On Dominant Poles and Model Reduction of Second Order Time-Delay Systems

Transcription

1 On Dominant Poles and Model Reduction of Second Order Time-Delay Systems Maryam Saadvandi Joint work with: Prof. Karl Meerbergen and Dr. Elias Jarlebring Department of Computer Science, KULeuven ModRed 2010, Berlin December 4, 2010

2 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion On Dominant Poles and MOR of Second Order TDS Maryam Saadvandi

3 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

4 Introduction Why do we need model order reduction techniques? Size of the original system is too large. Solving the system is time consuming.

5 Introduction Second order dynamical system Large-scale second order dynamical system arise in Electrical circuit simulation Acoustics Vibrating systems Second order dynamical system { Mẍ(t) + Cẋ(t) + Kx(t) = fu(t) y(t) = d x(t)

6 Introduction Second order dynamical system { Mẍ(t) + Cẋ(t) + Kx(t) = fu(t) y(t) = d x(t) Matrices M, C, K R n n x(t), f, d R n Stiffness Matrix Damping Matrix Mass Matrix u(t), y(t) R State Output vector Input vector Output Input

7 Introduction Second order dynamical system { Mẍ(t) + Cẋ(t) + Kx(t) = fu(t) y(t) = d x(t) Second order time-delay systems To improve the stability of system add control term { Mẍ(t) + Cẋ(t) + Kx(t) + Gẋ(t τ) + Fx(t τ) = fu(t) y(t) = d x(t) where G, F R n n : Control matrices τ: delay

8 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

9 Frequency Response Function (FRF) FRF shows a relation between output and input H(iω) = Y(iω) U(iω) where Y(iω) and U(iω) are the Fourier transforms of y(t) and u(t). FRF is used for Simulation Stability analysis FRF H(iω) : C C H(iω) = d ( ω 2 M + iωc + K + iωge iτω + Fe iτω ) 1 f,

10 Frequency Response Function (FRF) Transfer function s = iω Change frequency domain to time domain Apply Laplace transform under the condition x(0) = 0 H(s) = d (s 2 M + sc + K + sge τs + Fe τs ) 1 f

11 Frequency Response Function (FRF) For simplicity The polynomial eigenvalue problem A(s) = s 2 M + sc + K + sge sτ + Fe sτ, and the derivative of A(s) respect to s is A (s) = 2sM + C + Ge sτ sτge sτ τfe sτ. Time-delay eigenvalue problem (TDEP) (λ i, x i, y i ) are eigentriplets (i N) { A(λi )x i = 0, x i 0 y i A(λ i) = 0, y i 0 Note Poles of FRF eigenvalues of the TDEP

12 Frequency Response Function (FRF) Modal expansion For the case that the system is linear If all eigenvalues are simple, then it is possible to express transfer function as: n R i H(s) =, s λ i where R i are the residues. i=1

13 Frequency Response Function (FRF) Theorem: Modal expansion Consider a system with simple eigenvalues. Suppose there is a sequence of matrices P i, i N such that A(λ) 1 P i =. λ λ i Then, P i = i=1 x i yi, i A (λ i )x i where (λ i, x i, y i ) is an eigentriplet. Moreover, H(s) = d A(s) 1 R i f =, s λ i where R i is called the residue y i=1 R i = (d x i )(yi f ) yi. A (λ i )x i

14 Frequency Response Function (FRF) Weighted residue ρ i is called the weighted residue if ρ i = R i Re(λ i ) where R i is the associated residue of eigenvalue λ i. Dominant pole The pole λ i of H(s) with corresponding weighted residue ρ i is called the dominant pole if and only if for all j i. ρ i > ρ j

15 Frequency Response Function (FRF) Transfer function If all eigenvalues are simple, then it is possible to express transfer function as: R j H(s) =, s λ j where R j are the residues. j=1 Idea Dominant Poles λ 1,..., λ k to be ordered ρ 1 ρ 2 ρ 3 Effective transfer function behavior: k R j H(s) =, s λ j hopefully k n, (λ j, R j ) are ordered by decreasing dominance. j=1

16 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

17 Dominant Pole Algorithm (DPA) DPA for time-delay system If λ is pole Define lim H(s) =. s λ Q(s) = 1 H(s). Poles of H(s) Roots of Q(s) Apply Newton method for finding root of Q(s) s k+1 = s k Q(s k) Q(s k ) = s k + = s k + d A(s k ) 1 f d A(s k ) 1 A(s k ) A(s k ) 1 f d v k w k A(s k ) v k

18 Dominant Pole Algorithm (DPA) DPATD Input: System (M, C, K, G, F, f, d, τ), initial value s, TOL 1 Output: Dominant pole s, x, y right and left eigenvectors 1 Repeat 2 Solve v C from A(s)v = f 3 Solve w C from A(s) w = c 4 5 x = v v, y = 6 Until w w s = s d v w A (s)v max( A(λ)x, A(λ) y ) < TOL Note DPATD computes only one dominant pole.

19 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

20 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = orthogonal(v, v), W = orthogonal(w, w)] 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS

21 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = orthogonal(v, v), W = orthogonal(w, w)] 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS eigenvalue problem of TDS ( λ 2 i ( λ 2 i M + λ i C + K + λ Ge λ i τ + F e λ i τ ) x i = 0, x i 0, M + λ i C + K + λ Ge λ i τ + F e λ i τ ) ỹ i = 0, ỹ i 0, where M = W MV, C = W CV, K = W KV, G = W GV, F = W FV

22 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = orthogonal(v, v), W = orthogonal(w, w)] 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS 4 [ Λ, X, Ỹ ] = solve eigenvalue problem 5 [ Λ, X, Ỹ ] = sort the [ Λ, X, Ỹ ] in decreasing R i Re(λ i ) order eigenvalue problem of TDS ( λ 2 i ( λ 2 i M + λ i C + K + λ Ge λ i τ + F e λ i τ ) x i = 0, x i 0, M + λ i C + K + λ Ge λ i τ + F e λ i τ ) ỹ i = 0, ỹ i 0, Note The method for solving second order TDEP based on first order TD

23 Subspace Projection First order time-delay system { (λi E A 0 A 1 e λ i τ ) ϕ i = 0, where[ I 0 E = 0 M (λ i E A 0 A 1 e λ i τ ) ψ i = 0, ] [ 0 I, A 0 = K C ] [ 0 0, A 1 = F G ], where the eigentriplet is ( [ ] [ xi 1/ λi (K λ i, ϕ i =, ψ λ i x i = + F e λ i τ )y i i y i ]).

24 Subspace Projection First order time-delay system { (λi E A 0 A 1 e λ i τ ) ϕ i = 0, (λ i E A 0 A 1 e λ i τ ) ψ i = 0, where the eigentriplet is ( [ xi λ i, ϕ i = λ i x i ] [ 1/ λi (K, ψ i = + F e λ i τ )y i y i ]). equivalent eigentriplet for second order TD (λ i, x i = ϕ i (1 : n, 1), ỹ i = ψ i (n + 1 : 2n, 1)) sort them by largest R i / Re(λ i ).

25 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = orthogonal(v, v), W = orthogonal(w, w)] 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS 4 [ Λ, X, Ỹ ] = solve eigenvalue problem 5 [ Λ, X, Ỹ ] = sort the [ Λ, X, Ỹ ] in decreasing R i Re(λ i ) order 6 Dominant approximate of original system (λ, x, y)

26 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = orthogonal(v, v), W = orthogonal(w, w)] 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS 4 [ Λ, X, Ỹ ] = solve eigenvalue problem 5 [ Λ, X, Ỹ ] = sort the [ Λ, X, Ỹ ] in decreasing R i Re(λ i ) order 6 Dominant approximate of original system (λ, x, y) Approximate eigentriplet of the original problem λ = λ i, x i = V x i, y i = W ỹ i

27 Subspace Projection Subspace projection DPA for TDS (SPDPATD) 1 Solve A(s)v = f and A(s) w = d 2 Let V = [V, v], W = [W, w] and orthogonal V and W 3 Petrov-Galerkin leads to projected eigenvalue problem of TDS 4 [ Λ, X, Ỹ ] = solve eigenvalue problem 5 [ Λ, X, Ỹ ] = sort the [ Λ, X, Ỹ ] in decreasing R i Re(λ i ) order 6 Dominant approximate of original system (λ, x, y) 7 If max( A(λ)x, A(λ) y ) < TOL 8 End (λ, x, y) is dominant triplet X = [X x], Y = [Y y]

28 Subspace Projection Reduced second order time-delay system Keep the right and left dominant eigenvectors in matrices X, Y. { ˆM ˆx(t) + Ĉ ˆx(t) + ˆK ˆx(t) + Ĝ ˆx(t τ) + ˆF ˆx(t τ) = ˆf u(t) ŷ(t) = ˆd ˆx(t) ˆM = Y MX, Ĉ = Y CX, ˆK = Y KX R k k Ĝ = Y GX, ˆF = Y FX R k k ˆf = Y f, ˆd = X d R k 1 k n

29 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

30 Deflation Why do we need deflation? DPATD computes one dominant pole. SPDPATD computes several dominant poles. Deflation prevents the SPDPATD from recomputing the found dominant poles.

31 Deflation Why do we need deflation? DPATD computes one dominant pole. SPDPATD computes several dominant poles. Deflation prevents the SPDPATD from recomputing the found dominant poles. Idea of deflation λ i is dominant pole residue R i is largest one. R i = (d x i )(y i f ) y i Z (λ i )x i deflation R i = 0 Then we are sure in coming iterations λ i is not dominant anymore. How? Deflation replaces f and d by f and d s.t. y i f = d x i = 0

32 Deflation Modified System { Mẍ(t) + Cẋ(t) + Kx(t) + Gẋ(t τ) + Fx(t τ) = f u(t) y(t) = d x(t) The idea of deflation can be controlled by playing with f and d so that: the residue of the deflated poles become zero the residues of the other poles remain unchanged

33 Deflation Find the formula for deflation We determine a m = [α 1,..., α m ] T and b m = [β 1,..., β m ] T { f (s) = f A(s)X m a m d(s) = d A(s) Y m b m where X m = [x 1, x 2,, x m ], Y m = [y 1, y 2,, y m ] are right and left dominant eigenvectors matrices, so that Y m f (s) = 0 and d (s)x m = 0 These conditions leads to { am = (Y ma(s)x m ) 1 (Y mf ) b m = (Y ma(s)x m ) (X md).

34 Deflation Second condition Theorem: The change of the residue R j corresponding to eigentriplet (λ j, x j, y j ) during the deflation is O(s λ j ).

35 Deflation Second condition Theorem: The change of the residue R j corresponding to eigentriplet (λ j, x j, y j ) during the deflation is O(s λ j ). For proof the Theorem we need following Lemma. Lemma: Let X m and Y m denote the right and left eigenvectors associated with λ 1, λ m, respectively, and (λ j, x j, y j ) be an eigentriplet of the original system with j > m. Then, for s C { Y m A(s)x j = (s λ j )Y ma (λ j )x j + O ( (s λ j ) 2) y j A(s)X m = (s λ j )y j A (λ j )X m + O ( (s λ j ) 2).

36 Deflation Note dominant pole λ i C λ i is also dominant pole. λ i and λ i are counted as one dominant pole. λ i can be deflated.

37 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

38 Numerical Results The Mass-Spring-Damper [Hu, 1998] x 1 x 2 x n k k k k m m m f c 1 c 2 c 1 c Figure: mass-dual system under the state feedback with time-delay

39 Numerical Results Matrices C = M = diag(m), m: mass, m = 1. c 1 + c 2 c 1 c 1 c 1 + c 2 c 2 c 2 c 1 + c 2 c c 2 c 1 + c 2 c 1 c 1 c 1 + c 2 c 1, c 2 : damper c 1 = 0.4 c 2 = 0.2., K = tridiag( k, 2k, k), k: spring constant, k = 1. f = d = [0,, 0, 1], τ = 2

40 Numerical Results Matrices F = , G =

41 Numerical Results 10 4 Original (n=1000) Reduce (k=2) H(iω) ω x 10 3 Figure: Catching one dominant pole by the DPA successfully, #iteration 4.

42 Numerical Results Original (n=1000) Reduce (k=20) Error ω Figure: 10 dominant poles are matched, #iteration 20.

43 Numerical Results Original (n=1000) Reduce (k=30) Error ω Figure: 15 dominant poles are matched, #iteration 28.

44 Outline 1 Introduction 2 Frequency Response Function (FRF) 3 Dominant Pole Algorithm (DPA) 4 Subspace Projection 5 Deflation 6 Numerical Results 7 Conclusion

45 Conclusion DPATD is based on original matrices. DPATD computes a dominant pole. SPDPATD computes several dominant poles. Deflation prevents the algorithm from recomputing the found dominant pole. Structure of the system is preserved.

46 Conclusion Thank you for your attention!