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Some Asymptotic Properties of Multivariable Models Identified by Equation Error Techniques By P.M.J. Van den Hof and P.H.M. Janssen

EUT Report 85-E-153 ISBN 90-6144-153-6 ISSN 0167-9708 Novem ber 1985

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CESIUM-SEEDED

Hof. P.M.J. Van den and P.H.M. Janssen SOME ASYMPTOTIC PROPERTIES OF MULTIVARIABLE MODELS IDENTIFIED BY EQUATION ERROR TECHNIQUES. EUT Report 85-E-153. 1985. ISBN 90-6144-153-6

Eindhoven University of Technology Research Reports EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Electrical Engineering

Eindhoven

The Netherlands

SOME ASYMPTOTIC PROPERTIES OF MULTIVARIABLE MODELS IDENTIFIED BY EQUATION ERROR TECHNIQUES

by

P.M.J. Van den Hof and

P.H.M. Janssen

EUT Report 85-E-153 ISBN 90-6144-153-6 ISSN 0167-9708 Coden: TEUEDE

Eindhoven

November 1985

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Hof, P.M.J. Van den Some asymptotic properties of multivariable models identified by equation error techniques / by P.M.J. ,Van den Hof and P.H.M. Janssen. - Eindhoven: University of Technology_ - Tab. (Eindhoven University of Technology research reports / Department of Electrical Engineering, ISSN 0167-9708; 85-E-153) Met lit. opg., reg. ISBN 90-6144-153-6 S150656 UDC 519.71.001.3 UGI650 Trefw.: systeemidentificatie.

CONTENTS

Abstract

1- Introduction

2

2. The system and the model set

3

3. Equation error methods; some asymptotic results

5

4. Main result

9

5. Discussion

16

6. Conclusions

18

Appendix

20

References

23

SOME ASYMPTOTIC PROPERTIES OF MULTIVARIABLE MODELS IDENTIFIED BY EQUATION ERROR TECHNIQUES

P.M.J. Van den Hof P.H.M. Janssen

Dept. Electrical Engineering Eindhoven University of Technology The Netherlands

ABSTRACT

In this paper some interesting properties are derived for simple equation error identification techniques - least squares and basic instrumental variable methods-, applied to a class of linear time-invariant time-discrete multivariable models.

The system at hand is not supposed to be

contained in the chosen model set.

An analysis of the approximating

model is performed in the time-domain, relating the Markov parameters of the original system to the Markov parameters of the identified model.

The results are asymptotic in the sense that the number of data samples is supposed to be infinite; the input signals are supposed to be stationary zero mean white noise sequences with unit variance. The asymptotic results are derived for a general class of linear multivariable models in I/O form (matrix fraction descriptions), incorporating

most models currently used in system identification. In terms of approximation of systems, the results can also be applied in model reduction.

Mailing address of the authors: Eindhoven University of Technology Department of Electrical Engineering

P.O. Box 513 5600 MB

Eindhoven

The Netherlands Tel. (40)-473280

2

1.

INTRODUCTION

In system identification literature there is a growing interest in iden-

tification methods that give reliable results in situations where the process at hand is not necessarily contained in the chosen model set. This aspect is considered to be a valuable robustness property [1]. importance is indicated by realizing

t~at

Its

in many practical situations of

system identification, a model will be required that is of restricted

complexity, approximating the essential characteristics of the - possibly very complex - process, rather than a very sophisticated model that exactly models the process behaviour.

If the problem of system identification, or rather approximate modelling, is considered in this context, it comes very close to the problem of model reduction.

An important item now becomes: which criterion has to

be used to approximate the original process, resp. the higher order

model, and which model set has to be chosen.

It has been recognized that these choices highly determine the performance of the model, when used for specific purposes such as simulation, prediction, (minimum variance) control etc. [2], [3].

In many situations

the performance of an identified model is judged upon its ability to simulate the process under consideration.

However, output error methods,

being most appropriate if the simulation behaviour of the model is concerned, are much more complex than equation error methods.

It is there-

fore important to analyse the simulation behaviour of an equation error model.

A frequency domain analysis of this aspect of approximate models,

identified by prediction error methods, is given in [4],[5].

By con-

sidering the Markov parameters of the identified model, we will focus on

properties of the approximate model in the time domain.

In this paper an analysis is made of simple least squares and instrumental variable methods applied to equation error type models.

Earlier work

on this subject has been published in 1976 by Mullis and Roberts [6], who established a connection between results in model reduction and asymptotic results in least squares system identification.

An extension to

3

the multivariable case has been worked out by Inouye [7], but restricted to a very special parametrization (full polynomial ARMA form).

The pre-

vious work will be extended to a general class of multivariable models, containing both canonical and pseudo-canonical ones, and to simple instrumental variable· methods.

The analysis of the relation between the

original process and the identified model will be carried out in the time

domain, by means of the Markov parameters.

In section 2 the class of models is defined and notations are introduced. Asymptotic equation error results are presented in section 3.

In section

4 the main theorem is introduced and applied to a number of different parametrizations.

2.

A discussion on the results follows in section 5.

THE SYSTEM AND THE MODEL SET

We consider a discrete-time multivariable system having a p-dimensional

input signal u(t) and a q-dimensional output signal yet) at time instant t(t E Z).

At the outset we will keep the discussion quite general and

therefore only require that the input- and output signals are jointly wide-sense stationary and ergodic.

Later on we will specify our results

for white input signals with unit variance matrix, and for linear systems. Consider the problem of modelling the system approximately by using a parametrized set of linear time-invariant discrete-time multi variable I/O models, given by the following general MFD (Matrix Fraction Description)form:

P(z;9)y(t) = Q(z;9)u(t) + where

~(t;9)

( 1)

- z denotes the forward shift operator zy(t}:=y(t+1);

P(z,9):=[p .. (z)] ~J

Q(z,9):=[q .. (z)] ~J

are (qxq), resp.

qxq

and

qxp (qxp)-polynomial matrices where for ease of

notation the explicit dependency of the polynomial entries on the parameter 8 has been omitted.

- P(z,9) = Pd(z) + P*(z,9) where

(2 )

4

V diag.[z

1

v , ••• ,z q]

(3 )

and p*(z;9) = [p~,(z)] ~J

-

Pij*(z) = a, ,

~Jv ij

~

qij(z)

.. z ~Wij

z Il

v ij-1

ij

(4)

qxq

r, ,-1

+ ••• + a, , lJr

~JSj

The integer indices

V.

~

I

V ••

~ij'

I

~J

r

1'i, j'q

(5)

l~

(6)

ij

-1 + ••• + ~"

~J

z

z

ij

s,-1 J

i(.q,

1'j'p.

and Sj determine the struc-

ture of the model set (1). They are supposed to satisfy the following conditions:

- v ij ) 0,

po ij )

- r

Sj) 1,

ij

)

1,

0,

vi;o. 0,

(7)

if V < r then the polynomial pij(z) is equal to zero. ij ij if P.ij < Sj

then the polynomial qij{z) is equal to zero.

- As a restriction on the model set it will be required that

v j ) Vij

for

This means that the leading column coefficient matrix of P{z)

(i.e. the matrix consisting of the coefficients associated with the highest degree of z in each column) is equal to the identity matrix.

We will now consider the situation where the vector meters consists of all the coefficients a and nomial matrices p(z;e) and Q(z;9).

~

e

of unknown para-

occurring in the poly-

The residual

E(t;e) = p(z;9)y(t) - Q(z;9)u(t)

(9 )

is dependent on the parameter vector 8, but is not parametrized itself. It is called an equation error and can be computed from the available input and output samples; the residual is linear-in-the-parameter vector

e. Methods for estimating 8 in this context are often called equation error methods [1] and will be considered in the next section.

5

Remark 2.1 The model set (1) is very general and encompasses most uniquely identifi-

able MFD-forms, currently used in the identification of multi variable For most forms it will follow that r.. = 1, s. = 1~J J the model set is not necessarily restricted to causal models.

systems.

Note that

In section 4 it will be illustrated how all specific forms fit in the

general context.

3.

•

EQUATION ERROR METHODS; SOME ASYMPTOTIC RESUITS

Equation error methods are very popular in system identification.

The

main reason for this is the simplicity of the corresponding identifi-

cation algorithm, due to the linearity-in-the-parameters of the model. In this section some asymptotic results will be presented for least squares and basic instrumental variable estimators.

A common equation error method for obtaining an estimate of 9 in (1) is the simple least-squares estimator, minimizing N N

I t=1

with respect to 9.

E(t:9)

T

( 10)

E(t;9)

N denotes the number of data samples.

Being inter-

ested only in asymptotic results, the asymptotic analogon of this problem will be considered, minimizing T V(e) = E E(t;e) E(t;e)

( 11 )

with respect to 9, under assumption of stationary and ergocidity of the input and output signals. (E denotes the expectation-operator).

Remark 3.1 In our theoretical analysis we do not impose the condition that the parameter vector 9 minimizing V(9) is unique.

With respect to the identifi-

cation algorithm, however, one would like to use model sets which guarantee uniqueness.

Examples of these will be given in section 4.

analysis, there is no objection to the non-uniqueness of 9.

For the

•

With respect to the determination of 9, minimizing V(9), we can now state the following:

6

Proposi tion 3. 1 The asymptotic least squares estimator

E E.(t;S) z

,t -1

1

e

satisfies

y.(t) = 0 J

(12a)

and , l-1 EE.(t;S)z U.(t) 1 J

o

(

i

( q

(

j ( P

( 12b)

Proof V(S) is quadratic in S.

A standard necessary and sufficient condition

for S to be a minimum of V(S) is given by:

~~s

(13 )

= 0 for each component Ss of S.

[v(e)lle=e

Equivalently,

-~­ ~S

dt;e)

s

1

o for

each component

as .

( 14)

9=9

Since E(t;9) is a linear function in all the components of 9:

and

•

the result

Now we define P(z): = P(z;9)

(15a)

Q(z;9)

( 15b)

and

Q( z):

The output signal y(t) of the estimated model, when excitated by the original input signal, is given by:

p(z) y(t) = Q(z) u(t)

-=0