Model reduction of coupled systems

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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

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

Outline 3 Coupled linear systems Stability of coupled systems Model order reduction problem Error bounds Coupled systems with unstable subsystems Numerical examples Conclusion

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

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

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 2..... G k K k1 2 K kk 2

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

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

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.

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 )

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+1 +... + ξ n ) Disadvantages: need to work with the system [ E, A, B, C ] of order n = n 1 +...+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!

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

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

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)?

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 +1 +... + ) ξ(j) n j < 1, then ( (j) G cl G cl 2 min{ c 1, c 2 } max ξ 1 j k l j +1 +... + ) ξ(j) n j

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 +1 +... + ) ξ(j) n j < 1, then ( (j) G cl G cl 2 min{ c 1, c 2 } max ξ 1 j k l j +1 +... + ) ξ(j) n j

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

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

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

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

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)

Example 1: coupled string-beam system 22 G1(iω) G1(iω) 2 10 5 10 6 10 7 Absolute error and bound for the semidiscretized string equation error system error bound G2(iω) G2(iω) 2 10 4 10 5 10 6 10 7 Absolute error and bound for the semidiscretized beam equation error system error bound 10 8 100 50 0 50 100 Frequency ω 10 8 100 50 0 50 100 Frequency ω Gcl(iω) 2 and Gcl(iω) 2 10 0 10 5 10 10 10 15 full order reduced order Gcl(iω) Gcl(iω) 2 10 3 10 4 10 5 10 6 10 7 10 8 error system error bound 10 20 100 50 0 50 100 Frequency ω 10 9 100 50 0 50 100 Frequency ω

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 0000000000000 1111111111111 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)

Example 2: heated beam with a PI-controller 24 ) G2,0(iω) 2 and G2,0(iω) 2 1.005 1 0.995 0.99 0.985 0.98 2nd extented subsystem full order reduced order Gcl(iω) 2 and Gcl(iω) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Closed loop system full order reduced order 0.975 10 2 10 1 10 0 10 1 10 2 10 3 Frequency (rad/sec) 5 x 10 7 10 2 0 10 2 10 1 10 0 10 1 10 2 10 3 Frequency (rad/sec) G2,0(iω) G2,0(iω) 2 4.5 4 3.5 3 2.5 error system error estimate Gcl(iω) Gcl(iω) 2 10 3 10 4 10 5 error system error estimate 2 10 2 10 1 10 0 10 1 10 2 10 3 Frequency (rad/sec) 10 6 10 2 10 1 10 0 10 1 10 2 10 3 Frequency (rad/sec)

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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