Discrete adaptive regulation of not-necessarilyminimum-phase systems

C. Samson, Ing, E.S.E., Dr.-lng., and J-J. Fuchs, Ing. E.S.£., M.Sc, M.I.T., Dr.-lng., Mem.I.E.E.E.

Indexing terms: Algorithms, Adaptive control, Modelling

Abstract: A systematic procedure to adaptively regulate any SISO linear system is proposed. It is onlyrequired to know an upper bound of the system's order. The philosophy of the method is based on anextension of the indirect adaptive control approach. Any reasonable identification algorithm fits within theproposed scheme and a lemma states the conditions to be fulfilled by the regulation algorithm. One suchalgorithm is presented and simulations show this adaptive regulator - although applicable to a broader classof systems - to be as efficient as the existing ones in their own fields of applications.

1 Introduction

In this paper, the problem of the adaptive regulation of a not-necessarily minimum-phase system is considered. Considerableinterest has recently led to major contributions in the adaptivecontrol field. In the discrete, deterministic case the problem isnow fully solved [1—4] under the minimum-phase assumptionon the system. Several authors have considered the moregeneral problem of controlling nonminimum-phase systems[5—9]. The proposed approaches are mainly based on adaptivepole-placement schemes, but to our knowledge no completesolution has yet been obtained.

To control nonminimum-phase systems in a stable manner,simple methods must be discarded, and even in the knownparameter case a polynomial identity or a Riccati equationmust be solved. These complexities seem to forbid directadaptive control for nonminimum-phase systems.

We consider here SISO, discrete time, deterministic linearsystems, and it is important to realise that discretising acontinuous model often leads to a nonminimum-phase discreterepresentation even though the actual continuous system isminimum phase, and that, moreover, the nature of theobtained discrete model depends on the sampling time.

The proposed approach is based on the following verysimple idea which stems from the usual indirect controlscheme. Given a system of known structure, use an identifi-cation procedure which allows you to obtain a strictly parallelmodel of the system and ensures the convergence of the errorbetween the system's and model's output to go to zero for anyinput sequence, then one is just left with the problem of con-trolling the output of the known, but time-varying, parallelmodel, since the identification algorithm guarantees thesystem's output to follow the model's output.

This idea is certainly not new, but two major difficulties areyet to be removed:

(i) how to find an identification procedure with theannounced property

(ii) how to simply control or regulate a known but time-varying system, the identified model, the future variations ofwhich are unknown.As will be shown, the first difficulty can easily be removed fora given model structure, i.e. instead of looking for an adequateidentification algorithm, we propose a model structure whichcan be seen as parallel and for which all the current identifi-cation procedures satisfy the required property. Finally wepresent a control algorithm which allows us to regulate thetime-varying system constituted by the model of the system.

Paper 1307D, first received 30th November 1980 and in revised form10th February 1981The authors are with the Institut de Recherche en Informatique etSystemes Aleatoires, Laboratoire d'Automatique, Campus de Beaulieu,35042 Rennes Cedex, France

2 Statement of problem

2.1 Process modelOur basic assumption is that the process to be regulated can bemodelled by a linear discrete time transfer function:

= Q ut (1)

where we assume an upper bound n of the degrees of the Aand B polynomials to be known and the process to be stabil-isable.

Notice that we do not require the exact number of delaysto be known, the polynomials to be coprime, nor the systemto be minimum-phase.

Eqn. 1 can also be written:

yt =

...-anbi . . .

.ut.t.n(2)

(3)

or, in observable-canonical state form,

Xt+l = AXt+BUt

yt = CXt

v/ithA=S + KC

BT = [bx ...bn]

KT = [-at . . . - « „ ]

C = [1 0 . . .0]

and

S =

The adaptative regulation problem is to generate a sequence{ut} of bounded inputs such that yt-* 0 as t -•«», without apriori knowledge of the parameters of the system.

2.2 Phi/osphy of approachUsing an indirect approach to the adaptive regulation problem,we identify the process parameters and obtain a sequence ofestimates 6t. Let us assume that the identification algorithmallows us to build a strictly parallel model of the system withinput ut and output zt and guarantees that the output error

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e, =yt —zt converges to zero for any control sequence {ut}applied to the system and the model.

The regulation of the system will thus be achieved if we areable to generate a bounded control sequence {ut} whichregulates the time-varying model. While the same sequence ofinputs will, of course, be applied to the process, the analysis ofthe regulation can be done as if the model were perfectlyautonomous.

We now face two major difficulties:(i) how to find an identification procedure with the

announced property(ii) how to simply regulate a time-varying system: the

parallel model.Let us comment about these difficulties. Control of a time-varying system in a simple and stable way is not a trivial task.Moreover, and due to the context, the parallel model isdefined recursively in time; its evolution is not known inadvance. To be able to regulate it we shall require some kindof smoothness in its variations, besides requiring it to be

process

time-varying model

(i

Zt

Fig. 1 Block diagram

uniformly stabilisable at least. While smoothness in its vari-ations can be translated into constraints on the identificationalgorithm, uniform stabilisability or controllability of theparallel model obviously belongs to a different range ofpreoccupations, and will remain a mere wish.

Thus, it already appears that, besides our basic assumptionon the identification procedure (convergence of the 'output'error to zero), we shall impose other requirements on it.

These additional constraints will be removed as well as diffi-culty (i) above, by using a somewhat special model, i.e. insteadof looking for an identification procedure which fits into ourframework we shall propose in the next Section a modelstructure which, though not strictly parallel, will allow us toeasily obtain identification algorithms satisfying the requiredconstraints.

2.3 'Parallel' model: an adaptive observer [ 12—13]Based on the observable canonical form of the system, weconsider the following 'parallel' model in state-space form(see eqn. 3):

Xt+1 = AtXt+Btut+Kt(yt-yt)

yt = CXt (4)

with

ut — [Kt Bt J

At = S + KtC

It can be seen as an adaptive observer of the system, has theadvantage of being immediately deducable from any sequenceof estimates {6t} and is thus fully independent of the identifi-cation algorithm itself. It can also be defined by

XU1 = SXt+Btut+Ktyt

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where it appears that it is asymptotically stable, and that

9t = 0?-i0f-i

This equality is not strictly rigorous, but property P3,defined below, allows us to take easily into account the errorso introduced.

Thus

(5)

with

0^0,-0

This quantity is precisely the 'output' error — the errorbetween the system's and model's output — and from anidentification point of view it appears to be the a prioriprediction error. It also appears in eqn. 4 as an additionalinput — besides ut — to our model, so that strictly speakingthe observer is not a parallel model. However, since we requirethe output error to converge to zero we shall be able to handlethis divergence from the originally proposed scheme.

3 Identification algorithm

Owing to our choice of an adaptive observer as a parallelmodel, we can use any identification algorithm. We havealready indicated that besides the basic requirement (conver-gence of the output error to zero) we shall have to imposeother constraints on the identification scheme, mainly allow-ing us to regulate the time-varying parallel model.

Let us forget this preoccupation for a while and see whatkind of properties we can expect from an identificationalgorithm operating on an adaptively controlled system. Thishas been an active domain of research recently mainly in theadaptive control literature f 1 —4, 10].

Remember that the system to be identified can be describedby eqn. 2:

yt == 0 (6)

Thus the identification algorithm will be of the following form[1-41:

ut — Vt-i + Gt-iVt-iCt K')

where et is given by eqn. 5, and Gt is a 'gain' to be defined.This is the usual form for prediction error identification

methods, and Lyapunov's theory allows us to establish thefollowing properties:

Pl:\et\ <

lima, = lim/3, = 0 as t^-°°\

a.t and |3( uniformly bounded

P2:\\6t\\ < M V t

P3. lim 110,-0,-j || = 0

These results hold for constant gain identification algorithms[1-3] , as well as for decreasing gain algorithms [2— 3] .

Since these results are now well known, the proofs will notbe given here.

Property PI does not meet our basic requirement (e, -*• 0);however, we shall see how to handle this further divergencefrom the original scheme, so that all reasonable identificationalgorithms will be suitable for our purpose.

Notice that we do not require the estimates to converge tothe true parameters; this is sometimes required in indirectadaptive control schemes [13].

103

Let us briefly comment about P2 and P3 seen from thepoint of view of the time-varying model to be regulated. P2guarantees that the matrices appearing in the state equationsare uniformly bounded. P3 ensures that the speed of variationof the model converges to zero; this will be of great help in thesequel.

4 Main lemma

We are now ready to formalise the already implicity statedlemma: 'any way to stabilise the strictly parallel time-varyingmodel solves the adaptive regulator problem'. Let us specialiseit for

(i) the adaptive observer(ii) a time-varying state-feedback control law on the ob-

server.

LemmaGiven

(a) the system defined by eqn. 1(b) an identification algorithm satisfy ing PI -P3(c) the adaptive observer defined by eqn. 4

then the control

ut = -LjXt (8)

with {Lt} a uniformly bounded sequence of feedback gainsapplied to both system and observer, ensures boundedness ofall signals and convergence of the system's output to zero, ifthe following system:

zt+l = (At-BtLj)zt (9)

is exponentially stable.The proof of this lemma is given in Appendix 9.1.We have specialised the lemma for a state-feedback control

law; however, any stable way to obtain an exponentially stablecontrolled observer would solve the problem. Besides being anatural way to regulate a system with known state vector, thisapproach allows us to control in a stable manner a system, theadaptive observer, which might be nonminimum-phase. Sincewe do not require the estimates to converge to the true value— nor to a constant vector — the observer can be nonminimum-phase at some steps even if the constant system to be regulatedis minimum-phase.

5 Regulation algorithm

Having started from a quite general scheme we have beenobliged to choose a very special 'parallel' model: an adaptiveobserver written in observable canonical form. This, in turn,led us to specialise the lemma to a control law in state-feedbackform. As mentioned, any other means to obtain an exponen-tially stable transition matrix for the controlled observer aresuitable.

However, one should keep in mind that the observer isdefined recursively in time, i.e. the future variations of itsevolution are unknown, so that no classical regulation method,e.g. optimal control, dead-beat control etc., is appropriate.With the help of property P3, one can imagine that some ofthem can be adapted; for instance, pole placement at eachstep, or asymptotically in a sense to be defined. We shall giveone such way in the following Sections.

Notice that, in any case, an 'assumption' will be requiredon the sequences {At, Bt] — uniform stabilisability, forinstance. No matter what this 'assumption' might be, it willremain as a lack in the proof around which only heuristicarguments can be developed. Thus, it appears that not toomuch effort should be devoted to weaken this assumptionfurther than to a given point; one might define this as follows:

the assumed property must almost always be satisfied (mustbe generic).

We shall come back to this point in relation to the proposedmethod.

5.1 Brief review of Kalman filter propertiesLet us consider the following system:

Xt+1 = AjXt + CTvt

yt = BjXt + wt

where {vt} and {wt} are independent white noise sequenceswith variances 1 and X, respectively.

The equations of the associated Kalman filter are

Xt+l = AfXt+Lt(yt-BjXt)

RtBtBfRtt+i = A; \ R t -

+ B}RtB

=AjRtBt

+ BfRtBt

(10)

(11)

If the pair {Aj, CT) is uniformly stabilisable (ST)' [14] andthe pair {Bj ,Aj} uniformly detectable (DT)' it is shown [14]that the system

zt+i = (Aj -LtBj)zt (12)

is exponentially stable and {Rt} (and thus {Lt}) uniformlybounded. For completeness we give, in Appendix 9.2, thedefinitions of uniform stabilisability and detectability used inReference 14.

5.2 Connection with our problemUnder the announced assumptions on {Aj, CT, Bj}, we havethus given a means to generate a bounded sequence {Lt}ensuring exponential stability for the system. Remember thatwe are interested in exponential stability of the system definedin eqn.9:

zt+i = (At —BtLt )zt

which is quite different (matrix products do not commute, ingeneral!). However, we show in Appendix 9.3 that, under P2and P3, exponential stability of eqn. 12 implies exponentialstability of eqn. 9, which is what we are looking for.

5.3 Proposed regulation algorithmWe thus propose the following regulation algorithm (see eqns.10 and 11):

Lt - \+BjRtBt

+ I - At \Rt

T In RtBtBtRl

RQ > o

The control to be applied to both observer and system is

ut = — Lt Xt(13)

where Xt is the state vector of the observer, eqn. 4. This state-feedback control law satisfies the conditions of the lemma andthus solves the adaptive regulator problem under the followingassumption:

Al: {Bf, Aj} is uniformly detectable (it is trivial to verifythat {Aj, CT} is uniformly stabilisable)

104 IEEPROC, Vol. 128, Pt. D, No. 3, MAY 1981

If the identification procedure is such that the estimatesconverge to a constant vector 6,*, (recusive least-squarealgorithm, for instance) the previous 'assumption' can bechanged into {Aoc, £«} stabilisable [10, 11]. It is shown inReference 15 that for identification algorithms satisfying P2and P3, the 'assumption' can be changed into

Al': {At, Bt} uniformly stabilisable (which might seemmore appropriate to our problem)

5.4 Validity of 'assumptions'Thus we are left with the assumption onAl, which clearly is alack in the proof. Owing to the iteration of the 'Riccati'equation and the 'mixing' effect of this iteration, a pointwhere Al would not be satisfied is certainly not a stable'stationary' point, so that the assumed property is certainlygeneric. Moreover, since the estimates become better andbetter, and since we assume the (constant) system to bestabilisable, it seems that the proposed approach should alwayswork.

Notice that this type of assumption on the identifiedsystem has also been considered in Reference 12.

5.5 Possible interpretationLet us just mention that if the estimates converge to a constantvector 0oo which is such that the asymptotically constantobserver is stabilisable, the iteration on Rt will converge to aconstant matrix solution of the stationary Riccati equationassociated with 0« [10,11,15].

R = CTC + Al \ R -

The order (« = 1) of the system is assumed to be known; theadaptive observer given by eqn. 4 is of the form

A i Boo

which is also the equation associated with the followingoptimal quadratic control problem:

(14){ut\ t=o

for the system

,ut

Moreover, if #«, is equal to the true system's parameters 8,eqn. 2, the proposed adaptive control converges to the optimalcontrol of the system, eqn. 1, for the quadratic cost definedby eqn-. 14.

6 Experimental results

We consider two simple systems. In both cases we build theadaptive observer from estimates given by the recursive least-square identification algorithm and apply the proposed regu-lation algorithm given by eqns. 10,11 and 13. We compare theobtained results, denoted COMAD, with those obtained usingthe self-tuning regulator [16], denoted STURE.

Finally we present results obtained for a time-varyingsystem.

t+ibltut+au(yt-yt)

where au, bu are the RLS estimates with initial covariancematrix Po equal to identity anda10 = 0,blo = 1.

Further initial conditions are taken as follows:

x0 = 0, Ro = I, X = 1, ^o = l

Figs. 2a and b depict input ut and output yt of the system forboth STURE and COMAD.

, COMAD

' ' 2 b *^x

COMAD

STURE

Fig. 2 Minimum-phase system

a Evolution of outputb Evolution of input

For STURE, the control is taken equal to

1ut = -—auyt

bit

The evolutions are similar; for smaller X (see eqn. 14) COMADbecomes closer to STURE.

6.2 Nonminimum-phase systemWe consider the system

yt+l = l.2yt + 0.5ut +ut.l

The system is of order n = 2. Using previously given notations,the initial conditions are:

for the system: j>0 = "o = u-i = 0

6.1 Minimum-phase systemThe simulated system to be regulated is

yt+l = l.2yt + 0.5ut

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for the RLS identification algorithm: Po = I,

dl = [0 1 0]

for the regulation algorithm: Ro =1, \ = 1

105

Figs. 3a and b present the results. As expected, the input givenby STURE diverges rapidly.

6.3 Time-varying systemThe following time-varying system is simulated:

yt+1 = 1.2yt

bt+i = bt-0.0l,

y0 = 1

bo = 0.5

Initialisations are as in Section 6.1, above. The RLS identifi-cation algorithm is modified in order to be able to followtime-varying parameters by adding a constant diagonal matix(Q = 0.01 / ) in the usual recursion:

i! + QPt =

In Fig. 4a the evolution of the input ut and output yt of theadaptively controlled system is depicted. Fig. 4b presents theevolution of the parameter estimates.

2ry,

STURE

-31

Fig. 3 Nonminimum-phase systema Evolution of outputb Evolution of input

7 Conclusions: further extensions

We have presented an approach allowing us to adaptivelyregulate a SISO linear system without assuming it to beminimum-phase. In fact, we have formalised a well-knownphilosophy, which led us to a lemma stating that any way to(exponentially) stabilise a time-varying system, the future ofwhich is unknown but satisfies smoothness conditions, solvesthe adaptive regulation problem. An algorithm is proposedwhich solves this nontrivial problem under a 'minor' hypoth-esis. Experimental results are presented. It appears that theperformances of the proposed approach are similar to thoseobtained by known adaptive regulation schemes for minimum-phase systems, but are still good for nonminimum-phasesystems for which the other schemes diverge.

We have developed the analysis around the regulationproblem; it is clear that the approach can be readily extendedto the adaptive control problem, i.e. the servoproblem wherethe control object is to make the plant output follow a givenreference signal. We shall not develop this possible extensionhere. Examples can be found in Reference 15.

Heuristically all adaptive control schemes can be applied toslowly time-varying systems; however, it is our feeling thatthe proposed approach is, by its philosophy and especially forthe chosen adaptive regulation algorithm, better fitted for suchapplications. Experimental results are proposed in References10 and 15. One such application is presented in this paper.

Finally, it might be possible to handle stochastic systemsusing this approach; work in this direction is under progress.

-5L a

-0-5

Fig. 4 Time-varying system

a Evolution of output yf and input ut

b True and estimated parameters evolution

8 References

1 GOODWIN, G.C., RAMADGE, P.J., and CAINES, P.E.: 'Discretetime multivariable adaptive control', IEEE Trans., 1980, AC-25,pp. 449-456

2 EGARDT, B.: 'Stability analysis of discrete time adaptive controlschemes', ibid., 1980, AC-25, pp. 710-717

3 FUCHS, J.J.: 'Discrete adaptive control: a sufficient condition forstability and applications', ibid., 1980, AC-25, pp. 940-946

4 NARENDRA, K.S., and LIN, Y.H.: 'Stable discrete adaptive con-trol', ibid., 1980, AC-25, pp. 456-461

5 JEANNEAU, J.L., and DE LARMINAT, P.: 'Une methode deregulation pour les systemes a phase non minimale'. 3e congresonational, 1975, Informatica y Automatica, Madrid

6 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning controller',Proc. IEE, 1975, 122, (9), pp. 929-934

7ASTROM, K.J., WESTERBERG, B., and WITTENMARK, B.: 'Selftuning controllers based on pole-placement design'. Lund reportLUTFD/1-052/, 1978

8 WELLSTEAD, P.E., PRAGER, D., and ZANKER, P.: 'Pole assign-ment self-tuning regulator', Proc. IEE, 1979, 126, (8), pp. 781-787

9 CLARKE, D.W., and GAWTHROP, P.J.: 'Self-tuning control', ibid.,1979, 126,(6), pp. 633-640

10 FUCHS, J.J., and SAMSON, C: 'Methodes de commandes adapt-atives de systemes lineaires', IRISA, 1979, Rapport final contratDRET 77/545

11 SAMSON, C: 'Regulation adaptative des systemes a non-minimumde phase', Revue due CETHEDEC, 1980, 62, pp. 1-30

12 MORSE, A.S.: 'Global stability of parameter-adaptive controlsystems', IEEE Trans., 1980, AC-25, (3), pp. 433-440

106 IEE PROC, Vol. 128, Pt. D, No. 3, MA Y1981

13 KRAFT, L.G., III: 'A control structure using an adaptive observer',ibid., 1979, AC-24, (5), pp. 804-806

14 HAGER, W.W., and HOROWITZ, L.L.: 'Convergence and stabilityproperties of the discrete Riccati operator equation', SIAM J.Control & Optimiz., 1976,14, pp. 295-312

15 SAMSON, C: 'Commande adaptative a critere quadratique dessystemes lineaiies a minimum de phase ou non'. Univ. de Rennes,1980, These de Docteur-Ingenieur

16 ASTROM, K.J., BORISSON, U., LJUNG, L., and WITTENMARK,B.: 'Theory and applications of self-tuning regulators', Automatica,1977, 13, pp. 467-476

9 Appendixes

9.1 Pro of of lemmaUsing eqns. 4 and 5, one has

Xt+1 = AtXt+Btut+Ktet

Thus, with the input defined in Section 3.1,

Xt+l = (At-BtLj)Xt-Ktet (15)

Now, remembering the definition of (j)t in eqn. 2, we shallexplicit the evolution equation of 4>t with the help of thefollowing relations (eqns. 8,4 and 5, respectively):

ut = -LfXt

yt = y

h = cxt

Eqn. 5 =>

yt = CXt-et

Thus,

4>t = F fc . , +DtXt+Vt (16)

with F a (2n, 2ri) constant exponentially stable matrix, Dt a(2n, rt)-matrix and Vt a (2n, 1)-vector equal to

F =

0 0

4-!Dt =

C

0

V, =

0

Now define the (3n, 1)-vector Zt:

yT _ r iT vT l

z*+i - [<t>tXt+i\

then, by eqns. 15 and 16,

Zt+l = il,(t+l,t)Zt + Wt

(17)

(18)

transition matrix notiations:

\\\jj(t + n,t)\\ < M{exp(-cn) V

Mx > 0, c > 0

which implies that

W(t + n,t)\\ <MX Yt,n > 0

and V 5 > 0 3 p finite such that

p,t)\\ < 8Yt

(19)

(20)

Consider now the vector Wt: since by property P2, {Kt} isuniformly bounded, and by PI

with at and |3( uniformly bounded positive reals and lilim j3f = 0, we have

\\Wt\\ < atUt^W+ti

Eqn. 17=»

\\Wt\\ < a't\\Zt\\+& (21)

with

lima^ = lim& = 0

and there exists M2 such that

\a't\ <M2, | ^ | < M2 V t

Take now 5 strictly smaller than 1 in eqn. 20; for the corre-sponding finite and fixed integer p, one establishes, usingeqns. 18, 19 and 21, that, for/G [0,p], there exists finiteM3,M4 such that

\\Zt+i\\ <M3\\Zt\\+M4

Now, from eqn. 18,

Zt+P = \l>(t+p,t)Zt +

(22)

i=t

Taking norms, and using eqns. 19, 20

\\Zt+p\\ < 5 | |Z t | |+max(l , i l f f)

Eqn. 21

Eqn.22

+ M4a't+i-1 (23)

with i | / ( ? + U ) a ( 3 n , 3w)-block-upper-triangular matrix and Since lim a't = 0, there exists N such thatWt a(3«, 1)-vector:

F Dt

0 At-BtLj

a', < e Yt > N

where e can be chosen to satisfy

5 +pM^M3e < 1

By hypothesis, {Lt} and thus {Dt} is uniformly bounded. One then concludes from eqn. 23 that \\Zt\\Then, since the matrices of \p(t + 1 , 0 are both exponentially bounded and lim \\Zt\\ = 0. Hence limj^ = 0 andstable, \j/(t + 1, t) is exponentially stable, i.e. with the usual uniformly bounded.

is uniformlyall signals are

IEEPROC, Vol. 128, Pt. D, No. 3, MA Y 1981 107

9.2 Definitions of uniform stabilisability and detectability[14]

Consider the time-vary ing linear system:

Xt+1 = AtXt +Btut

yt = ctxt

It is said to beuniformly stabilisable if there exist an integer r > 1, a con-stant q and a sequence {Lt} of uniformly bounded vectorssuch that

fc+r-lU(At-BtLj)\ < q < 1t=k

uniformly detectable if there exist integers s, k > 0 andconstants 0<d<l, 0<b<°° such that, for all t>0,whenever

\\<P(t + k,t)X\\ > d\\X\\

then

bXTX

9.3 PropositionIf the system

Xt+i = AtXt

is exponentially stable, and

\\m\\At-At-i\\ = 0

then the system

Z,+ 1 = AjZt

is exponentially stable.

ProofLet M be such that

\\At\\ <M Vt

(24)

Consider 5 strictly smaller than 1; by hypothesis there existsfinite p such that

thus

\\Aj^,...,AuP\\ < 5 Vr

We shall establish the existence of TV such that

\\AlP,...,A] < 5' < 1 Vt > N (26)

by eqn. 25, the proposition is then established.Define A(s, t) by

Aj =

By eqn. 24, for fixed /,

lim A(t + i,t) = 0t

(27)

Now,

[Aj=2 +A(t+p-l,t

AJ+1AJ+2

(28)

where for fixed, finite, p, 2 is the sum of a finite number ofterms having each p factors, at least one of which is aA(*, •). Thus, using eqns. 25 and 27 one easily verifies that

lim ||X (f + 1 ,t + p) || = 0 as f -> °°

Choose, then, TV such that

where e satisfies

5 +e < 1

(25) Taking norms in eqn. 25 then establishes eqn. 26.

108 IEEPROC, Vol. 128, Pt. D, No. 3, MA Y 1981