On the bounds of the modelling errors of black-box MIMO transfer function estimates

On the bounds of the modelling errors of black-box MIMO

transfer function estimates

Citation for published version (APA):

Zhu, Y. (1987). On the bounds of the modelling errors of black-box MIMO transfer function estimates. (EUT

report. E, Fac. of Electrical Engineering; Vol. 87-E-183). Eindhoven University of Technology.

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Published: 01/01/1987

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On the Bounds of the

Modelling Errors of

Black-Box MIMO

Transfer Function Estimates

by

ZHU Yu-Cai

EUT Report 87-E-183 ISBN 90-6144-183-8 November 1987

ISSN 0167- 9708

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Electrical Engineering

Eindhoven The Netherlands

ON THE BOUNDS OF THE MODELLING ERRORS

OF BLACK-BOX MIMO TRANSFER FUNCTION ESTIMATES

by

ZHU Yu-Cai

EUT Report 87-E-183

ISBN 90-6144-183-8

Eindhoven

November 1987

Zhu Yu-Cai

On the bounds of the modelling errors of black-box MIMO

transfer function estimates / by Zhu Yu-Cai. - Eindhoven:

University of Technology, Faculty of Electrical Engineering.

-Fig. - (EUT report, ISSN 0167-9708; 87-E-183)

Met lit. opg.,

reg.

ISBN 90-6144-183-8

SISO 656 UDC 519.71.001.3 NUGI 832

Trefw.: systeemidentificatie.

1 • 2. 3.

4.

5. Abstract Introduction

The asymptotic theory

2.1 Markov parameter model set 2.2 Spectral analysis

The algorithms 3.1 Algorithm 1 3.2 Algorithm 2 3.3 Numerical test Other forms of Bounds Conclusions

Acknowledgement References

Addres of the author:

ir. ZHU Yu-cai

Measurement and Control Group Faculty of Electrical Engineering Eindhoven University of Technology P.O. Box 513

5600 MB Eindhoven The Nether lands

2 4 5 11 14 14 16 18 19 25 26 27

ON THE BOUNDS OF THE MODELLING ERRORS OF BLACK-BOX MIMO TRANSFER FUNCTION ESTIMATES

Y.C. Zhu

Abstract: This work attempts to solve the following problem: Derive a suitable description of the modelling errors (model uncertainty) of MIMO transfer function models which are obtained by black-box identification; this description must supply the quantitative information required for robustness study of the feedback system where the black-box model is used for the controller design.

An

upper bound of modelling errors will be estimated. We shall refer to the asymptotic theory on the properties of black-box transfer function estimates, developed recently by Ljung and Yuan. Their theory shows that the transfer function estimates are consistent, the errors of the esti-mates are asymptotically jOint normal, with a very simple expression for their covariances. In this paper, their results will be extended to cases where spectral analysis is used. Based on this theory, a bound of (additive) modelling errors is defined as the sum of the absolute value of the bias part and the 3cr bound of the variance (random) part of the modelling errors. Algorithms are proposed for the computations; a numer-ical test is performed to validate.the theory. The bounds for other forms of the modelling errors will also be derived from the bound matrix we have obtained.

INTRODUCTION

Robustness of a feedback controlled system has significant importance for engineering applications of identification and control theory. In order to apply the theory for robust controller design, one needs not only a nominal model of the process (plant), but also a suitable description of the modelling errors (model uncertainty).

Denote

cfJ

(:iw) as the true frequency responses of a input multi-output (MIMO) process, and G(iw) as the

g, ,(iw) as the (i,j) elements of GJ (iw) ~J

nominal model; denote ~ . ( iw )

• ~J

Then,

cfJ

(iw) can be represented as

•

GJ(iw)

=

G(iw) +l> G(iw) a

and G(iw) respectively.

where!J. G(.:iw) are the additive modelling errors (perturbation): a

or as

( 1. 1 )

( 1 .2)

where!J. G(iw) are the multiplicative modelling errors (perturbation).

m ,

and

There are other ways to define the modelling errors, but (1.1) and (1.2) are most frequently used. Similarly, for the element of

GP(iw),

we have

where 11 g .. (:iw) are called element additive modelling errors and a ~J

cjl,

,(iw) = ; (iw)[ 1

+

l> g ,(iw)] (1.4)

~J ~J m ~J

where 11 g .. (iw) are called element multiplicative modelling errors. m ~J

Note that l> g, ,(iw) are precisely the elements of'" G(iw); but. "'mg'J' (iw)

a ~J a .

are not the elements of '" G( iw ) • m

In general, it is difficult to determine the modelling errors exactly. Recent theoretical results have shown that an upper bound of the

model-ling errors is needed for studying the robustness of a feedback system;

cf.

Doyle and stein (1981), OWens and Chotai (1984) and Vidyasagar

(1985).

But how to obtain this bound is still an open problem.

This work attempts to tackle the problem: we estimate the upper bounds of

the modelling errors when the black-box identification technique has been used to obtain the nominal model of a MIMO process which is assumed to be linear time-invariant.

The work will concentrate on deriving a bound for element additive model-ling errors.

Briefly, the idea is the following:

The modelling error!J. g .. (jw) can be considered as a random

vari-a 1J

able, due to the assumption that the disturbance of the real process (plant) is a stochastic process.

6 g"

(jw)

is the sum of the bias part and the random (variance)

a 1J

part

6 a g

'J' (jw)

=

E{ 6 g"

(jw

»)

+

[6 g"

(jw )

-E{ 6 g"

(jw

»)

1

~

a

1J

a

1J

a

1J

( 1.5)

Prove that the variance part is asymptotically normal, with

a

vari-ance

(J (Ul ) •

Then

w.p. 99.7%

( 1.6)

Denote

UB, , (Ul)

as an upper bound of 6 g"

(jw )

such that

1) a 1J

( 1 .7)

and finally, let

UB 'J' (Ul) =

I

E{ 6 g" (jw

»)

I

+ 3a (Ul )

..L a 1.)

( 1 .8)

In section 2, the asymptotic theory of black-box identification is pres-ented. Based on this theory, in section 3, a bound of 11 g .. (:iw) is

de-a 1J

fined and computing algorithms are proposed. In section 4, other forms of bounds are derived. Conclusions and remarks are given in section 5.

2 THE ASYMPTOTIC THEORY

In recent years, many sophisticated time domain parametric identification methods have been developed. Most of these techniques can be classified

in the family of prediction error methods. Very recently, Ljung and Yuan

developed an asymptotic theory on the frequency properties of the predic-tion error models; cf. Ljung and Yuan (1985), Ljung (1985), and Yuan and

Ljung (1984). This theory not only gives us much insight into

identifi-cation, but also has practical importance, due to the fact that the

re-sults have very simple expressions. We shall present this theory for the

Markov parameter (impulse response) model set, and then extend the theory for spectral analysis.

Consider a process with m inputs and p outputs. A general linear time-invariant discrete model for the relationship between inputs and outputs can be written

yet) =

I

G(k) u(t-k)

+

vet) k=1

( 2. 1 )

where y(t) is a p-dimensional column output vector at time t i u(t) is an m-dimensional column input vector at time t i {G(k)} is a sequence of p x m matrices; and {v(t)} is assumed to be a p-dimensional stationary stochastic process with zero mean values.

When the delay operator q-l is introduced as q-l u(t)

=

u(t-1)

the model (2.1) can also be written

where

yet)

=

G(q-l )u(t)

+

vet)

I

k=1

-k

G(k)q

The transfer function matrix for the model is then

(2.2)

.iw

G(e

)

I

k=1

G(k) •

e -ik!l (2.4)

.iw

Note that we use G{e ) instead of G(iw) to indicate discrete models.

The real process is assumed to be linear time-invariant, with

m

dl(e.iw) =

I

dl(k).e-i)q,j

k=1

being the true transfer function matrix.

(2.5)

Suppose that the input-output data have been collected from the real

process until time N:

ZN:{ y( 1), u( 1), ••• , y(N) ,U(N») (2.6) Then the problem is twofold: firstly to estimate the transfer fUnction

N

matrix from data set Z ; secondly to assess the modelling errors defined

in section 1.

2.1

Markov Parameter Model Set

Basically the estimation of a transfer function is a non-parametric prob-lem. In practice, however, the estimation is carried out via a finite-dimensional parameter vector, so the techniques are parametrical.

But the parameters are only the vehicles for arriving at a transfer func-tion estimate. The validation is done in the frequency domain.

One of the parametrizations is the Markov parameter model (finite impulse response), given by n G (q,e) =

,\q

-k

k=1

n

I

(2.7)

so

G

k

= G(k) ,

k=1, ••• ,n

is a parametrization of model (2.1); and

where

en

is the parameter matrix and n is the order of the model. If n is big enough, this model set will give a good transfer function esti-mate, in the sense of a small bias, which is

n

L

k=l -ill", ~.e (2.9) n ~ n

If the objective is to estimate GN(e ), rather than to determine

e ,

i t is natural to let the order n depend on N, i.e. n=n(N}.

For black-box models, the "best" order will tend to infinity as N tends to infinity. If we introduce ~ (t) u(t-1) u(t-2) u(t-n)

the model can be rewritten as

where (;(t») ... (v(t»)

as n ... 00 •

Let us introduce the parameter vector n

«mn) x

1

vector)

«mpn) x

1

vector)

(2.10)

(2.11)

(2.12)

where co1(.) denotes the column operator which stacks the columns of the matrix on top of each other.

Then

(2.13)

where

®

denotes Kronecker products; I is a p x p identity matrix. p

Let A

=

product (a

ij), B (bij) be m x nand p x r matrices, then the Kronecker of A and B is defined as

[

:,,'

a 12B

.

. . .

a 1n B A®B = _(2.14) a m1B

. .

.

. .

a mn B

and the second equality in (2.13) is the property of Kronecker products.

The parameters of

e

can be estimated by the well known prediction error

method, which minimizes the sum of the squared prediction errors associ-ated with a parametrization of the model (2.1). If the parametrization

is given in (2.7) and (2.8), and no further assumption about (vet») is

included in the model except that which is given in (2.1), then the pre-diction error method will take the form

On N

_T

_{n n} nN = arg min _N

L

8 (t,6 )8 (t,6 ) t=l n 8 (t,6 ) yet) - G(q,6 )u(t) = yet) -

[~

(t)4i)IJ

.n

n

The

estimate;~

is the well-known linear least squares estimate

=[R(N)]-l n where and

~

(t) =

~

(t)4i)I P R (N) = n N

~

L

$'

(t)y(t) t=l

The corresponding transfer function estimate is n

I

G k k=l (2.15) (2.16) (2.17) (2.18)

where the parameters of

e~

are determined by (2.17) and W(ol) = -:iw e .1 m -2:iw e • I m -n:iw e • I m Formal assumptions: (2.19)

The results on the properties of the estimate (2.18) are given under the following conditions:

C1:

The real process is given

by (2.5)

and

C2:

C3:

I

< "'.

V

i , j

k=1

(The real process is stable).

R (T),

V

T =-IJO

IT

1.11

R (T)II

< '"

v

where E denotes the mean value expectation, and 11.11 denotes the

matrix norm. ill (ol) =

I

v T=....oo

'"

R (t)e-

iTOl

v Hill v (Ol)1I

<

c VOl.

C4: vet) =

I

H(k)e(t-k) k=O

where {e(t)} is white noise with unitary covariance matrix, and uniformly bounded fourth moment, and

I

Ih .. (k)1

<'"

Let N ~ (w,n) u N N

I

t=1 n

I

T=-n T u(t)u (t-l'r) -iTW e

C5: (U(t») is independent of (v(t») and II u(t)1I ( C

1 1ft C6: C7: C8: C9: N ~ (w, n (N» .. ~ (w) as N + 00 U U 1

I

n(N) 2n(N)

I

T=-2n(N) A . (~ (w»

>

<I

>

0 m1n u as N + 00 (2.20) (2.21)

where

A .

(A) denotes the smallest eigenvalue of the matrix A. m1n

C10: n(N) + IX) as N + 00

C11:

:r1-

(N)/N + 0 as N" 00

C12:

I

(n(~

)/kJ2

<

00

Theorem 2.1: Consider the transfer function estimate

~n(eiw)

defined in N

(2.17) and (2.18). Denote

Assume that C1 - C12 hold. Then

jw • .

lid!

(e ) -

E{G~(eJW)JII

+ 0 as N -+ 00 (2.22)

(Consistent estimate)

where 11.11 means the matrix norm.

as N -+ 00 (2.23)

(Asymptotic covariance)

< As N( 0,

~

-1 _u (w

~

T _v (w ) ) (2.24)

(Asymptotic normal distribution)

The proof of the asymptotic covariance can be found in Yuan and Ljung (1984); the consistency and asymptotic normality will follow the 8180 case analogously as in Ljung and Yuan (1985).

We now need the properties of each transfer function estimate; these follow directly from Theorem 2.1. Let

"'n:iw "'n iw

J

GN(e ) =

(g ..

(e )

1J

V

i , j

Corollary 2.1: Consider the estimate gij(e "'n .i.w ), as in (2.25). that C1 - C12 hold. Then

as N -+ 00

(2.25)

Assume

N

r[

-n Jw

- ( N) va gi . ( e )

J

+

l

~ -I (w )

J .

j • ~ (w )

n ) U J v,

. 1

as N ... 00

where [~-I (w )

1 ..

is the (j, j) element of ~ -1 (w) •

U JJ U As N( 0,[ ~ -1 (w )

1 ...

~ (w) U JJ Vi (2.28) From (2.27) we have (2.29)

This remarkably simple expression will have much practical importance. In this paper, for instance, this result is used to obtain a bound of the modelling errors.

2.2 Spectral analysis

For the estimation of transfer functions, the spectral analysis technique can be used. The transfer function matrix of a MIMO process can be

esti-mated row by row. The i-th row of the transfer matrix can be estimated by the following steps:

Calculate the auto- and cross-correlation functions:

N N

I

t=l N =

~

I

t=l T u(t) u (t+T) u(t) Yi(t+T) (m x m matrix) T = -n, ... ,-1,O,1, . . . ,n (2.30) (m x 1 vector)

Fourier transform n

l:

'[=-n m

L

T=-m

-iTW e -iTW e

Calculate the transfer function estimate

(m x m matrix)

(2.31) (m x 1 vector)

(2.32)

N

N

Here R (T) I R (T ), T = -n, ... , 0, •• • ,n, are the parameters of the

spec-U

uY

i

tral analysis. The order of this model is 2n+1. Note that here the n is not necessarily equal to the n in the Markov parameter model set (2.7).

The computations of RN(T) and

U RN _uYi may involve inputs and outputs prior

to the first, and after the N-th sample moment. These values are suppos-ed to be unknown. For the analysis, these values can either be replaced by zeros or by other bounded numbers. They will not influence the asymp-totie result. In practice, however, these values are taken to be the

real measurements.

This can be realized by collecting the data set Z N+2n

N

instead of Z in (2.6):

N~n{ )

Z : y(-n+l),u(-n+l), ••• y(N+m),n(N+m)

N

and computing R (T) and

u according to (2.30).

An:iw [ An :iw An :iw

1

T

Theorem 2.2: Consider the estimate Gi(e ) = gil(e ) ••• gim(e )

ob-tained from (2.30)-(2.32). Assume that Cl-Cll hold.

II

E{

~~(e:iw»)

-

~

(e:iw)1I

->

0

~ ~

as N .. o:J

(Consistent estimate)

where

cJ!.

(e:iw) is the i -th row of

cJ!

(e:iw ) •

~ N

2n(N)+1

(Asymptotic covariance) as N -+- 00 <

As

N ( 0 ,[

~

T (w ) ]-1

~

(w ) ) U vi

(Asymptotic normal distribution)

(2.33)

(2.34)

(2.35)

This is the straightforward

MIMO

extension of theorem

2.2

in Zhu

(1987a). Note that C12 is not necessary for spectral analysis. We found that the time domain estimation technique and the spectral analysis are asymptot-ically equivalent for the transfer function estimates.

"n :iw

Corollary 2.2: Consider the estimate gij(e

) obtained from (2.32).

Assume that C1-C11 hold.

Then

as N .. 00

(2.36)

as N -+- CD (2.37)

<

As N ( 0,[

oj)

-1

(w )] . . . oj) (w

»

u JJ Vi

(2.38)

The same result has been proved by Ljung (1985) for the general predic-tion error models, e.g. ARMA models, ARMAX models. Zhu (1987b) has

prov-ed the MIMO extension of the theory.

3. The algorithms

Based on the asymptotic theory, there are several possibilities for esti-mating a bound of the modelling errors. We will present two of them.

3.1 Algorithm 1:

According to Corollary 2.1, we propose

1) Perform Markov parameter estimation as in (2.17), with the order n sufficiently large. In practice, n will have to be determined ac-cording to the engineering knowledge about the process.

2) Calculate

~~(e:iw)

as in (2.18). Then, Corollary 2.1 tells us that 'n

(gij - ~ij) is asymptotically normal and

var(

~~j

- <f .

J

~

!!

[$

-1 (w )

J . .

~

(w )

~J

N

u )) v.

~

'n

Hence, asymptotically, we have the 3cr bound for (gij - gPij):

3) Estimate $ (w)

u and $ v (w ) by n

[~

f

u(t) uT(t+r)] e-:iwT N

~ (w )

l:

= ~ (w) (3.2) u u T=-n t~1 and n

[~

N ;T(t+r)] e-·llin N ~ (w)

_L

L

e(t) ~q,,{w) (3.3) v '[=-n t=1 e

4)

e

(t)

=

[rfJ

(q-l) -

G~(q-l

)] u(t)

+

vet) (3.4)

"n .iw

Perform a model reduction on GN(e ) to obtain the low order model "t :iw

GN{e ) I where R. denotes the low order. In practice, it is the low

order model that is used as the nominal model for the controller design. The theory and techniques of model reduction are well-esta-blished (see Glover (1984».

5) Define the modelling error as

(3.5)

"t :iw "t .

where gij(e ) is the (i,j) element of GN(e"').

Then

We call the first term the bias part and the second term the random :iw

(variance) part of the modelling error 11 g .. (e ). a ~J

Now

From (3.1) we have

where [.; -1] .. and

~

(w) can be obtained from (3.2) and (3.3).

u JJ vi

:iw

Denote UB .. (w) as an upper bound of 11 g .. ( e ), then

+

3J* [;

-I (w )

1 ..• ;

(w )

U JJ vi (3.6 )

Remarks:

From (3.4) we have

From Theorem 2.1 and the large number theorem,

<l> (w) + <Il (w)

v v as N .... ()O I n + 00

This means that 41 (w) is a consistent estimate of 41 (w).

v v

Algorithm 1 is a combination of identitification and model

reduc-tion. This method is proposed not only for obtaining a bound of

modelling error; i t is a new method for finding a

black-box-low-order-nominal model.

We have at least two reasons for stating this.

Firstly, we have the theories which give us clear insight into

iden-tification and model reduction - the asymptotic theory of Ljung and

Yuan, and the model approximation theory of Glover and others;

secondly, numerically reliable algorithms are available. See

Wahlberg (1985) for more discussions on the point.

If one already has a low-order-nominal-model, the computations can be

done by spectral analysis, as shown in the following algorithm.

3.2 Algorithm 2:

1) Calculate the output residual from the low order model,

e(t)

e(t) = I!. G(q-I) u(t)

+

vet) a

(3.7)

2)

where

Note that e(t) here is different from the one in (3.4) where e(t)

was the output residual from the high order model

;~(q-l).

Perform spectral analysis to estimate!J.. G(eiLu) I

a

"'n iLu

t.

g. (e )

a :un

(3.9)

where <l>N. (w) is calculated as in (2.30) and (2.31) by replacing

ue

i

yi(t) byei(t).

We note that (3.8) is a MIMO process with u(t) being the input, e(t) the output and vet) the disturbance. Therefore, Theorem 2.2 can be applied to the estimate (3.9). Then, according to Corollary 2.2, we

have "'n i1JJ iw

[t.

g . . (e ) -

t.

g . . (e )] a 1J a 1J < As N(O, 2n+ 1 [-1 ] - N - <I> (w) ..• <1> (w

»

U ]J vi (3.10) and 3

J

2Nn

+

1 [<I> -1 (w )] .. u JJ <I> V. (w ) 1 w.p. 99.7

%

3) Estimate the noise spectrum.

we can have a consistent estimate of 4> (w) by

v

4) Calculate the estimate of the bound,

Again, we call the first term random (variance) part of the

:iw

It,

g .. (e )

I

~ a ~)

From (3.11) we have

the bias part and the second modelling error t, g .. (e:iw ) •

a ~) (3.12) term the

J

2n+1 N - •

+

3 - N - [( 4> (w») 1

1 ..•

4> (w ) U JJ Vi (3.13) "'n i.w

where t, g .. (e ) is obtained from (3.9) and 4> (w) is determined by

a ~J vi

(3.12).

Now, if we denote US, ,(w) as an upper bound of the modelling error

.

~)

It,

g .. (elW)

I.

we have a 1)

3.3 Numerical Test

3 (3.14)

The numerical test is performed for Algorithm 1 using the Markov para-meter estimates of an industrial process. This is a process of 2 inputs and 2 outputs. The length of the Markov parameters is 50, and we will consider i t to be the real process ~ (t). We let

dl

(t) = 0 for

t

>

50.

the input u(t) and the output disturbance both consist of white noise signals. The power of the disturbances is about 11~of the power of the outputs.

Then a 50-th order parameter model is estimated by using 1000 samples of the I/O data from the simulation. The Markov parameters of the process and of the model are shown in Fig. 1. According to Algorithm 1, the upper bound is computed. Because the order of the model is equal to the

order of the process, the bias part of the modelling errors is zero. The estimated 3a bounds and the errors are plotted in Fig. 2, and the 20 bounds and the errors are in Fig. 3. We find that the 2a bounds (w.p. 95.5

%)

are good enough for the testing example.

More information about the numerical test can be found in van Beuningen (1988).

4.

OTHER FORMS OF BOUNDS

So far, we have derived an upper bound of the element additive modelling error /!, g .. (e:iw), which can be calculated by (3.6) or by (3.14). It is

a

1J

easy to verify that for the element multiplicative modelling error :iw /!, g .. (e ),

m

1J UB .. (w ) 1J 'y'i,j ( 4. 1 )

Hence we can take the right-hand side of (4.1) as an upper bound of :iw

/!,mgij(e ).

The bounds of the element (additive and multiplicative) modelling errors are also called structured model uncertainty, and they can be used to analyse the robust stability of a closed-loop system (see Owens and Chotai (1984».

The recently developed so-called singular value analysis has proven to be effective for studying the robust properties of a MIMO feedback control-led system (See Doyle and Stein (1981) and Vidyasagar (1985». This technique needs an upper bound of the matrix norm of the modelling error

...

••

...

.

.

"

...

"

....

..."

... I;;!

...

"

-II. It

...

...

"

"

,.

..

..

..

•

"

,.

..

..

tv 0

.

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f>, G(e.iw) or f>, G(e.iw) (see (1.1) and (1.2». Following Doyle, we denote

a

m

t

(w) and

t

(w) as the bounds such that

a m

'v'w

(4.2)

- .iw

(J (f>, G(

e

»

~

t

(w)

m

m

Vw

(4.3)

where (J ( . ) denotes the matrix norm, t (w) and

t

(w) are the additive and

a

m

multiplicative bounds respectively. For a matrix A with complex vari-ables, the matrix norm is defined as

f>,

(J (A) max II Axil _

II xii =1

II.

[

A*A]

max (4.4)

where 11.11 is the Euclidean norm, A[.] denotes eigenvalues, and

*

denotes

conjugate transpose. a(.) is also called the maximum singular value of a matrix.

Denote UB(w) as the matrix of the upper bounds of the element additive modelling errors which are obtained from section 3,

UB (w) = {UB .. (w

»)

l.J (4.5)

Then we can ask ourselves the question: can we derive

t

(w) and

t

(w)

a m

from UB(W)? Let us derive t (w) first. a

.iw

Theorem 4.1: For f>, G(e ) defined in (1.1) and UB(w) defined in (4.5), we a

have

Therefore, we can let

t (w) =

crr

UB(w )

1

a

\tw

where

t

(w)

is called unstructured uncertainty. a

(4.6)

The proof of Theorem 4.1:

Denote A as a matrix with complex variables, B as a matrix with real

positive variable. Assume that A and 8 have the same dimensions, and

IAijl ( 8_{i j} i , j

From ( 4.4)

we have

(jl.

(A) max IIAxl 2

II xII =1 max E (E _Aij X i j )2 II xII =1 i j But, E (E _Aij X i j )2 ( 1: (E IAijl

.

I

Xj

I

)2 i j i j

"

l: (l: _{B .. IXj} 1)2 .K€cn i j ~J

Because B . . ) 0, there exists a x with x. ;;. 0 I U xiI

~J J II Bxil = 0 (ll) Hence E (l: B .. x. )2 ( _{l: (E}

-

2

₌ 0 2 (B) i j ~J J i j BijXij ) From (4.9) and (4.10)

(jl.

(A) (

(jl.

(B) Therefore a(A) " a(B) 1 such that n x<C (4.8) (4.9) (4.10) (4.11) (4.12) :iw

If we sUbstitute A and B by r:, G(e ) and UB(w), we will obtain (4.6). a

Now let us try to derive the upper bound iw

error /:, G( e

) •

Assume that the nominal

m

for multiplicative modelling • iw

model

Gee

)

is square and in-vertible, then

(4.13)

It is easy to show that

(4.14)

and

(4.15)

where

t

(w)

is from (4.7), UB(w) is from (4.5), and

1.1

means taking the

a

absolute value of each element of the matrix.

iw Therefore we can determine the upper bound of A G( e ) by

m

t (w)

m

5. CONCLUSIONS

(4.16)

The problem of defining and estimating the bounds of model uncertainty

(modelling errors) of the transfer function estimates has been studied. We have shown that i t is possible to represent the uncertainty of black-box transfer function models stochastically by using the asymptotic the-ory developed by Ljung and Yuan. This new theory has been extended here for the spectral analysis technique and applied to obtain a bound of the

element additive errors of the transfer function models. Two algorithms have been proposed for the computations. The result of the numerical

test has shown that the theory is adequate.

The bounds of the matrix

norm of (additive and multiplicative) modelling errors have been derived

and they can readily be used to analyse the robustness of the feedback

system.

Note that we can only derive the mUltiplicative modelling error ~ G from m

6 i f G is square and invertible. a

OUr result here is based on the assumption that the real process is lin-ear, which is sometimes impractical.

The modelling and estimation of the model uncertainty for the processes with some non-linearity requires further research.

ACKNOWLEDGEMENT

I would like to express my thanks to my supervisor, Professor Eykhoff,

for

his encouragement, guidance and valuable help throughout the work;

to my colleagues Dr. Damen and Dr. van den Boom for the stimulating

dis-cussions we had. I would also like to thank Professor Hautus of the

Department of Mathematics and Informatics, Eindhoven University of

Tech-nology, who provided the proof of Theorem 4.1. The numerical test was

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Hlt;HEH LP.VELS OF A SlLICCN CO~PILER.

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AN OPTIMAL CMOS STRUCTURE FOR THE DESIGN OF A CELL LIBRARY. EUT Report 87-E-166. 1987. ISBN 90-6144-166-8

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Coden: TEUEDE

IMPLEMENTATION AND EVALUATION OF A COMBINED TEST-ERROR CORRECTION PROCEDURE FOR MEMORIES WITH DEFECTS. EUT Report 87-E-169. 1987. ISBN 90-6144-169-2

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DASM: A tool for decomposition and analysis of sequential machines. EUT Report 87-E-170. 1987. ISBN 90-6144-170-6

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MULTIPLE-BEAM GROUNDSTATION REFLECTOR ANTENNA SYSTEM: A preliminary study. EUT Report 87-E-171. 1987. ISBN 90-6144-171-4

(172) Bastlaans, M.J. and A.H.M. Akkermans

ERROR REDUCTION IN TwO-DIMENSIONAL PULSE-AREA MODULATION, WITH APPLICATION TO COMPUTER-GENERATED TRANSPARENCIES.

EUT Report 87-E-172. 1987. ISBN 90-6144-172-2

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011 A BOUND OF THE MOllEL1ANG ERRORS OF BI.ACK-BOX TRANSFER FUNCTION ESTIMATES.

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TECHNOLOGY MAPPING FROM BOOLEAN EXPRESSIONS TO STANDARD CELLS.

Ell'!" H .. purt 1:17-E-174. l'jfl7. ISBN :JG··Ll<44-17<:·';I

!I "I'») <Janss'~n, P.H.M.

FURTHER RESULTS ON THE McMILLAN DEGREE AND THE KRONECKER INDICES OF ARMA MODELS. EUT Report 87-E-l75. 1987. ISBN 90-6144-175-7

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~UDEL STR;]CTl'RE SELECTION FOR MlJLTIVARIABLE SYSTEMS BY CROSS-VALIDATION METHODS. ELTT Rt:port 87-£-176. 1987. ISBN 90-6144-176-5

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AIlCS IN CESIl:M SEEDED NOBLE GASES RESULTING FROM A MAGNETICALLY INDUCED ELECTRIC FIELD.

1;IJT P,,[·,;rt !-j7-L:-177. 1987. ISBN 90-6144-177

: 17B) J,m,;sen, P.II.:-I. ann P. Stoic<l

0:1 '1'HE EXPECTATION OF THE PRODUCT OF FOUR MATRIX-VALUED GAUSSIAN RANDOM VARIABLES. EU! Report 87-E-1-;'8. 1987. ISBN 90-6144-178-1

(17')) Lie.shout, G.J.P. van dnd L.P.P.P. van Ginneken

GM; A gat'· matrix IJyout Cjcllerator.

EUT R\~port 87-£-179. 1987. ISBN 90-6144-179-X (J80) Glnneken, L.P.P.P. van

GRIDLESS ROUTING FOR GENERALIZED CELL ASSEMBLIES: Report and user manual. EUT Re(-,ort 87-E-I130. 1987. ISBN 90-6144-180-3

(J 81) _{Bollen, M.H.J. and P.T.M. Vaessen}

FREQUENCY SPECTRA FOR ADMITTANCE AND VOLTAGE TRANSFERS