Abstract

The standard adaptive optics system can be viewed as a sampled-data feedback system with a continuous-time disturbance (the incident wavefront from the observed object) and discrete-time measurement noise. A common measure of the performance of adaptive optics systems is the time average of the pupil variance of the residual wavefront. This performance can be related to that of a discrete-time system obtained by lifting the incident and residual wavefronts. The corresponding discrete-time model is derived, and the computation of the adaptive optics system residual variance is based on that model. The predicted variance of a single mode of an adaptive optics system is shown to be the same as that obtained via simulation (as expected). The discrete-time prediction is also shown to be superior to a continuous-time approximation of the adaptive optics system.

© 2007 Optical Society of America

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References

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  1. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).
  2. F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
    [CrossRef]
  3. M. C. Roggemann, and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).
  4. D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
    [CrossRef]
  5. C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. Viaris de Lesegno, "Optimal control, observers and integrators in adaptive optics," Opt. Express 14, 7464-7476.(2006).
    [CrossRef] [PubMed]
  6. Y. Yamamoto, "New approach to sampled-data control systems--a function space method," in Proceedings of the 29th IEEE Conference on Decision and Control, 1990 (IEEE, 1990), pp. 1882-1887.
    [CrossRef]
  7. T. Chen, and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).
  8. T. Chen, "A simple derivation of the H2-optimal sampled-data controllers," Syst. Control Lett. 20, 49-56 (1993).
    [CrossRef]
  9. B. A. Bamieh, and J. B. Pearson, "The H2 problem for sampled-data systems," Syst. Control Lett. 19, 1-12 (1992).
    [CrossRef]
  10. B. A. Bamieh, and J. B. Pearson, "A general framework for linear periodic systems with applications to H∞ sampled-data control," IEEE Trans. Autom. Control 37, 418-435 (1992).
    [CrossRef]
  11. P. P. Karghonekar, and N. Sivashankar, "H2 optimal control for sampled-data systems," Syst. Control Lett. 17, 425-436 (1991).
    [CrossRef]
  12. T. Chen and B. A. Francis, "H2 Optimal sampled-data control," IEEE Trans. Autom. Control 36, 387-397 (1991).
    [CrossRef]
  13. K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
    [CrossRef]
  14. K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).
  15. K. J. Åström, and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1977).
  16. R. J. Vaccaro, Digital Control: a State Space Approach (McGraw-Hill, 1995).
  17. H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).
  18. C. F. Van Loan, "Computing integrals involving the matrix exponential," IEEE Trans. Autom. Control AC-23, 395-404 (1978).
    [CrossRef]
  19. D. P. Atherton, Nonlinear Control Engineering, student ed. (Van Nostrand Rheinhold, 1982).
  20. H. K. Khalil, Nonlinear Systems (Prentice-Hall, 2002).

2006

2005

D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
[CrossRef]

2004

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

1993

T. Chen, "A simple derivation of the H2-optimal sampled-data controllers," Syst. Control Lett. 20, 49-56 (1993).
[CrossRef]

1992

B. A. Bamieh, and J. B. Pearson, "The H2 problem for sampled-data systems," Syst. Control Lett. 19, 1-12 (1992).
[CrossRef]

B. A. Bamieh, and J. B. Pearson, "A general framework for linear periodic systems with applications to H∞ sampled-data control," IEEE Trans. Autom. Control 37, 418-435 (1992).
[CrossRef]

1991

P. P. Karghonekar, and N. Sivashankar, "H2 optimal control for sampled-data systems," Syst. Control Lett. 17, 425-436 (1991).
[CrossRef]

T. Chen and B. A. Francis, "H2 Optimal sampled-data control," IEEE Trans. Autom. Control 36, 387-397 (1991).
[CrossRef]

1978

C. F. Van Loan, "Computing integrals involving the matrix exponential," IEEE Trans. Autom. Control AC-23, 395-404 (1978).
[CrossRef]

Åström, K. J.

K. J. Åström, and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1977).

Atherton, D. P.

D. P. Atherton, Nonlinear Control Engineering, student ed. (Van Nostrand Rheinhold, 1982).

Bamieh, B. A.

B. A. Bamieh, and J. B. Pearson, "A general framework for linear periodic systems with applications to H∞ sampled-data control," IEEE Trans. Autom. Control 37, 418-435 (1992).
[CrossRef]

B. A. Bamieh, and J. B. Pearson, "The H2 problem for sampled-data systems," Syst. Control Lett. 19, 1-12 (1992).
[CrossRef]

Chen, T.

T. Chen, "A simple derivation of the H2-optimal sampled-data controllers," Syst. Control Lett. 20, 49-56 (1993).
[CrossRef]

T. Chen and B. A. Francis, "H2 Optimal sampled-data control," IEEE Trans. Autom. Control 36, 387-397 (1991).
[CrossRef]

T. Chen, and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

Choi, Y. H.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Conan, J.-M.

Francis, B. A.

T. Chen and B. A. Francis, "H2 Optimal sampled-data control," IEEE Trans. Autom. Control 36, 387-397 (1991).
[CrossRef]

T. Chen, and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).

Karghonekar, P. P.

P. P. Karghonekar, and N. Sivashankar, "H2 optimal control for sampled-data systems," Syst. Control Lett. 17, 425-436 (1991).
[CrossRef]

Khalil, H. K.

H. K. Khalil, Nonlinear Systems (Prentice-Hall, 2002).

Kim, N. H.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Kulcsár, C.

Kwakernaak, H.

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

Li, Z.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Looze, D. P.

D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
[CrossRef]

Ogata, K.

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).

Park, J. B.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Park, K. S.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Pearson, J. B.

B. A. Bamieh, and J. B. Pearson, "The H2 problem for sampled-data systems," Syst. Control Lett. 19, 1-12 (1992).
[CrossRef]

B. A. Bamieh, and J. B. Pearson, "A general framework for linear periodic systems with applications to H∞ sampled-data control," IEEE Trans. Autom. Control 37, 418-435 (1992).
[CrossRef]

Petit, C.

Raynaud, H.-F.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Sivan, R.

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

Sivashankar, N.

P. P. Karghonekar, and N. Sivashankar, "H2 optimal control for sampled-data systems," Syst. Control Lett. 17, 425-436 (1991).
[CrossRef]

Vaccaro, R. J.

R. J. Vaccaro, Digital Control: a State Space Approach (McGraw-Hill, 1995).

Van Loan, C. F.

C. F. Van Loan, "Computing integrals involving the matrix exponential," IEEE Trans. Autom. Control AC-23, 395-404 (1978).
[CrossRef]

Viaris de Lesegno, P.

Welsh, B. M.

M. C. Roggemann, and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Wittenmark, B.

K. J. Åström, and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1977).

Yamamoto, Y.

Y. Yamamoto, "New approach to sampled-data control systems--a function space method," in Proceedings of the 29th IEEE Conference on Decision and Control, 1990 (IEEE, 1990), pp. 1882-1887.
[CrossRef]

Automatica

D. P. Looze, "Realization of systems with CCD-based measurements," Automatica 41, 2005-2009 (2005).
[CrossRef]

IEEE Trans. Autom. Control

B. A. Bamieh, and J. B. Pearson, "A general framework for linear periodic systems with applications to H∞ sampled-data control," IEEE Trans. Autom. Control 37, 418-435 (1992).
[CrossRef]

C. F. Van Loan, "Computing integrals involving the matrix exponential," IEEE Trans. Autom. Control AC-23, 395-404 (1978).
[CrossRef]

T. Chen and B. A. Francis, "H2 Optimal sampled-data control," IEEE Trans. Autom. Control 36, 387-397 (1991).
[CrossRef]

Opt. Express

Real-Time Syst.

K. S. Park, J. B. Park, Y. H. Choi, Z. Li, and N. H. Kim, "Design of H2 controllers for sampled-data systems with input time delays," Real-Time Syst. 26, 231-260 (2004).
[CrossRef]

Syst. Control Lett.

T. Chen, "A simple derivation of the H2-optimal sampled-data controllers," Syst. Control Lett. 20, 49-56 (1993).
[CrossRef]

B. A. Bamieh, and J. B. Pearson, "The H2 problem for sampled-data systems," Syst. Control Lett. 19, 1-12 (1992).
[CrossRef]

P. P. Karghonekar, and N. Sivashankar, "H2 optimal control for sampled-data systems," Syst. Control Lett. 17, 425-436 (1991).
[CrossRef]

Other

D. P. Atherton, Nonlinear Control Engineering, student ed. (Van Nostrand Rheinhold, 1982).

H. K. Khalil, Nonlinear Systems (Prentice-Hall, 2002).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford, 1998).

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

M. C. Roggemann, and B. M. Welsh, Imaging through Turbulence (CRC Press, 1996).

Y. Yamamoto, "New approach to sampled-data control systems--a function space method," in Proceedings of the 29th IEEE Conference on Decision and Control, 1990 (IEEE, 1990), pp. 1882-1887.
[CrossRef]

T. Chen, and B. A. Francis, Optimal Sampled-Data Control Systems (Springer-Verlag, 1995).

K. Ogata, Discrete-Time Control Systems (Prentice-Hall, 1987).

K. J. Åström, and B. Wittenmark, Computer-Controlled Systems (Prentice-Hall, 1977).

R. J. Vaccaro, Digital Control: a State Space Approach (McGraw-Hill, 1995).

H. Kwakernaak and R. Sivan, Linear Optimal Control Systems (Wiley-Interscience, 1972).

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Figures (9)

Fig. 1
Fig. 1

Standard AO system with no anisoplanatism. To emphasize the sampled-data system, continuous-time signals are shown with solid lines, while discrete-time signals are shown with dotted lines.

Fig. 2
Fig. 2

Block diagram of the AO system from Fig. 1 in a canonical format with exogenous inputs ζ ( t ) and θ ̃ k , performance output e ( t ) , control input a ̃ k , and measurement y ̃ k .

Fig. 3
Fig. 3

The AO system that is equivalent to Fig. 1 in which the feedback compensator G ̃ K uses the measurement y ̃ k to compute the control a ̃ k . (b) The discrete-time model that corresponds to the AO system.

Fig. 4
Fig. 4

Solid curve, estimated power spectrum of data obtained from a simulated Kolmogorov phase screen that is representative of the Paranal Observatory site, Chile, with wind speed 17.7 m s . Dashed curve, PSD of the incident wavefront model G a in the example.

Fig. 5
Fig. 5

Hybrid Simulink representation of the example AO system. The last (gain) block of the WFS samples the continuous-time frame image at the end of the CCD frame to produce a discrete-time signal.

Fig. 6
Fig. 6

Histogram showing the average residual variance of the simulation example with controller (43), no measurement noise, and incident wavefront model (41) (the bin size is 20 nm 2 ). The discrete-time model prediction of the residual average ( 1152 nm 2 ) is essentially the same as the simulation residual average ( 1149 nm 2 ; the 95% error bar of the simulation average is also shown). The continuous-time model prediction of the average residual variance ( 1048 nm 2 ) is significantly lower than the simulation average.

Fig. 7
Fig. 7

Histogram showing the average residual variance of the simulation with controller (56), 10 nm rms measurement noise, and no incident wavefront (the bin size is 20 nm 2 ). The discrete-time model prediction of the residual average ( 406.5 nm 2 ) is essentially the same as the simulation residual average ( 405.8 nm 2 ; the 95% error bar of the simulation average is also shown). The continuous-time model prediction of the average residual variance ( 495.0 nm 2 ) is significantly higher than the simulation average.

Fig. 8
Fig. 8

Residual variance as a function of gain for the example AO system predicted by the discrete-time model (solid) and the continuous-time model (dashed).

Fig. 9
Fig. 9

Lifted sampled-data AO system (with lifted signals as dashed lines) shown in terms of (a) the system operator and (b) its component operators.

Equations (139)

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x ̇ m ( t ) = A m x m ( t ) + B m a ( t ) ,
s m ( t ) = C m x m ( t ) + D m a ( t ) ,
G m ( s ) = [ A m B m C m D m ] ,
A m R n m × n m , B m R n m × m , C m R q × n m , D m R q × m .
x ̇ a ( t ) = A a x a ( t ) + B a ζ ( t ) ,
w ( t ) = C a x a ( t ) ,
G a ( s ) = [ A a B a C a 0 ] ,
A a R n a × n a , B a R n a × m a , C a R q × n a .
e ( t ) = w ( t ) s m ( t ) R q .
θ ̃ k D θ θ ̃ n k , D θ D θ T = Θ , D θ R p × p .
y ̃ k = g ̃ k + θ ̃ k = g ̃ k + D θ θ ̃ n k ,
g ̃ = S T f g , g ̃ k = g ( k T f ) ,
g = δ τ c G WFS e , g ( t ) = 1 T f H t T f τ c t τ c e ( σ ) d σ .
x ̃ K , k + 1 = A K x ̃ K , k + B K g ̃ k ,
a ̃ k = C K x ̃ K , k + D K g ̃ k ,
G ̃ K ( z ) = [ A K B K C K D K ] ,
A K R n K × n K , B m R n K × p , C m R m × n K , D m R m × p
a = H T f a ̃ R m , a ( t ) = a ̃ k for k T k t < ( k + 1 ) T f .
[ e ( t ) y ̃ k ] = G [ ζ ( t ) θ ̃ k a ̃ k ] = [ G 11 0 G 12 G 21 D θ G 22 ] [ ζ ( t ) θ ̃ n k a ̃ k ] = [ G a 0 G m H T f S T f δ τ c G WFS G a D θ S T f δ τ c G WFS G m H T f ] [ ζ ( t ) θ ̃ n k a ̃ k ] .
J = lim W 1 W 0 W E { e ( σ ) T e ( σ ) } d σ ,
lim k P e ( k T f + τ ) = P e , ss ( τ ) , 0 τ < T f .
P ̱ e , ss = 1 T f 0 T f P e , ss ( τ ) d τ .
J = lim W 1 W 0 W tr [ E { e ( σ ) e ( σ ) T } ] d σ = lim W 1 T 0 W tr [ P e ( σ ) ] d σ ,
J = lim N 1 N k = 0 N tr [ 1 T f 0 T f P e ( σ ) d σ ] = tr [ 1 T f 0 T f lim N 1 N k = 0 N P e ( k T f + τ ) d τ ] = tr [ 1 T f 0 T f P e , ss ( τ ) d τ ] = tr [ P ̱ e , ss ] .
J d = lim N 1 N k = 0 N E { e ̃ k T e ̃ k } .
A = [ A d m 0 0 A d a ] = [ A m d 0 0 0 0 0 A m 2 0 0 0 0 0 A m 1 I 0 0 0 0 0 0 0 A a d 0 0 0 0 0 A a 2 0 0 0 0 0 A a 1 I 0 ] R n aug n aug ,
A m d = e A m T f , A a d = e A a T f ,
A m 2 = C m T f τ c T f e A m ν d ν , A a 2 = C a T f τ c T f e A a ν d ν ,
A m 1 = C m 0 T f τ c e A m ν d ν , A a 1 = C a 0 T f τ c e A a ν d ν .
B 2 d = [ B m 2 d 0 ] = [ B m d B 22 B 21 0 0 0 ] R n aug × m ,
B m d = 0 T f e A m σ B m d σ ,
B 22 = C m T f τ c T f 0 γ e A m σ d σ d γ B m + τ c D m ,
B 21 = C m 0 T f τ c 0 γ e A m σ d σ d γ B m + ( T f τ c ) D m .
B 1 d = [ 0 B a 1 d ] R n aug × m ζ ,
M = B a 1 d B a 1 d T = [ M 11 M 12 M 13 M 12 T M 22 M 23 M 13 T M 23 T M 33 ] R ( n a + 2 p ) × ( n a + 2 p ) ,
Γ a ( t ) = C a 0 t e A a ( t σ ) d σ ,
M 11 = 0 T f e A a γ B a B a T e A a T γ d γ ,
M 22 = Γ a ( τ c ) 0 T f τ c e A a γ B a B a T e A a T γ d γ Γ a T ( τ c ) + 0 τ c Γ a ( γ ) B a B a T Γ a T ( γ ) d γ ,
M 33 = 0 T f τ c Γ a ( γ ) B a B a T Γ a T ( γ ) d γ ,
M 12 = e A a τ c 0 T f τ c e A a γ B a B a T e A a T γ d γ Γ a T ( τ c ) + 0 τ c e A a γ B a B a T Γ a T ( γ ) d γ ,
M 13 = e A a τ c 0 T f τ c e A a γ B a B a T Γ a T ( γ ) d γ ,
M 23 = Γ a ( τ c ) 0 T f τ c e A a γ B a B a T Γ a T ( γ ) d γ .
A ̱ = [ A m 0 B m 0 A a 0 0 0 0 ] R ( n m + n a + m ) × ( n m + n a + m ) ,
C ̱ = [ C m C a D m ] R q × ( n m + n a + m ) .
C 1 d = [ C m 1 d 0 0 C a 1 d 0 0 ] R p e × n aug ,
C 2 d = [ 0 0 1 T f H 0 0 1 T f H ] R q × n aug ,
N = [ C m 1 d C a 1 d D 12 d ] T [ C m 1 d C a 1 d D 12 d ] = 0 T f e A ̱ T γ C ̱ T C ̱ e A ̱ γ d γ .
G ̃ c l ( z ) = [ A c l B c l C c l D c l ] = [ A + B 2 d D K C 2 d B 2 d C K B 1 d B 2 d D K D θ B K C 2 d A K 0 B K D θ C 1 d + D 12 d D K C 2 d D 12 d C K 0 D 12 d D K D θ ] ,
A c l R ( n aug + n K ) × ( n aug + n K ) , B c l R ( n aug + n K ) × ( m ζ + p ) ,
C c l R p e × ( n aug + n K ) , D c l R p e × ( m ζ + p ) .
P c l = A c l P c l A c l T + B c l B c l T ,
J d = tr { C c l P c l C c l T + D c l D c l T } .
J = 1 T f ( J 0 + J d ) ,
J 0 = tr { C a 0 T f 0 σ e A a γ B a B a T e A a T γ d γ d σ C a T } .
C ̂ s m = [ 0 C m 0 0 A m B m 0 0 0 ] , C ̂ s a = [ 0 C a 0 A a ] .
X s m ( Δ ) = e C ̂ s m Δ = [ I X s m 12 ( Δ ) X s m 13 ( Δ ) 0 X s m 22 ( Δ ) X s m 23 ( Δ ) 0 0 I ] ,
X s a ( Δ ) = e C ̂ s a Δ = [ I X s a 12 ( Δ ) 0 X s a 22 ( Δ ) ] .
A m d = X s m 22 ( T f ) , A a d = X s a 22 ( T f ) , B m d = X s m 23 ( T f ) ,
A m 1 = X s m 12 ( β ) , A a 1 = X s a 12 ( β ) ,
B 21 = X s m 13 ( β ) + β D m ,
A m 2 = X s m 12 ( T f ) A m 1 , A a 2 = X s a 12 ( T f ) A a 1 ,
B 22 = X s m 13 ( T f ) B 21 + T f D m .
C ̂ a = [ A a I 0 0 0 A a B a B a T 0 0 0 A a T C a T 0 0 0 0 ] .
X a ( Δ ) = e C ̂ a Δ = [ X a 11 ( Δ ) X a 12 ( Δ ) X a 13 ( Δ ) X a 14 ( Δ ) 0 X a 22 ( Δ ) X a 23 ( Δ ) X a 24 ( Δ ) 0 0 X a 33 ( Δ ) X a 34 ( Δ ) 0 0 0 I ] .
Γ a ( τ c ) = X a 34 T ( T f ) X a 34 T ( β ) ,
W a ( Δ ) = C a X a 33 T ( Δ ) X a 14 ( Δ ) + X a 14 T ( Δ ) X a 33 ( Δ ) C a T ,
M 11 = X a 33 T ( T f ) X a 23 ( T f ) ,
M 22 = Γ a ( τ c ) X a 33 T ( β ) X a 23 ( β ) Γ a T ( τ c ) + W a ( T f ) W a ( β ) ,
M 33 = W a ( β ) ,
M 12 = X a 33 T ( T f ) X a 23 ( β ) Γ a T ( τ c ) + X a 33 T ( T f ) X a 24 ( T f ) X a 33 T ( β ) X a 24 ( β ) ,
M 13 = X a 33 T ( T f ) X a 24 ( β ) ,
M 23 = Γ a ( τ c ) X a 33 T ( β ) X a 24 ( β ) .
C ̂ e = [ A ̱ T C ̱ T C ̱ 0 A ̱ ] , X e ( Δ ) = e C ̂ e Δ = [ X e 11 ( Δ ) X e 12 ( Δ ) 0 X e 22 ( Δ ) ] .
N = [ C m 1 d C a 1 d D 12 d ] T [ C m 1 d C a 1 d D 12 d ] = X e 22 T ( T f ) X e 12 ( T f ) .
J 0 = tr { C a X a 33 T ( T f ) X a 13 ( T f ) C a T } .
G m ( s ) = 1 , G a ( s ) = 450 0.5 s + 1 = [ 2 1 900 0 ] .
G WFS ( s ) = 1 e s T f s T f , T f = 0.001 s , τ c = 0.0006 s .
G K ( z ) = γ K 1.8 z 2 ( z 1 ) ( z + 0.8 ) , γ K = 0.6 .
A = [ A d m 0 0 A d a ] , A d m = [ 0 0 1 0 ] ,
A d a = [ 0.998 0 0 0.539 0 0 0.360 1 0 ] .
B 1 d = [ 0 B a 1 d ] , B a 1 d = [ 0.0316 0 0 0.0119 0.00566 0 0.002275 0.00206 0.00280 ] ,
B 2 d = [ B m 2 d 0 ] , B m 2 d = [ 0.6 0.4 ] .
C 1 d = [ 0 0 28.4 0 0 0 0 0 0 0 ] , D 12 d = [ 0.0316 1.83 × 10 5 ] ,
C 2 d = [ 0 1000 0 0 1000 ] , D θ = 10 .
J d = 1.91 , J 0 = 0.404 , J = 2310 .
S T f I , H T f G ZOH , G ZOH ( s ) = 1 e s T f s T f ,
δ τ c ( s ) = e s τ c .
G K ( z ) = 1 1 z 1 G K 1 ( z ) .
G K c ( s ) = G ZOH ( s ) G K ( e s T f ) = 1 s T f G K 1 ( e s T f ) .
G K c ( s ) = 1 s T f G K 1 ( 1 + T f 2 s 1 T f 2 s ) .
θ c = D θ c θ c n , D θ c = 1 T f D c .
L c ( s ) = G m ( s ) G K c ( s ) H e s τ c ( 1 e s T f s T f ) .
J c = 1 π 0 ( trace { T e ζ ( j ω ) T e ζ T ( j ω ) } + trace { T e θ c ( j ω ) T e θ c T ( j ω ) } ) d ω .
G K ( z ) = γ K 1.9 z 2 ( z 1 ) ( z + 0.9 ) , γ K = 0.9 .
L : L 2 m ( R ) l 2 m ( Z , ) : v ¯ L v
with v ¯ k = v ( k T f + σ ) for 0 σ < T f .
G L = [ G L , 11 0 G L , 12 G L , 21 D θ G L , 22 ] = [ L G a L 1 0 L G m H T f S T f δ τ c G WFS G a L 1 D θ S T f δ τ c G WFS G m H T f ] .
A m d = e A m T f , A a d = e A a T f , B m d = 0 T f e A m σ B m d σ ,
B ¯ a 1 : K m a R n a , ( B ¯ a 1 ζ ¯ ) ( t ) = 0 T f e A a ( T f σ ) B a ζ ¯ ( σ ) d σ ,
C ¯ m 1 : R n m K q , ( C ¯ m 1 x m ) ( t ) = C m e A m t x m ,
C ¯ a 1 : R n a K q , ( C ¯ a 1 x a ) ( t ) = C a e A a t x a ,
D ¯ 11 : K m a K q , ( D ¯ 11 ζ ¯ ) ( t ) = C a 0 t e A a ( t σ ) B a ζ ¯ ( σ ) d σ ,
D ¯ 12 : R m K q , ( D ¯ 12 a ̃ ) ( t ) = ( D m C m 0 t e A m σ B m d σ ) a ̃ .
G L , 22 ( s ) = [ A m d 0 0 B 2 d A m 2 0 0 B 22 A m 1 I 0 B 21 0 0 1 T f H 0 ] ,
A m 2 = C m T f τ c T f exp ( A m σ ) d σ ,
A m 1 = C m 0 T f τ c exp ( A m σ ) d σ ,
B 22 = C m T f τ c T f 0 γ exp ( A m σ ) d σ d γ B m + τ c D m ,
B 21 = C m 0 T f τ c 0 γ exp ( A m σ ) d σ d γ B m + ( T f τ c ) D m .
G L , 21 = [ A a d 0 0 B ¯ 1 d A a 2 0 0 B ¯ 12 A a 1 I 0 B ¯ 11 0 0 1 T f H 0 ] ,
A a d = e A a T f , A a 2 = C a T f τ c T f exp ( A a ν ) d ν ,
A a 1 = C a 0 T f τ c exp ( A a ν ) d ν ,
B ¯ 1 d : K m a R n a , B ¯ 1 d ζ ¯ = 0 T f exp ( A a ( T f σ ) ) B a ζ ¯ ( σ ) d σ ,
B ¯ 12 : K m a R p ,
B ¯ 12 ζ ¯ = C a T f τ c T f 0 γ exp ( A a ( γ σ ) ) B a ζ ¯ ( σ ) d σ d γ ,
B ¯ 11 : K m a R p ,
B ¯ 11 ζ ¯ = C a 0 T f τ c 0 γ exp ( A a ( γ σ ) ) B a ζ ¯ ( σ ) d σ d γ .
ζ ( t ) = { ζ 0 ( t ) 0 t < T f 0 otherwise } , ζ 0 K .
e ( t ) = 0 t C a exp ( A a ( t σ ) ) B a ζ ( σ ) d σ ,
g ̃ k = 1 T f H ( k 1 ) T f k T f e ( τ τ c ) d τ .
e ( t τ c ) = { 0 t < τ c C a 0 t τ c exp ( A a ( t τ c σ ) ) B a ζ 0 ( σ ) d σ τ c t < T f + τ c C a e A a ( t T f τ c ) 0 T f exp ( A a ( T f σ ) ) B a ζ 0 ( σ ) d σ T f + τ c t } .
g ̃ k = 1 T f H ( A a 1 + A a 2 A a d ) A a d k 3 B ̱ 1 d ζ 0 .
B ¯ 1 = [ 0 B ¯ a 1 ] = [ 0 0 0 B ¯ 1 d B ¯ 12 B ¯ 11 ] .
C ¯ 1 = [ C ¯ m 1 C ¯ a 1 ] = [ C ¯ m 1 0 0 C ¯ a 1 0 0 ] ,
C ¯ 1 = [ C ¯ m 1 C ¯ a 1 ] ,
C ¯ m 1 : R n m K q , ( C ¯ m 1 x m ) ( t ) = C m e A m t x m ,
C ¯ a 1 : R n a K q , ( C ¯ a 1 x a ) ( t ) = C a e A a t x a .
D ¯ 11 : K m a K q ,
( D ¯ 11 ζ ¯ ) ( t ) = C a 0 t exp ( A a ( t σ ) ) B a ζ ¯ ( σ ) d σ ,
D ¯ 12 : R m K q ,
( D ¯ 12 a ̃ ) ( t ) = ( D m C m 0 t exp ( A m σ ) B m d σ ) a ̃ ,
D ¯ 1 = [ D ¯ 11 D ¯ 12 ] , D θ = Θ .
B ¯ 1 d * ( t ) = B a T e A a T ( T f t ) ,
B ¯ 11 * ( t ) = 1 ( β t ) β T f B a T e A a T ( γ t ) C a T d γ + 1 ( t β ) t T f B a T e A a T ( γ t ) C a T d γ ,
B ¯ 12 * ( t ) = 1 ( β t ) t T f B a T e A a T ( γ t ) C a T d γ ,
B 1 d B 1 d T = B ¯ 1 B ¯ 1 * = [ 0 0 0 B ¯ a 1 B ¯ a 1 * ] ,
B ¯ a 1 B ¯ a 1 * = [ B ¯ 1 d B ¯ 1 d * B ¯ 1 d B ¯ 12 * B ¯ 1 d B ¯ 11 * B ¯ 12 B ¯ 1 d * B ¯ 12 B ¯ 12 * B ¯ 12 B ¯ 11 * B ¯ 11 B ¯ 1 d * B ¯ 11 B ¯ 12 * B ¯ 11 B ¯ 11 * ] = M .
M 22 = 0 β β T f C a e A a ( γ σ ) B a β T f B a T e A a T ( α σ ) C a T d α d γ d σ + β T f σ T f C a e A a ( γ σ ) B a σ T f B a T e A a T ( α σ ) C a T d α d γ d σ = 0 β ( 0 T f β C a e A a ν d η ) e A a ( β σ ) B a B a T e A a T ( β σ ) ( β T f e A a T ρ C a T d ρ ) d σ + 0 β ( T f ν T f C a e A a ( γ T f + ν ) d γ ) B a B a T ( T f ν T f e A a T ( α T f + ν ) C a T d α ) d ν .
C ̱ 1 = C ¯ 1 = [ C m C a ] exp [ A m 0 0 A a ] t ,
D ̱ 11 = D ¯ 11 , D ̱ 12 = D ¯ 12 .

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