rufus fraanje p.r.fraanje@tudelftmverhaegen/n4ci/slides_ao/lecture7.pdf · lecture 7: deformable...
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Lecture 7: Deformable mirror modeling and control
Rufus [email protected]
TU Delft — Delft Center for Systems and Control
Jun. 20, 2011
1 / 38
Outline
1 Adjustable optics in nature and engineering;
2 Deformable mirrors — modeling;
3 Control of AO systems;
4 Exams
2 / 38
Outline
1 Adjustable optics in nature and engineering;
2 Deformable mirrors — modeling;
3 Control of AO systems;
4 Exams
2 / 38
Outline
1 Adjustable optics in nature and engineering;
2 Deformable mirrors — modeling;
3 Control of AO systems;
4 Exams
2 / 38
Outline
1 Adjustable optics in nature and engineering;
2 Deformable mirrors — modeling;
3 Control of AO systems;
4 Exams
2 / 38
Adjustable optics
1 Adjustable pupils (size/shape);
2 Adjustable lenses (focus);
3 Adjustable pointing direction (tip/tilt);
4 Deformable mirrors.
3 / 38
Adjustable optics
1 Adjustable pupils (size/shape);
2 Adjustable lenses (focus);
3 Adjustable pointing direction (tip/tilt);
4 Deformable mirrors.
3 / 38
Adjustable optics
1 Adjustable pupils (size/shape);
2 Adjustable lenses (focus);
3 Adjustable pointing direction (tip/tilt);
4 Deformable mirrors.
3 / 38
Adjustable optics
1 Adjustable pupils (size/shape);
2 Adjustable lenses (focus);
3 Adjustable pointing direction (tip/tilt);
4 Deformable mirrors.
3 / 38
Adjustable optics — EngineeringAdjustable pupils: (Source: Wikipedia)
“Minolta” lens, f/1.5 − f/16
Numerical aperture (NA):
NA =fD
f/NA = D
NA of E-ELT: 17.7, NA of human eye?5 / 38
Adjustable optics — EngineeringAdjustable pupils: (Source: Wikipedia)
“Minolta” lens, f/1.5 − f/16 f/32 (top-left) and f/5 (bottom-right)
Numerical aperture (NA):
NA =fD
f/NA = D
NA of E-ELT: 17.7, NA of human eye?5 / 38
Adjustable optics — EngineeringAdjustable pupils: (Source: Wikipedia)
“Minolta” lens, f/1.5 − f/16 f/32 (top-left) and f/5 (bottom-right)
Numerical aperture (NA):
NA =fD
f/NA = D
NA of E-ELT: 17.7, NA of human eye?5 / 38
Adjustable optics — EngineeringAdjustable pupils: (Source: Wikipedia)
“Minolta” lens, f/1.5 − f/16 f/32 (top-left) and f/5 (bottom-right)
Numerical aperture (NA):
NA =fD
f/NA = D
NA of E-ELT: 17.7, NA of human eye?5 / 38
Adjustable optics — EngineeringAdjustable pupils: (Source: Wikipedia)
“Minolta” lens, f/1.5 − f/16 f/32 (top-left) and f/5 (bottom-right)
Numerical aperture (NA):
NA =fD
f/NA = D
NA of E-ELT: 17.7, NA of human eye? Answer:≈ 1.70.7 = 2.4
5 / 38
Adjustable optics — EngineeringAdjustable lenses (Source: http://www.nedinsco.nl):
7 / 38
Adjustable optics — Engineering
Adjustable lenses (Source: Jeong et al. (2005), c.f. http://biopoems.berkeley.edu)
8 / 38
Adjustable optics — EngineeringTelescope pointing (Source:
http://www.dailygalaxy.com/photos/uncategorized/2007/08/12/alma_2.jpg):
ALMA: Atacama Large Millimeter Array (66 telescopes when completed)
10 / 38
Adjustable optics — NatureMirrors in scallops (from: Animal Eyes, Land&Nilsson, 2002)
(Source: Wikipedia – Sint-Jakobsschelp)
11 / 38
Adjustable optics — Engineering
Adjustable mirrors (Electrostatic deformable mirror, Source: Flexible Optical, BV, Rijswijk):
12 / 38
Deformable mirrors
• Segmented facesheet (piston):
Liquid Crystals, DLP’svery high actuator density
• Segmented facesheet (piston/tip/tilt):
very large (M1’s) or MEMS deviceshigh actuator density
• Continuous facesheet:various types (electrostatic, reluctance,piezo, ...)medium sized diameters (cm range)‘no’ photon loss, low spatial aliasing
13 / 38
Deformable mirrorsDiffraction effects of hexagonal segmented mirrors: Point spread functions:
14 / 38
Deformable mirrorsDiffraction effects of hexagonal segmented mirrors: Obtained images:
Circular aperture
Hexagonal aperture (70% fill)
Hexagonal aperture (100% fill)
Hexagonal aperture (90% fill)
14 / 38
Deformable mirrors
Hexagonal grids
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
−8 −6 −4 −2 0 2 4 6 8
−8
−6
−4
−2
0
2
4
6
8
−15 −10 −5 0 5 10 15
−15
−10
−5
0
5
10
15
6 corner points:
[
±10
]
,
[
±12
±
√
32
]
hex- to rect.-coord.:[
xr
yr
]
=
[ 32
32
−
√
32
√
32
] [
xh
yh
]
rotationally-symmetric over60o angles.
15 / 38
Deformable mirrors — modeling
Let w(x , y , t) be the out-of-plane deformation and p(x , y , t) the distributedpressure from, e.g., actuators.
• Membrane mirror:
( 1c2
∂2
∂t2−∇2
)w(x , y , t) = p(x , y , t)
+boundary/initial conditions, where ∇2 = ∂2/∂x2 + ∂2/∂y2
• Thin plate mirror:
(ρh
∂2
∂t2+ EI∇4
)w(x , y , t) = p(x , y , t)
+boundary/initial conditions, where ∇4 = ∂4/∂x4 + 2∂4/∂x2∂y2 + ∂4/∂y4.
c.f. Timoshenko et al., Theory of Plates and Shells, McGraw-Hill, 1959.
16 / 38
Deformable mirrors — modeling
Let w(x , y , t) be the out-of-plane deformation and p(x , y , t) the distributedpressure from, e.g., actuators.
• Membrane mirror:
( 1c2
∂2
∂t2−∇2
)w(x , y , t) = p(x , y , t)
+boundary/initial conditions, where ∇2 = ∂2/∂x2 + ∂2/∂y2
• Thin plate mirror:
(ρh
∂2
∂t2+ EI∇4
)w(x , y , t) = p(x , y , t)
+boundary/initial conditions, where ∇4 = ∂4/∂x4 + 2∂4/∂x2∂y2 + ∂4/∂y4.
c.f. Timoshenko et al., Theory of Plates and Shells, McGraw-Hill, 1959.
16 / 38
Deformable mirrors — modelingSolutions obained by (dynamic or static case):
• finite difference approximation:
∂w∂x
∣∣∣∣x,y,t
≈w(x + δx , y , t)− w(x − δx , y , t)
2δx
• finite element approximation:
w(x , y , t) =n∑
k=1
uk(t)vk(x , y), vk(x , y) defined over ’finite element’
• modal approximation:
w(x , y , t) =n∑
k=1
uk(t)φk(x , y), φk(x , y) a solution of homogeneous PDE
• analytical (only for specific forcing functions p(x , y , t) and boundaryconditions), e.g., by separation of variables.
17 / 38
Deformable mirrors — modelingSolutions obained by (dynamic or static case):
• finite difference approximation:
∂w∂x
∣∣∣∣x,y,t
≈w(x + δx , y , t)− w(x − δx , y , t)
2δx
• finite element approximation:
w(x , y , t) =n∑
k=1
uk(t)vk(x , y), vk(x , y) defined over ’finite element’
• modal approximation:
w(x , y , t) =n∑
k=1
uk(t)φk(x , y), φk(x , y) a solution of homogeneous PDE
• analytical (only for specific forcing functions p(x , y , t) and boundaryconditions), e.g., by separation of variables.
17 / 38
Deformable mirrors — modelingSolutions obained by (dynamic or static case):
• finite difference approximation:
∂w∂x
∣∣∣∣x,y,t
≈w(x + δx , y , t)− w(x − δx , y , t)
2δx
• finite element approximation:
w(x , y , t) =n∑
k=1
uk(t)vk(x , y), vk(x , y) defined over ’finite element’
• modal approximation:
w(x , y , t) =n∑
k=1
uk(t)φk(x , y), φk(x , y) a solution of homogeneous PDE
• analytical (only for specific forcing functions p(x , y , t) and boundaryconditions), e.g., by separation of variables.
17 / 38
Deformable mirrors — modelingSolutions obained by (dynamic or static case):
• finite difference approximation:
∂w∂x
∣∣∣∣x,y,t
≈w(x + δx , y , t)− w(x − δx , y , t)
2δx
• finite element approximation:
w(x , y , t) =n∑
k=1
uk(t)vk(x , y), vk(x , y) defined over ’finite element’
• modal approximation:
w(x , y , t) =n∑
k=1
uk(t)φk(x , y), φk(x , y) a solution of homogeneous PDE
• analytical (only for specific forcing functions p(x , y , t) and boundaryconditions), e.g., by separation of variables.
17 / 38
Deformable mirrors — modelingStatic approximations: influence functions
w(x , y) =
n∑
k
h(x , y , sk , tk)uk
• Cubic influence function:
h(x , y , s, t) = A(1 − 3(x − s)2 + 2(x − s)3
)(1 − 3(y − t)2 + 2(y − t)3
)
• Gaussian influence function:
h(x , y , s, t) = Aer2/σ2
, r =√
(x − s)2 + (y − t)2
• Circular clamped edge faceplate1 (polar coordinates):
h(r , φ, ρ, ψ) = A(1 − r 2)(1 − ρ2)(r 2 + ρ2 − 2rρ cos(φ− ψ))
x lnr 2 + ρ2 − 2rφ cos(φ− ψ)
1 + r 2ρ2 − 2rρ cos(φ− ψ)
1Loktev et al., Comparison study of the performance of piston, thin plate and membranemirrors for correction of turbulence-induced phase distortions, Optics Communications 192,2001
18 / 38
Deformable mirrors — modelingStatic approximations: influence functions
w(x , y) =
n∑
k
h(x , y , sk , tk)uk
• Cubic influence function:
h(x , y , s, t) = A(1 − 3(x − s)2 + 2(x − s)3
)(1 − 3(y − t)2 + 2(y − t)3
)
• Gaussian influence function:
h(x , y , s, t) = Aer2/σ2
, r =√
(x − s)2 + (y − t)2
• Circular clamped edge faceplate1 (polar coordinates):
h(r , φ, ρ, ψ) = A(1 − r 2)(1 − ρ2)(r 2 + ρ2 − 2rρ cos(φ− ψ))
x lnr 2 + ρ2 − 2rφ cos(φ− ψ)
1 + r 2ρ2 − 2rρ cos(φ− ψ)
1Loktev et al., Comparison study of the performance of piston, thin plate and membranemirrors for correction of turbulence-induced phase distortions, Optics Communications 192,2001
18 / 38
Deformable mirrors — modelingStatic approximations: influence functions
w(x , y) =
n∑
k
h(x , y , sk , tk)uk
• Cubic influence function:
h(x , y , s, t) = A(1 − 3(x − s)2 + 2(x − s)3
)(1 − 3(y − t)2 + 2(y − t)3
)
• Gaussian influence function:
h(x , y , s, t) = Aer2/σ2
, r =√
(x − s)2 + (y − t)2
• Circular clamped edge faceplate1 (polar coordinates):
h(r , φ, ρ, ψ) = A(1 − r 2)(1 − ρ2)(r 2 + ρ2 − 2rρ cos(φ− ψ))
x lnr 2 + ρ2 − 2rφ cos(φ− ψ)
1 + r 2ρ2 − 2rρ cos(φ− ψ)
1Loktev et al., Comparison study of the performance of piston, thin plate and membranemirrors for correction of turbulence-induced phase distortions, Optics Communications 192,2001
18 / 38
Deformable mirrors — modeling
(Continued)
• Supported edge faceplate (c.f. Loktev et al., 2001)
• Free edge faceplate (idem)
19 / 38
Deformable mirrors — modeling
Loktev et al. 2001, various actuator configurations and plate models arecompared:
Hexagonal- Square- Ring-layout
20 / 38
Deformable mirrors — modelingResidual wavefront aberations for various type of mirrors each with 37 actuators (Credit:
Loktev et al, 2001, Table 1 (adjusted))
21 / 38
Deformable mirrors — modelingExample: finite difference discretization over hexagonal grid (Fraanje, 2010)
Thin plate model:(
EI∇4(1 + η∂
∂t) + ρh
∂2
∂t2
)
w(x , y , t) = p(x , y , t)
Actuator model at location (i , j):
p(xi , yi , t) =1A
f (i , j , t)− c(1 + ζ∂
∂t)w(xi , yi , t)
︸ ︷︷ ︸
spring/damper parallel to force
Surface area of one hexagon A = 3√
38 ∆2, where ∆ distance between two
neighboring actuators.22 / 38
Deformable mirrors — modelingExample: finite difference discretization over hexagonal grid (Fraanje, 2010)
Thin plate model:(
EI∇4(1 + η∂
∂t) + ρh
∂2
∂t2
)
w(x , y , t) = p(x , y , t)
Actuator model at location (i , j):
p(xi , yi , t) =1A
f (i , j , t)− c(1 + ζ∂
∂t)w(xi , yi , t)
︸ ︷︷ ︸
spring/damper parallel to force
Surface area of one hexagon A = 3√
38 ∆2, where ∆ distance between two
neighboring actuators.22 / 38
Deformable mirrors — modeling
Discretization of biharmonic operator ∇4 over hexagonal grid:
G(S1,S2) =49
(
42 − 10(S1 + S−11 + S1S−1
2 + S−11 S2 + S2 + S−1
2 ) +
2(S21S−1
2 + S−21 S2 + S1S2 + S−1
1 S−12 + S1S−2
2 + S−11 S2
2) +
(S21 + S−2
1 + S21S−2
2 + S−21 S2
2 + S22 + S−2
2 ))
where S1 and S2 unit-shifts along principal axes of hexagonal grid.
Then PDE will reduce to interconnected ODE’s:(
AEI∆4
G(S1,S2)(1 + η
∂
∂t
)+ c
(1 + ζ
∂
∂t
)+ ρAh
∂2
∂t2
)
w(i , j , t) = f (i , j , t)
(Note: I = h3/12 for thin plates.)
23 / 38
Deformable mirrors — modeling
Discretization of biharmonic operator ∇4 over hexagonal grid:
G(S1,S2) =49
(
42 − 10(S1 + S−11 + S1S−1
2 + S−11 S2 + S2 + S−1
2 ) +
2(S21S−1
2 + S−21 S2 + S1S2 + S−1
1 S−12 + S1S−2
2 + S−11 S2
2) +
(S21 + S−2
1 + S21S−2
2 + S−21 S2
2 + S22 + S−2
2 ))
where S1 and S2 unit-shifts along principal axes of hexagonal grid.
Then PDE will reduce to interconnected ODE’s:(
AEI∆4
G(S1,S2)(1 + η
∂
∂t
)+ c
(1 + ζ
∂
∂t
)+ ρAh
∂2
∂t2
)
w(i , j , t) = f (i , j , t)
(Note: I = h3/12 for thin plates.)
23 / 38
Deformable mirrors — modeling
Let w = ∂w/∂t and w = ∂2w/∂t2, then
[w(i , j , t)w(i , j , t)
]
=
[0 1
−(
Eh2G(S1,S2)12ρ∆4 + 8c
3√
3ρ∆2h
)
−(ηEh2G(S1,S2)
12ρ∆4 + 8ζc3√
3ρ∆2h
)
][w(i , j , t)w(i , j , t)
]
+
[08
3√
3ρ∆2h
]
f (i , j , t)
w(i , j , t) =[
1 0][
w(i , j , t)w(i , j , t)
]
Stacking over all N nodes (i , j) yields:
x(t) = (IN ⊗ Aa,c + PN ⊗ Ab,c)x(t) + (IN ⊗ Ba,c)f (t)
w(t) = (IN ⊗ Ca,c)x(t)
where PN a sparse pattern matrix and ...
24 / 38
Deformable mirrors — modeling
Let w = ∂w/∂t and w = ∂2w/∂t2, then
[w(i , j , t)w(i , j , t)
]
=
[0 1
−(
Eh2G(S1,S2)12ρ∆4 + 8c
3√
3ρ∆2h
)
−(ηEh2G(S1,S2)
12ρ∆4 + 8ζc3√
3ρ∆2h
)
][w(i , j , t)w(i , j , t)
]
+
[08
3√
3ρ∆2h
]
f (i , j , t)
w(i , j , t) =[
1 0][
w(i , j , t)w(i , j , t)
]
Stacking over all N nodes (i , j) yields:
x(t) = (IN ⊗ Aa,c + PN ⊗ Ab,c)x(t) + (IN ⊗ Ba,c)f (t)
w(t) = (IN ⊗ Ca,c)x(t)
where PN a sparse pattern matrix and ...
24 / 38
Deformable mirrors — modeling
Stacking over all N nodes (i , j) yields:
x(t) = (IN ⊗ Aa,c + PN ⊗ Ab,c)x(t) + (IN ⊗ Ba,c)f (t)
w(t) = (IN ⊗ Ca,c)x(t)
where PN a sparse pattern matrix and
Aa,c =
[0 1
− cm − ζc
m
]
Ab,c =
[
0 0
− Eh2
12ρ∆4 − ηEh2
12ρ∆4
]
Ba,c =[
0 1m
]TCa,c =
[1 0
]
where m = ρh 3√
38 ∆2 the mass of one hexagon.
25 / 38
Control of AO systems
ZOH
ZOH
AMP
AMP
DM
H
u(k)
n(k)
φDM(k)
φDM(k) = HDM (LAMPu(k) + n(k))
• Bandwidth of amplifiers;
• Noise-level of amplifiers;
• Dynamics of deformable mirror.
28 / 38
Control of AO systemsBlockscheme calculus:
+
+
++
d
u
ψd
ψu
ψr
s
S
H G
−C
[ψr
s
]
=
[S H
GS GH
] [du
]
u = −Cs
Solve for ψr29 / 38
Control of AO systems
[ψr
s
]
=
[S H
GS GH
] [du
]
u = −Cs
Solving for ψr yields:u = − CGψr
⇐⇒ψr = Sd − HCGψr
⇐⇒(I + HCG)ψr = Sd
⇐⇒ψr = (I + HCG)−1Sd
30 / 38
Control of AO systems
[ψr
s
]
=
[S H
GS GH
] [du
]
u = −Cs
Solving for ψr yields:u = − CGψr
⇐⇒ψr = Sd − HCGψr
⇐⇒(I + HCG)ψr = Sd
⇐⇒ψr = (I + HCG)−1Sd
30 / 38
Control of AO systems
Closed-loop residual wavefront phase:
ψr = (I + HCG)−1Sd
Objective: minimize ‖ ψr ‖22
How to design C?|C| → ∞ ⇒‖ ψr ‖
22→ 0
High-gain feedback.
But, ... (I + HCG)−1 need to be stable!
31 / 38
Control of AO systems
Closed-loop residual wavefront phase:
ψr = (I + HCG)−1Sd
Objective: minimize ‖ ψr ‖22
How to design C?|C| → ∞ ⇒‖ ψr ‖
22→ 0
High-gain feedback.
But, ... (I + HCG)−1 need to be stable!
31 / 38
Control of AO systems
Closed-loop residual wavefront phase:
ψr = (I + HCG)−1Sd
Objective: minimize ‖ ψr ‖22
How to design C?|C| → ∞ ⇒‖ ψr ‖
22→ 0
High-gain feedback.
But, ... (I + HCG)−1 need to be stable!
31 / 38
Control of AO systems
Example: C = c, H = az−1, G = S = 1, such that
ψr (k) = d(k) + au(k − 1)
u(k) = −cψr (k)
Closed-loop system:ψr (k) = d(k)− acψr (k − 1)
which is stable if and only if |ac| < 1,→ |c| < 1|a| .
32 / 38
Control of AO systems
Example: C = c, H = az−1, G = S = 1, such that
ψr (k) = d(k) + au(k − 1)
u(k) = −cψr (k)
Closed-loop system:ψr (k) = d(k)− acψr (k − 1)
which is stable if and only if |ac| < 1,→ |c| < 1|a| .
32 / 38
Control of AO systems
Example: C = c, H = az−1, G = S = 1, such that
ψr (k) = d(k) + au(k − 1)
u(k) = −cψr (k)
Closed-loop system:ψr (k) = d(k)− acψr (k − 1)
which is stable if and only if |ac| < 1,→ |c| < 1|a| .
32 / 38
Control: PID for MIMO systemsControl system:
s(k) = Gψd(k) + GH︸︷︷︸
=P
u(k)
u(k + 1) = −C(z)s(k)
Design strategy:
• Diagonalize system using SVD of P = UΣV T ;
• Design SISO controller.
Decoupled system:
s′(k) = UT Gψd (k) + Σu′(k)
u′(k + 1) = −C′(z)s′(k)
where s′(k) = UT s(k), u′(k) = V T u(k) and C(z) = VC′(z)UT .C′(z) is designed as diagonal PI controller.
34 / 38
Control: PID for MIMO systemsControl system:
s(k) = Gψd(k) + GH︸︷︷︸
=P
u(k)
u(k + 1) = −C(z)s(k)
Design strategy:
• Diagonalize system using SVD of P = UΣV T ;
• Design SISO controller.
Decoupled system:
s′(k) = UT Gψd (k) + Σu′(k)
u′(k + 1) = −C′(z)s′(k)
where s′(k) = UT s(k), u′(k) = V T u(k) and C(z) = VC′(z)UT .C′(z) is designed as diagonal PI controller.
34 / 38
Control: PID for MIMO systemsControl system:
s(k) = Gψd(k) + GH︸︷︷︸
=P
u(k)
u(k + 1) = −C(z)s(k)
Design strategy:
• Diagonalize system using SVD of P = UΣV T ;
• Design SISO controller.
Decoupled system:
s′(k) = UT Gψd (k) + Σu′(k)
u′(k + 1) = −C′(z)s′(k)
where s′(k) = UT s(k), u′(k) = V T u(k) and C(z) = VC′(z)UT .C′(z) is designed as diagonal PI controller.
34 / 38
Control: H2 optimal control+
+
++
d
u
ψd
ψu
ψr
s
S
H G
−C
One “big” state-space model:
x(k + 1) = Ax(k) + Bu(k) + Bd d(k)
ψr (k) = Cψx(k) + Dψ,uu(k) + Dψ,d d(k)
s(k) = Csx(k) + Ds,d d(k)
Minimize transfer from d(k) to ψr (k): MSE(ψr )/MSE(d)35 / 38
Control: H2 optimal control+
+
++
d
u
ψd
ψu
ψr
s
S
H G
−C
One “big” state-space model:
x(k + 1) = Ax(k) + Bu(k) + Bd d(k)
ψr (k) = Cψx(k) + Dψ,uu(k) + Dψ,d d(k)
s(k) = Csx(k) + Ds,d d(k)
Minimize transfer from d(k) to ψr (k): MSE(ψr )/MSE(d)35 / 38
Control: H2 optimal control+
+
++
d
u
ψd
ψu
ψr
s
S
H G
−C
One “big” state-space model:
x(k + 1) = Ax(k) + Bu(k) + Bd d(k)
ψr (k) = Cψx(k) + Dψ,uu(k) + Dψ,d d(k)
s(k) = Csx(k) + Ds,d d(k)
Minimize transfer from d(k) to ψr (k): MSE(ψr )/MSE(d)35 / 38
Control: H2 optimal controlSolution:
• Kalman filter:
s(k) = Csx(k)
x(k + 1) = Ax(k) + Bu(k) + K (s(k)− s(k))
where K the Kalman gain.
• State-feedback:u(k + 1) = − Fx(k + 1)
where F the state-feedback gain.
For static deformable mirrors ψDM(k) = Hu(k)2:
• Controller based on predicted wavefront:
u(k + 1) = − H†ψ(k + 1)
where H the Moore-Penrose pseudo-inverse and ψ(k + 1) = Cψ x(k + 1)2c.f., Hinnen, Data-Driven Optimal Control for Adaptive Optics, 2007, page 110
36 / 38
Control: H2 optimal controlSolution:
• Kalman filter:
s(k) = Csx(k)
x(k + 1) = Ax(k) + Bu(k) + K (s(k)− s(k))
where K the Kalman gain.
• State-feedback:u(k + 1) = − Fx(k + 1)
where F the state-feedback gain.
For static deformable mirrors ψDM(k) = Hu(k)2:
• Controller based on predicted wavefront:
u(k + 1) = − H†ψ(k + 1)
where H the Moore-Penrose pseudo-inverse and ψ(k + 1) = Cψ x(k + 1)2c.f., Hinnen, Data-Driven Optimal Control for Adaptive Optics, 2007, page 110
36 / 38
Control: H2 optimal controlSolution:
• Kalman filter:
s(k) = Csx(k)
x(k + 1) = Ax(k) + Bu(k) + K (s(k)− s(k))
where K the Kalman gain.
• State-feedback:u(k + 1) = − Fx(k + 1)
where F the state-feedback gain.
For static deformable mirrors ψDM(k) = Hu(k)2:
• Controller based on predicted wavefront:
u(k + 1) = − H†ψ(k + 1)
where H the Moore-Penrose pseudo-inverse and ψ(k + 1) = Cψ x(k + 1)2c.f., Hinnen, Data-Driven Optimal Control for Adaptive Optics, 2007, page 110
36 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38
Exams
1 Sparse matrix techniques for wavefront reconstruction;
2 Extended object and wavefront estimation by phase-diversity;
3 Pyramide wavefront sensor;
4 Adaptive optics for the eye;
5 Quantum noise effects in imaging;
6 Data-driven H2 control;
7 Modeling and validation of deformable mirrors.
38 / 38