Computing Transfer Function Dominant Poles of Large Second-Order Systems

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1 Computing Transfer Function Dominant Poles of Large Second-Order Systems Joost Rommes Mathematical Institute Utrecht University joint work with Nelson Martins March 22, Joost Rommes 1/24

2 1 Introduction 2 Transfer functions and dominant poles 3 Quadratic Dominant Pole Algorithm 4 Results 5 Conclusions Joost Rommes 2/24

3 Introduction Large-scale second-order dynamical systems arise in electrical circuit simulation structural engineering acoustics Transfer function is used for simulation stability analysis controller design Relatively few transfer function poles of practical importance Which poles, and how to compute? Joost Rommes 3/24

4 Second-order dynamical systems Second-order SISO dynamical system { Mẍ(t) + Cẋ(t) + Kx(t) = bu(t) y(t) = c x(t) + du(t), where u(t), y(t), d R, input, output, direct i/o x(t), b, c R n, state, input-to-, -to-output, M R n n mass matrix, C R n n damping matrix, K R n n stiffness matrix. Joost Rommes 4/24

5 Transfer function Second-order SISO dynamical system (d = 0): { Mẍ(t) + Cẋ(t) + Kx(t) = bu(t), y(t) = c x(t). Second-order transfer function: H(s) = c (s 2 M + sc + K) 1 b. Poles are λ C for which det(λ 2 M + λc + K) = 0 Joost Rommes 5/24

6 First-order transfer function H(s) = c (se A) 1 b Can be expressed as where residues R i are H(s) = n i=1 R i s λ i, R i = (c x i )(y i b), and (λ i, x i, y i ) are eigentriplets (i = 1,..., n) Ax i = λ i Ex i, right eigenpairs y i A = λ i y i E, left eigenpairs y i Ex i = 1, normalization y j Ex i = 0 (i j), E-orthogonality Joost Rommes 6/24

7 Dominant poles of first-order transfer functions H(s) = n i=1 Pole λ i dominant if R i s λ i with R i = (c x i )(y i b) R i Re(λ i ) large Dominant poles cause peaks in Bode-plot (ω, H(iω) ) Effective transfer function behavior: H k (s) = k i=1 R i s λ i, where k n and (λ i, R i ) ordered by decreasing dominance Joost Rommes 7/24

8 35 Bodeplot Gain (db) Frequency (Hz) Figure: Bode plot (ω, H(iω) ). Pole λ j dominant if R j Re(λ j ) large. Joost Rommes 8/24

9 2nd-order transfer function H(s) = c (s 2 M + sc + K) 1 b Can be expressed as where residues R i are H(s) = 2n i=1 R i s λ i, R i = (c x i )(y i b)λ i, and (λ i, x i, y i ) eigentriplets of QEP (i = 1,..., 2n) (λ 2 i M + λ i C + K)x i = 0, right eigenpairs y i (λ 2 i M + λ i C + K) = 0, left eigenparis y i Kx i + λ 2 i y i Mx i = 1, normalization Joost Rommes 9/24

10 Dominant poles of second-order transfer functions H(s) = 2n i=1 Pole λ i dominant if R i s λ i with R i = (c x i )(y i b)λ i R i Re(λ i ) large Dominant poles cause peaks in Bode-plot (ω, H(iω) ) Effective transfer function behavior: H k (s) = k i=1 R i s λ i, where k 2n and (λ i, R i ) ordered by decreasing dominance Joost Rommes 10/24

11 Quadratic Dominant Pole Algorithm (QDPA) QDPA [R., Martins (2007)] computes dominant poles of H(s) = c (s 2 M + sc + K) 1 b 1 Newton scheme 2 Subspace acceleration and selection 3 Deflation Joost Rommes 11/24

12 Quadratic Dominant Pole Algorithm Dominant pole λ of H(s) = c (s 2 M + sc + K) 1 b: Apply Newton to 1/H(s): s k+1 = s k + 1 H 2 (s k ) H(s k ) H (s k ) lim s λ 1 H(s) = 0 c (sk 2 = s k M + s kc + K) 1 b c (sk 2M + s kc + K) 1 (2s k M + C)(sk 2M + s kc + K) 1 b c v = s k w (2s k M + C)v, where v = (s 2 k M + s kc + K) 1 b and w = (s 2 k M + s kc + K) c. Joost Rommes 12/24

13 Quadratic Dominant Pole Algorithm [R., Martins (2007)] 1: Initial pole estimate s 1, tolerance ɛ 1 2: for k = 1, 2,... do 3: Solve v k C n from (s 2 k M + s kc + K)v k = b 4: Solve w k C n from (s 2 k M + s kc + K) w k = c 5: Compute the new pole estimate c v k s k+1 = s k wk (2s km + C)v k 6: The pole λ = s k+1 with x = v k and y = w k has converged if 7: end for (s 2 k+1 M + s k+1c + K)v k 2 < ɛ Joost Rommes 13/24

14 Subspace acceleration and selection Keep approximations v k and w k in search spaces V and W Petrov-Galerkin leads to projected QEP where (θ 2 M + θ C + K) x = 0, ỹ (θ 2 M + θ C + K) = 0, M = W MV, C = W CV, and K = W KV C k k. Gives 2k approximations (θ i, ˆx i = V x i, ŷ i = W ỹ i ) in iteration k Select approximation with largest residue: (c ˆx i )(ŷi s k+1 = argmax b)θ i i Re(θ i ) (with ˆx i = ŷ i 2 = 1) Joost Rommes 14/24

15 Deflation for H(s) = c (se A) 1 b Triplet (λ, x, y): Ax = λx and y A = λy E New search spaces: V E y and W Ex Deflate via (every iteration) v k (I xy E)v k w k (I yx E )w k More efficient: deflate only once b d (I Exy )b v k = (s k E A) 1 b d E y c d (I E yx )c w k = (s k E A) c d Ex Note that y b d = c d x = 0 Joost Rommes 15/24

16 Deflation for H(s) = c (s 2 M + sc + K) 1 b Eigenvectors can be linearly dependent: no direct deflation Consider linearization of second-order system: { Eẋ(t) = Ax(t) + bl u(t) y(t) = c l x(t), where (if K nonsingular) [ ] 0 K A =, E = K C [ ] K 0, b 0 M l = [ ] 0, c b l = [ ] [ ] (λ 2 x x M + λc + K)x = 0 A = λe λx λx Idea [Meerbergen, SISC (2001)]: deflate via linearization [ ] c. 0 Joost Rommes 16/24

17 Deflation via linearization 1/2 Triplets (λ, x, y) and (λ, φ = [x T, λx T ] T, ψ = [y T, λy T ] T ) Deflate via [ ] [ ] [ ] [ ] b 1 b d = d = (I Eφψ 0 c 1 ), and c b d = d = (I E ψφ c ) 0 b 2 d Solve expansion vectors v k and w k from ( [ ] [ ]) [ ] [ ] K 0 0 K vk b 1 s k = d 0 M K C e and ( [ ] K 0 s k 0 M [ 0 K K C c 2 d ]) [ ] wk = f b 2 d [ ] c 1 d c 2 d Joost Rommes 17/24

18 Deflation via linearization 2/2 Solve v k from ( [ ] K 0 s k 0 M [ 0 K K C ]) [ ] vk = e With e = (s 2 k M + s kc + K) 1 (s k b 2 d + b1 d ) follows v k = ( K 1 b 1 d + e)/s k [ ] b 1 d Similarly w k = ( K c 1 d + f)/ s k Solve projected QEP Select largest approximate linearized residue: [ ] (c ˆxi d )( [ ŷ θ s k+1 = argmax i iˆx i θ i ŷ ] i bd ) i Re(θ i ) b 2 d Joost Rommes 18/24

19 Butterfly gyro (Oberwolfach) (n = 17361) Modal approximation constructed by using left and right eigenspaces Y and X : M k = Y MX, C k = Y CX, K k = Y KX, b k = Y b, c k = X c Krylov reduced order model uses Y = X with colspan(x ) = K k (K 1 M, K 1 b) Krylov model is cheaper: (20 s, 40 iters) vs. (1345 s, 233 iters) Modal approximation preserves poles Joost Rommes 19/24

20 Bodeplot k=30 (QDPA) k=35 (QDPA) k=40 (PRIMA) Exact 140 Gain (db) Frequency (Hz) Figure: Butterfly gyro (n = 17361). Exact response (solid), 40th order Krylov model (dash), 35th (dash-dot) and 30th (dot) order modal approximations. Joost Rommes 20/24

21 60 40 k=40 (RKA) k=10 (QDPA) k=50 (RKA+QDPA10) Exact 20 Gain (db) Frequency (rad/sec) Figure: Breathing sphere (n = 17611). Exact transfer function (solid), 40th order SOAR RKA model (dot), 10th (dash-dot) order modal equivalent, and 50th order hybrid RKA+QDPA (dash). Joost Rommes 21/24

22 Breathing sphere ([Lampe, Voss (2006)]) (n = 17611) Two-sided rational SOAR [Bai and Su, SISC (2005)] model (k = 40, shifts 0.1, 0.5, 1, 5) misses peaks Small QDPA model (k = 10) matches some peaks, misses global response 500 s (SOAR, 80 iters) vs s (QDPA, 108 iters) Hybrid: Y = [Y QDPA, Y SOAR ] and X = [X QDPA, X SOAR ] Larger SOAR models: no improvement More poles with QDPA: expensive Use imag parts of poles as shifts for SOAR! Shifts σ 1 = 0.65i, σ 2 = 0.78i, σ 3 = 0.93i, and σ 4 = 0.1 Two-sided rational SOAR, 10-dimensional bases Joost Rommes 22/24

23 Gain (db) k=70 (RKA) Exact Rel Error Frequency (rad/sec) Figure: Breathing sphere (n = 17611). Exact transfer function (solid), 70th order SOAR RKA model (dash) using interpolation points based on dominant poles, and relative error (dash-dot). Joost Rommes 23/24

24 Concluding remarks QDPA for computation of dominant poles Subspace acceleration and efficient deflation Applications in model order reduction: Construction of modal approximations Interpolation points for rational Krylov Preservation of poles Generalizations: Higher-order systems MIMO systems Computation of dominant zeros z: H(z) = 0 For preprints and more info, see Joost Rommes 24/24