Model reduction of coupled systems
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1 Model reduction of coupled systems Tatjana Stykel Technische Universität Berlin ( joint work with Timo Reis, TU Kaiserslautern ) Model order reduction, coupled problems and optimization Lorentz Center, Leiden, September 19-23, 2005
2 Motivation 2 Elastic structures SIMPACK NASTRAN Mechanical systems Simulation Control Optimization Heat transfer Electrical circuits FEMLAB FLOMERICS Subsystems influence each other ( solvability, stability,... ) Subsystems have large dimension model reduction
3 Outline 3 Coupled linear systems Stability of coupled systems Model order reduction problem Error bounds Coupled systems with unstable subsystems Numerical examples Conclusion
4 Coupled linear systems 4 Consider a system of k linear time-invariant descriptor subsystems E j ẋ j (t) = A j x j (t) + B j u j (t), y j (t) = C j x j (t), where x j (t) states, u j (t) internal inputs, y j (t) internal outputs, E j, A j R n j,n j, B j R n j,m j, C j R p j,n j, λe j A j are regular. Transfer functions: G j (s) = C j (se j A j ) 1 B j Interconnection relations: u j (t)=k j1 y 1 (t)+...+k jk y k (t)+h j u(t) y(t)= R 1 y 1 (t) R k y k (t) u u 1 y 1 y 2 G 1 G 2 u 2 y
5 Closed-loop systems 5 The closed-loop system is given by E ẋ(t) = A x(t) + B u(t), y(t) = C x(t), where E = E, A = A + BKC, B = BH, C = R C, E = diag(e 1,..., E k ), A = diag(a 1,..., A k ), B = diag(b 1,..., B k ), C = diag(c 1,..., C k ), K = [ K jl ], H = [H T 1,..., H T k ]T, R = [R 1,..., R k ]. Transfer function: G cl (s) = R ( I G(s)K ) 1 G(s)H = R G(s) ( I KG(s) ) 1 H with G(s) = diag ( G 1 (s),..., G k (s) ) u u 1 y 1 y 2 G cl G 1 G 2 u 2 y
6 Stability 6 System Eẋ(t) = A x(t) + B u(t) is asymptotically stable λ E A is stable ( Sp f (E, A) C ) G cl (s)=r ( I G(s)K ) 1 G(s)H has no poles in C + 0 = C + ir and [ E, A, B, C ] is minimal G(s)H has no poles in C + 0, GK := sup G(iω)K 2 < 1 ω R and [ E, A, B, C ] is minimal [ Vidyasagar 81 ] ρ(ψ) < 1 with Ψ = G 1... K 11 2 K 1k G k K k1 2 K kk 2
7 Model reduction problem 7 Given a descriptor system E ẋ(t) = A x(t) + B u(t), y(t) = C x(t) with E, A R n,n, B R n,m, C R p,n and n m, p. Find a reduced-order system Ẽ x(t) = Ã x(t) + B u(t), ỹ(t) = C x(t) with Ẽ, Ã Rl,l, B R l,m, C R p,l and l n. Given G cl (s) = C (s E A ) 1 B, s.t. G cl G cl min. Find G cl (s) = C (s Ẽ Ã ) 1 B s.t. G cl G cl min. preserve system properties (stability, passivity,...) small approximation error numerically stable and efficient methods
8 Balanced truncation (E nonsingular) 8 Let λe A be stable ( Sp(E, A) C ). [ E, A, B, C ] is balanced, if the solutions of the Lyapunov equations satisfy E P A T + A P E T = B B T, E T Q A + A T Q E = C T C P = Q = diag(ξ 1,..., ξ n ). P and Q are controllability and observability Gramians, {ξ 1,..., ξ n } are Hankel singular values Idea: balance the descriptor system and truncate the states corresponding to small Hankel singular values Ẽ =WT E T, Ã=W T A T, B=W T B, C =C T, W, T R n,l
9 Balanced truncation algorithm 9 1. Compute P = RR T and Q = LL T ; 2. Compute the SVD L T ER = [U 1, U 2 ] [ Σ 1 Σ2 ] [V 1, V 2 ] T, with Σ 1 = diag(ξ 1,..., ξ l ), Σ 2 = diag(ξ l+1,..., ξ n ); ξ 1... ξ l > ξ l+1... ξ n ; 3. Compute the reduced-order system [ Ẽ, Ã, B, C ] = [ W T ET, W T AT, W T B, CT ] with W = LU 1 Σ 1/2 1 R n,l, T = RV 1 Σ 1/2 1 R n,l.
10 Generalized Lyapunov equations 10 EPA T + APE T = BB T P = RR T E T QA + A T QE = C T C Q = LL T Hammarling method [ Hammarling 86, Penzl 98, St 02 ] ( small / medium, dense ) Sign function method [ Roberts 71, Benner/Quintana-Ortí 99 ] ( large, dense ) H-matrices [ Grasedyck/Hackbush/Khoromskij 03, Benner/Baur 04 ] ( large, sparse ) Krylov subspace methods [ Saad 90, Jaimoukha/Kasenally 94 ] ( large, sparse ) low rank ADI method [ Wachspress 88, Penzl 99, Li/White 02, St 05 ] ( large, sparse )
11 Properties 11 Advantages: [ Ẽ, Ã, B, C ] is balanced, minimal, stable error bound: ỹ y 2 G cl G cl u 2 G cl G cl 2(ξ l ξ n ) Disadvantages: need to work with the system [ E, A, B, C ] of order n = n n k do not use the subsystem properties (structure, multi-scale,... ) the interconnection structure is not preserved block diagonal projection matrices [ Vandendorpe/Van Dooren 04, 05]... but no stability and accuracy results!
12 Subsystem model reduction approach 12 acements Coupled system E j ẋ j (t)= A j x j (t) + B j u j (t) y j (t)= C j x j (t) u j (t)=k j1 y 1 (t)+...+k jk y k (t)+h j u(t) y(t)= R 1 y 1 (t) R k y k (t) G cl u u 1 y 1 G 1 y y 2 G 2 u 2
13 Subsystem model reduction approach 13 acements Coupled system E j ẋ j (t)= A j x j (t) + B j u j (t) y j (t)= C j x j (t) u j (t)=k j1 y 1 (t)+...+k jk y k (t)+h j u(t) y(t)= R 1 y 1 (t) R k y k (t) Reduced-order coupled system Ẽ j xj (t) = Ãj x j (t) + B j ũ j (t) ỹ j (t) = C j x j (t) ũ j (t) =K j1 ỹ 1 (t)+...+k jk ỹ k (t)+h j u(t) ỹ(t) = R 1 ỹ 1 (t) R k ỹ k (t) G cl G cl u u 1 y 1 G 1 y u ũ 1 ỹ 1 G 1 ỹ y 2 G 2 u 2 ỹ 2 G 2 ũ 2
14 Subsystem model reduction approach 14 acements Coupled system E j ẋ j (t)= A j x j (t) + B j u j (t) y j (t)= C j x j (t) u j (t)=k j1 y 1 (t)+...+k jk y k (t)+h j u(t) y(t)= R 1 y 1 (t) R k y k (t) Reduced-order coupled system Ẽ j xj (t) = Ãj x j (t) + B j ũ j (t) ỹ j (t) = C j x j (t) ũ j (t) =K j1 ỹ 1 (t)+...+k jk ỹ k (t)+h j u(t) ỹ(t) = R 1 ỹ 1 (t) R k ỹ k (t) G cl G cl u u 1 y 1 G 1 y u ũ 1 ỹ 1 G 1 ỹ y 2 G 2 u 2 ỹ 2 G 2 ũ 2 Problem: How well does Gcl (s) approximate G cl (s)?
15 A priori error bounds 15 G cl (s) = R ( I G(s)K ) 1 G(s)H = R G(s) ( I K G(s) ) 1H G cl (s) = R ( I G(s)K ) 1 G(s)H = R G(s) ( I KG(s) ) 1H If 2 K(I GK) 1 max 1 j k G j G j < 1, then G cl G cl min{ c 1, c 2 } max 1 j k G j G j with c 1 = 2 R(I GK) 1 ( H 2 + K(I GK) 1 GH ), c 2 = 2 (I KG) 1 H ( R 2 + (I KG) 1 K R G ). If 4 K(I GK) 1 ( (j) max ξ 1 j k l j ) ξ(j) n j < 1, then ( (j) G cl G cl 2 min{ c 1, c 2 } max ξ 1 j k l j ) ξ(j) n j
16 A posteriori error bounds 16 G cl (s) = R ( I G(s)K ) 1 G(s)H = R G(s) ( I K G(s) ) 1H G cl (s) = R ( I G(s)K ) 1 G(s)H = R G(s) ( I KG(s) ) 1H. If 2 K(I GK) 1 max 1 j k G j G j < 1, then G cl G cl min{ c 1, c 2 } max 1 j k G j G j with c 1 = 2 R(I GK) 1 ( H 2 + K(I GK) 1 GH ), c 2 = 2 (I K G) 1 H ( R 2 + (I K G) 1 K R G ). If 4 K(I GK) 1 ( (j) max ξ 1 j k l j ) ξ(j) n j < 1, then ( (j) G cl G cl 2 min{ c 1, c 2 } max ξ 1 j k l j ) ξ(j) n j
17 Coupled system with unstable subsystems 17 Unstable subsystems: E j ẋ j (t)=a j x(t) + B j u j (t) y j (t)=c j x j (t), j =1,...,q Stable subsystems: E j ẋ j (t)=a j x(t) + B j u j (t) y j (t)=c j x j (t), j =q+1,...,k u G 3 G 1 G 2 y G 4
18 Coupled system with unstable subsystems 18 Extended stable subsystems: E j ẋ j (t)=(a j + B j F j )x(t) + B j g j (t) [ ] [ ] vj (t) Cj = x w j (t) F j (t), j =1,..., q j Interconnection relations: g j (t) = w j (t) + u j (t), y j (t) = v j (t) R 0 ( I Gj,0 (s)k 0 ) 1Gj,0 (s) = G j (s) u G 1,0 G 3 G 2,0 G 2 y G 1 G 4
19 Coupled system with unstable subsystems 19 Extended stable subsystems: Ẽ j xj (t)=(ãj + B j Fj ) x(t) + B j g j (t) [ ] [ ] ṽj (t) Cj = x w j (t) F j (t), j =1,..., q j Stable subsystems: Ẽ j xj (t)=ãj x(t) + B j ũ j (t) ỹ j (t)= C j x j (t), j =q+1,...,k u G 1,0 G 3 G2,0 ỹ G 4
20 Coupled system with unstable subsystems 20 Unstable subsystems: Ẽ j xj (t)=ãj x(t) + B j ũ j (t) ỹ j (t)= C j x j (t), j =1,...,q Stable subsystems: Ẽ j xj (t)=ãj x(t) + B j ũ j (t) ỹ j (t)= C j x j (t), j =q+1,...,k u G 1 G 3 G2 ỹ G 4
21 Example 1: coupled string-beam system 21 u κ 1 κ 2 Semidiscretized string equation: E 1 ẋ 1 (t) = A 1 x 1 (t) + B 1 u 1 (t) y 1 (t) = C 1 x 1 (t) n 1 = 1006, m 1 = 3, p 1 = 2 Semidiscretized beam equation: E 2 ẋ 2 (t) = A 2 x 2 (t) + B 2 u 2 (t) y 2 (t) = C 2 x 2 (t), n 2 = 1006, m 2 = 2, p 2 = 2 Interconnection [ ] relations: [ ] I 0 u 1 (t) = y 2 (t) + u(t) 0 I u 2 (t) = y(t) = y 1 (t) y 1 (t)
22 Example 1: coupled string-beam system 22 G1(iω) G1(iω) Absolute error and bound for the semidiscretized string equation error system error bound G2(iω) G2(iω) Absolute error and bound for the semidiscretized beam equation error system error bound Frequency ω Frequency ω Gcl(iω) 2 and Gcl(iω) full order reduced order Gcl(iω) Gcl(iω) error system error bound Frequency ω Frequency ω
23 Example 2: heated beam with a PI-controller 23 u + u 1 y 1 =u 2 y 2 y PI-controller G 1 (s)=κ P +κ I /s PI-controller: E 1 ẋ 1 (t) = A 1 x 1 (t) + B 1 u 1 (t) y 1 (t) = C 1 x 1 (t) n 1 = 2, m 1 = 1, p 1 = 1 Semidiscretized heated beam: E 2 ẋ 2 (t) = A 2 x 2 (t) + B 2 u 2 (t) y 2 (t) = C 2 x 2 (t), n 2 = 1000, m 2 = 1, p 2 = 1 Interconnection relations: u 1 (t) = y 2 (t) + u(t) u 2 (t) = y(t) = y 1 (t) y 2 (t)
24 Example 2: heated beam with a PI-controller 24 ) G2,0(iω) 2 and G2,0(iω) nd extented subsystem full order reduced order Gcl(iω) 2 and Gcl(iω) Closed loop system full order reduced order Frequency (rad/sec) 5 x Frequency (rad/sec) G2,0(iω) G2,0(iω) error system error estimate Gcl(iω) Gcl(iω) error system error estimate Frequency (rad/sec) Frequency (rad/sec)
25 Conclusion 25 (iω Stability of coupled systems Model reduction of the closed-loop system Subsystem model reduction preserving the interconnection structure preserving the subsystem properties existence of a priori and a posteriori error bounds parallelization Model reduction of coupled systems with unstable subsystems
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