Abstract

In adaptive optics (AO) the deformable mirror (DM) dynamics are usually neglected because, in general, the DM can be considered infinitely fast. Such assumption may no longer apply for the upcoming Extremely Large Telescopes (ELTs) with DM that are several meters in diameter with slow and/or resonant responses. For such systems an important challenge is to design an optimal regulator minimizing the variance of the residual phase. In this contribution, the general optimal minimum-variance (MV) solution to the full dynamical reconstruction and control problem of AO systems (AOSs) is established. It can be looked upon as the parent solution from which simpler (used hitherto) suboptimal solutions can be derived as special cases. These include either partial DM-dynamics-free solutions or solutions derived from the static minimum-variance reconstruction (where both atmospheric disturbance and DM dynamics are neglected altogether). Based on a continuous stochastic model of the disturbance, a state-space approach is developed that yields a fully optimal MV solution in the form of a discrete-time linear-quadratic-Gaussian (LQG) regulator design. From this LQG standpoint, the control-oriented state-space model allows one to (1) derive the optimal state-feedback linear regulator and (2) evaluate the performance of both the optimal and the sub-optimal solutions. Performance results are given for weakly damped second-order oscillatory DMs with large-amplitude resonant responses, in conditions representative of an ELT AO system. The highly energetic optical disturbance caused on the tip/tilt (TT) modes by the wind buffeting is considered. Results show that resonant responses are correctly handled with the MV regulator developed here. The use of sub-optimal regulators results in prohibitive performance losses in terms of residual variance; in addition, the closed-loop system may become unstable for resonant frequencies in the range of interest.

© 2010 Optical Society of America

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References

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    [CrossRef]

2009

2008

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express 16, 87-97 (2008).
[CrossRef] [PubMed]

A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515-525 (2008).
[CrossRef]

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Globally optimal minimum variance control in adaptive optical systems with mirror dynamics,” Proc. SPIE 7015, 70151F (2008).
[CrossRef]

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

2007

2006

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

2004

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]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman-filter-based control for adaptive optics,” Proc. SPIE 5490, 1414-1425 (2004).
[CrossRef]

1995

1994

E. Gendron and P. Lena, “Astronomical adaptive optics. 1: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

1992

1976

1974

Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and separation in stochastic control,” IEEE Trans. Autom. Control 19, 494-500 (1974).
[CrossRef]

1961

P. D. Joseph and J. T. Tou, “On linear control theory,” AIEE Trans. Appl. Industry 80, 193-196 (1961).

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

Arsenault, R.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Bar-Shalom, Y.

Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and separation in stochastic control,” IEEE Trans. Autom. Control 19, 494-500 (1974).
[CrossRef]

Beghi, A.

Biasi, R.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Bihan, D. L.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Cenedese, A.

Conan, J.-M.

Conzelmann, R.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Correia, C.

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Globally optimal minimum variance control in adaptive optical systems with mirror dynamics,” Proc. SPIE 7015, 70151F (2008).
[CrossRef]

de Lesegno, P. V.

Doelman, N.

Donaldson, R.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Dournaux, J. L.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Duchateau, M.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Fedrigo, E.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Fusco, T.

Gallieni, D.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Gasmi, R.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Gendron, E.

E. Gendron and P. Lena, “Astronomical adaptive optics. 1: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Gikhman, I. I.

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

Hammer, F.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Herrmann, J.

Hinnen, K.

Hubin, N.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Jagourel, P.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Joseph, P. D.

P. D. Joseph and J. T. Tou, “On linear control theory,” AIEE Trans. Appl. Industry 80, 193-196 (1961).

Kulcsár, C.

Lazzarini, P.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Le Roux, B.

Lena, P.

E. Gendron and P. Lena, “Astronomical adaptive optics. 1: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

Looze, D. P.

Macintosh, B. A.

Madec, P.-Y.

Masiero, A.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

Mugnier, L. M.

Noll, R. J.

Oberti, S.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Paschall, R.

R. Paschall, M. Von Bokern, and B. Welsh, “Design of a linear quadratic Gaussian controller for an adaptive optics system,” in Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, (IEEE, 1991), pp. 1761-1769.
[CrossRef]

Petit, C.

C. Petit, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, and T. Fusco, “First laboratory validation of vibration filtering with LQG control law for adaptive optics,” Opt. Express 16, 87-97 (2008).
[CrossRef] [PubMed]

C. Kulcsár, H.-F. Raynaud, C. Petit, J.-M. Conan, and P. V. de Lesegno, “Optimal control, observers and integrators in adaptive optics,” Opt. Express 14, 7464-7476 (2006).
[CrossRef] [PubMed]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman-filter-based control for adaptive optics,” Proc. SPIE 5490, 1414-1425 (2004).
[CrossRef]

C. Petit, “Etude de la commande optimale en OA et OAMC, validation numérique et expérimentale,” Ph.D. thesis (E. D. Galilée, Univ. Paris XIII, 2006).

Poyneer, L. A.

Quiros-Pacheco, F.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman-filter-based control for adaptive optics,” Proc. SPIE 5490, 1414-1425 (2004).
[CrossRef]

Raynaud, H.-F.

Riccardi, A.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Roddier, F.

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

Rousset, G.

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman-filter-based control for adaptive optics,” Proc. SPIE 5490, 1414-1425 (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]

Sedghi, B.

B. Sedghi, “E-ELT main axis control analysis,” Issue 3, Tech. Rep. (European Southern Observatory, 2007).

Sinquin, J. C.

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

Skorokhod, A. V.

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

Söderström, T.

T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
[CrossRef]

Stroebele, S.

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

Tou, J. T.

P. D. Joseph and J. T. Tou, “On linear control theory,” AIEE Trans. Appl. Industry 80, 193-196 (1961).

Tse, E.

Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and separation in stochastic control,” IEEE Trans. Autom. Control 19, 494-500 (1974).
[CrossRef]

Véran, J.-P.

Verhaegen, M.

Von Bokern, M.

R. Paschall, M. Von Bokern, and B. Welsh, “Design of a linear quadratic Gaussian controller for an adaptive optics system,” in Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, (IEEE, 1991), pp. 1761-1769.
[CrossRef]

Welsh, B.

R. Paschall, M. Von Bokern, and B. Welsh, “Design of a linear quadratic Gaussian controller for an adaptive optics system,” in Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, (IEEE, 1991), pp. 1761-1769.
[CrossRef]

AIEE Trans. Appl. Industry

P. D. Joseph and J. T. Tou, “On linear control theory,” AIEE Trans. Appl. Industry 80, 193-196 (1961).

Astron. Astrophys.

E. Gendron and P. Lena, “Astronomical adaptive optics. 1: Modal control optimization,” Astron. Astrophys. 291, 337-347 (1994).

IEEE Trans. Autom. Control

Y. Bar-Shalom and E. Tse, “Dual effect, certainty equivalence, and separation in stochastic control,” IEEE Trans. Autom. Control 19, 494-500 (1974).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Proc. SPIE

R. Gasmi, J. C. Sinquin, P. Jagourel, J. L. Dournaux, D. L. Bihan, and F. Hammer, “Modeling of a large deformable mirror for future E-ELT,” Proc. SPIE 7017, 70171Z (2008).
[CrossRef]

R. Arsenault, R. Biasi, D. Gallieni, A. Riccardi, P. Lazzarini, N. Hubin, E. Fedrigo, R. Donaldson, S. Oberti, S. Stroebele, R. Conzelmann, and M. Duchateau, “A deformable secondary mirror for the VLT,” Proc. SPIE 6272, 62720V (2006).
[CrossRef]

C. Correia, H.-F. Raynaud, C. Kulcsár, and J.-M. Conan, “Globally optimal minimum variance control in adaptive optical systems with mirror dynamics,” Proc. SPIE 7015, 70151F (2008).
[CrossRef]

C. Petit, F. Quiros-Pacheco, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, T. Fusco, and G. Rousset, “Kalman-filter-based control for adaptive optics,” Proc. SPIE 5490, 1414-1425 (2004).
[CrossRef]

Other

C. Petit, “Etude de la commande optimale en OA et OAMC, validation numérique et expérimentale,” Ph.D. thesis (E. D. Galilée, Univ. Paris XIII, 2006).

Front matter, Proc. SPIE 7018, 701801 (2008).

R. Paschall, M. Von Bokern, and B. Welsh, “Design of a linear quadratic Gaussian controller for an adaptive optics system,” in Proceedings of the 30th IEEE Conference on Decision and Control, Vol. 2, (IEEE, 1991), pp. 1761-1769.
[CrossRef]

T. Söderström, Discrete-time Stochastic Systems, Advanced Textbooks in Control and Signal Processing (Springer-Verlag, 2002).
[CrossRef]

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Dover, 1995).

I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes (Dover, 2006).

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

B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods (Dover, 1995).

B. Sedghi, “E-ELT main axis control analysis,” Issue 3, Tech. Rep. (European Southern Observatory, 2007).

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Figures (14)

Fig. 1
Fig. 1

General scheme of AO feedback control loop (closed-loop). Measurements are assumed to be affected by additive uncorrelated Gaussian zero-mean white noise w. Continuous/dashed lines for continuous/discrete-time variables.

Fig. 2
Fig. 2

Simplified temporal diagram of operations in AO. Controls u k are applied at t = k T s over the interval t [ k T s , ( k + 1 ) T s [ . Any other sources of delay are neglected in order to have a total delay of two frames. Example: light is integrated from t [ ( k 2 ) T s , ( k 1 ) T s [ . From t = ( k 1 ) T s onward the detector is read-out and measurements computed. Wavefront reconstruction and control take place together in the same time slot at the end of which the controls u k are ready for application and kept constant by a zero-order hold through t [ k T s , ( k + 1 ) T s [ . The system is said to have a two-frame delay.

Fig. 3
Fig. 3

Complete structure of the LQG regulator. A model of the real system is used by the optimal regulator to find the control decisions that minimize a predefined cost functional. The switch allows for choosing whether the complete-state information (no measurement) or the incomplete-state information (state estimated from noisy measurements) is used.

Fig. 4
Fig. 4

End-to-end simulator configurations. The input disturbance can be either the output of an identified model or a time series with specified PSD.

Fig. 5
Fig. 5

Comparison of the temporal PSD of the atmospheric TT with the temporal PSD of wind-induced TT on the E-ELT M4-M5. Spectra were obtained with the following custom parameters: D = 42 m , seeing = 1.0 mas at 0.5 μ m , thus r 0 0.1 m ; outer scale of turbulence L 0 = 50 m , with three layers of turbulence profile with relative weights {0.67, 0.22, 0.11}, average wind speed V = 12.5 m s and directions θ i = { 0 ° , 45 ° , 90 ° } .

Fig. 6
Fig. 6

Wind-buffeting auto-correlation curve and that of a second-order model that fits it. The auto-correlation of the atmospheric TT is shown for comparison.

Fig. 7
Fig. 7

Bode diagram of the tip/tilt obtained from the finite-element mechanical model. Frequency abscissa numbers are in hertz for the DM with natural frequencies located at f n = { 50 , 150 , 250 } Hz and damping coefficient ξ = 0.01 .

Fig. 8
Fig. 8

Optimal regulator’s rejection transfer functions with the disturbance and the DM models.

Fig. 9
Fig. 9

Suboptimal regulator’s rejection transfer functions with the disturbance and the DM models. T s = 500 Hz .

Fig. 10
Fig. 10

Performance comparison of the optimal and sub-optimal regulators. Analysis versus Monte Carlo simulation. The double arrow indicates the stability region of the sub-optimal solution. Outside these bounds the sub-optimal regulator cannot operate. T s = 500 Hz , δ = 200 nm at the edges of the telescope.

Fig. 11
Fig. 11

DM ringing effect. The difference in managing the strong oscillations likely to appear in weakly damped DM can be taken into account by the optimal regulator (light dashed curve). Sub-optimal solutions tend to present oscillatory responses of large amplitude (dark dashed curve). T s = 500 Hz and f n = 160 Hz .

Fig. 12
Fig. 12

Performance of the optimal regulator for two levels of noise. Modeling errors account for less than 1 mas rms in both cases. The bottom curve with crosses is the same as the bottom curve in Fig. 10. Ninety seconds of simulation are considered.

Fig. 13
Fig. 13

Performance of the optimal regulator with the choice of the disturbance parameters. The bottom curve with crosses is the same as the bottom curve in Fig. 10. Ninety seconds of simulation are considered.

Fig. 14
Fig. 14

Temporal auto-correlation of the various disturbance models used to derive the regulators: nominal, slower, and faster cases. To be compared with the disturbance auto-correlation of the wind-buffeting series WB.

Equations (103)

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J c ( u ) lim τ + 1 τ 0 τ ϕ res ( t ) 2 d t = lim τ + 1 τ 0 τ ϕ tur ( t ) ϕ cor ( t ) 2 d t ,
J c ( u ) k 1 T s k T s ( k + 1 ) T s ϕ res ( t ) 2 d t = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) ϕ cor ( t ) 2 d t .
J c ( u ) = lim M + 1 M k = 0 M J c ( u ) k .
J c ( u ) = a.s. E ( J c ( u ) k ) .
{ x k + 1 = A d x k + B d u k + Γ v k z k = C d x k + D d u k + w k } .
J d ( u ) = lim M + 1 M k = 0 M 1 ( x k T Q x k + u k T R u k + 2 x k T S u k ) ,
P ( Q S S T R ) 0 .
u k = K opt x k ,
u k = K opt x ̂ k | k 1 ,
x ̂ k | k 1 = A d x ̂ k 1 | k 2 + B d u k 1 + L opt ( z k 1 z ̂ k 1 | k 2 ) ,
L opt = A d Σ C d T ( C d Σ C d T + Σ w ) 1
Σ = A d Σ A d T + Σ v A d Σ C d T ( C d Σ C d T + Σ w ) 1 C d Σ A d T .
ϕ cor ( t ) = N u k , k T s t < ( k + 1 ) T s .
J ifm d ( u ) = lim M + 1 M k = 0 M 1 ϕ ¯ k + 1 tur N u k 2 ,
J c ( u ) = J ifm d ( u ) + δ J ( T s ) ,
δ J ( T s ) = lim M + 1 M k = 0 M 1 ( 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) ϕ ¯ k + 1 tur 2 d t ) .
u k = ϴ ϕ ¯ k + 1 = ( N T N ) 1 N T ϕ ¯ k + 1 tur ,
y k = D ϕ ¯ k 1 res + w k = D ϕ ¯ k 1 tur D N u k 2 + w k ,
u k = ϴ ϕ ¯ ̂ k + 1 | Y k tur ,
z k = y k + 1 = D ϕ ¯ k tur D N u k 1 + w k
ϕ ¯ k + 1 tur = A tur ϕ ¯ k tur + v k .
( ϕ ¯ k + 2 tur ϕ ¯ k + 1 tur u k ) = ( A tur 0 0 I 0 0 0 0 0 ) ( ϕ ¯ k + 1 tur ϕ ¯ k tur u k 1 ) + ( 0 0 I ) u k + ( I 0 0 ) v k ,
z k = D ( 0 I N ) x k + w k .
u k = ϴ ( I 0 0 ) x ̂ k | k 1 ,
Q = H T H 0 , S = N H T , R = N T N > 0 ,
J ifm d ( u ) = lim M + 1 M k = 0 M 1 ( ϕ ¯ k + 1 tur u k ) T ( I N N T N T N ) ( ϕ ¯ k + 1 tur u k ) .
ϕ cor = N p ( t ) ,
{ x ̇ m ( t ) = A m x m ( t ) + B m u ( t ) p ( t ) = C m x m ( t ) + D m u ( t ) } ,
ϕ tur ( t ) ϕ cor ( t ) 2 = ϕ tur ( t ) T ϕ tur ( t ) + p ( t ) T N T N p ( t ) 2 ϕ tur ( t ) T N p ( t ) .
p ( t ) = C m e ( t k T s ) A m x k m + ( I + C m e ( t k T s ) A m A m 1 B m ) u k .
J c ( u ) = J tur + J dyn d ( u ) .
arg min U J c ( u ) = arg min U J dyn d ( u ) .
z k crit ( ϕ ¯ k + 1 tur φ ¯ k + 1 tur x k m ) ( 1 T s 0 T s ϕ tur ( k T s + s ) d s 1 T s 0 T s e s A m T C m T N T ϕ tur ( k T s + s ) d s x m ( k T s ) ) ,
J dyn d ( u ) = lim M + 1 M k = 0 M ( z k crit u k ) T ( Q ¯ 1 S ¯ S ¯ T R ¯ ) ( z k crit u k ) .
Q ¯ 1 ( 2 ε N N T 0 0 0 1 ε ( I + 2 B m T A m T A m 1 B m ) I 0 I Q 0 ) 0 ,
S ¯ ( N A m B m S 0 ) , R ¯ = R 0 ,
R 0 1 T s 0 T s ( I + C m e s A m A m 1 B m ) T N T N ( I + C m e s A m A m 1 B m ) d s ,
S 0 1 T s 0 T s e s A m T C m T N T N ( I + C m e s A m A m 1 B m ) d s ,
Q 0 1 T s 0 T s e s A m T C m T N T N C m e s A m d s .
{ x ̇ tur ( t ) = A tur x tur ( t ) + v ( t ) ϕ tur ( t ) = C tur x tur ( t ) } .
u k = K opt x k ,
K opt = ( R + B d T P B d ) 1 ( B d T P A d + S ) ,
P = Q 1 + A d T P A d ( A d T P A d + S ) × ( R + B d T P B d ) 1 ( B d T P A d + S ) .
y k = D 1 T s ( k 2 ) T s ( k 1 ) T s ( ϕ tur ( t ) N p ( t ) ) d t + w k
= D ϕ ¯ k 1 tur D N [ C m T s ( e T s A m I ) A m 1 x k 2 m + ( C m T s ( e T s A m I ) A m 2 B m + I ) u k 2 ] + w k .
x k ( x k + 1 tur ϕ ¯ k + 1 tur φ ¯ k + 1 tur x k m ϕ ¯ k tur x k 1 m u k 1 ) A d ( e T s A tur 0 0 0 0 0 0 Ξ 0 0 0 0 0 0 Φ 0 0 0 0 0 0 0 0 0 e T s A m 0 0 0 0 I 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 ) ,
B d ( 0 0 0 A m 1 ( e T s A m I ) B m 0 0 I ) Γ ( I 0 0 0 I 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 ) ,
C d ( 0 0 0 0 D D N C m T s ( e T s A m I ) A m 1 D N ( C m T s ( e T s A m I ) A m 2 B m + I ) ) , D d = 0 ,
C d crit ( 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 I 0 0 0 ) .
u k = K opt x ̂ k | k 1 .
lim τ + 1 τ 0 τ ϕ tur ( t ) 2 d t = a.s. E ( ϕ tur ( t ) 2 ) ,
lim τ + 1 τ 0 τ ϕ tur ( t ) 2 d t = a.s. trace ( C tur C tur T Σ x ) ,
J c ( u ) = a.s. E ( ϕ tur ( t ) 2 ) + E ( ( z k crit ) T Q ¯ z k crit + u k T R u k 2 ( z k crit ) T S ¯ u k ) = a.s. E ( ϕ tur ( t ) 2 ) + E ( x k T Q x k + x ̂ k | k 1 T K T R K x ̂ k | k 1 + 2 x k T S K x ̂ k | k 1 ) .
ζ k + 1 = A ζ k + η k
Σ ζ = A Σ ζ A T + Σ η ,
E ( ζ T P ζ ) = trace ( P Σ ζ ) .
x k + 1 = A d x k + B d u k + Γ v k = A d x k B d K x ̂ k | k 1 + Γ v k .
x ̂ k + 1 | k = A d x ̂ k | k 1 + B d u k + L ( z k C d x ̂ k | k 1 ) = A d x ̂ k | k 1 + B d u k + L ( C d x k C d x ̂ k | k 1 + w k ) = ( A d B d K L C d ) x ̂ k | k 1 + L C d x k + L w k .
( x k + 1 x ̂ k + 1 | k ) = A ( x k x ̂ k | k 1 ) + ( Γ 0 0 L ) ( v k w k ) ,
A ( A d B d K L C d A d B d K L C d ) .
Σ ζ = ( Σ x Σ x k , x k | k 1 Σ x k , x k | k 1 Σ x k | k 1 )
Σ η ( Γ Σ v Γ T 0 0 L Σ w L T ) ,
J c ( u ) = a.s. E ( ζ T W ζ ) = a.s. E ( ( x k x ̂ k | k 1 ) T ( C tur C tur T + Q S K K T S T K T R K ) ( x k x ̂ k | k 1 ) ) = a.s. trace ( W Σ ζ ) .
p ̈ ( t ) + 2 ξ ω n p ̇ ( t ) + ω n 2 p ( t ) = ω n 2 u ( t ) ,
A m = ( 0 1 ω n 2 2 ξ ω n ) , B m = ( 0 ω n 2 ) ,
C m = ( 1 0 ) , D m = 0 .
p ( t ) = C m e ( t k T s ) A m x k m + ( 0 t k T s C m e v A m B m d v + D m ) u k .
p ( t ) = C m e ( t k T s ) A m x k m + ( I + C m e ( t k T s ) A m A m 1 B m ) u k ,
1 T s k T s ( k + 1 ) T s p ( t ) T N T N p ( t ) d t = x k m , T Q 0 x k m + 2 x k m , T S 0 u k + u k T R 0 u k ,
1 T s k T s ( k + 1 ) T s p ( t ) T N T ϕ tur ( t ) d t = u k T N T ϕ ¯ k + 1 tur + ( x k m + A m 1 B m u k ) T φ ¯ k + 1 tur .
J 0 d ( u ) = lim M + 1 M k = 0 M 1 ( z k crit ) T Q ¯ z k crit + 2 ( z k crit ) T S ¯ u k + u k T R ¯ u k ,
Q ¯ ( 0 0 0 0 0 I 0 I Q 0 ) , S ¯ ( N A m 1 B m S 0 ) , R ¯ R 0 > 0 ,
P ( Q ¯ S ¯ S ¯ T R ¯ ) 0 .
Q ¯ 1 Q ¯ + Q ¯ ε ( 0 0 0 0 0 I 0 I Q 0 ) + ( 2 ε N N T 0 0 0 1 ε ( I + 2 B m T A m T A m 1 B m ) 0 0 0 0 ) 0 ,
P 1 ( Q ¯ 1 S ¯ S ¯ T R ¯ ) 0 .
J c ( u ) k = 1 T s k T s ( k + 1 ) T s ϕ res ( t ) 2 d t = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) 2 d t + ( z k crit u k ) T ( Q ¯ S ¯ S ¯ T R ¯ ) ( z k crit u k ) = 1 T s k T s ( k + 1 ) T s ϕ tur ( t ) 2 d t + J 0 + ( z k crit u k ) T ( Q ¯ 1 S ¯ S ¯ T R ¯ ) ( z k crit u k ) .
P 0 ( Q 0 S 0 S 0 T R 0 ) 0 .
p k = p ( k T s ) = C m x k m + D m u k = 0 ,
i > 0 , d i p d t i ( k T s ) = C m A m i x k m + C m A m i 1 B m u k = 0 .
x ̇ ( t ) = A c x ( t ) + B c u ( t ) + ν ( t ) .
x k + 1 = A d x k + B d u k + v k ,
A d e A c T s , B d 0 T s e A c B c ,
Σ v 0 k T s e t A c Σ ν e t A c T d t .
( x ̇ tur r ̇ tur ) = ( A tur 0 C tur T s 0 ) ( x tur r tur ) + ( η 0 ) ,
( x k + 1 tur r k + 1 tur ) = ( e T s A tur 0 C tur T s ( e T s A tur I ) I ) ( x k tur r k tur ) + ( v k v k r ) .
( x k + 1 tur ϕ k + 1 tur ) = ( e T s A tur 0 C tur T s ( e T s A tur I ) 0 ) ( x k tur ϕ k tur ) + ( v k v k r ) .
( x ̇ tur x ̇ 2 ) = ( A tur 0 1 T s C m T N T C tur A m T ) ( x tur x 2 ) + ( η 0 ) .
A c ( A tur 0 0 1 T s C tur 0 0 1 T s e T s A m T C m T N T C tur 0 A m T ) .
Ξ ( I 0 0 ) e T s A c ( 0 I 0 ) T ,
Φ ( I 0 0 ) e T s A c ( 0 0 I ) T .
Σ v 0 T s e s A c ( I 0 0 ) Σ η ( I 0 0 ) T e s A c T d s .
ϕ cor ( t ) = j = 1 n e v j d j ( t ) ,
d ̈ j ( t ) + 2 ξ j ω j d ̇ j ( t ) + ω j 2 d j ( t ) = ω j 2 e j ( t ) .
ϑ ̇ ( t ) = ( ϑ ̇ 1 ( t ) ϑ ̇ 2 ( t ) ) = ( 0 1 ω j 2 2 ξ j ω j ) ( ϑ 1 ( t ) ϑ 2 ( t ) ) + ( 0 ω j 2 ) e j ( t ) ,
d j ( t ) = ( 1 0 ) ( ϑ 1 ( t ) ϑ 2 ( t ) ) .
x ̇ m ( t ) = ( 0 I Ω 2 2 Ω Λ ) x m ( t ) + ( 0 Ω 2 ) e ( t ) ,
d ( t ) = ( I 0 ) x m ( t ) .
A m = ( 0 I Ω 2 2 Ω Λ ) , B m = ( 0 Ω 2 ) ( V T V ) 1 V T N ,
C m = ( N T N ) 1 N T V ( I 0 ) , D m = 0 .
A m = ( 0 I Ω s 2 2 Ω s Λ s ) ,
B m = ( 0 Ω s 2 ) ( I 0 ) ( V T V ) 1 V T N ,
C m = ( N T N ) 1 N T V ( I 0 0 0 ) ,
D m = ( N T N ) 1 N T V ( 0 I ) .

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