Stability preserving post-processing methods applied in the Loewner framework

Transcription

1 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 1 / 20 Stability preserving post-processing methods applied in the Loewner framework Ion Victor Gosea and Athanasios C. Antoulas Jacobs University Bremen and Rice University Houston SPI 2016 Torino, Italy May 11, 2016

2 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 2 / 20 Outline 1 Introduction and Motivation 2 Loewner Framework 3 Post-Processing Methods 4 Numerical Examples

3 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 3 / 20 Introduction Model order reduction (MOR) is used to transform large, complex models of time dependent processes into smaller, simpler models that are still capable of representing accurately the behavior of the original process under a variety of conditions. Consider interpolatory model reduction methods: we seek reduced models whose transfer function matches that of the original system at selected interpolation points. Reduce dimension from n = dim(σ) to k = dim(ˆσ) { Eẋ(t) = Ax(t) + Bu(t) n k {Ê ˆx(t) = ˆx(t) + ˆBu(t) Σ : ˆΣ : y(t) = Cx(t) y(t) = Ĉ ˆx(t)

4 Main tool: Loewner framework data driven MOR method. Uses measured or computed data (e.g. measurements of the frequency response of a to-be approximated system) instead of the system matrices (E, A, B and C). Constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 4 / 20

5 Magnitude Main tool: Loewner framework data driven MOR method. Uses measured or computed data (e.g. measurements of the frequency response of a to-be approximated system) instead of the system matrices (E, A, B and C). Constructs reduced models based on a rank revealing factorization of appropriately constructed matrices. In some cases the resulting reduced systems may not be stable! Frequency response Original model Loewner model Frequency (ω) Poles of the Loewner reduced model Very good approximation but unstable model find a stable model which is an optimal or sub-optimal approximant of the original reduced Loewner model (in the H norm). Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 4 / 20

6 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 5 / 20 Loewner Framework Measured data: {(ω k, S k ) k = {1,, N}}, ω k C, S k C p m. { right data : (λ i, r i, w i ), i = {1,, ρ} Partition the data: ; left data : (µ j, l j, v j ), j = {1,, ν} Objective: Find H(s) C p m s.t. H(λ i )r i = w i, lj H(µ j) = vj

7 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 5 / 20 Loewner Framework Measured data: {(ω k, S k ) k = {1,, N}}, ω k C, S k C p m. { right data : (λ i, r i, w i ), i = {1,, ρ} Partition the data: ; left data : (µ j, l j, v j ), j = {1,, ν} Objective: Find H(s) C p m s.t. H(λ i )r i = w i, lj H(µ j) = vj The Loewner matrix L C ν ρ : v 1 r 1 l 1 w 1 µ 1 λ 1... L = v ν r 1 l ν w 1 µ ν λ 1... v 1 r ρ l 1 w ρ µ 1 λ ρ v ν r ρ l ν w ρ µ ν λ ρ V = ( v 1 v 2 v ν ) The shifted Loewner matrix L σ C ν ρ : µ 1 v 1 r 1 λ 1 l 1 w 1 µ 1 λ 1... L σ = µ ν v ν r 1 λ 1 l ν w 1 µ ν λ 1... W = ( w 1 w 2... w ρ ) µ 1 v 1 r ρ λ ρl 1 w ρ µ 1 λ ρ µ ν v ν r ρ λ ρl ν w ρ µ ν λ ρ

8 on Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 6 / 20 Theorem If (L σ, L) is regular, then {Ê = L, Â = L σ, ˆB = V, Ĉ = W } is a realization of the data. Hence, H(z) = W (L σ zl) 1 V is the required interpolant.

9 Theorem If (L σ, L) is regular, then {Ê = L, Â = L σ, ˆB = V, Ĉ = W } is a realization of the data. Hence, H(z) = W (L σ zl) 1 V is the required interpolant. Minimal amount of data: ( ) ( E A - L - Lσ B C V W ) Redundant data: ( ) ( E A -Y LX -Y L σ X B C Y V W X ) H(s) = W (L σ sl) 1 V Ĥ(s) = WX [Y (L σ sl)x ] 1 Y V Truncate at k = rank(l) SVD : [L L σ ] = Y Σ l X, [ L L σ ] = Ỹ Σ r X Theorem The quadruple {Ê = -Y LX, Â = -Y L σ X, ˆB = Y V, Ĉ = WX }, is the realization of an approximate data interpolant. Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 6 / 20

10 on Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 7 / 20 Reduction to order 2: L = L σ = [ H(µ1) H(λ ] 1) H(µ 1) H(λ 2 µ 1 λ 1 µ 1 λ 2 H(µ 2) H(λ 1) H(µ 2) H(λ 2) µ 2 λ 1 µ [ 2 λ 2 µ1h(µ 1) λ 1H(λ 1) µ 1H(µ 1) λ 2H(λ 2 µ 1 λ 1 µ 1 λ 2) µ 2H(µ 2) λ 1H(λ 1) µ 2H(µ 2) λ 2H(λ 2) µ 2 λ 1 µ 2 λ 2 [ ] H(µ1 ) V = H(µ 2 ) W = [ H(λ 1 ) H(λ 2 ) ] ] Interpolate 2k moments of the initial linear system (k=2): H(µ 1 ), H(µ 2 ), H(λ 1 ), H(λ 2 )

11 Post-Processing Methods Consider the following classes of linear systems Σ 0 n,p,m = {(E, A, B, C, D) Σ n,p,m ρ(e, A) ir = } Σ + n,p,m = {(E, A, B, C, D) Σ n,p,m ρ(e, A) C + } Σ n,p,m = {(E, A, B, C, D) Σ n,p,m ρ(e, A) C, } Σ + n,p,m and Σ n,p,m are the sets of the stable and antistable systems. Let Σ Σ 0 n,p,m. Find two systems so that Σ (Σ + Σ ) where: Σ + = (E +, A +, B +, C +, D) Σ + n +,p,m, Σ = (E, A, B, C, 0) Σ n,p,m. Define the space of all rational matrix transfer functions with bounded H norm (where G = sup ω R G(iω) 2 ):. RH = {G R(s) p m, ir D(G), G < } Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 8 / 20

12 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016 Houston 9 / 20 Post-Processing Methods Definition (The (AP ) Problem) Given an unstable system Σ Σ 0 n,p,m with transfer function G, find a stable system ˆΣ whose transfer function is the best ñ N Σ ñ,p,m approximation of G in the space RH. G(Σ) G(ˆΣ) = inf G Σ G Σ Σ Σ ñ N ñ,p,m solving (AP ) for an unstable system Σ is equivalent to solving (AP ) for its antistable part Σ +. For standard systems Σ, the reversed is also known as an Nehari Problem.

13 Theorem (Main Result) The unstable reduced order Loewner model is decomposed: Σ L = Σ }{{} {E, A, B, C } Σ }{{} + {E +, A +, B +, C +} Σ 0 k,p,m Let the infinite gramians P + and Q + and σ 1 the largest Hankel singular value (i.e. σ 1 = max ( eig(p + Q + ) ). Take γ σ 1. R + = Q + E + P + E T + γ 2 I, Ê + = E T + R +, ˆB + = E + Q + B + Ĉ + = C + P + E T +, Â + = A T +R + C T + Ĉ+ It follows that: ˆΣ L = Σ }{{} {E, A, B, C } ˆΣ+ }{{} {Ê +, Â +, ˆB +, Ĉ +} Σ ˆk,p,m and the inequality holds: σ 1 Σ L ˆΣ L min(γ, Σ + ) Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 10 / 20

14 1 Σ = stable part of Σ L. 2 Σ opt = optimal stable approx. of Σ L in RH. 3 Σ γ sopt = sub-optimal stable approx. of Σ L in RH (for γ > σ 1 ). 4 Σ flp = flipping the unstable poles approx. of Σ L. Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 11 / 20

15 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 11 / 20 1 Σ = stable part of Σ L. 2 Σ opt = optimal stable approx. of Σ L in RH. 3 Σ γ sopt = sub-optimal stable approx. of Σ L in RH (for γ > σ 1 ). 4 Σ flp = flipping the unstable poles approx. of Σ L. The Loewner model transfer function is given (σ 1 = σ 2 = 1 6 ) H L (s) = } s {{ + 3 } s 2 1 }{{ s 1 } H (s) H +(s) H (s) = 1 s+3, Σ L Σ H = 1 3 H opt (s) = 1 s+3 1 6, Σ L Σ opt H = 1 6 H flp (s) = 1 s s+2 1 s+1, Σ L Σ flp H = 2 3 H sopt (s) = 1 s+3 s 6(8s 2 +25s+16), Σ L Σ sopt H = , γ = 7 6

16 on Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 12 / 20 Scattering parameters from EM simulation (Three coupled lines) Data acquired by courtesy of Prof. Dr. Grivet-Talocia, Politecnico di Torino (presentation held at MORML 16, Sttutgart Causality Verification for Data-Driven Macromodeling ) given sample pairs {(ω k, S k })

17 Scattering parameters from EM simulation (Three coupled lines) Data acquired by courtesy of Prof. Dr. Grivet-Talocia, Politecnico di Torino (presentation held at MORML 16, Sttutgart Causality Verification for Data-Driven Macromodeling ) given sample pairs {(ω k, S k }) 10 0 Singular values of the Loewner matrix Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 12 / 20

18 Scattering parameters from EM simulation (Three coupled lines) Data acquired by courtesy of Prof. Dr. Grivet-Talocia, Politecnico di Torino (presentation held at MORML 16, Sttutgart Causality Verification for Data-Driven Macromodeling ) given sample pairs {(ω k, S k }) 10 0 Singular values of the Loewner matrix Poles of the Loewner reduced model Original data Loewner model Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 12 / 20

19 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 13 / 20 Scattering parameters from EM simulation (Three coupled lines) Use truncation order k = 14. In this case, the Loewner model has 6 unstable poles (4 real and 2 complex conjugates) G(iω) 0.04 Magnitude Original data Loewner: Σ L Optimal H : Σ opt Suboptimal H : Σ γ sopt Flipped: Σ flp Frequency(ω)

20 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 14 / 20 Scattering parameters from EM simulation (Three coupled lines) Σ L Σ opt Σ L Σ γ sopt Σ L Σ Σ L Σ flp Frequency response of the error systems ΣL Σopt ΣL Σ γ sopt ΣL Σflp Magnitude Frequency(ω) Notice that as ω, the norm of the optimal error system stays constant while the norm of the sub-optimal error system 0.

21 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 15 / 20 Scattering parameters from EM simulation (Three coupled lines) Vary the offset γ σ 1 within (10 5, 10 1 ); 10 1 H norm of the error systems 10 2 Σ Σ opt H Σ Σ H Σ Σ γ sopt H Σ Σ flp H γ σ 1 lim γ σ1 Σ L Σ γ sopt = Σ L Σ opt = σ 1 lim γ Σ L Σ γ sopt = Σ L Σ = Σ +

22 7135 th order power system [FRM08] For the II nd experiment, consider a stable descriptor power system of dimension n = 7135 with two inputs and two outputs, i.e. m = p = Singular values of the Loewner matrix By analyzing the decay of the singular values, choose for instance reduction order k = 82. The reduced Loewner model has 9 unstable poles. Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston / 20

23 7135 th order power system [FRM08] 80 Number of unstable poles for different reduction orders Reduction order (k) σ1(g(iω)) σ2(g(iω)) Magnitude 10 2 Magnitude Frequency(ω) σ3(g(iω)) Original samples Loewner: ΣL Suboptimal H : Σ γ sopt Flipped: Σflp 10 1 Frequency(ω) σ4(g(iω)) Magnitude Frequency(ω) Magnitude Frequency(ω) on Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 17 / 20

24 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 18 / th order power system [FRM08] Magnitude 10 0 σ1 Magnitude100 σ Frequency(ω) σ3 ΣL Σ γ sopt H ΣL Σflp H 10 1 Frequency(ω) σ4 Magnitude 10 2 Magnitude 10 1 Frequency(ω) 10 1 Frequency(ω)

25 Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston 19 / 20 M. Koehler, On the closest stable descriptor system in the respective spaces RH 2 and RH. Linear Algebra and its Applications, 443(0):34-49, F. Freitas, J. Rommes, N. Martinsem, Gramian-Based Reduction Method Applied to Large Sparse Power System Descriptor Models IEEE Transactions on Power Systems, Vol. 23, Issue 3, August 2008, pp

26 Thank you for your attention! Any questions? Ion Victor Gosea and Athanasios C. Antoulas (Jacobs University Bremen and Rice University May 11, 2016Houston / 20