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

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 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

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. 2/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 2/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 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 − 1 B ∗ X z H . 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 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 − 1 B ∗ X z H . 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 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 − 1 B ∗ X z H . 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 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 − 1 B ∗ X z H . – z H := Pz , with P : Helmholtz projector � div z H ≡ 0. 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

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 − ω z + ∇ p = 0 in Q ∞ , div z = 0 in Q ∞ , z = bu in Σ ∞ , z (0) = z 0 in Ω , � ∞ J ( z , u ) = 1 � Pz , Pz � L 2 (Ω) + ρ u T u dt . 2 0 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 − 1 B ∗ X z H . – z H := Pz , with P : Helmholtz projector � div z H ≡ 0. – X = X ∗ : unique nonnegative semidefinite weak solution of τ + ρ − 1 B n B ∗ 0 = I + ( A + ω I ) ∗ X + X ( A + ω I ) − X ( B τ B ∗ n ) X . 3/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution Linearized NSE: ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 div z = 0 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution DAE: Linearized NSE: M ˙ z = A ˜ ˜ z + Gp ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 0 = G T ˜ div z = 0 z 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution DAE: State space system: Linearized NSE: M ˙ z = A ˜ ˜ z + Gp M ˙ z = A z + B u ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 with M = M T ≻ 0 0 = G T ˜ div z = 0 z 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution DAE: State space system: Linearized NSE: M ˙ ˜ z = A ˜ z + Gp M ˙ z = A z + B u ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 with M = M T ≻ 0 0 = G T ˜ div z = 0 z Generalized algebraic Riccati equation: R ( X ) = M + A T X M + M X A − M X BB T X M = 0 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution DAE: State space system: Linearized NSE: M ˙ ˜ z = A ˜ z + Gp M ˙ z = A z + B u ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 with M = M T ≻ 0 0 = G T ˜ div z = 0 z Generalized algebraic Riccati equation: R ( X ) = M + A T X M + M X A − M X BB T X M = 0 Newton iteration: X k +1 = X k + ˜ N k is solution of N k , where ˜ ( A − BB T X k M ) T ˜ N k ( A − BB T X k M ) = −R ( X k ) N k M + M ˜ 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems

Project: SPP1253 Keys to Numerical Solution Keys to Numerical Solution DAE: State space system: Linearized NSE: M ˙ z = A ˜ ˜ z + Gp M ˙ z = A z + B u ∂ t z − ν ∆ z + ( z · ∇ ) w + ( w · ∇ ) z + ∇ p = 0 with M = M T ≻ 0 0 = G T ˜ div z = 0 z Generalized algebraic Riccati equation: R ( X ) = M + A T X M + M X A − M X BB T X M = 0 Newton iteration: X k +1 = X k + ˜ N k is solution of N k , where ˜ ( A − BB T X k M ) T ˜ N k ( A − BB T X k M ) = −R ( X k ) N k M + M ˜ Lyapunov equation ⇒ ADI-Method: A T k ˜ N k M + M ˜ N k A k = −W T k W k 4/6 E. B¨ ansch, P. Benner, J. Saak, M. Stoll, H. Weichelt Feedback Stabilization of Multi-Field Flow Problems