Abstract

The electronics, computing hardware, and computing used to provide real-time modal control for a laser guide-star adaptive optics system are presented. This approach offers advantages in the control of unobserved modes, the elimination of unwanted modes (e.g., tip and tilt), and automatic handling of the case of low-resolution lens arrays. In our two-step modal implementation, the input vector of gradients is first decomposed into a Zernike polynomial mode by a least-squares estimate. The number of modes is assumed to be less than or equal to the number of actuators. The mode coefficients are then available for collection and analysis or for the application of modal weights. Thus the modal weights may be changed quickly without recalculating the full matrix. The control-loop integrators are at this point in the algorithm. To calculate the deformable-mirror drive signals, the mode coefficients are converted to the zonal signals by a matrix multiply. When the number of gradients measured is less than the number of actuators, the integration in the control loop will be done on the lower-resolution grid to avoid growth of unobserved modes. These low-resolution data will then be effectively interpolated to yield the deformable-mirror drive signals.

© 1998 Optical Society of America

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References

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  1. J. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
    [CrossRef]
  2. A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.
  3. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  4. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  5. A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.
  6. J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).
  7. G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1988).
  8. G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
    [CrossRef]
  9. J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.
  10. VxWorks is a product of Wind River Systems, Inc., 1010 Atlantic Avenue, Alameda, Calif. 94501.
  11. EPICS is free to nonprofit and government sites from Los Alamos National Laboratories and is commercially available from Kinetic Systems Corp., 7308 South Alton Way, Bldg. 2, Engelwood, Colo. 80112.
  12. Both Tcl and Tk are freely available in the public domain. For further information on these languages, see B. B. Welsh, Practical Programming in Tcl and Tk (Prentice-Hall, Upper Saddle River, N.J., 1995).

1993

J. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

1980

1976

Beckers, J.

J. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Beuzit, J. L.

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Beuzit, J.-L.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Boyer, C.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Bruno, T.

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

Castanon, D.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

Chazallet, F.

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Demailly, L.

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Denis, N.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

Emami-Naeini, A.

G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1988).

Franklin, G. F.

G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1988).

Gaffard, J. P.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Gendron, E.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Gigan, P.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Glindemann, A.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

Hippler, S.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

Huang, J.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

Hubin, N.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Lacombe, F.

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Landers, F.

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

Lena, P. J.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Looze, D. P.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

Madec, P.-Y.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

McCaughran, M. J.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

Navetta, J.

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

Noll, R. J.

Powell, J. D.

G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1988).

Rabaud, D.

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Richard, J.-C.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Rohloff, R.-R.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

Rousset, G.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

Silva, D. E.

Trvalik, B.

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

Vittot, M.

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

Wagner, C.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

Wang, J. Y.

Welsh, B. B.

Both Tcl and Tk are freely available in the public domain. For further information on these languages, see B. B. Welsh, Practical Programming in Tcl and Tk (Prentice-Hall, Upper Saddle River, N.J., 1995).

Wirth, A.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

Annu. Rev. Astron. Astrophys.

J. Beckers, “Adaptive optics for astronomy: principles, performance, and applications,” Annu. Rev. Astron. Astrophys. 31, 13–62 (1993).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

Other

A. Wirth, F. Landers, B. Trvalik, J. Navetta, T. Bruno, “A laser guide star atmospheric compensation system for the 3.5m Calar Alto telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MA3-1.

A. Glindemann, M. J. McCaughran, S. Hippler, C. Wagner, R.-R. Rohloff, “CHARM: a tip/tilt tertiary system for the Calar Alto 3.5m telescope,” presented at the OSA summer topical meeting on Adaptive Optics, Maui, 8–12 July 1996.

J. Huang, D. P. Looze, N. Denis, D. Castanon, A. Wirth, “Modeling and identification of adaptive optics systems,” Int. J. Control (to be published).

G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems (Addison-Wesley, Reading, Mass., 1988).

G. Rousset, J.-L. Beuzit, N. Hubin, E. Gendron, P.-Y. Madec, C. Boyer, J. P. Gaffard, J.-C. Richard, M. Vittot, P. Gigan, P. J. Lena, “Performance and results of the COME-ON+ adaptive optics system at the ESO 3.6m telescope,” in Adaptive Optics in Astronomy, M. A. Ealey, F. Merkle, eds., Proc. SPIE2201, 1088–1095 (1994).
[CrossRef]

J. L. Beuzit, N. Hubin, E. Gendron, L. Demailly, P. Gigan, F. Lacombe, F. Chazallet, D. Rabaud, G. Rousset, “ADONIS: a user-friendly adaptive optics system for the ESO 3.6m telescope,” in Adaptive Optics, Vol. 23 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper MC1-1.

VxWorks is a product of Wind River Systems, Inc., 1010 Atlantic Avenue, Alameda, Calif. 94501.

EPICS is free to nonprofit and government sites from Los Alamos National Laboratories and is commercially available from Kinetic Systems Corp., 7308 South Alton Way, Bldg. 2, Engelwood, Colo. 80112.

Both Tcl and Tk are freely available in the public domain. For further information on these languages, see B. B. Welsh, Practical Programming in Tcl and Tk (Prentice-Hall, Upper Saddle River, N.J., 1995).

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Figures (21)

Fig. 1
Fig. 1

Functional layout of the electrical system for the MPIA AO system.

Fig. 2
Fig. 2

Shack–Hartmann wave-front sensor configurations for MPIA AO.

Fig. 3
Fig. 3

Superposition of the wave-front sensor view of the actuator positions (*), subaperture centers for the finest lenslet array (+), and aperture for the geometry of the ALFA optical system. The axis scales are in image pixels.

Fig. 4
Fig. 4

Logical control architecture of the ALFA AO system.

Fig. 5
Fig. 5

Logical control architecture of the tip–tilt control loop.

Fig. 6
Fig. 6

Logical control architecture of the DM control system.

Fig. 7
Fig. 7

Block diagram of the feedback loop of the DM. D/A, digital–analog converter.

Fig. 8
Fig. 8

Mapping from compensated modes to reconstructed modes.

Fig. 9
Fig. 9

Dynamic model of the DM.

Fig. 10
Fig. 10

Dynamic model of the wave-front sensor.

Fig. 11
Fig. 11

AO dynamic models.

Fig. 12
Fig. 12

AO dynamic models after manipulation.

Fig. 13
Fig. 13

Simplified dynamic model.

Fig. 14
Fig. 14

Compensation loop block diagram.

Fig. 15
Fig. 15

Magnitudes of the sensitivity function for 300- and 1200-Hz designs.

Fig. 16
Fig. 16

Top pair: open loop (left) and closed loop (right) images obtained at a wavelength of 2.2 μm of the 0.24-arc min binary ϕ Ursa Majoris (HR 3894, m v = 4.6). Observation was made through clouds in 1.7 arc min seeing on 1 March 1997. The AO system operated at a 900-Hz frame rate and corrected 20 modes. The data represent the sum of several exposures for a total exposure time of 50 s. The two components of the binary are clearly resolved. Bottom pair: open loop (left) and closed loop (right) images obtained at a wavelength of 2.2 μm of the star 14Peg (m v = 5.04) obtained on 21 July 1997. For the data 17 modes were corrected at a loop frequency of 100 Hz. The Strehl ratio improved from 2.4% open loop to 20% closed loop. The seeing was 0.85 arc min. The data represent the sum of 100 exposures at 0.2 s. No image restoration techniques have been applied to these data.

Fig. 17
Fig. 17

Spectra of focus-mode coefficients with the AO loop open and closed.

Fig. 18
Fig. 18

Transfer function computed from observed data.

Fig. 19
Fig. 19

Wave-front control processor box at lower left, along with one of the authors (JN).

Fig. 20
Fig. 20

Top level of software control between the RTC and the entire AO system. GUI, graphic user interface; LGS, laser guide star; TCS, telescope control system.

Fig. 21
Fig. 21

Data flow diagram for obtaining off-line data from the wave-front control processor.

Tables (3)

Tables Icon

Table 1 Summary of Requirements for the MPIA AO System

Tables Icon

Table 2 Compensation Parameters and Disturbance Rejection Bandwidths for Various Wave-Front Sensor Camera Frame Rates

Tables Icon

Table 3 Allocation of Wave-Front Control Functions in a DSP-Based Processor

Equations (26)

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

e x ,   y = s x ,   y + d x ,   y .
a ,   b = A a x ,   y b x ,   y d x d y .
g = W e
s = u
s x ai ,   y ai = α u i     i = 1 , ,   N a ,
d ρ ,   θ = n , m   d n , m Z n , m ρ ,   θ .
s ρ ,   θ = n , m   s n , m Z n , m ρ ,   θ .
u n , m = Z n , m ρ 1 ,   θ 1 Z n , m ρ N a ,   θ N a ,
s n , m = u n , m Z n , m .
u = - n = 2 N c m   d n , m u n , m ,
e = n = 2 N c m   d n , m Z n , m - s n , m + n = N c + 1 m   d n , m Z n , m .
μ = μ 2 , - 2 μ 2 , 0 μ 2 , 2 μ 3 , - 3 μ 3 , - 1 M = u 2 , - 2 u 2 , 0 u 2 , 2 u 3 , - 3 u 3 , - 1   .
u = M μ .
s = γ u n , m = γ Z n , m - Z n , m - u n , m γ Z n , m ,
e = s γ Z n , m .
g n , m = γ W u n , m .
e = s = n = 2 N c m   μ n , m u n , m n = 2 N c m   μ n , m Z n , m .
g = n = 2 N c m   μ n , m W u n , m = n = 2 N c m   μ n , m g n , m .
μ = μ 2 , - 2 μ 2 , 0 μ 2 , 2 μ 3 , - 3 μ 3 , - 1 G = g 2 , - 2 g 2 , 0 g 2 , 2 g 3 , - 3 g 3 , - 1   .
g = G μ .
μ = G T G - 1 Gg Hg ,
H W · M = I .
P z = 1 - δ 1 - γ z - γ z l z - δ I ,
γ = exp - T s / τ a - exp τ c / τ a / 1 - exp τ c / τ a , δ = exp - T s / τ a .
m = 1 1 + p z k z Id = S z Id ,
k z = k 0 z - α z - 1 z - b z - a 1 - a 1 - α 1 - b .

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