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Some Results on Identifying Linear Systems Using Frequency Domain Data

Lennart Ljung

Department of Electrical Engineering Linkoping University

S-581 83 Linkoping, Sweden

Abstract

The usefulness of frequency domain interpretations in linear systems is well known. In this contribution the connenctions between frequency domain and time domain expressions will be discussed. In particular, we consider some aspects of using frequency domain data as primary observations.

1 Introduction

For linear systems the connections and interplay be- tween time-domain and frequency domain aspects have proved to be most fruitful in all applications.

We shall in this contribution discuss some aspects in applications to linear system identication.

There are two sides of this interplay. One is to consider the primary observation to be in the time- domain, and then to interpret corresponding identi- cation criteria, algorithms and properties in the fre- quency domain. There are many early results of this character, e.g. 9], 2], 1], 4]. More recently such results have been exploited and developed in 6].

The other side of the interplay is to consider the pri- mary observations to be in the frequency domain.

That is, the Fourier transforms of the measured sig- nals (or certain ratios of them) are treated as the ac- tual measurements. This view has been less common in the traditional system identication literature, but has been of great importance in the Mechanical En- gineering community, vibrational analysis and so on.

An early reference is 5]. An excellent recent account, with many references, of this view is given in the book

7].

This contribution will deal with a few questions of the latter view from a more traditional System Iden- tication background.

2 Parameterized models

We shall throughout this paper consider linear mod- els in discrete or continuous time, parameterized as follows:

y(t) =G(q)u(t) +H(q)e(t) (1) (Discrete time)

y(t) =G(p)u(t) +H(p)e(t) (2) (Continuous time)

Hereyuandeare the output, the input and the noise source, respectively. eis supposed to be white noise with variance (intensity) . q is the shift operator and pis the dierentiation operator.

A typical parameterization, both in continuous and discrete time could be as a rational function

G(p) = b1pn;1+ +bn

p

n+f1pn;1+ +fn =B(p)

F(p) (3)

= (b1:::bn f1:::fn):

In discrete time we could, e.g. also use parameter- izations that originate from an underlying continu- ous time state space model, discretized under the as- sumption that the input is piecewise constant over the sampling interval:

G(q) =C(qI;eA()T);1

Z

T

0 e

A()

B()d (4) See, e.g. 6] for many more examples of the parame- terization (1).

3 Time domain data

Suppose input-output data in the time domain are given:

z

N=fy(t)u(t)t=T2T:::NTg (5) 1

(2)

pp ( ) form the corresponding predictions

^

y(tj) =G(q)u(t)+(I;H;1(q))(y(t);G(q)u(t)) and the associated prediction errors: (6)

"(t) =y(t);y^(tj)(tj) =H;1(q)(y(t);G(q)u(t)) and then compute (7)

^



N = argmin

 N

X

t=1

"

2(t) (8) Most frequency domain interpretations of this time domain method go back to the application of Parse- val's relationship to the right hand side of (8):

^



N

argmin

 Z



;

jE(!)j2d! (9) whereE is the Fourier transform of ":

E(!) =H;1(ei!)Y(!);G(ei!)U(!)] (10)

Y(!) = 1p

N N

X

t=1

y(t)e;i!t (11) and similarly for U(!). If we introduce the "Empiri- cal Transfer Function Estimate",

^^

G(ei!) =Y(!)

U(!) (12)

(8) - (11) can be rewritten

^



N

argmin

 Z



;

jG^^(ei!);G(ei!)j2 jU(!)j2

jH(ei!)j2d!

(13)

4 Frequency domain data

Suppose now that the original data are supposed to be

Z

N =fY(!k)U(!k)k= 1:::Ng (14) whereY(!k) andU(!1) either are the discrete Fourier transforms ofy(t) andu(t) as in (11) or are considered as Fourier transforms of the underlying continuous signals:

Y(!) =

Z

1

;1

y(t)e;i!tdt (15) (or a normalized version). Which interpretation is more suitable depends of course of the signal charac- ter, sampling interval and so on.

( ) ( ) ( )

(13) it would be tempting to use

^



N= argmin

 V()

V() =XN

k =1

jY(!k);;G(ei!kT)U(!k)j2 1

jH(ei!kT)j2 (replacingei!kT byi!kfor the continuous-time model(16) (2).)

If H in fact does not depend on  ( xed or known noise model) experience shows that (16) works well.

Otherwise the estimate ^N may not be consistent.

To nd a better estimator we turn to the maximum likelihood (ML) method for advice: (We give the ex- pressions for the continuous time case in the case of (1), just replacei!k byei!kT)

If the data were generated by

y(t) =G(p)u(t) +H(p)e(t) the Fourier transforms would be related by

Y(!) =G(i!)U(!) +H(i!)E(!) (17) To be true, (17) should in many cases contain an error term that accounts for the fact that the measured data Y(!k) often are not exact realizations of (15).

For periodic signals, observed over an integer number of periods, (17) may however hold exactly.

Now, ife(t) is white noise, its Fourier transform (suit- ably normalized) will have a (complex) Normal dis- tribution:

E(!)2N(0I) complex (18) This means that the real and imaginary parts are each normally distributed, with zero means and variances

. The real and imaginary parts are independent and, moreover,E(!1) andE(!2) are independent for!16=

!

2. This implies that

Y(!k)2N(G(i!k)U(!k)jH(i!k)j2) (19) according to the model, so that the negative loga- rithm of the likelihood function becomes

V

N() =XN

k =1

f2logjH(i!k)j+ 1



jY(!k);G(i!k)U(!k)j2 1

jH(i!k)j2



+Nlog The ML estimate is (20)

^



N= argmin

 V

N() (21)

(3)

p y , obtain

^



N= argmin



"

N logWN() + 2XN

k =1

logjH(i!k)j

#

(22)

W

N() = 1

N N

X

k =1

jY(!k);G(i!k)U(!k)j2 1

jH(i!k)j2

^ (23)



N =WN(^N) (24) Compared to (16) we thus have an additional term

N

X

k =1

logjH(i!k)j2 (25) We may note that for any monic, stable and inversely stable transfer functionH(q) we have

Z



;

logjH(ei!)j2d!0 (26) This is the reason why (25) is missing from criteria that use dense, equally spaced frequencies!k for dis- crete time models (like (13)).

In fact (25) is the determinant from the change of variables from Y to E (outputs to innovations).

In the discrete time domain this transformation is a triangular operator with 1's along the diagonal (e(t) = y(t)-past data). Hence this transformation has a determinant equal to 1, so it does not aect the ML criterion.]

It is apparently often assumed (as in 7]) that the noise model is given or known. Then of course the term (25) is again not essential.

5 Asymptotic properties

The asymptotic properties (as N ! 1) of the esti- mate (20)-(21) can be developed in a rather straight- forward fashion, using the standard techniques. We conne ourselves below to the case of a xed noise model H(i!) =H(i!) and a known . Suppose, as N ! 1 the frequencies !k cover the frequency interval ;] with a density functionW(!). That is, letwN(12) be the number of observed frequen- cies in the interval 1 to 2 when the total number of frequencies is N. Then

lim

N!1

1

N w

N(12) =

Z

2

1

W(!)d!:]

p p ( ) g ,

formly in and with probability 1 to



V() =

Z



;

jG

0(i!);G(i!)j2u(!)W(!)

jH

(i!)j2 d!

where G0 is the true transfer function, and u(!(27)) is the input spectrum. Hence

^



N

!argmin





V()w:p:1 asN !1 (28) If there exists a value0such thatG0(i!) =G(i!0) and u(!)W(!) is dierent from zero at su ciently many frequencies it will follow that

^



N

!

0

as N !1

In that case the covariance matrix of ^N will be, asymptotically,

Cov^N

"

N

X

k =1 G

0

(i!k0)G0(i!k0)u(!k)

jH

(i!k)j2

#

;1

Here G0is the gradient ofG(i!) with respect to(29) and superscript denotes complex conjugation and matrix transpose.

6 Some practical aspects

There are several distinct features with the direct fre- quency domain approach that could be quite useful.

We shall list a few (see also, e.g., 7])

Preltering is known as quite useful in the time-domain approach. For frequency domain data it becomes very simple: It just corresponds to assigning dierent weights to dierent fre- quencies, which in turn is the same as using a fre- quency dependent cheating on the assumed noise levels. It is of course particularly easy to implement perfect band-pass ltering eects in the frequency domain approach.

Condensing large data sets. When dealing with systems with a fairly wide spread of time constants, large data sets have to be collected in the time domain. When converted to the fre- quency domain they can easily be condensed, so that, for example, logarithmically spaced fre- quencies are obtained. At higher frequencies one would thus decimate the data, which involves av- eraging over neighbouring frequencies. Then the noise level (k) is reduced accordingly.

Combining experiments. Nothing in the ap- proach of Section 4 says that the frequency re- sponse data at dierent frequencies have to come

(4)

p ,

quencies involved (!kk= 1:::N) all have to be dierent. It is thus very easy to combine data from dierent experiments.

Periodic inputs. The main drawback with the frequency domain approach in that the underly- ing frequency domain model (17) is strictly cor- rect only for a periodic input and assuming all transients have died out. On the other hand, typical use of the time domain method (6) - (8) assumes inputs and outputs prior to time 0 to be zero. Whichever assumption about past be- haviour is closer to the truth should thus aect the choice of approach.

Band-limited signals. If the actual input sig- nals are band-limited, (like no power above the Nyquist frequency) the continuous time Fourier transform (15) can be well computed from sam- pled data. It is then possible to directly build continuous-time models without any extra work.

Continuous-time models. The comment above shows that direct continuous-time system identication from "continuous-time data" can be dealt with in a much more relaxed way than in the time-domain, with all its mathematical in- tricacies.

Trade-o noise/frequency resolution. The approach also allows for a more direct and fre- quency dependent trade-o between frequency resolution and noise levels. That will be done as the original Fourier transform data are dec- imated to the selected range of frequencies

!

k

k= 1:::N.

7 Some algorithmic questions

The criterion (22) to be minimized is non-quadratic inin most cases. This calls for iterative search pro- cedures for the calculation of ^N. This in turn raises two questions:

1. What method should be applied for the itera- tions?

2. At what parameter values should the search be initialized?

We shall deal with these questions in order.

Iterative minimization

If the noise modelH is xed (-independent), the re- maining criterion to be minimized inWN(), which is

deal with such a function minimization is the dampedq Gauss-Newton method 3]. This apparently is still the best approach around, and is the basic method used in System Identication. Indeed, the MATLAB Sig- nal Processing Toolbox commands for solving (22) for a xed noise model (invfreqzand invfreqs) inple- ment this approach.

Unfortunately, it turns out that the additional term (25) may seriously deteriorate the performance of the damped Gauss-Newton proacedure. This is, not unexpectedly, most pronounced for continuous time models and for very unequally spaced frequency sam- ples. One probably then has to go to full Newton- methods, which however puts greater demands on the line search. Also, it is important to scale the param- eterization, so that the criterion remains reasonably well conditioned.

Initial parameter estimates

Also in the time-domain approach it is very impor- tant to provide the Gauss-Newton iterative scheme with good initial conditions. In 6] (Section 10.5) several steps to achieve such initial estimations are described. They are based on the Instrumental Vari- able (IV) method and the so called repeated Least Squares (rLS) method (i.e. estimating a high order ARX-model, then compute the innovations from this and use them as measured inputs in the next step).

Fortunately these methods can be more or less di- rectly carried over to direct frequency domain meth- ods. The IV method (see also 8]) can be described as follows: The problem is to nd an initial estimate

^

G

(0)(ei!) =B(ei!)

A(ei!)

Step i): Solve min

aibi X

k

jA(ei!k)Y(!k);B(ei!k)U(!k)j2 forAs,Bs. Let ^Gs= BAss

Step ii): Solve 0 =X

k

(A(ei!k)Y(!k);B(ei!k)U(!k)) (!k) (30) forAand B where

(!k) =

2

6

6

6

6

6

6

6

4

^ ...

G

s(ei!k) ei`!kU(!k) ...

e i`!k

U(!k) ...

3

7

7

7

7

7

7

7

5

(31)

(5)

g p tively. The vector (31) is the vector of instruments.

The rLS method. is as follows in the frequency do- main. The problem is to nd A(q) and C(q) in an ARMA model

A(q)y(t) =C(q)e(t):

Step 1). Solve min

 X

R

j (ei!k)Y(!k)j2 for ^ (ei!) for a "high order" polynomial .

Step 2). Treat

^

E(!k) = ^ (ei!k)Y(!k) as measured input and solve

min

AC X

k

jA(ei!k)Y(!k);(C(ei!k);1) ^E(!k)j2 (32) for ^A, ^C. It is my experience that these start-up procedures work well.

8 Conclusions

We have in this contribution discussed various aspects of frequency domain methods for linear system identi-

cation. Generally speaking, it could be said that the direct frequency domain approach has been underuti- lized in conventional system identication. The con- tribution has been partly of tutorial character, sum- marizing some main points. In addition the author's experiences with various implementations of the al- gorithms have been described.

References

1] H. Akaike. Maximum likelihood identication of gaussian autoregregressive moving average mod- els. Biometrika, 1973.

2] E. J. Hannan. Multiple Time Series. Wiley, New York, 1970.

3] J.E.Dennis and R. B. Schnabel. Numerical meth- ods for unconstrained optimization and nonlinear equations. Prentice-Hall, 1983.

4] P. V. Kabaila and G. C. Goodwin. On the estima- tion of the parameters of an optimal interpolator when the class of interpolators is restricted. SIAM J. Control and Optimization, 1980.

 ] p g

Automatic Control, 1959.

6] L. Ljung. System Identi cation - Theory for the User. Prentice-Hall, Englewood Clis, N.J., 1987.

7] J. Schoukens and R. Pintelon. Identi cation of Linear Systems: A Practical Guideline to Accu- rate Modeling. Pergamon Press, London (U.K.), 1991.

8] A. van den Bos. Identication of continuous-time systems using multiharmonic test signals. Identi - cation of Continuous-Time Systems, 1992. Edited by Sinha and Rao, Kluwer Academic, Dordrecht (The Netherlands).

9] P. Whittle. Hypothesis Testing in Time Series Analysis. PhD thesis, Uppsala University, 1951.

/ingegerd/lennart/cdc93.tex

References

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