TangentiaL Interpolation

7/28/2019 TangentiaL Interpolation

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Modeling Multi-Port Systems from FrequencyResponse Data via Tangential Interpolation

Sanda Lefteriu and Athanasios C. AntoulasDepartment of Electrical and Computer Engineering

Rice University

Houston, TX, USA

[email protected], [email protected]

AbstractSystem identification from frequency domain dataarises in many areas, e.g., in control, electronics, mechanical andcivil engineering and many other fields. Currently available tech-niques work well for the single input single output case. However,for the case of large numbers of inputs and outputs, present

methods are expensive. This paper proposes a new approachwhich is based on the concept of tangential interpolation. Ourapproach allows the identification of the underlying system usingsmall CPU times. The numerical results we present show thatour algorithms yield more accurate models in less time, whencompared to the column-wise implementation of vector fitting.

I. INTRODUCTION AND MOTIVATION

Measuring the frequency response of a system, be it electri-cal, mechanical, structural, etc, over a desired frequency range

provides data which can be used to identify the underlyingsystem. This work employs measured scattering parameters

as frequency domain data, but the approach is general and

can be applied to any kind of system identification. The

problem of building a macromodel which approximates givenmeasurements of the response at various frequencies is known

as the rational interpolation problem and has been studiedthoroughly (see [1] for a survey). Most approaches are based

on least-squares approximations, for instance vector fitting [2],[3], [4], which is widely used in the electronics community.

Some other algorithms, like [5], [6], [7], enforce passivity by

construction.Our approach is based on the concept of tangential interpo-

lation, using, as a main tool, the Loewner matrix pencil. We

are able to construct models of low complexity using a small

CPU time and are mainly addressing the case of systems with

a large number of inputs and outputs.

II . THEORETICAL ASPECTS

We start with the simple case of rational approximation

from scalar data: (si, i), i = 1, . . . , P, si = sj , i = j,and si, i C. We aim at finding H(s) = n(s)d(s) , n, d coprimepolynomials, such thatH(si) = i, i = 1, . . . , P . This alwayshas a solution, e.g., the Lagrange interpolating polynomial.

Our main tool, however, is the Loewner matrix which is

constructed by partitioning the data in disjoint sets:

(i, wi), i = 1, . . . , k and (j , vj), j = 1, . . . , h ,

where h, k P2 such that k + h = P, using the formula:Li,j =

vi wji k

, L

Chk.

There are many reasons to use this tool. The degree ofthe minimal interpolant is determined from the rank of the

Loewner matrices constructed using all possible partitions.

Moreover, the Loewner matrix has a system theoretic inter-

pretation in terms of observability and controllability matrices.Finally, for data consisting of a single point and derivatives at

that point, the Loewner matrix has Hankel structure. Thus, the

Loewner matrix generalizes the Hankel matrix.

A. Tangential interpolation

Sampling matrix data directionally on the left and on the

right leads to the concept of tangential interpolation. The right

interpolation data is given as

{(i, ri,wi) | i C, ri Cm1,wi Cp1, }, (1)for i = 1, . . . , k, or, more compactly,

= diag [1, , k] Ckk, (2)R = [r1, , rk] Cmk, (3)W = [w1, , wk] Cpk, (4)

while the left interpolation data is given as

{(j , j,vj) | j C, j C1p,vj C1m, }, (5)for j = 1, . . . , h, or, more compactly,

M = diag [1, , h] Chh, (6)

L =

1...

h

Chp, V =

v1

...

vh

Chm. (7)

The rational interpolation problem consists of finding a

realization in descriptor form [E,A,B,C,D], such that theassociated transfer function H(s) = C(sE A)1B + D,satisfies the right and left constraints

H(i)ri = wi, jH(j) = vj . (8)

The key tools we use for addressing this are the Loewner and

the shifted Loewner matrices, associated with the data. We

refer to [8] for more details on these concepts.

B. The Loewner and the shifted Loewner matrices

Given a set Z = {z1, , zP} of points in the complexplane and the rational matrix function H(s) at those points:{H(z1), ,H(zP)}, we can partition Z as:

Z ={

1,

, k} {

1,

, h}

,

978-1-4244-4489-2/09/$25.00 2009 IEEE SPI 2009


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where h, k P2 and k + h = P. We build the right and leftdata by selecting appropriate sampling directions ri and j.

The Loewner and the shifted Loewner matrices are defined

in terms of the data (1) and (5) as

L=

v1r11w1

11 v1rk1wk

1k...

. . ....

vhr1hw1h1

vhrkhwkhk

Chk, (9)

L=

1v1r111w111

1v1rkk1wk1k

.... . .

...hvhr11hw1

h1 hvhrkkhwk

hk

Chk.(10)

Each entry above is scalar, as it is obtained by taking inner

products of the left and right data. The following lemma [8]

provides the solution to the tangential interpolation problemin a simplfied case.

Lemma II.1. Assume that k = h, that the matrix pencil

(L,L) is regular, and thatj , i / (L,L). Then E = L,A = L, B = V, C = W and D = 0 is a minimalrealization of an interpolant of the data. Thus, the associated

transfer function H(s) = W(L sL)1V satisfies both leftand right interpolation conditions.

III . MODELING SCATTERING PAR AM ET ER S

Modeling multi-port systems from frequency-domain data

(for instance, scattering parameters) is formulated as a rational

approximation problem as follows. An LTI system approxi-

mately models the data set containing k measurements of theS-parameters of a device with p input and output ports

fi,S(i) := S(i)11 . . . S

(i)1p

... ... ...

S(i)p1 . . . S

(i)pp

, i = 1, , k,if the value of the associated transfer function evaluated at

j 2fi = ji, H(ji), is close to the measured S(i), i.To obtain a real system, the S-parameters at the complex

conjugate values of the sample points ji are set equal thecomplex conjugates of the measurements S(i), namely S(i).

A. The Loewner matrix pencil in the complex implementation

We use columns and rows of the identity matrix of dimen-sion p as sampling directions ri and i, respectively.Remark. The fact that the p2 entries of the matrix can becollapsed into one vector of dimension

pmakes this method

suitable for devices with a large number of ports.

The right interpolation data can be chosen asi = ji, ri,wi = S

(i)ri

. (11)

for i = 1, , k2 . As right directions, we use ri = em Rp1, with m = p for i = p c1 and m = 1, , p 1 for

i = p c1 + m, for some c1 Z, where em denotes the m-thcolumn of the indentity matrix Ip. Hence, the right data wiare columns ofS(i). More compactly,

= diag [j1, , jk] Ckk, (12)R = [r1, , rk] Cpk, (13)W = [w1,

, wk]

Cpk, (14)

while the left interpolation data are constructed asi = ji, i,vi = iS(i)

, (15)

with i = rTi and the left data vi are rows ofS(i). Compactly,

M = diag [

j1,

,

jk]

Ckk, (16)

L =

1 k

, L Ckp, (17)

V =

v1 vk

, V Ckp. (18)

After the tangential data have been identified, the Loewnerand shifted Loewner matrices are built as in (9)-(10).

B. The Loewner matrix pencil in the real implementation

To guarantee that the resulting system is real, the right

interpolation data can be chosen asji, ji; ri, ri;wi = S(i)ri,wi = S(i)ri

, (19)

for i = 1, , k2 , with ri as in Sect. III-A. More compactly, = diag j1, j1, , j k2 , j k2 , (20)R =

r1, r1, , rk

2

, r k2

Cpk, (21)

W =w1, w1, , wk

2

, w k2

Cpk, (22)

while the left interpolation data is constructed asji+ k

2

, ji+ k2

; i, i;vi = iS(i+ k

2),vi = iS

(i+ k2)

.

(23)for i = 1, , k2 , with i as in Sect. III-A. More compactly,

M = diagj1+ k

2

, j1+ k2

, , jk, jk

, (24)

L =

1

1 k

2

k2

, L Ckp, (25)

V =

v1 v

1 vk

2

vk2

, V Ckp. (26)

Without loss of generality, we assumed an even number of

samples. Next, the Loewner and shifted Loewner matrices are

built using Eq. (9)-(10). As a last step, a change of basis is

to be performed to ensure real matrix entries: = ,M = M, L = L, V = V, R = R, W = WL = L, L = L, where

= diag [, . . . , ] Ckk, = 12

1 j1 j

.

IV. IMPLEMENTATION

We use all measurements to construct the Loewner matrix

pencil, in the complex (Sect. III-A) or real implementation(Sect. III-B). Lemma II.1 assumes that the resulting Loewner

matrix pencil is regular. However, when too many measure-

ments are available, the pencil is singular, so one needs to

project out the singular part. Assuming that x {i}{i},rank(xL L) =: n, one can perform the singular valuedecomposition:

xL L = Y1X1, (27)where Y1 Ckn, X1 Cnk and n is the dimension ofthe regular part of xL L. Furthermore, it is precisely theorder of the underlying system. Using the singular vectors asprojectors, the realization is given as E = Y1LX1, A =

Y

1

LX1, B = Y

1

V, C = WX1 with D = 0 [8].


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This approach is computationally expensive for data sets

with a large number of samples k, as the cost of the SVDof the matrix xL L scales with k3. This is overcome byadaptive approaches presented in [9], [10], [11].

A. Remarks on the D-term, stability and passivity

Our realizations have a zero D-term. Nonetheless, systems

may have D = 0. Suppose the order of the underlying systemwith p ports is n and D = 0. This representation is equivalentto a system of order n +p with E singular, A invertible andD = 0. The poles of the new system are the n poles ofthe original system together with p infinite ones. Given such arealization, it can be written in the original form by recovering

the D-term. This is illustrated by the examples in Sect. V.

Our algorithms are able to identify the underlying system,

therefore, for data sets obtained from real-world systems, theresulting models are stable (after extracting the necessary D-

term). Passivity is not enfored by construction, thus out ofband passivity violations may occur. Like in the VF approach,

these can be corrected by an a posteriori passivation enforce-

ment based on first-order perturbations, for example [12], [13].

Passivity can also be enforced by contruction in the currentframework, as in [14].

V. NUMERICAL RESULTS

We compare our approach to vector fitting on an a-priorigiven system and on one where only measurements are avail-

able, in terms of accuracy and CPU time required to produce

a macromodel. The accuracy was assessed using:

the normalized

H-norm of the error system, defined as:

H error = maxi=1...k 1H(ji) S(i)

maxi=1...k 1S(i)

,where 1() denotes the largest singular value of ().

the normalized H2-norm of the error system:

H2 error =k

i=1

H(ji) S(i)2Fki=1

S(i)2F

,

where 2F stands for the Frobenius-norm, namely thesum of the magnitude squared of all p2 entries.

We used the column-wise implementation of fast, relaxedvector fitting [2], [15], [16] with the following options:

the starting poles are complex conjugate pairs with weak

attenuation, distributed over the frequency band each column was fitted using 5 iterations.

The experiments were performed on a Pentium Dual-Core

at 2.2GHz with 3GB RAM.

A. A-priori given system with 2 ports, 14 poles andD = 0We consider a system of order 14 with p = 2 ports and

D = 0 [9], [10]. We take k = 608 samples of the transferfunction between 101 and 101 rad/sec (Fig. 1(a)).

Fig. 1(b) shows the first 30 normalized singular values ofthe Loewner and shifted Loewner matrices (the rest are zero).The Loewner matrix has rank 14, while the shifted Loewnermatrix has rank 16, so, based on the drop of singular values

(Eq. (27)), we generate models of order 16 with D = 0. Toyield a realization of order 14, VF was given 7 starting poles.

101

100

101

80

70

60

50

40

30

20

10

0

Frequency (rad/sec)

Magnitude(dB)

Singular Value Plot

(a) Original system

0 5 10 15 20 25 30

1015

1010

105

100

index

logarithmic

Normalized Singular Values

LLsLL

(b) Singular value drop

Fig. 1. Original system and singular value drop of the Loewner matrix pencil

Table I presents the CPU time and the errors for the resultingmodels. All proposed algorithms identified the original system,

while VF did not. If VF is given N = 14 starting poles, theresulting errors are similar to ours. Each column is fit by the

same poles, so the realization has order n = 28 and each polehas multiplicity 2. Recovering the original system requires anadditional compacting step [3].

Algorithm CPU (s) H error H2 error

VF 0.93 1.1324 5.8327e-002SVD Complex 0.88 1.3937e-010 5.8900e-022

SVD Real 1.82 1.3146e-012 9.4168e-026

TABLE IRESULTS FOR k = 608 NOISE-FREE MEASUREMENTS OF AN ORDER 14

SYSTEM WITH p = 2 PORTS

B. Example involving measurements

Measurements were provided by CST AG. For examples

with a larger number of ports, see [9]. This set contains k =

200 frequency samples between 5MHz and 1GHz of a systemwith p = 16 ports. Fig. 2(a) shows the normalized singularvalues of the Loewner and shifted Loewner matrices.

0 50 100 150 20010

16

1014

1012

1010

108

106

104

102

100

X: 27Y: 0.03802

index

logarithmic

Normalized Singular Values

X: 28Y: 0.0008019

X: 43Y: 0.0009489

X: 44Y: 7.034e005

LL

sLL

(a) SVD drop

0 50 100 150 200 250 300 350 40010

5

104

103

102

101

100

X: 27Y: 0.3705

Hankel SVs of the VF model

(b) HSVs of the VF model

Fig. 2. Drop of the singular values of the Loewner matrix pencil and drop

of the Hankel singular values of the VF model of size n = 352

We notice the same behaviour as in the previous example.

The singular values of the Loewner matrix decay 2 ordersof magnitude between the 27th and 28th, while those of the

shifted Loewner matrix decay 1 order of magnitude betweenthe 43rd and 44th. This plot allows to identify the order of the

system and suggests that there is an underlying D-term. We

build models of order n = 43 with D = 0, so after recoveringthe D-term, we have an order n = 43 16 = 27 model.Table II summarizes the results. Vector fitting was required to

produce an asymptotic D matrix, but the closest order modelto n = 27 was n = 32. To obtain comparable errors, VFneeds to built an order n = 352 model. Fig. 2(b) shows that


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the Hankel singular values of the VF n = 352 model exhibitno decay around the 27th singular value. Thus, reducing to

order n = 27 using balanced truncation (BT), as also donein [17], leads to unsatisfacorty results. One could also try a

compacting step, as presented in [3].

Algorithm CPU time (s) H error H2 errorVF (n=32, D = 0) 0.54 1.4315e+000 1.2178e-001

Complex (n=43, D = 0) 0.21 3.4795e-002 2.0945e-005Real (n=43, D = 0) 0.14 8.3858e-002 3.5407e-005VF (n=352, D = 0) 3.63 7.9734e-002 7.4647e-005

VF & BT (n=27, D = 0) 5.46 6.5170e-001 5.8586e-002

TABLE IIRESULTS FOR CONSTRUCTING A M ODEL FROM A DATA SET OBTAINED

FROM A DEVICE WITH p = 16 PORTS

Our model constructed with the complex approach and the

VF model of order n = 32 are shown in Fig. 3. We alsopresent plots of the magnitude and angle of two entries of the

S-parameters in Fig. 4. We compare the measured S3,2 andS7,15 entries to the model obtained with our complex approachand to the order n = 32 model obtained with VF. The reasonbehind the poor performance of vector fitting shown in Fig.

3 and 4 is the fact that each one of the 16 columns of theS-parameters are fit by 32/16 = 2 poles. Thus, each columnshares only two common poles, which is clearly too restrictive.

107

108

109

1010

4

3.5

3

2.5

2

1.5

1

0.5

0

0.5

Frequency (rad/sec)

Magnitude(dB)

Singular Value Plot

Data

Model

(a) Our Approach (n = 43, D = 0)

107

108

109

1010

35

30

25

20

15

10

5

0

5

Frequency (rad/sec)

Magnitude(dB)

Interpolating system obtained with VF

Data

Model

(b) VF (n = 32, D = 0)

Fig. 3. Models for a device with p = 16 ports

107

108

109

1010

50

45

40

35

30

25

20

15

10

Frequency (rad/sec)

Magnitude(dB)

Magnitude of S3,2

data

our model

VF model

(a) Magnitude ofS3,2

107

108

109

1010

200

150

100

50

0

50

100

150

200

Frequency (rad/sec)

Angle(degrees)

Angle of S3,2

data

our model

VF model

(b) Angle ofS3,2

107

108

109

1010

40

35

30

25

20

15

10

Frequency (rad/sec)

Magnitude(dB)

Magnitude of S7,15

data

our model

VF model

(c) Magnitude ofS7,15

107

108

109

1010

200

150

100

50

0

50

100

150

200

Frequency (rad/sec)

Angle(degrees)

Angle of S7,15

data

our model

VF model

(d) Angle ofS7,15

Fig. 4. Comparison of the performance in modeling different entries of themeasured S-parameters obtained from a device with p = 16 ports

VI. CONCLUSION

This paper summarizes some of the features of a new

approach to modeling multi-port systems from frequency

domain data. For details, see [9]. It is based on the concept

of tangential interpolation and empoys as a main tool theLoewner matrix pencil, which is motivated by a systemtheoretic consideration [8]. Tangential interpolation was also

adopted for model order reduction [18]. In this note, we

are adressing the issue of large number of ports. Our main

application used the scattering parameters but, due to the

generality of our approach, other kinds of frequency-domain

data can be considered. We compared the performance ofour method to state-of-the-art vector fitting on two numerical

examples and concluded that our approach is faster and,moreover, identifies the order of the underlying system.

REFERENCES

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[3] , A robust approach for system identification in the frequencydomain, IEEE Trans. Power Del., vol. 19, pp. 11671173, Jul. 2004.

[4] D. Deschrijver, Broadband macromodeling of linear systems by vectorfitting, Ph.D. dissertation, Universiteit Antwerpen, Oct. 2007.

[5] S.-H. Min and M. Swaminathan, Construction of broadband passivemacromodels from frequency data for simulation of distributed inter-connect networks, IEEE Trans. Electromagn. Compat., vol. 46, no. 4,pp. 544558, Nov. 2004.

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[7] A. Woo and A. Cangellaris, Real-part sufficiency and its application tothe rational function fitting of passive electromagnetic responses, IEEE

International Microwave Symposium, pp. 99102, June 2007.[8] A. J. Mayo and A. C. Antoulas, A framework for the solution of the

generalized realization problem, Linear Algebra and Its Applications,vol. 405, pp. 634662, 2007.

[9] S. Lefteriu and A. C. Antoulas, A new approach to model multi-portsystems from frequency response data, IEEE Trans. Comput.-Aided

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[13] D. Saraswat, R. Achar, and M. Nakhla, Fast passivity verification andenforcement via reciprocal systems for interconnects with large ordermacromodels, IEEE Trans. VLSI Syst., vol. 15, pp. 4859, Jan. 2007.

[14] A. C. Antoulas, On the construction of passive models from frequencyresponse data, Automatisierungstechnik, vol. 56, pp. 447452, Aug.2008.

[15] B. Gustavsen, Improving the pole relocation properties of vectorfitting, IEEE Trans. Power Del., vol. 21, no. 3, pp. 15871592, Jul.2006.

[16] D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, Macro-modeling of multiport systems using a fast implementation of the vectorfitting method, IEEE Microw. Wireless Compon. Lett., vol. 18, no. 6,pp. 383385, June 2008.

[17] F. Ebert and T. Stykel, Rational interpolation, minimal realization andmodel reduction, DFG Research Center MATHEON, Tech. Rep. 371-2007, 2007.

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TangentiaL Interpolation