Optimal Control-Based Feedback Stabilization in Multi-Field Flow - - PowerPoint PPT Presentation

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Optimal Control-Based Feedback Stabilization in Multi-Field Flow - - PowerPoint PPT Presentation

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Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems ansch 1 Peter Benner 2 , 3 Jens Saak 2 , 3 Martin Stoll 2 Eberhard B Heiko Weichelt 3 1Department of Applied Mathematics III Friedrich-Alexander-Universit at

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Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems

Eberhard B¨ ansch1 Peter Benner2,3 Jens Saak2,3 Martin Stoll2 Heiko Weichelt3

1Department of Applied Mathematics III Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg 2Computational Methods in Systems and Control Theory Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg 3Department of Mathematics, Research group Mathematics in Industry and Technology Chemnitz University of Technology

MAX−PLANCK−INSTITUT DYNAMIK KOMPLEXER TECHNISCHER SYSTEME MAGDEBURG

Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems Trogir, 12 October 2011

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Project: SPP1253 Keys to Numerical Solution

Project: SPP1253

Project Description

Multi-Field Flow Problems – Derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems. – Explore the potentials and limitations of feedback-based (Riccati) stabilization techniques. – Explore numerical solution of algebraic Riccati equations associated to special LQR problems for linearized Navier-Stokes/Oseen-like equations.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Project: SPP1253

Project Description

Multi-Field Flow Problems – Derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems. – Explore the potentials and limitations of feedback-based (Riccati) stabilization techniques. – Explore numerical solution of algebraic Riccati equations associated to special LQR problems for linearized Navier-Stokes/Oseen-like equations. Coupled PDEs – Navier-Stokes – Navier-Stokes coupled with Convection-Diffusion equation – Phase transition liquid/solid with convection – Stabilization of a flow with a free capillary surface

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt. Proposition [Raymond ’05, Bahdra ’09] The solution to the instationary NSEs with perturbed initial data is exponentially controlled to the steady-state solution w by the feedback law u = −ρ−1B∗XzH.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 6

Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt. Proposition [Raymond ’05, Bahdra ’09] The solution to the instationary NSEs with perturbed initial data is exponentially controlled to the steady-state solution w by the feedback law u = −ρ−1B∗XzH.

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  • E. B¨

ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 7

Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt. Proposition [Raymond ’05, Bahdra ’09] The solution to the instationary NSEs with perturbed initial data is exponentially controlled to the steady-state solution w by the feedback law u = −ρ−1B∗XzH.

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  • E. B¨

ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 8

Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt. Proposition [Raymond ’05, Bahdra ’09] The solution to the instationary NSEs with perturbed initial data is exponentially controlled to the steady-state solution w by the feedback law u = −ρ−1B∗XzH. – zH := Pz, with P: Helmholtz projector div zH ≡ 0.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 9

Project: SPP1253 Keys to Numerical Solution

Optimal Control-Based Stabilization for NSEs

Analytical Solution [Raymond ’05–’07]

Linearized Navier-Stokes Control System ∂t z − ν∆z + (z · ∇)w + (w · ∇)z − ωz + ∇p = 0 in Q∞, div z = 0 in Q∞, z = bu in Σ∞, z(0) = z0 in Ω, J(z, u) = 1 2 ∞ Pz, PzL2(Ω) + ρ uTu dt. Proposition [Raymond ’05, Bahdra ’09] The solution to the instationary NSEs with perturbed initial data is exponentially controlled to the steady-state solution w by the feedback law u = −ρ−1B∗XzH. – zH := Pz, with P: Helmholtz projector div zH ≡ 0. – X = X∗: unique nonnegative semidefinite weak solution of 0 = I + (A + ωI)∗X + X(A + ωI) − X(BτB∗

τ + ρ−1BnB∗ n)X.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 12

Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z State space system: M˙ z = Az + Bu with M = MT ≻ 0

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z State space system: M˙ z = Az + Bu with M = MT ≻ 0 Generalized algebraic Riccati equation: R(X) = M + AT XM + MXA − MXBBT XM = 0

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 14

Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z State space system: M˙ z = Az + Bu with M = MT ≻ 0 Generalized algebraic Riccati equation: R(X) = M + AT XM + MXA − MXBBT XM = 0 Newton iteration: X k+1 = X k + ˜ Nk, where ˜ Nk is solution of (A − BBT X kM)T ˜ NkM + M ˜ Nk(A − BBT X kM) = −R(X k)

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 15

Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z State space system: M˙ z = Az + Bu with M = MT ≻ 0 Generalized algebraic Riccati equation: R(X) = M + AT XM + MXA − MXBBT XM = 0 Newton iteration: X k+1 = X k + ˜ Nk, where ˜ Nk is solution of (A − BBT X kM)T ˜ NkM + M ˜ Nk(A − BBT X kM) = −R(X k) Lyapunov equation ⇒ ADI-Method: AT

k ˜

NkM + M ˜ NkAk = −WT

k Wk 4/6

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Linearized NSE: ∂tz − ν∆z + (z · ∇)w + (w · ∇)z + ∇p = 0 div z = 0 DAE: M ˙ ˜ z = A˜ z + Gp 0 = G T ˜ z State space system: M˙ z = Az + Bu with M = MT ≻ 0 Generalized algebraic Riccati equation: R(X) = M + AT XM + MXA − MXBBT XM = 0 Newton iteration: X k+1 = X k + ˜ Nk, where ˜ Nk is solution of (A − BBT X kM)T ˜ NkM + M ˜ Nk(A − BBT X kM) = −R(X k) Lyapunov equation ⇒ ADI-Method: AT

k ˜

NkM + M ˜ NkAk = −WT

k Wk

Saddle Point System: [Heinkenschloss, Sorensen, Sun ’08]

  • AT + pjM − K kBT

G G T Λ ∗

  • =
  • W
  • pj ... ADI shift parameter

K k := MX kB ... Feedback operator

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Project Description

Multi-Field Flow Problems – Derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems. – Explore the potentials and limitations of feedback-based (Riccati) stabilization techniques. – Explore numerical solution of algebraic Riccati equations associated to special LQR problems for linearized Navier-Stokes/Oseen-like equations. Coupled PDEs – Navier-Stokes – Navier-Stokes coupled with Convection-Diffusion equation – Phase transition liquid/solid with convection – Stabilization of a flow with a free capillary surface

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  • E. B¨

ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Project Description

K´ arm´ an vortex street Movie: → http://www-user.tu-chemnitz.de/~wehei/uni/DA/videos/ sym_kram_feed_sqrt.avi

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 19

Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Project Description

Multi-Field Flow Problems – Derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems. – Explore the potentials and limitations of feedback-based (Riccati) stabilization techniques. – Explore numerical solution of algebraic Riccati equations associated to special LQR problems for linearized Navier-Stokes/Oseen-like equations. Coupled PDEs – Navier-Stokes – Navier-Stokes coupled with Convection-Diffusion equation – Phase transition liquid/solid with convection – Stabilization of a flow with a free capillary surface

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  • E. B¨

ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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SLIDE 20

Project: SPP1253 Keys to Numerical Solution

Keys to Numerical Solution

Project Description

NSE Coupled with Transport of Reactive Substance Movie: → http://www-user.tu-chemnitz.de/~wehei/uni/DA/videos/ dce_vergleich.avi

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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Project: SPP1253 Keys to Numerical Solution

References

1

  • E. B¨

ansch and P. Benner Stabilization of Incompressible Flow Problems by Riccati-Based Feedback

  • G. Leugering, et al (eds.), Constrained Optimization and Optimal Control for Partial

Differential Equations, International Series of Numerical Mathematics, Vol. 160, Birkh¨ auser, Basel, 2011.

2

  • P. Benner and J. Saak

A Galerkin-Newton-ADI Method for Solving Large-Scale Algebraic Riccati Equations, Preprint SPP1253-090 (January 2010)

3

  • P. Benner, J.-R. Li, and T. Penzl.

Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems.

  • Numer. Lin. Alg. Appl., vol. 15, no. 9, pp. 755–777, 2008.

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ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems