
C ( z ) = ( k = 0 N a K k α k 1 α k z 1 ) ( 1 + z 1 D 1 + z 1 k = 1 N b C k 1 β k z 1 ) 1 ,
K k = Q 1 ( p N a + 1 , k + p N a + N b + 2 , k ) ,
D 1 = Q 1 k = 0 N a ( p N a + 1 , k + p N a + N b + 2 , k ) ,
C k = Q 1 ( p N a + 1 + k , N a + 1 * + p N a + N b + 2 , N a + 1 + k ) ,
Q = p N a + N b + 2 + σ v 2 .

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