Reduced-Order Model Development and Control Design

Transcription

1 Reduced-Order Model Development and Control Design K. Kunisch Department of Mathematics University of Graz, Austria Rome, January 6

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3 Laser-Oberflächenhärtung eines Stahl-Werkstückes [WIAS-Berlin]: workpiece laser beam heated zone moving direction Phasenübergänge: Ferrit Pearlit Bainit Martensit heizen Austenit kühlen Ferrit Pearlit Bainit Martensit 1. Laser-Oberflächenhärtung 1

4 Data: T >, Ω R d bounded, d = 2 or 3 ϱ, c, k, L positive constants θ H 1 (Ω), α L (Q) non-negativ Energy-Balance: ϱcθ t k θ = αu ϱla t in Q = (, T ) Ω θ n = on Σ = (, T ) Ω θ(, ) = θ in Ω Austenit-phasetransition: a t = f (θ, a) in Q, a(, ) = in Ω state-variables: θ and a control-parameter: u = u(t) L 2 (, T ) (function of time) c.f. S.Volkwein

5 FE solution θ at time t=t y axis x axis POD 1 solution θ at time t=t POD 2 solution θ at time t=t y axis 1 1 x axis y axis 1 2 x axis Fehlerabschätzungen für POD Approximationen 9 note the scales

6 m max Ψ i L = j m θj l θj FE L ( θ j (Ω), Ψ i l θj FE 2 max j m θj FE L L 2 = j= L 2 ) (Ω) 1/2 { i = 1 POD with DQ m (Ω) θ j FE 2 i = 2 POD without DQ j= L 2 (Ω) X = L 2 (Ω) X = H 1 (Ω) l Ψ 1 L Ψ 2 L Ψ 1 L 2 Ψ 2 L 2 Ψ 1 L Ψ 2 L Ψ 1 L 2 Ψ 2 L % 4.6% 11.3% 12.1% 21.% 4.1% 22.9% 11.8% % 38.4% 3.4% 6.8% 16.2% 37.8% 4.4% 6.7% 2 6.% 35.3% 1.8% 4.3% 13.5% 34.3% 2.2% 4.3% % 26.9%.6% 2.9% 4.% 24.6% 1.2% 2.9% error-quotient: E(l) = l d λ i/ λ i 1% 94% i=1 i=1 l = 5 l = 1 l = 15 l = 2 l = 25 l = 3 E(l), X = L 2 (Ω) E(l), X = H 1 (Ω) Fehlerabschätzungen für POD Approximationen 1

7 OS-POD min J l (x, ψ, u) over (x, ψ, u) L 2 (R l ) X l L 2 (, T ; R m ), subject to (POSP l ) E(ψ) ẋ(t) + A(ψ)x(t) + N(x(t), ψ) = B(ψ)u(t), t (, T ], E(ψ) x() = x (ψ), d dt y(t) + Ay(t) + N (y(t)) = Bu(t), t (, T ], y() = y, R(y)ψ i = λ i ψ i, i = 1,..., l, ψ i, ψ j = δ ij i, j = 1,..., l.

8 The following optimality system holds: { E(ψ) q(t) + (A(ψ) + N T x (x, ψ)) q(t) = J l x(x, ψ, u), q(t ) = ṗ(t) + Ap(t) + N (y(t)) p(t) = l i=1 y(t), µ i X I 1 ψ i + y(t), ψ i X I 1 µ i p(t ) = { ηi = 1 2 G i(x, ψ, u, q), ψ i X,X µ i = (R λ i I) 1 [2 η i ψ i + I G i (x, ψ, u, q)], for i = 1,..., l, and Ru(t) = B T (ψ) q(t) + B p(t).

9 TRUST REGION POD Minimize u Uad J(u) Assume that current iterate u k is given. Build quadratic model around u k. m k (u k + s) = J(u k ) + g T k s st H k s For the computation of the next step solve Minimize m k (u k + s), s 2 δ k where δ k is current trust region radius. c.f. M.Fahl, E.Sachs

10 Full where min J(u) = 1 2 T y(u; t, x) y d (t, x) 2 L 2 dt y(t, x) = y(u; t, x) solution to the PDE for given control u(x, t) y d (t, x) desired state Reduced min f POD (u) = 1 2 T y POD (u; t, x) y d (t, x) 2 L 2 dt where y POD = y POD (u) POD solution for given control u(t).? when update POD-basis? dimension of POD basis

11 Consider as local model for J around u k. m k (u k + s) = J POD (u k + s) Unlike common quadratic trust region model functions we have m k (u k + s) is not quadratic in s m k (u k ) J(u k ) g k := m k (u k ) J(u k )

12 Algorithm 1: Outline of TR-POD Alg. Let < η 1 < η 2 < 1, < γ 1 γ 2 < 1 γ 3 and u 1, δ 1 >, set k = Compute snapshot set corresponding to control u k 2. Compute POD basis and build POD control model 3. Compute minimizer s k of 4. Compute J(u k + s k ) and 5. Update trust region: min m k (u k + s) s 2 δ k ρ k = J(u k) J(u k + s k ) m k (u k ) m k (u k + s k ) If ρ k η 2 : set u k+1 = u k + s k and increase trust region radius δ k+1 = γ 3 δ k If η1 < ρ k < η 2 : set u k+1 = u k + s k and decrease trust region radius δ k+1 = γ 2 δ k If ρ k η 1 : set u k+1 = u k and decrease trust region radius δ k+1 = γ 1 δ k 6. Set k = k + 1 and GOTO 1

13 Algorithm 2: Step determination algorithm (Toint 88) Let < α < β < 1, < ν 1 1, ν 2 >, ν 3 >, < µ 1 be given. Phase 1: Find λ A k such that m k (u k λ A k g k) m k (u k ) αλ A k g k 2 and λ A k g k 2 δ k λ A k min{ δ k g k, ν 2} (there are alternatives) Phase 2: If δ k ν 3 : Choose step s k such that m k (u k ) m k (u k + s k ) µ(m k (u k ) m k (u k λ A k g k)) s k 2 δ k

14 Convergence Regularity assumption on f Assumptions on model m k each m k is differentiable g k f (u k ) g k ς for some ς > (Carter 91). Sufficient decrease condition for Algorithm 2 J(u k ) J(u k + s k ) c g k 2 min{δ k, g k 2 }

15 Theorem Let J satisfy the standard assumptions. Assume that {u k } is a sequence of iterates generated by Algorithm 1 with step determination according to Algorithm 2 and g k J(u k ) g k ς for all k for some ς with < ς < 1 η 2. Then lim k J(u k) =. Note: Consistency of gradients in POD framework true in practice, but no theory.

16 Example (Sachs et alt.) k J(u k ) m k (u k ) δ k s k 2 ρ k dim

17 BALANCED TRUNCATION and POD ẏ = Ay + Bu, z = Cy y() = y, C n A... stable, i.e.: Re λ(a) <. controllability operator Ψ c : L 2 (, ] C n, Ψ c (u) = e At Bu(t)dt controllability Gramian Z c = Ψ c Ψ c = e At B B e A t dt = eat B B e A t dt Z c pos. def. iff (A, B) controllable Liapunov equation AZ c + Z c A + BB = ref: Dullerud-Paganini, A Course in Robust Control; C.W. Rowley, Int.J. Bifurcation and Chaos

18 observability operator Ψ o : C n L 2 (, ), Ψ o (y o ) = C e At y o... t... t <. observability Gramian: Z o = Ψ oψ o = e A s C C e As ds Z o pos. def. iff. (C, A) observable, z 2 L 2 (o, ) = y o Z o y o. Liapunov equation A Z o + Z o A + C C =. {η i }... EVE (Z 1/2 o ), E o = {Z 1/2 o y o : y o C n 1}

19 recall control-liapunov equation A Z c + Z c A + BB =. E c = {Ψ c u : u L 2 1} = {Z 1/2 c y c : y c C n 1}

20 Change of basis ỹ = Ty, Zc = T Z c T, Zo = (T ) 1 Z o T 1 If (A, B, C) controllable and observable balanced realization, i.e. T. Z c = Z o = Σ = diag(σ 1,..., σ n ). (Hankel singular values) Z c Z o T = T Σ 2, or Z c = L L (Cholesky), < L Z o L = U Σ 2 U, U... orthogonal T = Σ 1/2 U L 1. balanced truncation: project onto first r columns of T. reduced state again balanced and stable

21 Error bounds input-output operator: (Gu)(t) = t CeA(t τ) Bu(τ) dτ transfer function: Ĝ(s) = C(s I A) 1 B Ĝ = max ω σ 1(Ĝ(iω)) = max u Gu L 2 u L 2 reduction to r dimensions: Ĝr G > σ r+1 balanced truncation: Ĝbal r G < 2 n j=r+1 σ j but not optimal (Hankel-norm reductions) 1 1 next we recall POD formulas

22 Y n : R n X, Y n υ = n j= β j υ j y(t j ), Y n z = ( z, y(t j ) X,.., z, y(t n ) X ) R n = Y n Y n = n j= β jy(t j ) y(t j ) = n j= β jy(t j ) y(t j ), X L(X) K n = Y n Y n = y(t i ), y(t j ) X R n n R n... {σ j }, {ϕ j }; K n... {σ j }, {ψ j }; ϕ k = 1 σk K = Y Y = Y : L 2 (, T ; R) X, Yv = R = YY = T T T y(t) y(t) dt n (ψ k ) j y j j= v(t)y(t, ) dt L(X) y(t, ), y(t, ) X dτ L(L 2, L 2 )... Hilbert-Schmidt X... finite dimensional, SVD: Y n ψ j = σ j ϕ j, Y n ϕ j = σ j ψ j R n ϕ j = Y n Y n ϕ j = σ j ϕ j, y j = n i= ϕ i, y j X ϕ i

23 POD + is concept for nonlinear systems high-energy modes are not necessarily most important to dynamics Balanced truncation + takes into consideration input - output information restricted to linear (stable) systems.

24 Connections POD Balanced Truncation controllability Gramian: Z c = eat B B e A t dt e.g.: B = [b 1,..., b m ] y i = e At b i, Z c = y 1(t)y 1 (t) t,..., y m (t)y m (t) t correlation matrix: R = Y Y = T y(t) y(t)t dt discretized Z c = Y c Y c, Z = Y o Y o R n n Y c R n m, rectangular! SVD of Y o Y c c.f. C.W. Rowley, Int. J. on Bifurcation and Chaos, to appear.

25 Connections POD Balanced Truncation controllability Gramian: Z c = eat B B e A t dt e.g.: B = [b 1,..., b m ] y i = e At b i, Z c = y 1(t)y 1 (t) t,..., y m (t)y m (t) t correlation matrix: R = Y Y = T y(t) y(t)t dt discretized Z c = Y c Y c, Z = Y o Y o R n n Y c R n m, rectangular! SVD of Y o Y c c.f. C.W. Rowley, Int. J. on Bifurcation and Chaos, to appear.

26 Connections POD Balanced Truncation controllability Gramian: Z c = eat B B e A t dt e.g.: B = [b 1,..., b m ] y i = e At b i, Z c = y 1(t)y 1 (t) t,..., y m (t)y m (t) t correlation matrix: R = Y Y = T y(t) y(t)t dt discretized Z c = Y c Y c, Z = Y o Y o R n n Y c R n m, rectangular! SVD of Y o Y c c.f. C.W. Rowley, Int. J. on Bifurcation and Chaos, to appear.

27 ( ) Y o Y c = UΣV, rectangular, Σ R n n, V R m n Theorem If rank Σ = n, then balanced truncation matrix is given by T = Y c V Σ 1/2, T 1 = Σ 1/2 UY o, and Σ contains Hankel singular values. weighted inner product on R n : (a, b) Zo = a t Z o b POD-modes: equivalently Y c Y c Z o ϕ = Z c Z o ϕ = λ ϕ Y c Z o Y c ψ = λ ψ ( ) = V Σ 2 V ψ EVE(Y c Z o Y c ) = V Σ 2 EVE(Z c Z o ) = Y c V Σ 1

28 CLOSED LOOP CONTROL BASED ON POD T min u Uad (P) e µt h(y) + β 2 ẏ = f (y) + Bu(t), y() = y T u 2 f linear u = B T Π y ; henceforth f nonlinear υ = υ(t, y)... value functional = y ( ) υ t + µυ = min u Uad (f (t, y) + B T u, υ) + h(y) + β 2 u 2 υ(t, ) = u = P Uad ( 1 β BT υ)

29 υ t + µυ = ( f (t, y), υ(t, y) ) + h(y) β 2 (HJB) P U( 1 β BT υ(t, y)) 2 υ(t, ) = close the loop u(t) = P Uad ( 1 β BT υ(t, y(t)))

30 1) 2 point BVP-solutions µ =, no control constraints ẏ = f (y) + Bu, y() = y (OS) λ + f (y)λ = h (y), λ(t ) = u = 1 β λ u(t) = 1 β BT y v(t, y(t)) = F T (t, y(t)) = 1 β λ(t; y ) = F T t (, y(t)) F T (, y(t)) = 1 β λ (; y(t)) finite dimensional realization (i) choose grid R n containing optimal trajectory (ii) calculate (y(, ȳ ), λ(, ȳ )) for all ȳ, thus F T (, ȳ ) available (iii) use interpolation for F T (, y), y R n arbitrary.

31 Remark: (OS) must be solved often! (gridpoints in one direction) # basis POD use for open loop optimal control to estimate values in. Example (Control-to-zero of Burgers equation). uncontrolled FE POD closed loop cost

32 2) POD-HJB Problem Setting (NSt) min J(u) = 1 2 { yt 1 Re y + (y )y + p = Bu div y =, y(, ) = y, B.C. m Bu = ˆϕ i u i (t) i=1 Ω e µt y y Stokes 2 dxdt+ β 2 subject to u U ad and (NSt). Streamline, Re=1 1 e µt u(t) 2 dt

33 Dynamic Programming Principle { min J(y, u) = L(y(t), u(t))e µt dt ẏ(t) = f (y(t), u(t)), y() = y, u U ad. value function υ(y ) = min u Uad J(y, u). (DPP) υ(y ) = min u Uad { T L(y(t; y, u), u(t)e µt dt + υ(y(t ; y, u))e µt } (HJB) µυ(y ) = min u Uad { υ(y ) f (y, u) + L(y, u)} replace y by y (t). u (t) = F(y (t)) where F(y (t)) argmin u Uad { υ(y (t)) f (y (t), u) + L(y (t), u)} close the loop

34 t-discretisation t j = jh { yj+1 = y j + h f (y j, u j ) y given J h (y, u h ) = h 2 [L(y, u ) + υ h (y ) = e µhj (L(y j, u j 1 ) + L(y j, u j ))]. j=1 inf u h U h ad J h (y, u h ) (HJB h ) υ h (y ) = inf { h u U ad h 2 [L(y, u)+e µh L(y +hf (y, u), u)]+e µh υ h (y +hf (y, u))} numerically still infeasible yj+1 = y j + hf (yj, S h (yj )), j

35 POD-Galerkin Ansatz: y l (t, x) = l α i (t)ϕ i (x), i=1 u(t, x) = u i (t) ˆϕ i (x), where { ˆϕ i } POD basis from {y (t k ) y stokes } y (t k )... no control ˆϕ i injection on top filtered out. α = Aα α T Hα + G(u) A ij = 1 Re ϕ i, ϕ j, H ijk = ϕ j ϕ k, ϕ i, G i = u, ϕ i

36 Implementation Aspects for NS: finite difference upwind scheme on staggered nonuniform grid, fractional ϑ-scheme for time-integration. (HJB h ) υ h (α) = inf { h ( ) L(α, u) + e µh L(α + hf (α, u), u) +e µh υ(α+hf (α, u))} u U ad h 2 hypercube Γ h in R n such that α (t) Γ h for all t, α j grid points in Γ h, α = λ j α j, λj = 1 υ h (α) = λ j υ h (α j ) Solve (HJB h ) as fixed point equation, (Bardi, Capuzzo-Dolcetta; Gonzales-Rofman) nonuniform grid decay of POD eigenvalues. Constrained nonlinear programming problem at each grid point of Γ h : in parallel. Validation

37 Reynolds number, Re 1 Time horizon, T 1 No. of basis functions 6 No. of controls 2 Bounds on controls [.5,.5] Grid system Discount rate, µ 2 Observe region, Ω o [ 1, 12] [, 1] Table: Parameter Settings

38 1 Nonuniform grid system y x 1 Streamline, Re= Stokes flow

39 1 Streamline, Re= Streamline, Re=

40 Boundary Control no injection on top construction of shape functions u = u(t) χ(x), x Γ c y ref = y stokes u 2 y stokes u ˆV = span { y(t k ) u, y(t k ) u 2 y ref } n k=1 V = { ˆV y m }, y m = 1 2n POD(V) = {ϕ i } Ansatz for controlled solution y n (t) = y m + 2n i=1 α i(t)ϕ i + u(t)y ref. 2n k=1 ˆV

41 Boundary Control 1 Streamlines, Re = Streamlines, Re =

42 Semi implicit solution of the Burgers equation (with opt. control) y(x) x axis 1 x axis t axis t axis! "$# % &(' *),+ -. /) -* "2143, ): & "; %) % <.- ),+ % -*) =?> "@"@+ A, <CB + "D E*)F1 "G>H I7 ),+;-. J"@) KL -M-* ;+ "@ %E.)N1 ' %E.&. I7IO y(x) Semi implicit solution of the Burgers equation (with opt. control) x axis 1 t axis x axis 1 2 t axis QP R % * S! " # T &U'C ),+ -. V) -. "W146,5 X.Y*7R )F & "$Z\[ ]^= > "_"D+ A, <CB + "@ %E.)N1 "G>H C7 *),+;-* `"_) Ka -8-. ;+ "@ %E.)214' E*&8 C7GO 3 4 5