IEEE TRANSACTIOXS ON AUTOMATIC CONTROL. VOL. AC-26. NO. 2, APRIL 1981 573

0018-9286/81/0400-0573S00.75 01981 IEEE

In (5) and (6) the integration path is the unit circle. It follows from [2, Lemma 11 that in the integrand of (6 ) all poles inside the uni t circle should be canceled by zeros. Then we obtain

where { k , ) are such that I ”‘ -‘“‘K( I - I ) is analytic inside the unit circle, Thus part 1) is concluded.

If n* = O then (7) implies G( z)=O and. therefore, R ( n u . ;e ) 1s nonsin-

To prove part 3). it is sufficient to give an example. Let us consider the

A 2 .

gular in this case. -

following simple case:

1 7 u = 2 , A ( Z ) = ( I + u I z ) ( 1 + u ~ z ) u j € ( - l , l ) j = l , 2

n c = O , i . e . . C ( z ) ~ l .

Then we have

R(I , I )=E 1

( l + u l q - ~ ) ( l + u ~ q - ~ ) e( t - 1)

- - - ( ~ ~ + u ? ) E e ( ? ) ~ ( I -uf)( I -.I)( 1 - u 1 u z ) ‘

If u I = - u l then R(1, I ) will be “singular.”

Renzurks 1) The first assertion of part 1) as well as part 2) of the previous lemma

were correctly anticipated in [4]. [5]. They are clearly related to our problem of testing the orders of (1).

2) The second assertion of I ) is of particular interest mrhen the AR parameters of ( 1 ) are estimated, prior to MA parameters, using a generdi- zation of the Yule- Walker equations, e.g., [4], [5]. Moreover, part 2) is a simple consequence of this assertion.

3) Part 3) is in contrast to what has been conjectured in [4]. Note, however, that the systems which give a singular R( nu, R C ) for some n*tO, are expected to occur with measure zero since their parameters must fulfil some nontrivial relations. cf. For instance (8). Furthermore, the existence of these ”pathological” systems is not too important for the testing procedure which can cope with such cases in an obvious way.

n A

I? n

In practice. the covariance matrix R( ;I;, i 7 i ) will be replaced by its

sample consistent estimate R ( I I U . ?le), e.g., [4], [5]. Then the hypothesis

det I?( hi. k ) = O should be tested in some way. In the case of MA processes (i.e.. nu=o) this can be easily done if we choose

” /> I\

A. ,A

I = I I <

as test quantity. see, e.g.. [3], [6]. Otherwise it is difficult to infer a confidence test on zero determinant.

Alternatively, we can consider

where It. I1 is some suitable matrix norm. It is expected that (9) will rapidly increase when the model becomes overparameterized.

Finally, let us note that the results of this section can be extended to control systems by means of the instrumental variable approach [7], [8 ] . It is. however, more difficult to prove that the matrix R ( . : ) corresponding to the case of control systems is nonsingular when the model has correct structure [7].

REFERENCES

[I] K. J. &mom, Inrroducrion Io Srochmric Conrrol Theoy. New York: Academic, 1970. 121 K. J. h r o m and T. SMerstriim, “Uniqueness of the maximum likelihood estimates of

the parameters of an ARMA model,” IEEE Trans. Auromar. Conrr.. vol. AC-19, pp. 769-773. k c . 1974.

[3] J. C. Chow, “On the estimation of the order of a moving-average process,” IEEE Trans. Auromr. Conrr., vol. AC-17. pp. 386-387, June 1972.

[4] -. “On estimating the orders of an autoregressive moving-average process with uncertain observations.” IEEE Trans. Auromur. Conrr.. vol. AC-17, pp. 707-709, Oct.

[5] N. A. Lindberger, “Comments on ‘On estimating the orders of an autoregressive 1972.

moving-average process with uncertain observations,”’ IEEE Trans. Automat. Conrr., vol. AC-18, pp. 689-691, Dec. 1973.

[6] P. Stoica. “Identification of the univariate time series by spectral factorization,” Problem of Conrr. Injorm. Theov, vol. 8, no. 2, pp. 115-122. 1979.

I71 T. Sdentrom and P. Stoica “Comparisons of some instrumental variable method-consistency and accuracy aspects,” Uppsala Univ., Inst. of Tech., Uppsala,

[81 P. E. Wellestead. “An instrumental product moment test for model order estimation,” Sweden. Rep. UPTEC 7888R, 1978.

Auromaricu, vol. 14. pp. 89-91. 1978.

A Method for Accelerating the First-Order Stochastic Approximation Algorithms

FRYDERYK Z. UNTON

A hstruct- The main problem considered is the method of choosing step coefficients in first-order stochastic approximation algorithms for system identification. The proposed method increases the efficency of the Saridis and Stein algorithm [4].

GENERAL CASE

Usually the first-order stochastic approximation algorithms for system parameter identification can be represented as follows:

where HA( B ) is a function depending on a random variable measured at the moment k . The sequence B, is convergent with probability one or, in the mean-square sense, to the optimal vector of parameters B*. The deterministic sequence ph of scalars satisfied the conditions

The most common practice is to assume that p, is a harmonic sequence

p k = u / ( h + k ) , a>O, h>O. (3)

The proposed method of accelerating the convergence consists of

I ) To choose pk, minimize HA( B ) in the direction choosing p , according to the following scheme.

( - grad HA( B ) ) (starting with B h ) .

2) Represent the chosen value by

Manuxript received June 3. 19x0: revised September 12. 1980. The author is u,ith the Systcms Research Institute. Polish Academy of Sciences. 01-447

Warsaw. Poland.

574 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-26. NO. 2. .4PRIL 1981

I t 200 400 6011 Bcx, 4WD 1200

ITERATIONS

F I ~ I

3) Choose p h according to the formula

PA + ( ' / ' A ) )

where uh is the k th term of the harmonic sequence (3) The random variable p r ( .) can be rewritten as

The coefficient (7) is identical to the well-hoan Kanmarz coefficient [I]. Finally. then. the coefficient ( 5 ) is as follows:

(5)

p r = l / (

If sI( .) is uniformly bounded, then p h given by ( 5 ) satisfies the conditions (2) and

Let us consider the identification of the dynamic system (see [2] and [4] and references therein). The system is given by

( 6 ) h = l

The condition (6) implies that the given algorithm has an identical asymptotic rate of convergence for the coefficients (3) and ( 5 ) . The : ( k ) = s , ( k ) + e , . proposed method ( 5 ) has the advantage of accelerating convergence in the initial iterations. The Saridis and Stein algorithm for estimating the true parameters

Let us now consider the form of p h ( 5 ) in some specified algorithms. w=[ 'v~- - ' . 3 w , ~ 1 T is given in [41

STATIC SYSTEM B , - , = B A + p ( , + I ) G ~ ~ - , ~ , ~ , , , , , , . k=l.~1-2,2~1-3 (8) k - 1

The system is given by where y =/( x) + e

where e is noise. The algorithm for minimizing the function

I ( B ) = E [ H , ( B ) ] w h e r e H , ( B ) = 2 ( ; ' - F T ( X ~ ) B ) * I Jf

Z ( k + ? l - I ) = [ , - ( k ) . . . . . ; ( k - n - l ) ] r . y ' = E ( e i ) . q ? = E ( w ; )

is given by 0 d#!

~ h , l = B , f p , F ( x , ) ( L . , - ~ r ( x , ) B h ) D = 1:' / _ l i [ 4" (all the conditions of convergence may be found in [4]).

(the other convergence conditions can be found, e.g., in [3]). It is trivial to d,,-l d,,-, dl prove that i,:, given by (4). has the form

@h=l/FT(Xh)F(x,j. (7) The algorithm (8) can be presented in the form

IEliE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. AC-26, NO. 2. APRIL 1981 575

I W *OD

ITERATIONS ZVOD 3ow

Fig. 2.

where

H , ( B ) = ~ ( z ( k + n ) - Z r ( k + n - l ) B ) 1 2 1 - -B7Byz 2

- - BrDD7Bq2 + B7q2d’. I 2

The function HA reaches an extremum in the direction (-grad,&H,( B ) ) for

EXAMPLES

The method increases the efficiency of algorithm with respect to the number of iterations (observations of the system).

REFERENCES

[ I ] S. Kaczman. “Approximate solution of system of linear equations” (in German), Bdl.

[2] L. Ljung. “Analysis of recursive stochastic algorithms,’‘ IEEE Tram. Automat. Conlr., Inl. Acud. Polon. So’. CI.. Sci. Math. Nat. Ser. A, 1937.

[3] J. M. Mendel and K S. Fu. Adaptioe Learning and Pattern Recognition Sprem. New vol. AC-22. Aug. 1977.

141 G. N . Saridis and G. Stein, “Stochastic approximation algorithms for linear discrete- York: Academic, 1970.

time system identification,” IEEE Tram. Automat. Conlr.. vol. AC-13, Oct. 1968.

On the Stability of Dynamic Models Obtained by Least-Squares Identification

TORSTEN SODERSTROM AND PETRE STOICA

Abstruct-The stability properties of the least-squares (LS) models fitted to stochastic dynamic systems described by difference equations are investigated. It is shown through a counterexample that the LS method can, for such systems, give unstable models. Sufficient conditions for the stability of the LS models in question are also included.

An example illustrating the identification of the static system is shown in Fig. 1. A uniform probability distribution was assumed for e,.

An example illustrating the identification of the dynamic system is The stability properties of models obtained by least-squares (LS) identi- shown in Fig. 2. Because dr=[O.O,O, 11, then DrD=O and d’=O. This fication have recently received a certain attention, see, e& [I]-[3], [5], [6]. simplifies the algorithm (8) and the formula (9). A uniform probability The main reason for this attention is that the asymptotic stability of LS distribution was assumed for e, and a,. models is an essential condition in many applications.

I. INTRODUCTION

CONCLUDING REMARKS Manuscript received September 24, 1980.

Improved convergence is obtained at a not too great expense of T. Werstrbm is with the Department of Automatic Control and Systems Analysis,

burden ( n f 3 inner products for coefficient (9) P. Stoica is with the Facultatea de Automatica. Institutul Politehnic Bucuresti, Splaiul Institute of Technology. Uppsala University. P. 0. Box 534, S-751 21, Uppsala. Sweden.

at each iteration). lndependentei 313, Bucharest. Romania

0018-9286/81/0400-0575$00.75 01981 IEEE