Sparse Matrix Methods for Large-Scale. Closed-Loop Adaptive Optics

Transcription

1 Sparse Matrix Methods for Large-Scale Closed-Loop Adaptive Optics Luc Gilles Michigan Technological University, ECE Department January 23, 24 L.Gilles, IPAM Workshop, UCLA p.1/37

2 Large-Scale Adaptive Optics Extreme Adaptive Optics (ExAO), e.g. coronographs, XAOPI (4K actuators),... Multi-Conjugate Adaptive Optics (MCAO), e.g. 32m TMT (CfAO, NOAO), 5m Euro5 (consortium), 1m OWL (ESO),... L.Gilles, IPAM Workshop, UCLA p.2/37

3 Basic Model Assumptions Geometrical optics propagation through atmos-ao Taylor Frozen flow hypothesis WFS measurements s(n) are linear in the input turbulent OPD profiles x(n) and DM OPD profiles ˆ a(n) DM residual profile ˆ e a (n) is linear in measurements s(n) Standard type-1 discrete integrator with 2-cycle lag Note: - Work on wave optics propagation in progress... - Formalism valid for both ExAO and MCAO - MCAO: block structured matrices, concatanated vecs L.Gilles, IPAM Workshop, UCLA p.3/37

4 Generic System Layout NGS (one or several) Science field - one or several - multiple λ s LGS - one or several - 3-dimensional Geometric or Fresnel propagation through atmosphere Turbulence - multi layer -frozen flow Temporal Filter M2 tip/tilt LGS tip/tilt mirror(s) Temporal Filter LGS pointing commands Reconstructor Temporal Filters Non-common path error Fast tip/tilt mirror Tip/tilt offload Deformable mirror(s) - Zonal actuators (square pattern) - 5 degrees of misregistration Common path error(s) LGS WFS NGS WFS Geometric or Fresnel propagation through optics - Shack-Hartmann - Square subap. geometry - N by N pixels, centroid algorithm - Photon statistics - Read noise - Pupil and relay distortion - Pixel blurring Non-common path errors Performance Evaluation Strehl ratio histories Mean PSF s L.Gilles, IPAM Workshop, UCLA p.4/37

5 Discrete-Space-Time Servo-Loop Standard Control Algorithm: n = ; ˆ a() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); ˆ e a (n) = R s(n); ˆ a(n + 1) = α ˆ a(n) + β ˆ a(n 1) + γ ˆ e a (n) + δ ˆ e a (n 1); n = n + 1; end cycles s o (n) : open-loop measurements G a : interaction (poke, registration) matrix R : DM residual reconstruction matrix L.Gilles, IPAM Workshop, UCLA p.5/37

6 Simulation-Sensor-Reconstruction Grids - Turbulence profile, x(n) Simulation Grids - Aperture-plane simulation OPD s for performance evaluation : ǫ(n) = Hx SCFoV,sim x(n) Ha SCFoV,sim ˆ a(n) Note: typically 8 finer resolution than sensor grids Sensor Grids - for MCAO, s(n) resolution typically D/16, D/32, or D/64 - for ExAO, s(n) resolution up to D/128 or D/256 L.Gilles, IPAM Workshop, UCLA p.6/37

7 SSR Grids (cont.) Reconstruction Grids - DM grids. Sampling typically matches sensor sampling. - for FE-type reconstructors, i.e. when R = F E: (1) reconstructed turbulence profile ˆ x(n) = E s(n) (2) reconstructed aperture-plane OPD H SCFoV x ˆ x(n) 2 finer resolution than DM grids L.Gilles, IPAM Workshop, UCLA p.7/37

8 Aperture-plane transfer matrices Based on bilinear influence functions: [ ] H SCFoV {iobs, ips} dr dr = h (l) ( r (k) ) k l [ = h ( r (k) r ] (l) (z))/dr where r (l) (z) = r (l) + z θ{iobs} is the coordinate of grid point l in the transverse plane at range z and r (l) is the corresponding aperture-plane coordinate. Bilinear splines: (1 x )(1 y ) if x < 1 and y < 1 h( r) = else L.Gilles, IPAM Workshop, UCLA p.8/37

9 Aperture-plane transfer matrices (cont.) Notation: H SCFoV,sim x maps simulated profile x(n) to aperture-plane simulation grid (perf eval) H SCFoV,sim a maps DM OPD ˆ a(n) to aperture-plane simulation grid (perf eval) For FE-type reconstructors: - Hx SCFoV maps ˆ x(n) = E s(n) to aperture-plane reconstruction grid - Ha SCFoV maps Hx SCFoV ˆ x(n) to DM grids L.Gilles, IPAM Workshop, UCLA p.9/37

10 Control Algorithm: Performance Eval n = ; ˆ a() = ; begin cycles = [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); ˆ e a (n) = R s(n); ˆ a(n + 1) = α ˆ a(n) + β ˆ a(n 1) + γ ˆ e a (n) + δ ˆ e a (n 1); n = n + 1; end cycles L.Gilles, IPAM Workshop, UCLA p.1/37

11 Performance Evaluation ǫ(n) = H SCFoV,sim x x(n) H SCFoV,sim a ˆ a(n) σol 2 SCFoV,sim (n) = Hx x(n) 2, σ 2 W sim cl (n) = ǫ(n) 2 W sim PSF(n) = F{e i ǫ(n) W A } 2, OTF(n) = F{PSF(n)} SR(n),FWHM(n),... W is the FoV averaging matrix (SPSD). Based on bilinear splines, it provides accurate modeling of aperture-plane OPD and edge effects. Includes piston removal. W = W v v T if zero SCFoV W = Diag ( ω (1) W,,ω (Nobs) W ) else L.Gilles, IPAM Workshop, UCLA p.11/37

12 Control Algorithm: Sensor Measurements n = ; ˆ a() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; = s(n) = s o (n) G aˆ a(n); ˆ e a (n) = R s(n); ˆ a(n + 1) = α ˆ a(n) + β ˆ a(n 1) + γ ˆ e a (n) + δ ˆ e a (n 1); n = n + 1; end cycles L.Gilles, IPAM Workshop, UCLA p.12/37

13 Sensor Measurements s(n) = s o (n) G aˆ a(n) s o (n) = G sim x x(n) + η(n) s{iwfs}(n) = ips G sim x {iwfs, ips} x{ips}(n) + η{iwfs} idm G a {iwfs, idm} ˆ a{idm}(n) Note: G sim x = Γ sim x Hx GSFoV and G a = Γ a Ha GSFoV, where the Γ sim x and Γ a map aperture-plane OPD s to WFS measurements L.Gilles, IPAM Workshop, UCLA p.13/37

14 Turbulence-to-WFS Influence Matrix [ G sim x {iwfs, ips} ] k l = d 2 r W (k) SA {iwfs}( r) α h (l) ( r) - h (l) ( r) = h = ξ2 ξ 1 h (l) ( r(ξ)) α n(ξ)dξ [( r r (l) (z))/dr ], dr = sim grids resolution - r (l) (z) = γ{iwfs} r (l) + z (l) θ{iwfs}, γ{iwfs} = (1 z (l) /z GS {iwfs}) - α = (α x,α y ) is the measurement direction - n(ξ) = (dy(ξ)/dξ, dx(ξ)/dξ) Same eqs for G a {iwfs, idm} but with dr = DM resolutions L.Gilles, IPAM Workshop, UCLA p.14/37

15 Control Algorithm: Reconstructor n = ; ˆ a() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); = ˆ e a (n) = R s(n); ˆ a(n + 1) = α ˆ a(n) + β ˆ a(n 1) + γ ˆ e a (n) + δ ˆ e a (n 1); n = n + 1; end cycles L.Gilles, IPAM Workshop, UCLA p.15/37

16 Filtered Least-Squares Reconstructor ˆ e a (n) = R s(n) = arg min e a (n) { s(n) G a e a (n) 2 Cη 1 ) 1 a G T a Cη 1 = ( G T a Cη 1 G a + P GSFoV + e a (n) 2 P GSFoV a s(n) Pa GSFoV = V a Va T, where V a = [ v a (1),, v a (p) ]. Range(V a ) spans constrained modes, e.g. waffle for SCAO, cancelling tip/tilt for MCAO NGS, cancelling tip/tilt, astigmatism and defocus for MCAO LGS, i.e. Ha GSFoV {iobs, idm} v a {idm} = Z 1 {iobs} or f{iobs} Null(Γ a ) idm Note: e a (n) 2 P GSFoV a = V a e a (n) 2 } L.Gilles, IPAM Workshop, UCLA p.16/37

17 Minimum Variance Reconstructor Open-loop reconstructor: - Estimation step: ˆ e x (n) = E s(n) = arg min e x (n) = ( G T x C 1 η G x + C 1 x { s(n) G x e x (n) 2 C 1 η + e x (n) 2 C 1 x ) 1 + Px GSFoV G T x Cη 1 s(n) + e x (n) 2 P GSFoV x } - Fitting step: ˆ e a (n) = F ˆ e x (n) = arg min = ( H SCFoV a e a (n) { T WH SCFoV a H SCFoV x + P SCFoV a ˆ e x (n) H SCFoV a ) 1 H SCFoVT a WH SCFoV x e a (n) 2 W + e a (n) 2 P SCFoV ˆ e x (n) a } L.Gilles, IPAM Workshop, UCLA p.17/37

18 Minimum Variance Reconstructor (cont.) E-step GSFoV-projector: P GSFoV x ips = V x V T x, V x = [ v (1) x,, v (q) x ] H GSFoV x {iwfs, ips} v x {ips} = Z 1 {iwfs} or f{iwfs} Null(Γ x ) F-step SCFoV-projector: P SCFoV a idm = V a V T a, V a = [ v (1) a,, v (r) a ] H SCFoV a {iobs, idm} v a {idm} = Z 1 {iobs} or f{iobs} Null(Γ a ) - MVR s P SCFoV a is identical to LSR s P SCFoV a - q >> r, i.e. dim Null(Γ x ) >> dim Null(Γ a ) if H SCFoV a = H GSFoV a - In practive, projectors Pa SCFoV, Px GSFoV not known exactly L.Gilles, IPAM Workshop, UCLA p.18/37

19 Approximate Closed-Loop Reconstructor - Estimation step: ˆ e x (n) = E s(n) = arg min e x (n) = ( G T x C 1 η G x + C 1 { s(n) G x e x (n) 2 C 1 η x d + Px GSFoV ) 1 G T x C 1 η + e x (n) 2 C 1 x d + e x (n) 2 P GSFoV x s(n) where x d (n) = x(n) x(n n τ ) models an n τ -cycle lag, and C xd = F 1 diag( (2π n τ v κ) 2 κ 11/3 )F If wind speed direction v is uniformly distributed in [,2π], (2π n τ v κ) 2 = (1/2)(2πn τ v ) 2 κ 2 } - Fitting step as for MVR L.Gilles, IPAM Workshop, UCLA p.19/37

20 Control Algorithm: Loop Filtering n = ; ˆ a() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); ˆ e a (n) = R s(n); = ˆ a(n + 1) = α ˆ a(n) + β ˆ a(n 1) + γ ˆ e a (n) + δ ˆ e a (n 1); n = n + 1; end cycles Loop filtering can be rewritten ˆ a(n + 1) = g(n) ˆ e a (n) with g(z) = (γz + δ)/(z 2 αz β). Integrator: α + β = 1 For FE-recons, filtering can be applied on ˆ e x or ˆ e a since gain coeffs are scalar multiples of the identity matrix L.Gilles, IPAM Workshop, UCLA p.2/37

21 Pseudo Open-Loop Reconstructor Inspired by open-loop MVR with loop filtering applied on ˆ e x : s o (n) = s(n) + G aˆ a(n) ˆ e x (n) = arg min e x (n) { s o (n) G x (ˆ x(n) + e x (n)) 2 Cη 1 } + ˆ x(n) + e x (n) 2 P GSFoV x = A 1 G T x C 1 η where ( s(n) + s (n)) A 1 (C 1 x + ˆ x(n) + e x (n) 2 C 1 x + Px GSFoV ) ˆ x(n) s (n) = (G a F G x ) ˆ x(n), A = G T x C 1 η G x + C 1 x + P GSFoV x Note: A 1 G T x C 1 η = E = MVR E-matrix L.Gilles, IPAM Workshop, UCLA p.21/37

22 POLR (cont.) Control Algorithm: Version 1 n = ; ˆ a() = ; ˆ x() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); s (n) = (G a F G x )ˆ x(n); ˆ e x (n) = E ( s(n) + s (n)) A 1 (Cx 1 + Px GSFoV )ˆ x(n); ˆ x(n + 1) = α ˆ x(n) + β ˆ x(n 1) + γ ˆ e x (n) + δ ˆ e x (n 1); ˆ a(n + 1) = F ˆ x(n + 1); n = n + 1; end cycles L.Gilles, IPAM Workshop, UCLA p.22/37

23 POLR (cont.) Control Algorithm: Version 2 n = ; ˆ a() = ; ˆ x() = ; begin cycles [PSF(n), OTF(n),σ 2 (n)] = perfevl[ x(n), ˆ a(n)]; s(n) = s o (n) G aˆ a(n); ˆ e x (n) = E s(n); ˆ x(n + 1) = [αi + γm] ˆ x(n) + [αi + δm] ˆ x(n 1) + γˆ e x (n) + δˆ e x (n 1); where M = E(G a F G x ) A 1 (Cx 1 ˆ a(n + 1) = F ˆ x(n + 1); n = n + 1; end cycles + P GSFoV x ) L.Gilles, IPAM Workshop, UCLA p.23/37

24 POLR (cont.) POLR is not a conventional matrix multiply reconstructor: requires real-time knowledge of ˆ x(n) POLR requires knowledge of interation matrix G a E = ( G T x Cη 1 G x + Cx 1 open-loop MVR E-matrix + P GSFoV x ) 1 G T x C 1 η = A 1 G T x C 1 η = Computational complexity identical to MVR G a F G x = Γ a H GSFoV a Ha SCFoV Hx SCFoV Γ x H GSFoV x L.Gilles, IPAM Workshop, UCLA p.24/37

25 Transfer Matrix and Loop Stability LSR: ˆ a o (z) = R s o (z) ˆ a(z) = T a (z) ˆ a o (z) T a (z) = g(z)(i + g(z)rg a ) 1 g(z) = (γz + δ)/(z 2 αz β) RG a = I P GSFoV a = I V a V T a - Given RG a = Udiag(λ i )U 1, T a (z) = Udiag[g(z)/(1 + g(z)λ i )]U 1 U 1ˆ a(z) = diag[g(z)/(1 + g(z)λi )]U 1 ˆ a o (z) - Representative filter: γ =,δ = β = α = 1/2, yielding g(z) = 1/[2(z 1)(z + 1/2)] L.Gilles, IPAM Workshop, UCLA p.25/37

26 Loop Stability (cont.) - Poles {z 1 + g(z )λ i = } are z = (1 ± 9 8λ i )/4 - System is stable if and only if all poles are located inside unit circle, i.e. z 1 - LSR is stable since λ i =,1 and thus z = 1, ±1/2 MVR - ACLR: - E-step: ˆ x o (z) = E s o (z) ˆ x(z) = T x (z) ˆ x o (z) T x (z) = g(z)(i + g(z)eg a F) 1 L.Gilles, IPAM Workshop, UCLA p.26/37

27 Loop Stability (cont.) - F-step: ˆ a(z) = F ˆ x(z) = F T x (z) ˆ x o (z) F T x (z) = g(z)f (I + g(z)eg a F) 1 = g(z)(i + g(z)rg a ) 1 F = ˆ a(z) = T a (z) ˆ a o (z) (as for LSR) T a (z) = g(z)(i + g(z)feg a ) 1, ˆ a o (z) = F ˆ x o (z) - Since FEG a has complex eigenvalues λ i that in general won t satisfy the stability criterion (1 ± 9 8λ i )/4 < 1, MVR-ACLR will be unstable. Stability recovered for SCAO by slaving edge actuators and keeping only subaps at least 5% illuminated. Demonstrated in simulations and on the PALAO. L.Gilles, IPAM Workshop, UCLA p.27/37

28 Loop Stability (cont.) POLR: M = EG a F EG x A 1 (C 1 x T x (z) = g(z)[i + g(z)(eg a F M)] 1 ˆ a(z) = F ˆ x(z) = F T x (z) ˆ x o (z) + P GSFoV x ) - M = yields MVR. Given EG a F M = Udiag(λ i )U 1, T x (z) = Udiag[g(z)/(1 + g(z)λ i )]U 1 - POLR doesn t have simple T a (z) s.t. F T x (z) = T a (z)f - Simulations demonstrate that POLR is efficient but becomes unstable with little uncertainty in G a - Real-time implementation worth considering L.Gilles, IPAM Workshop, UCLA p.28/37

29 Sparse MVR solver Sparse layer-oriented multigrid-preconditioned conjugate gradient (MGCG) solver for MVR s E- and F-steps: Layer-by-layer OPD estimation 1 BSGS smoothing iteration for layer decoupling LGS tip/tilt uncertainty leads to spatially colored measurement noise handled by rank-1 projector LGS focus uncertainty can be handled similarly with an additional rank-1 projector E- and F-systems solved using sparse plus low-rank inversion lemma (Sherman-Morrison) L.Gilles, IPAM Workshop, UCLA p.29/37

30 Sample 32m problem description - Cerro Pachon 6 layer median turbulence profile ranging from m to 15,46m scaled to yield r = θ = m phase screens. Reconstructed PS1-PS4 oversample 2 subaps; recons PS5-PS6 match subaps resolution. 7,942 grid points reconstructed - 5 LGS s at the four edges and center of a 1 GSFoV; subapertures each; 33, 32 measurements - 4 NGS s at mid distance between the four LGS s delimiting the GSFoV (8 tip/tilt measurements) SCFoV; Simpson weights - 3 DM s. DM s 1-2 in Fried config, DM3 (1,39m) undersamples 2 subaps. Total of 8, 449 actu L.Gilles, IPAM Workshop, UCLA p.3/37

31 32m open-loop convergence results 1 Estimation Step, σ=.2 arcsec 1 Fitting Step Relative Residual Norm no precon BSGS precon MG precon PCG Iteration Relative Residual Norm no precon BSGS precon MG precon PCG Iteration 1 Estimation Step, σ=.2 arcsec no precon 1 Fitting Step Relative Residual Norm BSGS precon MG precon Relative Residual Norm no precon BSGS precon MG precon PCG Iteration PCG Iteration L.Gilles, IPAM Workshop, UCLA p.31/37

32 8m closed-loop simulation results Same geometry and fields as for 32m but with LGS WFS subaps. 7,826 reconstructed phase screen grid points; 629 actu and 2,248 measurements. 1 7 MVR input 1 6 OPD Variance (nm 2 ) 1 5 POLR with.5% rms error in GA LSR POLR with 1% rms error in phase estimate POLR Simulation Cycle L.Gilles, IPAM Workshop, UCLA p.32/37

33 8m closed-loop poles 1 16 eig( A est ) 1 eig( A fit ) eig( F x E x G a ) 1 Poles L.Gilles, IPAM Workshop, UCLA p.33/37

34 Real part of sample offending modes λ = λ = λ = λ = λ = λ = λ = i x 1 3 λ = i λ = i L.Gilles, IPAM Workshop, UCLA p.34/37

35 Imag part of sample offending modes λ =.2152 λ =.2152 λ = λ = λ = λ = λ = i λ = i λ = i L.Gilles, IPAM Workshop, UCLA p.35/37

36 Constrained MVR Minimizes < σ 2 (n) >=< ǫ(n) 2 W constraints: > subject to 2 linear RG a = P a where P 2 a = P a and P T a = MP a M 1 with M = Ha SCFoVT WHa SCFoV (I P a )R = = R = R 1 + R 2 with R 1 = FE f ( ) 1 E f = G T x Cη 1 f G x + Cx 1 f G T x Cη 1 f η f (z) = η(z)g(z), x f (z) = xg(z) R 2 = (I R 1 G a )G a ( ) 1 G a = G T a Cs 1 f G a G T a Cs 1 f C sf = G x C xf G T x + C ηf L.Gilles, IPAM Workshop, UCLA p.36/37

37 Constrained MVR (cont.) Obviously stable since λ i =,1 Is there a sparse efficient way to compute it? - C 1 x f F 1 diag( κ 11/3 1 + v 2 κ 2 /ν c )F - C ηf C η - G a in R 2 most troublesome Optimized range of projector P a also troublesome L.Gilles, IPAM Workshop, UCLA p.37/37