Model reduction of large-scale systems

Transcription

1 Model reduction of large-scale systems Lecture IV: Model reduction from measurements Thanos Antoulas Rice University and Jacobs University URL: aca International School, Monopoli, 7-12 September 28 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 1 / 48

2 Outline 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 2 / 48

3 Outline Introduction S-parameters 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 3 / 48

4 Introduction S-parameters Recall: the big picture Physical System and/or Data Modeling ODEs discretization PDEs Model reduction Simulation reduced # of ODEs Control Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 4 / 48

5 Introduction S-parameters A motivation: electronic systems Growth in communications and networking and demand for high data bandwidth requires streamlining of the simulation of entire complex systems from chips to packages to boards, etc. Thus in circuit simulation, signal intergrity (lack of signal distortion) of high speed electronic designs require that interconnect models be valid over a wide bandwidth. An important tool: S-parameters They represent a component as a black box. Accurate simulations require accurate component models. In high frequencies S-parameters are important because wave phenomena become dominant. Advantages: S 1 and can be measured using VNAs (Vector Network Analyzers). Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 5 / 48

6 Introduction S-parameters Scattering or S-parameters Given a system in I/O representation: y(s) = H(s)u(s), the associated S-paremeter representation is where ȳ(s) = S(s)ū(s) = [H(s) + I][H(s) I] 1 ū(s), }{{} S(s) ȳ = 1 2 (y + u) are the transmitted waves and, ū = 1 2 (y u) are the reflected waves. S-parameter measurements. S(jω k ): samples of the frequency response of the S-parameter system representation. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 6 / 48

7 Introduction S-parameters Measurement of S-parameters VNA (Vector Network Analyzer) Magnitude of S-parameters for 2 ports Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 7 / 48

8 Outline Classical realization theory 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 8 / 48

9 Classical realization theory Model construction from data (at infinity): Classical realization Given h t R p m, t = 1, 2,, find A R n n, B R n m, C R p n, such that h t = CA t 1 B, t > Main tool: Hankel matrix h 1 h 2 h 3 h 2 h 3 h 4 H = h 3 h 4 h 5 = C CA CA 2 }. {{ } O [ B AB A 2 B ] }{{} R Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 9 / 48

10 Classical realization theory Classical realization Solvability rank H = n < Solution: Find R n n : det. Let σ R n n : shifted matrix. Let Γ R n m : first m columns, while Λ R p n : the first p rows. Then Consequence. In terms of the data: A = 1 σ, B = 1 Γ, C = Λ. H(s) = Λ(s σ ) 1 Γ Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 1 / 48

11 Outline Finite data points 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 11 / 48

12 Finite data points Model construction from data at finite points: interpolation Assume for simplicity that the given data are scalar: (s i, φ i ), i = 1, 2,, N, s i s j, i j Find H(s) = n(s) d(s) such that H(s i) = φ i, i = 1, 2,, N, and n, d: coprime polynomials. A solution always exists (e.g. Lagrange interpolating polynomial). Additional constraints for H: minimality, stability, bounded realness etc. Main tool: Loewner matrix. Divide the data in disjoint sets: (λ i, w i ), i = 1, 2,, k, (µ j, v j ), j = 1, 2,, q, k + q = N: L = v 1 w 1 v 1 w k µ 1 λ k µ 1 λ v q w 1 µ q λ 1 v q w k µ q λ k Cq k Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 12 / 48

13 Finite data points Model construction from data at finite points: interpolation Main result (1986). The rank of L encodes the information about the minimal degree interpolants: rank L or N rank L. Remarks. (a) In this framework the strict properness assumption has been dropped. Thus rational functions with polynomial part can be recovered from input-output data. (b) The construction of interpolants will be deferred until later. (c) If H(s) = C(sI A) 1 B + D, then C(λ 1 I A) 1 C(λ 2 I A) 1 [ L = (µ1 I A) 1 B (µ q I A) 1 B ]. }{{} C(λ k I A) 1 R }{{} O Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 13 / 48

14 Finite data points Scalar interpolation multiple points Special case. single point with multiplicity: (s ; φ, φ 1,, φ N 1 ), i.e. the value of the function and that of a number of derivatives is provided.the Loewner matrix becomes: L = φ 1 1! φ 2 2! φ 3 3! φ 4 4!. φ 2 2! φ 3 3! φ 4 4!. φ 3 3! φ 4 4! φ 4 4!... HANKEL MATRIX Thus the Loewner matrix generalizes the Hankel matrix when general interpolation replaces realization. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 14 / 48

15 Outline Problems 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 15 / 48

16 Problems General framework tangential interpolation Given: right data: (λ i ; r i, w i ), i = 1,, k left data: (µ j ; l j, v j ), j = 1,, q. We assume for simplicity that all points are distinct. Problem: Find rational p m matrices H(s), such that H(λ i )r i = w i Right data: λ 1 Λ =... C k k, Left data: µ 1 M =... µ q λ k l 1 C q q, L =. l j H(µ j ) = v j R = [r 1 r 2, r k ] C m k, W = [w 1 w 2 w k ] C p k l q C q p, V = v 1. v q C q m Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 16 / 48

17 Problems Problems Model Construction Data Λ, M, R, L, W, V Construct [E, A, B, C, D], such that H(λ i )r i = w i, l j H(µ j ) = v j. Model Reduction Data M, Λ, R, L and high-order system [Ê, Â, ˆB, Ĉ, ˆD] Contrsuct [E, A, B, C, D] of lower order such that Ĥ(λ i )r i = H i (λ i )r i, l j Ĥ(µ j ) = l j H(µ j ). Model Construction and Reduction (MR from data) Data Λ, M, R, L, W, V First step: construct high-order [E, A, B, C, D] Second step: reduce its dimension appropriately Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 17 / 48

18 Outline Tangential interpolation 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 18 / 48

19 Tangential interpolation The Loewner matrix General framework tangential interpolation Input-output data. The Loewner matrix is: v 1 r 1 l 1 w 1 µ 1 λ 1 L =..... v qr 1 l qw 1 µ q λ 1 Recall: v 1 r k l 1 w k µ 1 λ k v qr k l qw k µ q λ k Cq k H(λ i )r i = w i, l j H(µ j ) = v j Therefore L satisfies the Sylvester equation LΛ ML = VR LW Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 19 / 48

20 Tangential interpolation The Loewner matrix General framework tangential interpolation State space data. Suppose that E, A, B, C of minimal degree n are given such that H(s) = C(sE A) 1 B. Let X, Y satisfy the following Sylvester equations EXΛ AX = BR and MYE YA = LC If the generalized eigenvalues of (E, A) are distinct from λ i and µ j, X, Y are unique solutions of these equations. Actually x i = (λ i E A) 1 Br i X: generalized reachability matrix y j = l j C(µ j E A) 1 Y: generalized observability matrix. L = YEX Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 2 / 48

21 Tangential interpolation The Loewner matrix Construction of Interpolants Suggested construction procedure for an interpolant of McMillan degree n: 1 Factor L = YEX, so that E has rank n. 2 Construct A, B, C to satisfy the Sylvester equations above. 3 Define D := w j C(s j E A) 1 B. Steps 2 and 3 are easy; the first step is problematic: how do we choose E? If the system is proper, then size (E) = rank (E), and we could use, for example, the SVD to factor L proper systems are easy. If the system is singular, then size (E) > rank (E), and we re stuck. Solution: use the shifted Loewner matrix σl Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 21 / 48

22 Tangential interpolation The shifted Loewner matrix The shifted Loewner matrix The shifted Loewner matrix, σl, is the Loewner matrix associated to sh(s). µ 1 v 1 r 1 l 1 w 1 λ 1 µ µ 1 λ 1 1 v 1 r k l 1 w k λ k µ 1 λ k σl =..... Cq k µ qv qr 1 l qw 1 λ 1 µ q λ 1 σl satisfies the Sylvester equation σl can be factored as µ qv qr k l qw k λ k µ q λ k σlλ MσL = VRΛ MLW σl = YAX Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 22 / 48

23 Tangential interpolation Construction of interpolants Construction of Interpolants (Models) Assume that k = l, and let Then det (xl σl), x {λ i } {µ j } E = L, A = σl, B = V, C = W is a minimal realization of an interpolant of the data, i.e., the function interpolates the data. H(s) = W(σL sl) 1 V Proof. Multiplying the first equation by s and subtracting it from the second we get (σl sl)λ M(σL sl) = LW(Λ si) (M si)vr. Multiplying this equation by e i on the right and setting s = λ i, we obtain (λ i I M)(σL λ i L)e i = (λ i I M)Vr i (λ i L σl)e i = Vr i We i = W(λ i L σl) 1 V Therefore w i = H(λ i )r i. This proves the right tangential interpolation property. To prove the left tangential interpolation property, we multiply the above equation by e j on the left and set s = µ j : e j (σl µ j L)(Λ µ j I) = e j LW(µ j I Λ) e j (σl µ j L) = l j W e j V = l j W(σL µ j L) 1 V Therefore v j = l j H(µ j ). Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 23 / 48

24 Tangential interpolation Construction of interpolants Construction of interpolants: New procedure Main assumption: rank (xl σl) = rank ( L σl ) = rank ( L σl ) =: k, x {λ i } {µ j } Then for some x {λ i } {µ j }, we compute the SVD xl σl = YΣX with rank (xl σl) = rank (Σ) = size (Σ) =: k, Y C ν k, X C k ρ. Theorem. A realization [E, A, B, C], of an interpolant is given as follows: E = Y LX A = Y σlx B = Y V C = WX Remark. The system [E, A, B, C] can now be further reduced using any of the usual reduction methods. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 24 / 48

25 Tangential interpolation Example A = 1 1, E = 1 1, B = 1 1 1, C = [ ]. Thus the transfer function is H(s) = C(sE A) 1 B = [ s s ]. We now define the interpolation data. Λ = diag([1, 2, 3]), M = diag([ 1, 2, 3, 4]), while L = , R = [ ]. These imply: V = 4 1, W = [ ]. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 25 / 48

26 Tangential interpolation Example It follows that the tangential controllability and observability matrices associated with this data are: [ X = (E A) 1 BR(:, 1), (2E A) 1 BR(:, 2), (3E A) 1 BR(:, 3)] ] = , Y = L(1, :)C( 1E A) 1 L(2, :)C( 2E A) 1 L(3, :)C( 3E A) 1 L(4, :)C( 4E A) 1 = We notice that rank X = rank Y = 2. Thus the Loewner and shifted Loewner matrices are L = , σl = We check the assumption for δ = : rank (1 L σl) = 2, rank (2 L σl) = 2, rank (3 L σl) = 1 rank ( 1 L σl) = 2, rank ( 2 L σl) = 2 rank ( 3 L σl) = 2, rank ( 4 L σl) = 1 rank ([L σl]) = 2, rank ([L; σl]) = 2, Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 26 / 48

27 Tangential interpolation Example and for δ = 1: rank (L σl + L R) = 3 rank (2 L σl + L R) = 3 rank (3 L σl + L R) = 3 rank ( 1 L σl + L R) = 3 rank ( 2 L σl + L R) = 3 rank ( 3 L σl + L R) = 3 rank ( 4 L σl + L R) = 3 rank ([L σl L R]) = 3 rank ([L; σl L R]) = 3 Since the assumption is violated for δ =, but is not violated for δ = 1, we will compute interpolants with δ. For non-zero D, the dimension is 3. From L σl + LR, we obtain the projectors π l = , π r = The resulting transfer function is Ĥ(s) = (W δr)π r (π l (sl σl + δlr)π r ) 1 π l (V δl) δi 2 ) = δs 2 s 2 +δs+s 12 δs s 2 +δs+s 12 δs s 2 +δs+s 12 δ s 2 +δs+s 12 Notice that the original rational function is obtained for δ. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 27 / 48

28 Tangential interpolation Example It can be readily checked that all seven interpolation conditions are satisfied. A realization of this interpolant is as follows sê Â = π l (sl σl + δlr) π r = 2 d 4 d 2 d s d s d 1 5 d 13 d 3 s 8 d and thus Ê = π l L π r, Â = π l σl π r, turn out to be Ê = 1 1 1, Â = 2 δ 4 δ 2 δ 4 δ 4 δ 1 5 δ 13 δ δ. Furthermore ˆB = π l (V δl) and Ĉ = (W δr) πr : ˆB = 2 d 4 d 1 3 d 8 d, Ĉ = [ d d 1 d 2 d 1/3 d ]. Finally we need to check that the characteristic polynomial of the system is non zero at all interpolation points: χ(s) = det (sê Â) = 2δ(s2 + (δ + 1)s 12) Therefore χ(1) = 2δ(δ 1), χ(2) = 4δ(δ 3), χ(3) = 6δ 2 χ( 1) = 2δ(δ + 12), χ( 2) = 4δ(δ + 5) χ( 3) = 6δ(δ + 2), χ( 4) = 8δ 2 Thus δ must be different from 12, 5, 2,, 3, 1. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 28 / 48

29 Tangential interpolation Example: revisited H = s 1 1 s Λ =, M = 3 3 W = R = , L = /2 4/3 3/ /2, V = 2 2/3 5 5/4 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 29 / 48

30 Tangential interpolation Example: revisited Π = /4 1/6 1/ P =, Q = 1 1/ / Π PΠ , Π QΠ = Notice rank Π PΠ = 2 = McMillan degree of H. 1 W Π }{{} s Π PΠ Π QΠ }{{}}{{} Π V = C(sE A) 1 B = H(s) = s 1 }{{} 1 s 1 C E A B Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 3 / 48

31 which is much smaller than the discretization error which is of order 1 4. Tangential interpolation Constrained damped mass-spring system Example: mechanical system Consider the holonomically constrained damped mass-spring system illus- Mechanical trated inexample: Figure 3.4. Stykel, Mehrmann k 1 k i k i+1 k g 1 u m 1 m i m g d 1 d i d i+1 d g 1 κ 1 κ i κ g δ 1 δ i δ g Fig A damped mass-spring system with a holonomic constraint. The vibration of this system is described in generalized state space form as: The ith mass of weight m i is connected to the (i + 1)st mass by a spring and a damper with ṗ(t) constants = k i v(t) and d i, respectively, and also to the ground by a spring and a damper with constants κ i and δ i, respectively. Additionally, the M v(t) = Kp(t) + Dv(t) G λ(t) + B first mass is connected to the last one by a 2 u(t) rigid bar and it is influenced = by the Gp(t) control u(t). The vibration of this system is described by a descriptor system y(t) = C 1 p(t) ṗ(t) = v(t), Measurements: 5 M v(t) frequency = K p(t) + Dv(t) response G T λ(t) data + B 2 between u(t), [ 2i, (3.34) +2i]. = G p(t), Thanos Antoulas (Rice U. & Jacobs y(t) U.) = C 1 p(t), Model reduction of large-scale systems 31 / 48

32 Tangential interpolation Mechanical system: plots 2 Frequency responses of original n=97 and identified systems k=2,6,1,14,18 2 Error plots identified systems k=2,6,1,14, Left: Frequency responses of original system and approximants (orders 2, 6, 1, 14, 18) Right: Frequecy responses of error systems Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 32 / 48

33 Tangential interpolation Example: Four-pole band-pass filter 1 measurements between 4 and 12 GHz; S-parameters 2 2, MIMO (approximate) interpolation L, σl R 2 2. Singular values of 2 2 system vs 1 1 systems Magnitude of S(1,1), S(1,2) and 21st order approximants The singular values of L, σl The S(1, 1) and S(1, 2) parameter data 17-th order model Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 33 / 48

34 Tangential interpolation Example: co-axial cable 1 measurements between and 4 MHz; S-parameters 2 2, tangential interpolation L, σl R 1 1. Normalized Singular Values of Loewner and shifted Loewner Normalized Singular Values of Loewner and shifted Loewner LL rand LL rand sll rand 2 sll rand LL eye LL eye 5 sll eye LL eye alternative 4 sll eye LL eye alternative sll eye alternative 6 sll eye alternative logarithmic plot 1 15 logarithmic plot n n singular values of L, σl detail Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 34 / 48

35 Tangential interpolation Example: co-axial cable.3 Singular Value Plot S 11 Magnitude.5 S 12 Magnitude.2 2 db 4 db Magnitude(dB) Frequency.5 S 21 Magnitude Frequency 2 S 22 Magnitude.2 1st SV data 2nd SV data 1st SV interpolated 2nd SV interpolated log(f) singular values db Frequency db Frequency freq. resp. magnitude Error S 11 Magnitude Error S 12 Magnitude S 11 Angle S 12 Angle db 1 db 1 15 degrees 1 1 degrees Frequency Frequency Frequency Frequency Error S 12 Magnitude Error S 22 Magnitude S 21 Angle S 22 Angle db 1 db 5 1 degrees 1 1 degrees Frequency Frequency freq. resp. magnitude: error Frequency freq. resp. angle Frequency Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 35 / 48

36 Tangential interpolation Example: delay system where E, A, A 1 are 5 5 and B, C are 5-vectors. Eẋ(t) = A x(t) + A 1 x(t τ) + Bu(t), y(t) = Cx(t), Procedure: compute 1 frequency response samples. Then apply recursive/adaptive Loewner-framework procedure. (Blue: original, red: approximants.) 1!1 Adaptive/Recursive approximant N = 35; Hinf!error =.8 1!1 Non!adaptive/recursive approximant N = 5; Hinf!error =.18 1!2 1!2 1!3 1!3 1! ! th order adaptively constructed model; 5-th order non-adaptively constructed model; H norm of error:.8. H norm of error:.18. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 36 / 48

37 The data-to-model method when (E, A, B, C) is given Outline 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 37 / 48

38 The data-to-model method when (E, A, B, C) is given Motivation Motivation 2 Frequency response 1 Hankel singular values Frequency response and Hankel singular values Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 38 / 48

39 The data-to-model method when (E, A, B, C) is given Dominant poles Dominant system poles Goal: compute a reduced order model by evaluating the transfer function of the original high-order system, in as few frequencies as possible. Given a rational function H(s), let s C be a (simple) pole, The corresponding residue is defined as ρ = (s s )H(s) s=s. Concept of dominance: µ i := ρ i Re(s i ). Remark: for poles s i, s j which are not too close to each other, µ i, µ j, give the (local) maxima of the amplitude Bode plot; these are attained at the frequencies ω i = Im(s i ), ω j = Im (s j ), respectively. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 39 / 48

40 The data-to-model method when (E, A, B, C) is given The resulting procedure The procedure The proposed method (sometime referred to as FSS - Fast Frequency Sweep) consists of the following steps. 1 Compute some of the most dominant system poles s i = σ i + jω i. 2 Evaluate the transfer function of the system at the frequencies ω i, that is at the imaginary parts of the computed dominant poles. 3 Finally, compute a low order model of the underlying system using the Loewner matrix-based, data-to-reduced-model method described above. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 4 / 48

41 The data-to-model method when (E, A, B, C) is given Three examples Goal: construction of reduced models for three systems given by means of (E, A, B, C), using the proposed interpolation approach. A CD player model, with order n = 12. The model of a flexible beam, fixed at one, with oder n = 348. The model of a transmission line, with order n = 199. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 41 / 48

42 The data-to-model method when (E, A, B, C) is given CD player th order approx 18th order from Original data1 data2 data3 18th order compressed from data1 data2 1th order approx Left pane: amplitude Bode plot of the original model (blue), 8th order reduced (red) and 18th order reduced models. Right pane: amplitude Bode plots of the corresponding error systems. Notice the local character of the approximants: the first one captures the first main peak, while the second reproduces both. 5 x Left pane: the poles of the system (blue dots) and the dominant poles (red circles). Right pane: detail of the pole plot. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 42 / 48

43 The data-to-model method when (E, A, B, C) is given Clamped beam data1 data2 data3 data4 data data1 data2 data Left pane: amplitude Bode plots of original system (blue), 8th order approximant (red dash-dot), 17th order approximant (blue dash-dot), and 12th oder approximant obtained by balanced truncation (green dash-dot). The stars indicate the value of the dominance index µ i, for the first 5 dominant poles. Right pane: amplitude Bode plots of error systems for the 8th order approximant (blue), for the 17th order approximant (red), and for the 12th order approximant obatined by balanced truncation Left pane: system poles (blue) and some dominant poles (red). Right pane: close-up look. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 43 / 48

44 The data-to-model method when (E, A, B, C) is given Transmission line 2 Frequency response 1 Hankel singular values Frequency response Hankel singular values 2 Comparison of frequency responses 2 Comparison of frequency responses original bal 45 original new spec zeros original=blue, bal=red original=blue, new=red, spec-zeros=green Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 44 / 48

45 The data-to-model method when (E, A, B, C) is given Transmission line continued 2 Comparison of frequency responses Error plots bal new spec zeros h2 opt 2 bal new spec zeros h2 opt original=blue, new=red, H2-opt=mag Error plots H infinity norm 2 Poles: original and reduced systems bal new spec zeros h2 opt original bal new spec zeros h2 opt H2 norm bal new spec zeros h2 opt H 2 and H errors Poles Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 45 / 48

46 Outline Summary and conclusions 1 Introduction S-parameters 2 Classical realization theory 3 Finite data points 4 Problems 5 Tangential interpolation The Loewner matrix The shifted Loewner matrix Construction of interpolants 6 The data-to-model method when (E, A, B, C) is given Motivation Dominant poles The resulting procedure 7 Summary and conclusions Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 46 / 48

47 Summary and conclusions Summary and conclusions Given frequency response data, we can construct with no computation, a high order system in generalized state space form let the data reveal the underlying system The system is such that (A, E) is neither regular nor stable and hence (E, A, B, C) is not passive. An SVD of L, (σl), will produce a regular (and stable) system. At this stage all usual model reduction methods are applicable. Approach is the natural way to construct models and reduced models from data as it does not require inversion of E. For small systems described by (E, A, B, C), one can compute all poles s i, and their dominance indices µ i. The major issue for the applicability of the proposed method to large-scale systems is the determination of a few dominant poles of (A, E), without computing all the poles first. This can be achieved using the iterative method known as SADPA (subspace accelerated dominant pole algorithm) developed by Joost Rommes. The proposed method reduces systems at specific frequency ranges. This in not a Krylov method. It is an interpolation method. It exhibits similarities with Krylov methods, like the spectral zero method and the optimal H 2 method. Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 47 / 48

48 Summary and conclusions References Mayo, Antoulas, LAA 27 Lefteriu, Antoulas, Tech. Report 28 Vandendorpe, Van Dooren, SIMAX 25 Antoulas: SIAM book 25 Thanos Antoulas (Rice U. & Jacobs U.) Model reduction of large-scale systems 48 / 48