Approximation of the Linearized Boussinesq Equations

Transcription

1 Approximation of the Linearized Boussinesq Equations Alan Lattimer Advisors Jeffrey Borggaard Serkan Gugercin Department of Mathematics Virginia Tech SIAM Talk Series, Virginia Tech, April 22, 2014 Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

2 Motivation Outline 1 Motivation 2 Iterative Rational Krylov Method (IRKA) 3 Boussinesq Problem 4 Results 5 Future Work Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

3 Motivation Motivation Controlling large-scale non-linear dynamical systems resulting from partial differential equation models is prohibitive and thus requires simplification, i.e. model reduction. One technique is to linearize the full system and then apply the control to the linear system. This talk focuses on reducing the linear portion of the system using the Iterative Rational Krylov Algorithm (IRKA). Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

4 Iterative Rational Krylov Method (IRKA) Outline 1 Motivation 2 Iterative Rational Krylov Method (IRKA) 3 Boussinesq Problem 4 Results 5 Future Work Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

5 Iterative Rational Krylov Method (IRKA) Descriptor System Given the system Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) where x(t) R n, u(t) R m, y(t) R p, E singular, E, A R n n, B R n m, C R p n, and D R p m. We obtain ŷ(s) = G(s)û(s) where ŷ(s) and û(s) are the Laplace transforms of y(t) and u(t) and the transfer function is G(s) = C(sE A) 1 B + D G(s) is a rational function of degree n. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

6 Iterative Rational Krylov Method (IRKA) Reduced Model Our goal is to create a reduced order model of size r n such that Ẽ ẋ r (t) = Ãx r (t) + Bu(t) y r (t) = Cx r (t) + Du(t) with x r (t) R r, Ẽ, Ã Rr r, B R r m, C R p r, and D R p m. Similarly ŷ r (s) = G(s)û(s) where ŷ r (s) and û(s) are the Laplace transforms of y r (t) and u(t) and the transfer function is G(s) = C(sẼ Ã) 1 B + D G(s) is a rational function of degree r. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

7 Iterative Rational Krylov Method (IRKA) Main Goal For a wide range of inputs, we seek to minimize the error between the output of the full model y and the output of the reduced model y r. This result is equivalent to minimizing the error between the transfer functions of the full model and the reduced model. For descriptor systems, the transfer functions may be improper rational functions. That is G(s) = G sp (s) + P(s) G(s) = G sp (s) + P(s) where P and P are the polynomial parts of the transfer functions. In order to prevent the error from blowing up as s, we must enforce equivalence of the polynomial parts, P = P. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

8 Iterative Rational Krylov Method (IRKA) Visual Representation of Model Reduction n m n A B r m à B r p C D C D p Remember that n >> r. Although the dimension of the output and input remain the same, the full state space has been projected onto a much smaller subspace. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

9 Iterative Rational Krylov Method (IRKA) Projection Framework In order to project the model onto the reduced state space, n r matrices V and W must be constructed. Using V, the full-order state x(t) is approximated by Vx r (t). Using W, the Petrov-Galerkin condition is enforced as follows W T (EVẋ r (t) AVx r (t) Bu(t)) = 0, y r (t) = CVx r (t)+du(t). This results in a reduced order model where Ẽ = W T EV, B = W T B, Ã = W T AV, C = CV, D = D. The big question is How do we find V and W? Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

10 Iterative Rational Krylov Method (IRKA) The Optimal Reduced Order Model For simplicity, let us assume that we are dealing with a SISO system. That is m = p = 1. Once we solve this problem, we can easily extend to the MIMO setting where m 1 and/or p 1. The method for doing this is described in Gugercin, Antoulas, and Beattie [2]. We want to find G such that min G G H2 = dim( G)=r min dim( G)=r ( 1 2π G(iω) G(iω) ) 2 dω This is equivalent to the following min y ỹ Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

11 Iterative Rational Krylov Method (IRKA) Theorem Given the transfer function G(s) = C(sE A) 1 B, the optimal reduced order transfer function G(s) = C(sẼ Ã) 1 B satisfies G( λ i ) = G( λ i ) G ( λ i ) = G ( λ i ) where λ i, i = 1,, r are the eigenvalues of the pencil λẽ Ã. Note: This is simply a Hermite interpolation of the rational transfer function at the points { λ i } r i=1. Unfortunately, we have no way of knowing the points {λ i } r i=1 a priori. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

12 Iterative Rational Krylov Method (IRKA) Iterative Rational Krylov Algorithm (IRKA) IRKA is the method we use to find the points {σ i } r i=1 = { λ i} r i=1. At its core, IRKA is a fixed point algorithm. In short, we first guess {σ i } r i=1 points. and determine V and W using those While the relative change in σ > tol 1 Ẽ = WT EV and à = WT AV 2 σ i λ i (A r, E r ) for i = 1,..., r. 3 Assemble new V and W using σ i. For details see Gugercin, Stykel, and Wyatt [3]. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

13 Boussinesq Problem Outline 1 Motivation 2 Iterative Rational Krylov Method (IRKA) 3 Boussinesq Problem 4 Results 5 Future Work Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

14 Boussinesq Problem Natural Convection Cavity T / n = 0 T = T h Ω T = T c T / n = 0 Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

15 Boussinesq Problem Natural Convection Cavity Figure : Flow after 60 seconds Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

16 Boussinesq Problem Boussinesq Equation Model v t over Ω = (0, 1) (0, 1) R 2. + v v = p + 1 Re v v = 0 T t + v T = 1 T + Bu RePr Note: This is just incompressible Navier-Stokes coupled with the temperature convection-diffusion equation. For details see Borggaard, Burns, Surana, and Zietsman [1]. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

17 Boussinesq Problem Linearized Boussinesq Equations Linearizing the Boussinesq equations around an average velocity results in the following Stokes-type descriptor system of index 2 E 11 ẋ 1 (t) = A 11 x 1 (t) + A 12 x 2 (t) + B 1 u(t) 0 = A 21 x 1 (t) + B 2 u(t) y(t) = C 1 x 1 (t) + C 2 x 2 (t) + Du(t) where the state is x 1 = [v, T] T, x 2 = [p] T. Notes: 1 For this model B 2 = 0 and D = 0, although this is not a requirement. 2 The control u(t) sets the temperature difference between the hot and cold walls. 3 The output y(t) is the average vorticity for the cavity. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

18 Results Outline 1 Motivation 2 Iterative Rational Krylov Method (IRKA) 3 Boussinesq Problem 4 Results 5 Future Work Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

19 Results Results The full model was reduced from n = 7301 to r = 15. The transfer function response was compared over a wide frequency range. Both systems were then simulated for 10 seconds using the same control. Results were compared for the output and state space. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

20 Results Bode Plot Singular Values (db) Bode Plot G Reduced G Frequency (rad/s) Figure : System response over the frequency domain Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

21 Results Control Term u(t) 6 Control Function u(t) 4 Temperature Difference (K) Time (s) Figure : Temperature difference between hot and cold walls Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

22 Results Output y(t) and y r (t) - Full and Reduced Models Average Vorticity x Vorticity (1/s) Full Model Reduced Model Time (s) Figure : Average vorticity of the system Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

23 Results Output Error 3.5 Average Vorticity Error x Relative Error Time (s) Figure : Relative error of average vorticity between full and reduced models Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

24 Results State Space Results Although the model reduction technique is designed to minimize the output error between the full and reduced models, we were curious as to how well the reduced system approximated the full state. By projecting the reduced model back up to the full state space using our projection matrices, we obtained the following results. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

25 Results Velocity Field (Full Model) after 10 Seconds 1 Final Velocity Vector Field Figure : Velocity Field Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

26 Results Velocity Field (Reduced Model) after 10 Seconds 1 Final Velocity Vector Field (Reduced) Figure : Velocity Field Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

27 Results State Error Between Full and Reduced Models State Error x Relative Error Time (s) Figure : Error over time Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

28 Future Work Outline 1 Motivation 2 Iterative Rational Krylov Method (IRKA) 3 Boussinesq Problem 4 Results 5 Future Work Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

29 Future Work Future Work 1 Work with larger, more complex models. 2 Couple these results with POD (proper orthogonal decomposition) to reduce the full non-linear system. 3 Establish error bounds. 4 Establish controllability conditions. Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30

30 Appendix Bibliography [1] Jeff Borggaard, John A. Burns, Amit Surana, and Lizette Zietsman. Control, estimation and optimization of energy efficient buildings. In American Control Conference, 2009., pages IEEE, [2] Serkan Gugercin, Athanasios C. Antoulas, and Christopher Beattie. H 2 model reduction for large-scale linear dynamical systems. SIAM Journal on Matrix Analysis and Applications, 30(2): , [3] Serkan Gugercin, Tatjana Stykel, and Sarah Wyatt. Model reduction of descriptor systems by interpolatory projection methods. SIAM Journal on Scientific Computing, 35(5):B1010 B1033, Alan Lattimer (VT) Reduction of Boussinesq using IRKA VT SIAM / 30