Accurate estimation for the closed-loop robust control using global modes

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1/21 G. Tissot, GDR CDD 216. 1/21 Accurate estimation for the closed-loop robust control using global modes application to the Ginzburg-Landau equation Gilles Tissot, Jean-Pierre Raymond Institut de Mathématiques de Toulouse GDR CDD, Lyon, November 28th, 216

2/21 G. Tissot, GDR CDD 216. Context 2/21 Closed-loop flow control: y(t: measurements c(t f(t Flow z(t Compensator z perf (t y(t r(t c(t: control/actuation z(t: system state z perf (t: performance output f(t: disturbance r(t: measurement noise Goal: Design the compensator to reach an objective.

/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 3/21 Control strategy Linear state feedback control Low-order High-order c f dz dt Plant = Az + Bc + F f Compensator z perf = Cz r y = Hz + Dc + r

/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 3/21 Control strategy Linear state feedback control Low-order High-order c f K dz dt Controller ẑ r Plant = Az + Bc + F f dẑ r dt = A rẑ r + B r c + L(ŷ(z r, c y Estimator z perf = Cz r y = Hz + Dc + r

3/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 3/21 Control strategy Linear state feedback control Low-order High-order c f K dz dt Controller ẑ r Plant = Az + Bc + F f dẑ r dt = A rẑ r + B r c + L(ŷ(z r, c y Estimator z perf = Cz r y = Hz + Dc + r Which reduced-order basis? Ensure compatibility between y and ŷ(ẑ r, c.

Linear state feedback control Model reduction Gains computation 4/21 Control strategy Model reduction Separate unstable/stable part: dz u = A u z u + B u c + F u f dt dz s dt = A sz s + B s c + F s f y = H u z u + H s z s + Dc + r, such that: λi 1.5.5 1 Flow over a thick plate: 1.8.6.4.2.2 λ r unstable region z = Φ u z u + Φ s z s, with (Φ u, Φ s appropriate basis. z u small dimension (unstable modes + few stable modes. A s is exponentially stable. Airiau et al. (in prep. Strategy: keep z u (responsible of instability for performance construction and model z s. 4/21 G. Tissot, GDR CDD 216.

5/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 5/21 Control strategy Model reduction Sipp & Schmid (216 Usual procedure of modelling z s : Galerkin projection onto a reduced basis Φ r along Ψ r A s,r = (Ψ r, A s Φ r, B r,s = (Ψ r, B s, H r,s = H s Φ r, Estimator ( ( ( ( d ẑu Au ẑu Bu = + c dt ẑ s,r A s,r ẑ s,r B s,r + L(H u ẑ u + H r,s ẑ s,r + Dc y.

6/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 6/21 Control strategy Model reduction Usual procedure of modelling z s : Åkervik et al. (28 Ehrenstein et al. (211 Barbagallo et al. (29,211 Global modes Preserves system dynamics ( A11 A s =, A A s,r = A 11. 22 σ max H s ẑ s H r,s ẑ r,s can be large, poor I/O representativeness. can destabilise the system. Barbagallo 211, cavity flow, GM.

7/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 7/21 Control strategy Model reduction Usual procedure of modelling z s : Balanced truncation, BPOD, POD Reduces H s ẑ s H s,r ẑ s,r, by state representation optimality. Corrupts system dynamics ( A11 A A s = 12, A A 21 A s,r = A 11. 22 Antoulas (25 Rowley (25 Atwell & King (21 Spectrum different, approximated optimal control law. Barbagallo 29, cavity flow, BPOD.

8/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 8/21 Control strategy Model reduction Our procedure of modelling z s : Estimator dẑ u = dt Ãuẑ u + B u Kẑ u + L ((H u + DKẑ u + H s ẑ s y dẑ s dt = A sẑ s + B s Kẑ }{{} u, with Ãu = Remarks: c ( A u + 1 γ 2 F u F ux u, and γ the robustness parameter. Response of the (unperturbed stable part to actuation. ẑ s high-dimensional.

8/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 8/21 Control strategy Model reduction Our procedure of modelling z s : Estimator dẑ u dt = Ãuẑ u + B u Kẑ u + L ((H u + DKẑ u + H s ẑ s y H s ẑ s = H s e tas z s ( + with Ãu = Remarks: t H s e (t τas B }{{} s Kẑ u (τ dτ. R(t τ ( A u + 1 γ 2 F u F ux u, and γ the robustness parameter. R(t τ is low-dimensional! (and computationable H s ẑ s response of full stable part submitted to actuation.

Linear state feedback control Model reduction Gains computation 8/21 Control strategy Model reduction Our procedure of modelling z s : Estimator dẑ u dt = Ãuẑ u + B u Kẑ u + L ((H u + DKẑ u + H s ẑ s y H s ẑ s = H s e tas z s ( + with Ãu = Remarks: t H s e (t τas B }{{} s Kẑ u (τ dτ. R(t τ ( A u + 1 γ 2 F u F ux u, and γ the robustness parameter. Global modes are used (unstable + few stable modes. Response of stable part to perturbation can be improved by adding stable modes. 8/21 G. Tissot, GDR CDD 216.

9/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 9/21 Control strategy Gains computation Standard H control: Riccati equations (A u + Ω X u + X u (A u + Ω X u M u X u + CuC u = (small A u Y u + Y u A u Y u N u Y u + F u Q e Fu = (small A s Y s + Y s A s + F s Q e Fs =, (large, but decoupled ( with Ω=diag(ω 1,...,ω nu, M u= B ur 1 Bu 1 γ 2 FuQeF u, ( N u= Hu R 1 e H u 1 γ 2 (C u Cu+Ω X u+x uω. Zhou (1996

9/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 9/21 Control strategy Gains computation Standard H control: Riccati equations { (Au + Ω X u + X u (A u + Ω X u M u X u + C uc u = (small A u Y u + Y u A u Y u N u Y u + F u Q e F u =, ( with Ω=diag(ω 1,...,ω nu, M u= ( N u= H ur 1 e H u 1 γ 2 (C uc u+ω X u+x uω B ur 1 Bu 1 γ 2 FuQeF u,. (small Zhou (1996

Linear state feedback control Model reduction Gains computation 9/21 Control strategy Gains computation Standard H control: Riccati equations { (Au + Ω X u + X u (A u + Ω X u M u X u + C uc u = (small A u Y u + Y u A u Y u N u Y u + F u Q e F u =, ( with Ω=diag(ω 1,...,ω nu, M u= ( N u= H ur 1 e H u 1 γ 2 (C uc u+ω X u+x uω Feedback and Kalman gains K = R 1 B ux u, Remarks: B ur 1 Bu 1 γ 2 FuQeF u,. L = Stable and unstable part decoupled. (small ( I 1 γ 2 X uy u 1 Y u H ur 1 e. Zhou (1996 Optimal gains exact (unlike BT/BPOD. 9/21 G. Tissot, GDR CDD 216.

/21 G. Tissot, GDR CDD 216. Linear state feedback control Model reduction Gains computation 1/21 Control strategy Gains computation Design details: If only the unstable modes: minimal energy control Ω = ωi, C u =, If more stable modes kept: ( I Ω = ω, C u = (, ri Parameters: (ω, r, γ.

11/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 11/21 Coupling 3 systems Full information: dz dt = (A + BK φz + F f, with K φ = KΨ uw.

11/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 11/21 Coupling 3 systems Full information: dz dt = (A + BK φz + F f, Partial coupling: ẑ s =, ( ( d LHu ( ( z A BK z F = dt ẑ u LH Ãu + BuK + + ẑ u L with K φ = KΨ uw. ( f r.

11/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 11/21 Coupling 3 systems Full information: dz dt = (A + BK φz + F f, Partial coupling: ẑ s =, ( ( d LHu ( ( z A BK z F = dt ẑ u LH Ãu + BuK + + ẑ u L Full coupling: with integral term, ( ( ( d z A BK z ẑu = LH Ãu + BuK + LHu LHs ẑu + dt ẑ s B sk A s ẑ s with K φ = KΨ uw. ( f r ( F L. (f r.

12/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 12/21 Coupling 3 systems Defining e u = z u ẑ u and e s = z s ẑ s, Partial coupling: ẑ s = e s = z s, d dt ( zu z s e u = ( Au + B uk B uk B sk A s B sk LH s à u + LH u (zu z s e u + ( Fu (f F s r F u L Stability not guaranteed due to LH s (generally omitted..

12/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 12/21 Coupling 3 systems Defining e u = z u ẑ u and e s = z s ẑ s, Partial coupling: ẑ s = e s = z s, d dt ( zu z s e u = ( Au + B uk B uk B sk A s B sk LH s à u + LH u (zu z s e u + ( Fu (f F s r F u L Stability not guaranteed due to LH s (generally omitted. Full coupling: with integral term, z u d z s = dt e u e s A u + B uk B uk B sk A s B sk à u + LH u LH s A s z u z s e u e s + Stability guaranteed provided that A u + B u K and Ãu + LH u stable. F u F s F u L F s. ( f r.

3/21 G. Tissot, GDR CDD 216. 3 systems Performance measures 13/21 Coupling Performance measures Transfer functions Largest Hankel singular value: T f z (λ = σ max ( C CL (A CL λi 1 F f. Trefethen (25 Burke et al. (23 A CL, C CL, F f matrices of various coupling systems selecting I/O f z. ɛ-pseudospectra: domain T f z (λ > ɛ 1. Structural robustness, non-normality. H performance: T f z = sup λi T f z (iλ i. Worst harmonic forcing, robustness (α ɛ< T f z < 1 ɛ. H 2 performance: T f z 2 = ( + T f z (iλ i 2 dλ i 1 2. Response to stochastic forcing (Parseval s theorem.

Discretised using Chebyshev polynomials (weight matrix W. Boundary conditions enforced by Lagrange multipliers, eliminated by projection technique. 14/21 G. Tissot, GDR CDD 216. Formulation Open loop system Minimal energy control Robustification 14/21 Ginzburg-Landau Formulation Lauga & Bewley (23 ψ t + U ψ x = µ(xψ + ν 2 ψ Bagheri et al. (29 x 2 + c d(x, t + f d (x, t ψ(, t = c b (t, ψ(x, t C, ψ (l, t =, U = 6, x ν = 1 1i. ψ(x, =, y = (ψ(x o, t, ψ(x Γ o, t T + r µ x

Formulation Open loop system Minimal energy control Robustification 15/21 Ginzburg-Landau Open loop system λi 1.5 -.5 d c j = -1 -.5 d c j do j 2 18 16 14 12 1 8 6 4 2 λ r B jb j 2real(λ j, do j = H j H j 2real(λ j. ψ ψ.3.2.1 -.1 -.2 Real(Φ 1 Φ 1 Real( Ψ 1 Ψ 1 act. sens. -.3 2 4 6 8 1 12 14 16 x.4.3.2.1 -.1 -.2 mode 1. Real(Φ 7 Φ 7 Real( Ψ 7 Ψ 7 act. sens. -.3 2 4 6 8 1 12 14 16 x mode 7. 15/21 G. Tissot, GDR CDD 216.

16/21 G. Tissot, GDR CDD 216. Formulation Open loop system Minimal energy control Robustification 16/21 Ginzburg-Landau Minimal energy control Compensated systems ɛ-pseudospectra: 1 Minimal energy control no stable mode ω =.5λ r,max.5 λi.5 1.5 λ r Partial coupling vs FI.

Formulation Open loop system Minimal energy control Robustification 16/21 Ginzburg-Landau Minimal energy control Compensated systems ɛ-pseudospectra: 1 1 Minimal energy control no stable mode ω =.5λ r,max.5.5 λi λi.5.5 1.5 λ r Partial coupling vs FI. 1.5 λ r Full coupling vs FI. Integral term seems to preserve spectrum. 16/21 G. Tissot, GDR CDD 216.

Formulation Open loop system Minimal energy control Robustification 17/21 Ginzburg-Landau Minimal energy control Transfer function T f z (iλ i : Minimal energy control no stable mode ω =.5λ r,max Full information Partial coupling Full coupling Tf z(iλi 1 1 1.9.8.7.6.5 No clear improvement. Estimation not improved, response of stable modes to perturbations omitted. 17/21 G. Tissot, GDR CDD 216. λ i

8/21 G. Tissot, GDR CDD 216. Formulation Open loop system Minimal energy control Robustification 18/21 Ginzburg-Landau Robustification d c j do j 1 2 18 16.5 14 12 1 8 6.5 4 2 1.5 λ r d c j = B j Bj, 2real(λ j do j = λi H j H j 2real(λ j. Adding stable modes: Pseudospectrum mode s selection. Modes playing together (non-normality. Modes highly sensitive controllable. Truncation compensated by H s ẑ s. Other choices tested.

Formulation Open loop system Minimal energy control Robustification 19/21 Ginzburg-Landau Robustification Transfer function T f z (iλ i : 3 stable modes added ω =.5λ r,max, r = 1 4, γ =.2 Tf z(iλi 1 Full information stable mode 3 stable modes partial coupling 3 stable modes full coupling 1 1.9.8.7.6.5 λ i H performance improved. 19/21 G. Tissot, GDR CDD 216.

Formulation Open loop system Minimal energy control Robustification 2/21 Ginzburg-Landau Robustification E 2. 1 5 1.5 1 5 1. 1 5 5. 1 6 stable mode 3 stable modes noise. 1 1 2 3 4 5 t Response to stochastic forcing. Real(zu.2.15.1.5.5.1 3 stable modes added ω =.5λ r,max, r = 1 4, γ =.2.15 1 15 2 25 3 t Estimation mode 1. zu ẑu.4 y Huzu.4 y Huzu Huzu + Hszs Huzu + Hszs.2.2 y y z u z s.2.2.4 2 22 24 26 28 3 t y(x o = 47. 2 22 24 26 28 3 t y(x = 11. H 2 performance improved. Estimation drastically improved. H s ẑ s compensates truncation. 2/21 G. Tissot, GDR CDD 216. z

21/21 G. Tissot, GDR CDD 216. Conclusion 21/21 Summary: Strategy based on global modes. Integral term H s ẑ s : Consistency of measurement s estimation, Guarantee stability of compensated system, Full stable part submitted to actuation, Low order R(t τ. Adding well selected stable modes: Improves H /H 2 performances and estimation. Perspectives: Apply on flow over a thick plate, in progress...

2/21 G. Tissot, GDR CDD 216. Model reduction 22/21 Model reduction Projection onto unstable/stable subspaces: A u = (Ψ u, AΦ u ; A s = (Ψ s, AΦ s B u = (Ψ u, B ; B s = (Ψ s, B F u = (Ψ u, F ; F s = (Ψ s, F H u = HΦ u ; H s = HΦ s,

3/21 G. Tissot, GDR CDD 216. Model reduction 23/21 Model reduction Discretised Ginzburg-Landau equation: ( I d dt y = Hz h + r, ( ( zh A11 A = 21 λ A 21 ( zh + λ ( B1 B 2 c + ( F1 F 2 f.

4/21 G. Tissot, GDR CDD 216. Model reduction 24/21 Model reduction Projected system: z h = Π z h + (I Π z h, with Π = I A 21 (A 21A 21 1 A 21. z = Π z h respects: dz = Az + Bc + F f. dt y = Hz + Dc + r, with A = Π A 11 Π, D = HA 21(A 21 A 21 1 B 2, B = B 1 ΠA 11 A 21(A 21 A 21 1 B 2, F = F 1 ΠA 11 A 21(A 21 A 21 1 F 2. (I Π z h = A 21 (A 21A 21 1 B 2 c. Heikenschloss et al. (28

Model reduction 25/21 Model reduction E 2. 1 5 1.5 1 5 1. 1 5 5. 1 6 Partial coupling Full coupling noise. 1 1 2 3 4 5 t.8.4 Response to stochastic forcing. y Huẑu Huẑu + Hsẑs Real(zu.4.2.2.4 1 15 2 25 3 t.4.2 Estimation mode 1. Minimal energy control no stable mode zu ẑu ω =.5λ r,max y Huẑu Huẑu + Hsẑs y y.4.2 2 22 24 26 28 3 t 2 22 24 26 28 3 t y(x o = 47. y(x = 11. H 2 performance degraded. Estimation not improved. 25/21 G. Tissot, GDR CDD 216.

26/21 G. Tissot, GDR CDD 216. Model reduction 26/21 Model reduction Others mode s selections: 1 Tf z(iλi 1 Ref. full coupling Ref. FI 1 least stable mode (1 6,.1 1 stable mode (1 6,.9 3 stable modes (1 3,.1 7 stable modes (1 6,.2 BT (1 1,.8 1 1.9.8.7.6.5 λ i

27/21 G. Tissot, GDR CDD 216. Model reduction 27/21 Model reduction Balanced truncation: Tf z(iλi 1 Full information stable mode 3 stable modes FI BT partial coupling BT full coupling 1 1.9.8.7.6.5 λ i

Model reduction 28/21 Model reduction Compensated systems ɛ-pseudospectra: 1 1 Minimal energy control BT ω =.5.5.5 λi λi.5.5 1.5 λ r Partial coupling vs FI. 1.5 λ r Full coupling vs FI. Balanced truncation does not preserves spectra but approximates TF. 28/21 G. Tissot, GDR CDD 216.

29/21 G. Tissot, GDR CDD 216. Model reduction 29/21 Model reduction Summary: Model H 2 H Comment No stable mode full information 3.8 14.9 Perfect est. No stable mode partial coupling 4.9 47.6 worst No stable mode full coupling 6.2 42.1 worst 1 least stable mode full coupling 4.3 16.3 good choice 1 stable mode full coupling 4.5 19.4 bad choice 3 stable modes full coupling 4.2 15.8 best choice 7 stable modes full coupling 4.1 16.7 good choice Balanced truncation partial coupling 3.7 19.9 good for H 2 Balanced truncation full coupling 3.9 2.9 good for H 2

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