Vector FittingHeinzel

Vector fitting

Gerhard Heinzel

1Max-Planck-Institut fur Gravitationsphysik Hannover (Albert-Einstein-Institut)

2007/06/26 LTP data analysis meeting Barcelona

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Gerhard Heinzel Vector fitting

Problem

complex functions of frequency

transfer functions

loop gains

electrical impedance, admittance, S parameters

origin

These functions typically come from measurements and are usually shownas Bode diagrams.

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Problem


transfer functions

loop gains


origin


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Problem


transfer functions

loop gains


origin


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Problem


transfer functions

loop gains


origin


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Problem


transfer functions

loop gains


origin


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Problem


transfer functions

loop gains


origin


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Problem


transfer functions

loop gains


origin


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Modelling

An analytical model of a measured transfer function is often desirable:

simple, smooth, description for simulations

parameters may have physical meanings, e.g. gain factors, resonancefrequencies, quality factors, delay times.

general rational model

H(s) = A0a0 + a1s + a2s

2 + · · ·+ ansn

b0 + b1s + b2s2 + · · ·+ sme−s τ

H(s) is a real function in s, but s is a complex variable:

s = iω = 2π i f


Modelling





H(s) = A0a0 + a1s + a2s

2 + · · ·+ ansn

b0 + b1s + b2s2 + · · ·+ sme−s τ


s = iω = 2π i f


Modelling





H(s) = A0a0 + a1s + a2s

2 + · · ·+ ansn

b0 + b1s + b2s2 + · · ·+ sme−s τ


s = iω = 2π i f


Input data

The measured data is given as a set of pairs or triples

fk ,Hk [,wk ], k = 1, . . . ,Ndata

withfk the frequency of the data point, used as sk = 2π i fk ,Hk the value of the complex function,wk the weight of this data point, given as one of

wk = 1/σ2(Hk) measured, best if available,

wk = 1 absolute,

wk = 1/|Hk |2 relative,

wk = 1/|Hk | ‘semi’ relative


Three equivalent forms

polynomial

H0(s) =a0 + a1s + a2s

2 + · · ·+ ansn

b0 + b1s + b2s2 + · · ·+ sm

pole-zero (preferred)

H0(s) =(s − z0)(s − z1) . . . [(s − zj)(s − z∗j )] . . .

(s − p0)(s − p1) . . . [(s − pj)(s − p∗j )] . . .

partial fractions with residuals (used by vector fitting)

H0(s) =r0

(s − p0)+

r1(s − p1)

+ · · ·+

[rj

(s − pj)+

r∗j(s − p∗j )

]+ . . .


Linear fit of residuals

Partial fractions with residuals:

H0(s) =r0

(s − p0)+

r1(s − p1)

+ · · ·+

[rj

(s − pj)+

r∗j(s − p∗j )

]+ . . .

If the poles pj are known, the residuals can be found from a linearleast-squares fit to the data using the base functions

1

(s − p0),

1

(s − p1), . . . .

But: The poles are in general not known.


pole fitting

Enter vector fitting (Bjørn Gustavsen 1999), which could also be calledpole fitting: The aim is to find all rm, pm, d and e in the model

H(s) =N∑

m=1

rms − pm

+ d + se (1)

such as to minimize the (weighted) least-square deviation from the set ofmeasured data. Starting with a set of initial poles {qm}, write

σ(s)H(s) = x(s), (2)

where σ(s), and x(s) share the initial poles {qm}:(N∑

m=1

rms − qm

+ 1

)H(s) =

N∑m=1

rms − qm

+ d + se


pole fitting 2

(N∑

m=1

rms − qm

+ 1

)H(s) =

N∑m=1

rms − qm

+ d + se

This augmented equation is fitted to the given data, yielding rm (and rm,d , e). Rewriting (2)

σ(s)H(s) = x(s) =⇒ H(s) =x(s)

σ(s)

shows that the poles of H(s) are the zeroes of σ(s), and the latter can befound by explicit algebraic operations on

σ(s) =N∑

m=1

rms − qm

+ 1.

This yields a new set of poles {q′m}, and the procedure is iteratedtowards σ(s) ≡ 1 and {qm} = {pm}.


realization

One step of the VF iteration (to yield a new set of poles {q′m}) needs to:

Solve the overdetermined linear real system(N∑

m=1

rmsk − qm

+ 1

)Hk =

N∑m=1

rmsk − qm

+ d + ske

of size (2Ndata + 1)× (2N + 3) in the least-squares sense. Asuccessful method is DGELSY from NETLIB, which uses QRfactorization with column pivoting and is also used by MATLAB.

Solve a real asymmetric N × N Eigenvalue problem. A goodsubroutine is DGEES from NETLIB.

Solve another overdetermined linear real system (1) of size(2Ndata)× (N + 2) in the least-squares sense.


realization



m=1

rmsk − qm

+ 1

)Hk =

N∑m=1

rmsk − qm

+ d + ske





realization



m=1

rmsk − qm

+ 1

)Hk =

N∑m=1

rmsk − qm

+ d + ske





realization



m=1

rmsk − qm

+ 1

)Hk =

N∑m=1

rmsk − qm

+ d + ske





Is it worth the effort?

transfer function of a commercial line filter:

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14 poles to describe 660 data points are found in 0.3 seconds, fullyautomatically except choosing the number of poles and the weightingoption.



transfer function of a commercial line filter:

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14 poles to describe 660 data points are found in 0.3 seconds, fullyautomatically except choosing the number of poles and the weightingoption.



a GEO600 interferometric response function:

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16 poles to describe 16384 data points from 1 to 8192 Hz are found in2.5 seconds, fully automatically except choosing the number of poles andthe weighting option.



a GEO600 interferometric response function:

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16 poles to describe 16384 data points from 1 to 8192 Hz are found in2.5 seconds, fully automatically except choosing the number of poles andthe weighting option.


Software status

MATLAB

Vector fitting was developed in MATLAB by its inventor BjørnGustavsen.

His code is freely available:http://www.energy.sintef.no/produkt/VECTFIT/index.asp

Another version is available as function “rationalfit” in the RFtoolbox.

LISO

I have implemented the algorithm in LISO.


Software status

MATLAB




LISO



Software status

MATLAB




LISO



Software status

MATLAB




LISO



Software status

MATLAB




LISO



Software status

Pole relocation:

problemn*n Eigenvalue

working well

missing

working but needsrefinement

Yes

No

No

Select initial poles

Pole relocation:

linear fit2ND * 2n overdet.

residuals:2ND * n overdet.

linear fit

convergence ?


linear fit

Fit satisfactory ?

increase n

(Gustavsen)vectfit2.m

Yes

Postprocessing

Future work:

for fixed number of poles N,find the best criterion tostop the iteration

find an automatic method tochoose a suitable number ofpoles N

handle delays

handle data without phase(if possible)


Software status

Pole relocation:


working well

missing


Yes

No

No


Pole relocation:



linear fit

convergence ?


linear fit

Fit satisfactory ?

increase n


Yes

Postprocessing

Future work:



handle delays



Software status

Pole relocation:


working well

missing


Yes

No

No


Pole relocation:



linear fit

convergence ?


linear fit

Fit satisfactory ?

increase n


Yes

Postprocessing

Future work:



handle delays



Software status

Pole relocation:


working well

missing


Yes

No

No


Pole relocation:



linear fit

convergence ?


linear fit

Fit satisfactory ?

increase n


Yes

Postprocessing

Future work:



handle delays



Software status

Pole relocation:


working well

missing


Yes

No

No


Pole relocation:



linear fit

convergence ?


linear fit

Fit satisfactory ?

increase n


Yes

Postprocessing

Future work:



handle delays



Postprocessing

The VF result

H0(s) =r0

(s − p0)+

r1(s − p1)

+ · · ·+

[rj

(s − pj)+

r∗j(s − p∗j )

]+ · · ·+d + se

(3)

is converted into its pole-zero representation

H0(s) = A0

(s − z0)(s − z1) . . . [(s − zj)(s − z∗j )] . . .

(s − p0)(s − p1) . . . [(s − pj)(s − p∗j )] . . .

and written as LISO input file for further processing.


Literature

Gustavsen, B; Semlyen, A: Rational approximation of frequencydomain responses by vector fitting IEEE TRANSACTIONS ON POWER

DELIVERY, Volume 14, Issue 3, page 1052–1061, 1999

Grivet-Talocia, S; Bandinu, M: Improving the convergence ofvector fitting for equivalent circuit extraction from noisyfrequency responses IEEE TRANSACTIONS ON ELECTROMAGNETIC

COMPATIBILITY, Volume 48, Issue 1, page 104–120, 2006

Gustavsen, B: Improving the pole relocating properties ofvector fitting IEEE TRANSACTIONS ON POWER DELIVERY, Volume 21, Issue 3,

page 1587–1592, 2006

Hendrickx, W; Dhaene, T: A discussion of ”rationalapproximation of frequency domain responses by vectorfitting” IEEE TRANSACTIONS ON POWER SYSTEMS, Volume 21, Issue 1, page

441–443, 2006

http://www.energy.sintef.no/produkt/VECTFIT/index.asp


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