Lecture 7: Deformable mirror modeling and control

Rufus Fraanjep.r.fraanje@tudelft.nl

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 — NatureAdjustable pupils (from: Animal Eyes, Land&Nilsson, 2002)

4 / 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

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Adjustable optics — NatureAdjustable lenses (from: Animal Eyes, Land&Nilsson, 2002)

6 / 38

Adjustable optics — EngineeringAdjustable lenses (Source: http://www.nedinsco.nl):

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Adjustable optics — Engineering

Adjustable lenses (Source: Jeong et al. (2005), c.f. http://biopoems.berkeley.edu)

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Adjustable optics — NatureEye/head movement (from: Animal Eyes, Land&Nilsson, 2002)

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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)

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Adjustable optics — NatureMirrors in scallops (from: Animal Eyes, Land&Nilsson, 2002)

(Source: Wikipedia – Sint-Jakobsschelp)

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Adjustable optics — Engineering

Adjustable mirrors (Keck II, Source: Wikipedia):

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Adjustable optics — Engineering

Adjustable mirrors (Electrostatic deformable mirror, Source: Flexible Optical, BV, Rijswijk):

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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

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Deformable mirrorsDiffraction effects of hexagonal segmented mirrors: Apertures:

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Deformable mirrorsDiffraction effects of hexagonal segmented mirrors: Point spread functions:

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Deformable mirrorsDiffraction effects of hexagonal segmented mirrors: Obtained images:

Circular aperture

Hexagonal aperture (70% fill)

Hexagonal aperture (100% fill)

Hexagonal aperture (90% fill)

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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Deformable mirrors — modeling

(Continued)

• Supported edge faceplate (c.f. Loktev et al., 2001)

• Free edge faceplate (idem)

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Deformable mirrors — modeling

Loktev et al. 2001, various actuator configurations and plate models arecompared:

Hexagonal- Square- Ring-layout

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Deformable mirrors — modelingResidual wavefront aberations for various type of mirrors each with 37 actuators (Credit:

Loktev et al, 2001, Table 1 (adjusted))

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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.)

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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.)

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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 ...

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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.

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Deformable mirrors — modeling

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Deformable mirrors — modeling

27 / 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.

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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 of AO systems

1 PID for MIMO systems;

2 H2 optimal control;

3 Distributed control.

33 / 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

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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: Distributed control

(Fraanje, Massioni, Verhaegen 2011)

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Control: Distributed control(Fraanje, Massioni, Verhaegen 2011)

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Control: Distributed control(Fraanje, Massioni, Verhaegen 2011)

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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