Fluid flow dynamical model approximation and control

Transcription

1 Fluid flow dynamical model approximation and control... a case-study on an open cavity flow C. Poussot-Vassal & D. Sipp Journée conjointe GT Contrôle de Décollement & GT MOSAR Frequency response of an aerodynamical phenomena (Navier and Stokes equations) Open cavity flow closed loop full order model (at Re=75) 7 7 Open loop n=68974 Closed loop Performance weight Gain [db] 3 Gain [db] Reduced order model (2 states, obtained in h) Original model ( states) Optimal interpolation points ω [rad/s] ω [rad/s]

2 LARGE-SCALE DYNAMICAL MODELS... some motivating examples in the simulation & control domains Large-scale systems are present in many engineering fields: aerospace, computational biology, building structure, VLI circuits, automotive, weather forecasting, fluid flow... difficulties with simulation & memory management (e.g. ODE solvers) difficulties with analysis (e.g. frequency response, µ ssv and H computation... ) difficulties with controller design (e.g. robust, optimal, predictive,... )

3 Fluid flow dynamical models Complex phenomena describing the motion of fluid flows, described by Navier and Stokes equations, LARGE-SCALE DYNAMICAL MODELS... in fluid flow dynamical problems arising when modeling the weather, ocean currents, water flow in a pipe and air flow around a wing... Some challenges arising Modeling and simulation Control turbulences

4 OPEN-CAVITY FLOW AND HOPF BIFURCATION

5 OUTLINES Physical model and dynamical modeling Navier and Stokes equations and assumptions Linearisation and simplifications Reduce and control approach Large-scale dynamical model approximation Active closed-loop control design Conclusions

6 Navier and Stokes equations PHYSICAL MODEL AND DYNAMICAL MODELING Navier and Stokes equations and assumptions tu + u u = p + 1 u Re (1) u = (2) or in a condensed way ẋ(t) = f`x(t), Re (3) Existence of equilibrium points for a range of Reynolds numbers Family of base-flows parametrized by the Reynolds number: f`x, Re =

7 Navier and Stokes equations PHYSICAL MODEL AND DYNAMICAL MODELING Navier and Stokes equations and assumptions Existence of equilibrium points for a range of Reynolds numbers Family of base-flows parametrized by the Reynolds number: f`x, Re =

8 Linearisation for different Reynolds Numbers PHYSICAL MODEL AND DYNAMICAL MODELING Linearisation and simplifications - Eigenvalues where ɛ is small x(t) = x (Re) (t) + ɛx (Re) 1 (t) (1) ẋ (Re) 1 (t) = f x x x(re) 1 (t) = A(Re)x 1 (t) (2)

9 PHYSICAL MODEL AND DYNAMICAL MODELING Linearisation and simplifications - Eigenvectors Right and left eigenvectors Localization of sensor and actuator

10 Actuator/sensor PHYSICAL MODEL AND DYNAMICAL MODELING Linearisation and simplifications - Dynamical model and control setting Eẋ(t) = A(Re)x(t) + Bu(t) y(t) = Cx(t) (3) Actuator (volumic forcing in momentum equations) Sensor (shear stress)

11 Actuator/sensor PHYSICAL MODEL AND DYNAMICAL MODELING Linearisation and simplifications - Dynamical model and control setting Eẋ(t) = A(Re)x(t) + Bu(t) y(t) = Cx(t) (4) Two Reynolds cases (Re = 7 and Re = 75) Single Input Single Output Differential Algebraic Equations (SISO DAE) 8 unstable modes, order 65, states 5 Open cavity flow model (at Re=7) Open cavity flow model (at Re=75) Gain [db] 2 Gain [db] ω [rad/s] ω [rad/s]

12 PHYSICAL MODEL AND DYNAMICAL MODELING Reduce and control approach Proposed procedure Approximate the large-scale dynamical model Design a stabilizing active closed loop control strategy Frequency response of an aerodynamical phenomena (Navier and Stokes equations) Open cavity flow closed loop full order model (at Re=75) 7 7 Open loop n=68974 Closed loop Performance weight Gain [db] 3 Gain [db] Reduced order model (2 states, obtained in h) Original model ( states) Optimal interpolation points ω [rad/s] ω [rad/s] Challenge of simulating and controlling such high complexity system...

13 PHYSICAL MODEL AND DYNAMICAL MODELING Reduce and control approach Proposed procedure Approximate the large-scale dynamical model Design a stabilizing active closed loop control strategy Frequency response of an aerodynamical phenomena (Navier and Stokes equations) Open cavity flow closed loop full order model (at Re=75) 7 7 Open loop n=68974 Closed loop Performance weight Gain [db] 3 Gain [db] Reduced order model (2 states, obtained in h) Original model ( states) Optimal interpolation points ω [rad/s] ω [rad/s] Challenge of simulating and controlling such high complexity system...

14 OUTLINES Physical model and dynamical modeling Large-scale dynamical model approximation Projection-based approximation framework Approximation in the H 2, H 2,Ω and L 2 -norm MIMO IRKA (or ITIA) IETIA Balanced Truncation POD Fluid flow dynamical model approximation Active closed-loop control design Conclusions

15 Projection-based approximation framework Let H : C C ny nu be a n u inputs n y outputs, full order H ny nu 2 (or L ny nu 2 ) complex-valued function describing a LTI dynamical system as a DAE of order n, with realization H: j Eẋ(t) = Ax(t) + Bu(t) H : (5) y(t) = Cx(t) E,A B C

16 Projection-based approximation framework Let H : C C ny nu be a n u inputs n y outputs, full order H ny nu 2 (or L ny nu 2 ) complex-valued function describing a LTI dynamical system as a DAE of order n, with realization H: j Eẋ(t) = Ax(t) + Bu(t) H : (5) y(t) = Cx(t) the approximation problem consists in finding V, W R n r (with r n) spanning V and W subspaces and forming a projector Π V,W = V W T, such that j W Ĥ : T EV ˆx(t) = W T AV ˆx(t) + W T Bu(t) ŷ(t) = CV ˆx(t) (6) well approximates H. W T EV,W T AV W T B Ê, ˆB CV Π V,W = Ĉ

17 Projection-based approximation framework Let H : C C ny nu be a n u inputs n y outputs, full order H ny nu 2 (or L ny nu 2 ) complex-valued function describing a LTI dynamical system as a DAE of order n, with realization H: j Eẋ(t) = Ax(t) + Bu(t) H : (5) y(t) = Cx(t) the approximation problem consists in finding V, W R n r (with r n) spanning V and W subspaces and forming a projector Π V,W = V W T, such that j W Ĥ : T EV ˆx(t) = W T AV ˆx(t) + W T Bu(t) ŷ(t) = CV ˆx(t) (6) well approximates H. Small approximation error and/or global error bound Stability / passivity preservation Numerically stable & efficient procedure

18 H 2 model approximation Approximation in the H 2, H 2,Ω and L 2-norm 1 2 Ĥ := arg min ny nu G H2 rank(g) = r n H G H2 (7) Bode Diagram Magnitude (db) Frequency (rad/s) 1 S. Gugercin and A C. Antoulas and C A. Beattie, "H 2 Model Reduction for Large Scale Linear Dynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 3(2), June 28, pp K. A. Gallivan, A. Vanderope, and P. Van-Dooren, "Model reduction of MIMO systems via tangential interpolation", SIAM Journal of Matrix Analysis and Application, vol. 26(2), February 24, pp

19 H 2 model approximation Approximation in the H 2, H 2,Ω and L 2-norm 1 2 Ĥ := arg min ny nu G H2 rank(g) = r n H G H2 (7) Magnitude (db) Bode Diagram Frequency (rad/s) Energy to an impulse input Z «`H(iν)H(iν) dν 1 H 2 H 2 := trace 2π ««:= trace CPC T = trace B T QB nx «:= trace φ i H( λ i ) T i=1 1 S. Gugercin and A C. Antoulas and C A. Beattie, "H 2 Model Reduction for Large Scale Linear Dynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 3(2), June 28, pp K. A. Gallivan, A. Vanderope, and P. Van-Dooren, "Model reduction of MIMO systems via tangential interpolation", SIAM Journal of Matrix Analysis and Application, vol. 26(2), February 24, pp

20 H 2,Ω model approximation Approximation in the H 2, H 2,Ω and L 2-norm 3 4 Ĥ := arg min ny nu G H rank(g) = r n H G H2,Ω (8) Bode Diagram Magnitude (db) Frequency (rad/s) 3 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "A Spectral Expression for the Frequency-Limited H 2 -norm", Available as P. Vuillemin, C. Poussot-Vassal and D. Alazard, "Spectral expression for the Frequency-Limited H 2 -norm of LTI Dynamical Systems with High Order Poles", European Control Conference, 214, pp

21 H 2,Ω model approximation Approximation in the H 2, H 2,Ω and L 2-norm 3 4 Ĥ := arg min ny nu G H rank(g) = r n H G H2,Ω (8) Magnitude (db) Bode Diagram Frequency (rad/s) Energy (in a finite frequency) to an impulse input Z «1 H 2 H 2,Ω := trace `H(iν)H(iν) dν π Ω ««:= trace CP Ω C T = trace B T Q Ω B nx := trace φ i H( λ i ) T 2π «atan ωλi i=1 3 P. Vuillemin, C. Poussot-Vassal and D. Alazard, "A Spectral Expression for the Frequency-Limited H 2 -norm", Available as P. Vuillemin, C. Poussot-Vassal and D. Alazard, "Spectral expression for the Frequency-Limited H 2 -norm of LTI Dynamical Systems with High Order Poles", European Control Conference, 214, pp

22 H model approximation Approximation in the H 2, H 2,Ω and L 2-norm 5 Ĥ := arg min ny nu G H rank(g) = r n H G H (9) Bode Diagram Magnitude (db) Frequency (rad/s) 5 P. Vuillemin, C. Poussot-Vassal, D. Alazard, "Two upper bounds on the H -norm of LTI dynamical systems", 19th IFAC World Congress, pp , 214.

23 H model approximation Approximation in the H 2, H 2,Ω and L 2-norm 5 Ĥ := arg min ny nu G H rank(g) = r n H G H (9) 3 2 Bode Diagram Worst case to an impulse input (numerically complex to compute) Magnitude (db) H H := sup σ (H(jω)) ω R y 2 := max w L 2 u Frequency (rad/s) 5 P. Vuillemin, C. Poussot-Vassal, D. Alazard, "Two upper bounds on the H -norm of LTI dynamical systems", 19th IFAC World Congress, pp , 214.

24 Mismatch objective and eigenvector preservation Approximation in the H 2, H 2,Ω and L 2-norm 6 Ĥ := arg min ny nu G L2 rank(g) = r n λ k (G) λ(h) k = 1,..., q 1 < r H G H2 (1) More than a H 2 (sub-optimal) criteria Keep some user defined eigenvalues... e.g. the unstable/well known ones 6 C. Poussot-Vassal and P. Vuillemin, "An Iterative Eigenvector Tangential Interpolation Algorithm for Large-Scale LTI and a Class of LPV Model Approximation", European Control Conference, 213, pp

25 Approximation in the H 2, H 2,Ω and L 2-norm MIMO Iterative Krylov Interpolation Algorithm (or ITIA) H 2 -optimal, but still do not theoretically preserves stability Numerically very efficient (e.g. with sparse methods, Ax = b) Iterative Eigenvector Tangential Interpolation Algorithm (IETIA) H 2 sub-optimal, but still do not theoretically preserves stability Numerically very efficient (e.g. with sparse methods, Ax = b and AV = EV λ) Applicable to L 2 dynamical systems Balanced Truncation Proper Orthogonal Decomposition (BT POD) Provides a H -norm mismatch error (not tight), preserves stability Costly to compute, but a Matrix free version alleviate this problem by replacing by simulation (direct and adjoint)

26 MIMO IRKA (or ITIA) - H 2 optimality conditions (Tangential subspace approach) 7 8 Given H(s), let V C n r and W C n r be matrices of full column rank r such that W V = I r. If, for j = 1,..., r, i h i h(σ j E A) 1 Bˆb j span(v ) and (σ j E A T ) 1 C T ĉ j span(w ) (11) where σ j C, ˆb j C nu and ĉ j C ny, be given sets of interpolation points and left and right tangential directions, respectively. 7 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H 2 -optimal model reduction of MIMO systems", Applied Mathematics Letters, vol. 21(12), December 28, pp S. Gugercin and A C. Antoulas and C A. Beattie, "H 2 Model Reduction for Large Scale Linear Dynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 3(2), June 28, pp

27 MIMO IRKA (or ITIA) - H 2 optimality conditions (Tangential subspace approach) 7 8 Given H(s), let V C n r and W C n r be matrices of full column rank r such that W V = I r. If, for j = 1,..., r, i h i h(σ j E A) 1 Bˆb j span(v ) and (σ j E A T ) 1 C T ĉ j span(w ) (11) where σ j C, ˆb j C nu and ĉ j C ny, be given sets of interpolation points and left and right tangential directions, respectively. Then, the reduced order system Ĥ(s) satisfies the tangential interpolation conditions ĉ j H( ˆσ j )ˆb j = Ĥ( ˆσ j)ˆb j ĉ j H( ˆσ j) = ĉ j Ĥ( ˆσ j) d ds H(s) ˆbj = ĉ d (12) j Ĥ(s) ˆbj s= ˆσj ds s= ˆσj 7 P. Van-Dooren, K. A. Gallivan, and P. A. Absil, "H 2 -optimal model reduction of MIMO systems", Applied Mathematics Letters, vol. 21(12), December 28, pp S. Gugercin and A C. Antoulas and C A. Beattie, "H 2 Model Reduction for Large Scale Linear Dynamical Systems", SIAM Journal on Matrix Analysis and Applications, vol. 3(2), June 28, pp

28 Require: H = (E, A, B, C), {σ () 1,..., σ() q 2 } C q 2, {ˆb 1,..., ˆb q2 } C nu q 2, {ĉ 1,..., ĉ q2 } C ny q 2 and r N 1: Construct, span`v (σ () j, ˆb j ) and span`w (σ () j, ĉ ) j 2: Compute W W (V T W ) 1 3: while Stopping criteria do 4: k k + 1 5: Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV 6: Compute ÂR = Λ(Â, Ê)R and LÂ = Λ(Â, Ê)L 7: Compute {ˆb 1,..., ˆb r} = ˆB T L and {ĉ 1,..., ĉ r } = ĈR 8: Set σ (i) = Λ(Â, Ê) 9: Construct, span`v (σ (k) j, ˆb j ) and span`w (σ (k) j, ĉ ) j 1: Compute W W (V T W ) 1 11: end while 12: Construct Ĥ := (W T EV, W T AV, W T B, CV ) Ensure: V, W R n r, W T V = I r (13) (14)

29 Require: H = (E, A, B, C), {σ () 1,..., σ() q 2 } C q 2, {ˆb 1,..., ˆb q2 } C nu q 2, {ĉ 1,..., ĉ q2 } C ny q 2 and r N 1: Construct, span`v (σ () j, ˆb j ) and span`w (σ () j, ĉ ) j 2: Compute W W (V T W ) 1 3: while Stopping criteria do 4: k k + 1 5: Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV 6: Compute ÂR = Λ(Â, Ê)R and LÂ = Λ(Â, Ê)L 7: Compute {ˆb 1,..., ˆb r} = ˆB T L and {ĉ 1,..., ĉ r } = ĈR 8: Set σ (i) = Λ(Â, Ê) 9: Construct, span`v (σ (k) j, ˆb j ) and span`w (σ (k) j, ĉ ) j 1: Compute W W (V T W ) 1 11: end while 12: Construct Ĥ := (W T EV, W T AV, W T B, CV ) Ensure: V, W R n r, W T V = I r (13) (14)

30 IETIA - H 2 & spectral optimality conditions (Tangential subspace approach) 9 Given H(s), let V C n r and W C n r be matrices of full column rank r = q 1 + q 2 such that W V = I r. If, for i = 1,..., q 1 and j = 1,..., q 2, i h i hr i (σ j E A) 1 Bˆb j span(v ) and li (σ j E A T ) 1 C T ĉ j span(w ) (15) l i Cn and r i Cn are left and right eigenvectors associated to λ i C eigenvalues associated to A, E and σ j C, ˆb j C nu and ĉ j C ny, be given sets of interpolation points and left and right tangential directions, respectively. 9 C. Poussot-Vassal and P. Vuillemin, "An Iterative Eigenvector Tangential Interpolation Algorithm for Large-Scale LTI and a Class of LPV Model Approximation", European Control Conference, 213, pp

31 IETIA - H 2 & spectral optimality conditions (Tangential subspace approach) 9 Given H(s), let V C n r and W C n r be matrices of full column rank r = q 1 + q 2 such that W V = I r. If, for i = 1,..., q 1 and j = 1,..., q 2, i h i hr i (σ j E A) 1 Bˆb j span(v ) and li (σ j E A T ) 1 C T ĉ j span(w ) (15) li Cn and ri Cn are left and right eigenvectors associated to λ i C eigenvalues associated to A, E and σ j C, ˆb j C nu and ĉ j C ny, be given sets of interpolation points and left and right tangential directions, respectively. Then, the reduced order system Ĥ(s) satisfies the eigenvalue conditions, and the tangential interpolation conditions {λ 1,..., λ q 1 } Λ(Â, Ê) (16) ĉ j H( ˆσ j )ˆb j = Ĥ( ˆσ j)ˆb j ĉ j H( ˆσ j) = ĉ j Ĥ( ˆσ j) d ds H(s) ˆbj = ĉ d (17) j Ĥ(s) ˆbj s= ˆσj ds s= ˆσj 9 C. Poussot-Vassal and P. Vuillemin, "An Iterative Eigenvector Tangential Interpolation Algorithm for Large-Scale LTI and a Class of LPV Model Approximation", European Control Conference, 213, pp

32 Require: H = (E, A, B, C), {λ 1,..., λ q 1 } C q 1, {σ () 1,..., σ() q 2 } C q 2, {ˆb 1,..., ˆb q2 } C nu q 2, {ĉ 1,..., ĉ q2 } C ny q 2 and r = q 1 + q 2 N 1: Compute {l1,..., l q 1 } and {r1,..., r q 1 }, eigenvectors of {λ 1,..., λ q 1 } 2: Construct, span`v (li, σ() j, ˆb j ) and span`w (ri, σ() j, ĉ ) j 3: Compute W W (V T W ) 1 4: while Stopping criteria do 5: k k + 1 6: Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV 7: Compute ÂR = ÊΛ(Â, Ê)R and LÂ = Λ(Â)L 8: Compute {ˆb 1,..., ˆb q2 } = ˆB T L and {ĉ 1,..., ĉ q 2 } = ĈR 9: Set σ (i) = Λ(Â, Ê) 1: Construct, span`v (li, σ(k) j, ˆb j ) and span`w (ri, σ(k) j, ĉ ) j 11: Compute W W (V T W ) 1 12: end while 13: Construct Ĥ := (W T EV, W T AV, W T B, CV ) Ensure: V, W R n r, W T V = I r, {λ 1,..., λ q 1 } Λ(Â, Ê) (18) (19)

33 Require: H = (E, A, B, C), {λ 1,..., λ q 1 } C q 1, {σ () 1,..., σ() q 2 } C q 2, {ˆb 1,..., ˆb q2 } C nu q 2, {ĉ 1,..., ĉ q2 } C ny q 2 and r = q 1 + q 2 N 1: Compute {l1,..., l q 1 } and {r1,..., r q 1 }, eigenvectors of {λ 1,..., λ q 1 } 2: Construct, span`v (li, σ() j, ˆb j ) and span`w (ri, σ() j, ĉ ) j 3: Compute W W (V T W ) 1 4: while Stopping criteria do 5: k k + 1 6: Ê = W T EV, Â = W T AV, ˆB = W T B, Ĉ = CV 7: Compute ÂR = ÊΛ(Â, Ê)R and LÂ = Λ(Â)L 8: Compute {ˆb 1,..., ˆb q2 } = ˆB T L and {ĉ 1,..., ĉ q 2 } = ĈR 9: Set σ (i) = Λ(Â, Ê) 1: Construct, span`v (li, σ(k) j, ˆb j ) and span`w (ri, σ(k) j, ĉ ) j 11: Compute W W (V T W ) 1 12: end while 13: Construct Ĥ := (W T EV, W T AV, W T B, CV ) Ensure: V, W R n r, W T V = I r, {λ 1,..., λ q 1 } Λ(Â, Ê) (18) (19)

34 Balanced Truncation POD - Idea Assume a stable system, the impulse response, t such that h(t) = Ce At B Input-to-state map x c(t) = e At B State-to-output map x o(t) = Ce At = (e A t C ) Corresponding to Gramian: P = X t Q = X t x c(t)x c(t) = x o(t)x o(t) = Z e At BB e A t dt Z e A t C Ce At dt (2) solution of the Lyapunov equations, j AP + PA + BB = A Q + QA + C C = (21)

35 Balanced Truncation POD - in its Balanced basis T = [T 1,..., T n] Meaning of Gramians: x f P 1 x f, is the minimal energy required to steer the state from to x f as t. x Qx is the maximal energy produced by observing the output of the system corresponding to an initial state x when no input is applied. Balanced basis T = [T 1,..., T n]: P = Q = S = diag(σ 1,..., σ n) with σ 1 > σ 2 > > σ n (22) T1 P 1 T 1 = 1 (easily controllable) and T σ 1 1 QT 1 = σ 1 (easily observable) TnP 1 T n = 1 (weakly controllable) and T σ n nqtn = σn (weakly observable) moreover, stability is preserved error between original and reduced systems is upper-bounded by, σ r H Ĥ H 2(σ r σ n) (23) where σ i, i = 1,... n are the Hankel singular values.

36 Require: A R n n, B R n nu, C R ny n, r N 1: Solve AP + PA T + BB T = 2: Solve A T Q + QA + C T C = 3: P = UU T and Q = LL T 4: SVD decomposition: [Z, S, Y ] = SVD(U T L) 5: Set V = UZS 1/2 6: Set W = LY S 1/2 Balanced Truncation POD 7: Apply projectors V and W and obtain 2 A 11 A 12 B 1 3 H = 4 A 21 A 22 B 2 5 (24) C 1 C 2 D» A11 B 8: Approximation is obtained by Ĥ = 1 C 1 D Ensure: Small approximation error

37 Balanced Truncation POD - Factorization of Gramians with snapshot method Numerical approximation Solving Lyapunov equations is memory/time consuming. P = UU (25) usually done with Cholesky Factorization, but noticing that (i finite) Z P = e At BB e A t X x c(t i )x c (t i) t i ˆ x c(t 1 ) t x c(t 2 ) t x c(t 1 ) t x c(t 2 ) t (26) where x c(t) = e At B (here use of the linearized model, ẋ(t) = Ax(t), x() = B). Same process for Q with adjoint simulation ẋ(t) = A x(t), x() = C.

38 Balanced Truncation POD - Hankel singular values and bpod structures)

39 Balanced Truncation POD - Properties (lower/upper bounds & mismatch errors)

40 Fluid flow dynamical model approximation - Re=7 5 Open cavity flow reduced order models POD (at Re=7) 5 Open cavity flow reduced order models ITIA (at Re=7) Gain [db] 2 Gain [db] ROM r=16 ROM r=18 ROM r=2 Original n= ROM r=16 (in 1.683h) ROM r=18 (in h) ROM r=2 (in h) Original n= ω [rad/s] ω [rad/s] Open cavity flow reduced order models POD (at Re=7) Open cavity flow reduced order models ITIA (at Re=7) Phase [deg] 6 Phase [deg] ROM r=16 ROM r=18 ROM r=2 Original n= ROM r=16 ROM r=18 ROM r=2 Original n= ω [rad/s] ω [rad/s]

41 Fluid flow dynamical model approximation - Re=75 Open cavity flow reduced order models POD (at Re=75) Open cavity flow reduced order models ITIA (at Re=75) Gain [db] 3 Gain [db] ROM r=16 ROM r=18 ROM r=2 Original n= ω [rad/s] ROM r=16 (in h) ROM r=18 (in 1.464h) ROM r=2 (in h) Original n= ω [rad/s] Open cavity flow reduced order models POD (at Re=75) Open cavity flow reduced order models ITIA (at Re=75) Phase [deg] 4 5 Phase [deg] ROM r=16 ROM r=18 ROM r=2 Original n= ROM r=16 ROM r=18 ROM r=2 Original n= ω [rad/s] ω [rad/s]

42 OUTLINES Physical model and dynamical modeling Large-scale dynamical model approximation Active closed-loop control design Objectives Results Conclusions

43 Objectives and H control approach Stabilize the system Damp modes Potentially attenuate the H -norm ACTIVE CLOSED-LOOP CONTROL DESIGN Engineering appealing / structured in view of RT implementation A standard control approach: Objectives K = arg min K K F l(ĥ, K) H (27) P w(t) W i w(t) Ĥ z(t) W o z(t) u(t) K y(t)

44 ACTIVE CLOSED-LOOP CONTROL DESIGN Results - Spectral (ROM) Open loop Closed loop Imag Real

45 ACTIVE CLOSED-LOOP CONTROL DESIGN Results - Impulse (un-normalized ROM) Shear stress Time [s]

46 ACTIVE CLOSED-LOOP CONTROL DESIGN Results - Frequency-domain (un-normalized ROM) Open cavity flow reduced order models ITIA (at Re=75) 7 Open loop ROM r=18 (ITIA) Closed loop Performance weight 6 5 Gain [db] ω [rad/s]

47 ACTIVE CLOSED-LOOP CONTROL DESIGN Results - Frequency-domain (original LSS) Open cavity flow closed loop full order model (at Re=75) 7 Open loop n=68974 Closed loop Performance weight Gain [db] ω [rad/s]

48 OUTLINES Physical model and dynamical modeling Large-scale dynamical model approximation Active closed-loop control design Conclusions About fluid flow control About model approximation

49 CONCLUSIONS About fluid flow control About today s presentation Good performance of two model approximation techniques First attempt of H control synthesis (controller of order 6) Application on Navier and Stokes equations for an open cavity flow Some perspectives Extension to robust analysis / parameter dependent control Apply the realization-less approaches (e.g. handle delays) Include learning policy for (on-line) model accuracy enhancement?

50 Physical phenomena Model approximation Control design Conclusions C ONCLUSIONS About model approximation - MORE toolbox 1 I Successful application of advanced model approximation techniques I both full and sparse I on a complex unstable aerodynamical set of equations u(f ) = [u(f1 )... u(fi )] y(f ) = [y(f1 )... y(fi )] E x (t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) H(s) = e τ s u(x, t) t Σ (A, B, C, D)i Data Σ State x(t) Rn, n large or infinite more AP + P AT + BB T = =... PDEs Infinite order equations (require meshing) WTV Kr (A, B) model reduction toolbox DAE/ODE Σ (A, B, C, D )i Reduced DAE/ODE Reduced state x (t) Rr with r n (+) Simulation (+) Analysis (+) Control (+) Optimization 1 C. Poussot-Vassal and P. Vuillemin, "Introduction to MORE: a MOdel REduction Toolbox", IEEE Multi Systems Conference, pp , 212. C. Poussot-Vassal & D. Sipp [Onera] Fluid flow dynamical model approximation and control

51 Fluid flow dynamical model approximation and control... a case-study on an open cavity flow C. Poussot-Vassal & D. Sipp Journée conjointe GT Contrôle de Décollement & GT MOSAR Frequency response of an aerodynamical phenomena (Navier and Stokes equations) Open cavity flow closed loop full order model (at Re=75) 7 7 Open loop n=68974 Closed loop Performance weight Gain [db] 3 Gain [db] Reduced order model (2 states, obtained in h) Original model ( states) Optimal interpolation points ω [rad/s] ω [rad/s]