Asymptotically Optimal Smoothing of Stochastic Approximation Estimates for Regression Parameter Tracking

Stochastic Approximation Estimates for

Regression Parameter Tracking

Alexander V. Nazin and Lennart Ljung

Department of ElectricalEngineering

Linkoping University, SE-581 83Linkoping, Sweden

WWW: http://www.control.isy.l iu. se

Email: ljung@isy.liu.se

October2, 2001

REG

LERTEKNIK

AUTO

_{MATIC CONTR}

OL

LINKÖPING

Report no.: LiTH-ISY-R-2360

Stochastic Approximation Estimates

for Regression Parameter Tracking  Alexander V. Nazin, Institute of Control Sciences, Profsoyuznaya str., 65, 117997 Moscow, Russia nazine@ipu.rssi.ru LennartLjung, Departmentof Electrical Engineering,

Linkoping University,

S-581 83 Linkoping,

Sweden

ljung@isy.liu.se

April25, 2001

Abstract

ThesequenceofestimatesformedbytheLMSalgorithmforastandard

linear regression estimation problem are considered. It is known since

earlier that smoothing these estimates by simpleaveraging will lead to,

asymptotically,the recursiveleastsquares algorithm. Inthis paperit is

rstshownthat smoothing the LMSestimates usingamatrixupdating

willlead tosmoothedestimateswithoptimaltrackingproperties,alsoin

thecasethetrueparametersarechangingasarandomwalk. Thechoice

of smoothing matrixshould betailored tothe properties ofthe random

walk.Second,itisshownthatthesameaccuracycanbeobtainedalsofor

asimpliedalgorithm,SLAMS,whichisbasedonaveragesandrequires

muchlesscomputations.

1 Introduction

Trackingof time varying parametersis abasicproblem in many applications,

andthereis aconsiderableliteratureonthisproblem. See,amongmany

refer-ences,e.g. [12],[6],[5].

One of the most common methods is the least mean squares algorithm,

LMS, ([12]) which is a simple gradient based stochastic approximation (SA)



TheworkoftherstauthorhasbeencarriedoutwhilevisitingLinkopingUniversityas

theaccuracycouldinfactbequitebad. Itiswellknownthat forsuchsystems,

the recursiveleast squares (RLS) algorithm is optimal, but it is on the other

handconsiderablymorecomplex. Averyniceobservation,independentlymade

byB.T.PolyakandD.Ruppert,isthatthisoptimalaccuracycanasymptotically

alsobeobtainedbyasimpleaveragingoftheLMS-estimates. See[10],[11],and

[9][1]fortheanalysis.

In [3] it is shown that this asymptotic convergence of the averaged

LMS-algorithm to theRLS algorithm is obtainedalso for thetrackingcase, with a

moving true system and constant gain algorithms. This means that in

gen-eral theaveragedalgorithm will notgive optimal accuracy. Optimal tracking

propertiesthenwill beobtainedbyaKalman-lterbasedalgorithmwhere the

updatedirection is carefullytailoredto theregressor properties, thecharacter

ofthechangesinthetrueparametervectorandthenoiselevel.

Inthispaperweshallconsideramoregeneralpost-processingofthe

LMS-estimates, obtained from a constant gain, unnormalized LMS-method. The

generalversionof this algorithm we callSLAMS { Smoothed Averaged LMS

(allowing a metathesis for pronouncability). It consists of rst forming the

standardLMS-estimates ^

(t),andthenformingsimpleaveragesofthese

~ (t)= 1 m t X t m ^ (k)

and nally smoothingthese by asimple exponentialsmoother, applyinga

di-rectioncorrectioneverym:thsample:

 (t)= (  (t m) S(   (t m) e (t)); t=km;k=1;2;:::  (t 1) (  (t 1) ^ (t 1)); t6=km;k=1;2;:::

Thisalgorithmhasthedesignvariables(thegainoftheLMSalgorithm),S;m

and . By,for example,choosing masthedimensionof theaveragenumber

ofoperationsperupdate in the SLAMSalgorithm is still proportionalto dim

,just asin thesimpleLMSalgorithm.

The main result of this paperis to establishan asymptotic expression for

thecovariancematrixof thetrackingerror 

(t) (t) ((t) beingthetrue

pa-rametervalue). WeshowthatbythechoiceofSand wecanobtainthesame

asymptoticcovarianceas theoptimalKalmanlter gives,regardlessof mand

(aslongasithasacertainsize relationto ).

InSection2weformulatethetrackingproblemandstatethebasic

assump-tions. InSections3and4wetreataspecialcaseofSLAMS,viz. withmxed

to1. Theextensiontothegeneralalgorithm isdonein Section5.

2 Problem Statement and Basic Assumptions

Consideradiscrete-timelinearregressionmodelwithtime-varyingparameters.

y(t) =  T

(t)'(t)+e(t) (1)

(t) = (t 1)+w(t) (2)

wheree(t) and w(t)stand forobservation errorand parameterchange

respec-tively. Due to the followingassumptions, the equation(2) describesevolution

ofslowlydriftingparameter(t)2R n

asarandomwalk.

BasicAssumptions:

A1 Thesequencesfe(t)g, fw(t)g and f'(t)garei.i.d. mutuallyindependent

sequencesofrandomvariables.

A2 The observation error e(t) is unbiased and hasa nite variance, that is

Ee(t)=0andEe 2 (t)= 2 e 2(0;1).

A3 Theparameterchangew(t)isunbiasedvariablewithpositivedenite

co-variance matrix, i.e. Ew(t) = 0 and Ew(t)w T (t) = 2 R w > 0, where

>0representsmall parameteroftheproblemunderconsideration.

A4 Theregressorcovariancematrixisnon-singular,i.e. E'(t)' T

(t)=Q>0;

moreover,Ej'(t)j 4

<1.

A5 Theinitial parameter value(0) is supposed to bexed (for thesakeof

simplicity).

Considertheparametertrackingproblemwiththeperformanceevaluatedas

theasymptoticmeansquare error(MSE).That is,theproblemisto designan

estimationalgorithmwhichon-linedeliversanestimatesequencef 

(t)g onthe

basisofobservations(1)withaminimalasymptoticerrorcovariancematrixU:

U = lim t!1 U t (3) and U t =E(  (t) (t))(  (t) (t)) T (4)

As is known (see, e.g. [8] and the lower bound below), matrix U must be

proportionaltothesmallparameter ,thatis

U = U

0

+o( ) as !0: (5)

Moreover,fromthelowerboundformatrixU

0

provedin[8]follows,inparticular,

thatwithGaussianrandomvariablese(t)andw(t)

U

0 U

lb

(6)

foranyparameterestimatorwhere U

lb

isasymmetricsolutiontotheequation

asfollows U lb QU lb = 2 e R w (7)

U lb = e Q 1=2 (Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 (8) By(6)ismeantthatU 0 U lb

isapositivesemidenitematrix.

Wefurther call matrixU

0

(5) limitingasymptoticerror covariance matrix,

since U 0 = lim !0 1 U: (9)

Below we study how the matrix U

0

depends on the parameters both of the

problemandthealgorithmdescribedinthefollowingsection. Wethenminimize

thismatrixoverthedesignparametersin thealgorithm.

3 Parameter Tracking Algorithm

We aim at studying the following recursive constant gain SA-like procedure,

whichwecall SLMS,(SmoothedLMS):

^ (t) = ^ (t 1) + '(t)(y(t) ' T (t) ^ (t 1)) (10)  (t) =  (t 1) S(  (t 1) ^ (t 1)) (11)

Here >0is ascalarstepsize whileS representsannn-matrixgain. The

relation (10)is exactly the constant(scalar) gainSA-algorithm (LMS), while

recursiveprocedure(11)generatesasequenceofsmoothedSAestimates.

Specialinterestmightbeconnectedtotheparticularcaseofa\scalarmatrix"

SwhenS=I

n

withascalarstepsize>0andidentitynn-matrixI

n (see

subsection 4.1 below). In that case there are no matrix calculations in the

algorithm(10),(11),whichmakesitparticularlysimple.

Assumptionson theAlgorithm Parameters:

B1 =o(1)as !0

B2 =o()as !0

B3 The matrix ( S) is stable, i.e., the real part of any eigenvalue of S is

positive.

Note: Dueto assumptionsB1{B2,stochasticstabilityofequations(10),(11)

(in mean-square sense) is obviously ensured (for suÆciently small ). This

impliestheexistenceoflimitin (3)andfurther.

4 Main Results

Theorem 1 Let the assumptions A1{A5and B1{B3 hold, and consider the

estimates 

(t)generatedbythe algorithm (10), (11). Then the limiting

asymp-toticerrorcovariancematrix U

0

,denedby(5), isthe solutiontothe equation

SU 0 +U 0 S T =R w + 2 e SQ 1 S T : (12)

0

Hence,if( S)isstablethenauniquesolutionU

0 =U

0

(S)to(12)existswhich

is symmetric and positive denite. Furthermore, the relationship (12) might

be consideredasan algebraicRiccati equationwith respect to S. Due to well

knownpropertiesofRiccatiequation(see,e.g. [4],Lemma5.1,p.127)wearrive

atthefollowing

Corrolary: IfthematrixgainS issubjecttoassumptionB3thenthesolution

U

0

(S)to(12)hasthefollowinglowerbound

U 0 (S)U min = e Q 1=2 (Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 (13)

whichcoincides withU

lb

(8)andisattainedforS=S

opt with S opt = 2 e U min Q= 1 e Q 1=2 (Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 (14)

ThecorrolaryaboveisaspecialcaseofLemma5.1in[4]. However,itcanbe

easilyprovedindependently. Indeed,from(12)andanevidentmatrixinequality

( 2 e SQ 1=2 U 0 Q 1=2 )( 2 e SQ 1=2 U 0 Q 1=2 ) T 0

itdierctlyfollowsthat

U 0 QU 0  2 e R w

whereequalityisattainedfor

 2 e SQ 1=2 =U 0 Q 1=2 (15)

Consequently(13)and(14)hold true.

Note: Since bothU

min

and Qare positivedenite matrices, then itsproduct

U

min

Qhasonlyeigenvalueswithpositiverealparts. Hence,theoptimalmatrix

gainS

opt

(14)meetsthestabilityassumptionabove.

4.1 Scalar Smoothing Gain

Now consider the special case of \scalar matrix" gain S = I

n , > 0. Then equation(12)impliesU 0 =U 0 ()with U 0 ()= 1 2  1 R w + 2 e Q 1
: (16)

Hence,theoptimalinasense ofminimaltraceTrU

0 isasfollows  opt = 1 e  TrR w TrQ 1  1=2 (17)

andtheminimumtraceoflimitingasymptoticerrorcovariancematrix

TrU 0 ( opt )=min >0 TrU 0 ()= e ( TrR w ) 1=2 TrQ 1
1=2 (18)

min lb dependentmatricesR 1 w andQ,that is R 1 w =Q forsome2R; (19)

thetracescoincide

TrU 0 ( opt )=TrU lb (20)

which means that TrU

0

() attains its lower bound for  = 

opt

among all

possible estimators. The condition (19) is both necessary and suÆcient for

theequality(20). That followsdirectly from thewellknown properties of the

correspondingCauchy-Schwarzinequalityformatrixtraces[2],thatis

TrAB T
2  TrAA T
TrBB T
; (21)

withequalityiAandB arelinearlydependent. Theinequality(21)mightbe

appliedhereforA=Q 1=2 andB T =(Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 .

Note, nallythatunder condition(19)theoptimalmatrixgainS

opt (14)is reducedto S opt = 1  e p  I n

whichalsofollowsfrom (17).

Note: Ifthepropertiesoftheregressors'(t)canbechosenthen itispossible

to ensure the condition (19) { if the paramter variation R

w

is known { by

experiment design. Such an experiment would thus give oprimal parameter

trackingwiththesimplestpossiblealgorithm.

4.2 Proof of Theorem 1

Weprovean evenmoregeneraltheoremhavingitsown interest. The

general-izationconsistsinintroducingnon-singularmatrixgainA intoprocedure(10),

thatis ^ (t)= ^ (t 1)+A'(t)(y(t) ' T (t) ^ (t 1)) (22)

Hence,it will be provedthat the matrix U

0

(5) doesnot depend onA. This

result explains why only a scalar step size is enough for procedure (10), and

thatwearenotabletoin uence matrixU

0

byamatrixgainS in (22).

Proof:Lettheestimates ^

(t)begeneratedbythemoregeneralprocedure(22),

insteadof(10). LetthematrixgainAbenon-singularandassumethat( AQ)

isstable. Denotetherelatedestimationerrorcovariancematrixby

V t =E( ^ (t) (t))( ^ (t) (t)) T (23)

anditslimit

V = lim

t!1 V

t

AQV +VQA T = 2 e AQA T + 2  R w (25)

followsdirectlyfromwellknownpreviousresults(see, e.g. [5]). Then

assump-tionsB1,B2implytheLyapunovequation

AQV +VQA T = 2 e AQA T +o( ) (26)

fromwhatfollows,inparticular,that 1

kVk=O() as !0: (27)

Furthermore,forthecrosscovariancematrix

R t =E( ^ (t) (t))(  (t) (t)) T (28)

anditslimit

R= lim

t!1 R

t

(29)

weobtainfrom (22),(11)and(1),(2)that

R t =(I n AQ)  R t 1 (I n S)+ V t 1 S+ 2 R w  (30)

Lettingt!1andtakingassumptionsB1,B2intoaccount,weobtain

R=  1 Q 1 A 1 VS T +o( ): (31)

Inasimilarmanner,weevaluateU

t

(denedby(4))andU (denedby(3)):

U t = (I n S)U t 1 (I n S) T + 2 R w + 2 SV t 1 S T + (I n S)R T t 1 S T + SR t 1 (I n S) T

and,taking(27),(31)as wellasB1,B2intoaccount,wendthat

SU+US T = R w +R T S T +SR+ SUS T +o( ): (32)

Note, that (32)is aLyapunov-typeequationwith U entering linearly. Hence,

kUk=O( )as !0,andsubsituting (31)into(32)weobtain

SU+US T = R w +  1 SVA T Q 1 S T +SQ 1 A 1 VS T
+o( ) = R w +  1 S(AQ) 1 (AQV +VQA T )(AQ T ) 1 S T +o( )

Finally,using (26),wearriveat

SU+US T = R w +  2 e SQ 1 A 1 (AQA T +o(1))A T Q 1 S T +o( ) = R w +  2 e SQ 1 S T +o( ) (33)

Thus, thelimitmatrix U

0

dened by(5)meet the equation(12)anddoesnot

depend onA, andTheorem1isproved.

1

HereandfurtheronweusematrixnormkAk= TrAA T

1=2

whichcorrespondstoinner

producthA;Bi=TrAB T

Letusconsiderthefollowingmodicationoftheparametertrackingalgorithm

(10),(11). Itcontainsanaturalnumbermasaparameter.

^ (t) = ^ (t 1) +'(t)(y(t) ' T (t) ^ (t 1)) (34a) e (t) = 1 m t 1 X =t m ^ () (34b)  (t) = (  (t m) S(  (t m) e (t)); t=km;k=1;2;:::  (t 1) (  (t 1) ^ (t 1)); t6=km;k=1;2;::: (34c)

Sinceit isaSmoothingalgorithm basedandthe Averagedestimates from the

LMS procedure, we call it SLAMS. Evidently, this algorithm coincides with

(10), (11),when m =1. However, when m >1, the procedure (34a) {(34c)

takeslessarithmetic calculationspertimeunit beingcomparedto (10),(11).

Theorem2 Assumethatthe assumptions A1{A5andB1{B3 hold,and

con-diderthe estimates 

(t)generatedbythealgorithm (34a){(34c). Then forany

xednaturalnumber mthe asymptoticerrorcovariance matrix

U (m) = lim t!1 E(  (t) (t))(  (t) (t)) T (35)

isthe solutiontothe equation

SU (m) +U (m) S T =  mR w +  2 e m SQ 1 S T  +o( ) as !0: (36)

Hence, the lower boundtothe limitingasymptoticerrorcovariance matrix

U (m) 0 =lim !0 1 U (m) (37) coincides withU lb =U min

(see(8)and(13)), thatis

U (m) 0 (S)U min = e Q 1=2 (Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 (38)

Thus, the lower boundto U (m)

0

does not depend on m and isattainedfor S =

S (m) opt with S (m) opt =m 2 e U min Q=m 1 e Q 1=2 (Q 1=2 R w Q 1=2 ) 1=2 Q 1=2 : (39)

The proof of Theorem 2 is analogous to that of Theorem 1. Note, that

a comparison of the equation (36) with (33) shows that the modication of

thetrackingalgorithmsuggestedabovecorrespondsto asimultaneousm-times

increase in the drift covariance matrix R

w

and and m-times decrease in the

varianceof observation error 2

e

in therighthand side ofLyapunovequations

(12), (33). Since the optimal gain matrix (39) is balanced the in uence of

correspondent summands in the right hand side (36), this explains that the

Considerasubsequence of time instantst = km, k =1;2;::: and rst prove

theTheoremforthepartiallimit

U (m) = lim k !1 E(  (km) (km))(  (km) (km)) T : (40)

Theestimation error 

Æ(t)= 

(t) (t)fort=kmisrecursivelyrepresentedas

 Æ(t)=(I n S)  Æ(t m)+ S e Æ(t) (I n S) t 1 X =t m w(+1): (41)

Note,thattherstandthelastsummandsinther.h.s. of(41)areuncorrelated.

Furthermore, the correlation between the second and the last summands is

evaluatedasO( 2

)=o( 2

),sincebyCauchy-Schwarzinequality

kE e Æ(t)w T ()kkE e Æ(t) e Æ T (t)k
kEw()w T ()k=O( )=o( )

Hence,thecovariancematrix(40)meets theequation

SU (m) +U (m) S T =  mR w + e R T m S T +S e R m  +o( ) (42) where e R m = lim k !1 E e Æ(km)  Æ T (km m) (43)

WemovetrivialandratherbulkycalculationstotheAppendixwhichprovethat

e R m = lim k !1 E e Æ(km)  Æ T (km)+o( )= 2m  2 e Q 1 S T +o( ) (44)

Substituting(44)into(42),wearriveat (36)forthepartiallimitconsidered.

Now, consider subsequenceoftime instantst =km+1,k=1;2;:::; then

wehavefortheestimationerror 

Æ(t)thesimplierrecursive-likeequation

 Æ(t)=(1 )  Æ(t 1)+ ^ Æ(t 1) w(t); (45)

fromwhichfollowsthat

lim k !1 E  Æ(km+1)  Æ T (km+1)=U (m) +O( 2 ) (46) withU (m)

asdenedby(40);hence,thepartiallimit(46)alsomeettheequation

(36).

Continuingfurther thisniteinduction,weobtain

lim k !1 E  Æ(km+s)  Æ T (km+s)=U (m) +O( 2 ); s=1;2;:::;m 1 with U (m)

as dened by (40). This proves the equation (36) for any partial

limitsofthematrixsequencefU(t)g.

The rest of the Theorem is provedin completely the same manner as the

Fromtheobtainedresultsitfollowsthattheoptimallimitingasymptoticerror

covariancematrixU

min

(13)for theSLAMS algorithm(34).coincides with the

lowerboundU

lb

(8). Thus,Theorem1andthelowerbound(8)implythatunder

Gaussian distributions of e(t) and w(t) the algorithm (10), (11)with optimal

matrixgainS=S

opt

(14)deliversasymptoticallyoptimalestimatesamongall

possibleestimators.

An interestingtheoretic aspectof this isthat itis possibleto achieve

opti-malaccuracy with analgorithm that is considerably simplerthat theoptimal

Kalman-lterbasedalgorithm. Thismightalsoproveusefulincertainpractical

applications.

It mightbeseenas aparadoxthat theresultis independentofthe integer

m,whichalsogovernsthealgorithmcomplexity. Oneshouldbearinmindthat

theresultisasymptoticin!0. Forxed, non-zero,therewillbeanupper

limitof m forwhich thelimit expressionid agood approximationof the true

covariancematrix.

Appendix

Proof of (44): First,note that wemayusethe relations(23){(27)from

theproof of Theorem 1,since theprocedure (22) coincides withthat of (34a)

underA=I

n

. Thus,puttingA=I

n

andintroducingthetrackingerrorfor ^ (t) asfollows ^ Æ(t)= ^ (t) (t); (47)

weobtainthesolutionto (25)

V = lim t!1 E ^ Æ(t) ^ Æ T (t)=  2  2 e I n +o( ) (48) andconsequently lim t!1 kE ^ Æ(t) ^ Æ T (t)k=O() as !0 (49)

Note,that(49)canbeextendedto

lim t!1 kE ^ Æ(t) ^ Æ T (t+`)k=O() as !0 8`=1;:::;m: (50)

Indeed,for`=1weobtaintheequation

E ^ Æ(t) ^ Æ T (t+1)=E ^ Æ(t) ^ Æ T (t)(I n Q) (51)

whichfollowsfromthestrightforwardrecursiveequation

^ Æ(t)= I n '(t)' T (t)
 ^ Æ(t 1) w(t)  +'(t)e(t) (52)

Furthermore,from(50)and(34b)theevaluation lim t!1 kE e Æ(t) e Æ T (t)k=O() as !0; =!0 (53)

followsdirectlyforthesecondtrackingerror(i.e.,theonefor e (t)) e Æ(t)= e (t) (t)= 1 m t 1 X =t m ^ Æ() t 1 X s= w(s+1) ! (54)

sinceforeachsand underconsideration

kEw(s+1)w T (s+1)k=O( 2 )=o() and kE ^ Æ()w T (s+1)k  kE ^ Æ() ^ Æ T ()k
kEw(s+1)w T (s+1)k ! t!1 O( )=o()

Nowwearereadytodirectlyevaluate(43),that is

e R m = lim k !1 E e Æ(km)  Æ T (km m) (55)

Taking(54)intoaccount,weobtainfort=km

E e Æ(t)  Æ T (t m) = 1 m t 1 X =t m E ^ Æ() t 1 X s= w(s+1) !  Æ T (t m) ! k !1 (1+O()) lim k !1 E ^ Æ(km)  Æ T (km)+O( )U (m) (56)

sincedueto (52)weobtainfort, andsunder consideration

E ^ Æ()  Æ T (t m) = (I n Q)E ^ Æ( 1)  Æ T (t m) =


=(I n Q)  t+m E ^ Æ(t m)  Æ T (t m) ! k !1 (1+O()) lim k !1 E ^ Æ(km)  Æ T (km) andEw(s+1)  Æ T (t m)=0.

Thus,wehaveto evaluate

^ R m = lim k !1 E ^ Æ(km)  Æ T (km) (57) Denote e w(t)= t 1 X =t m w( +1): (58)

E ^ Æ(t)  Æ T (t) = E ^ Æ(t) h (I n S)  Æ(t m) w(t)e
+ S e Æ(t) i T (59) = E ^ Æ(t)  Æ(t m) w(t)e
T (I n S) T (60) + E ^ Æ(t) e Æ T (t)S T (61)

Consequentlyapplying(52)totheexpectationtermof(60)weobtain

E ^ Æ(t)  Æ T (t m) = (I n Q)E ^ Æ(t 1)  Æ T (t m) =


=(I n Q) m E ^ Æ(t m)  Æ T (t m) (62)

and,dueto(52)and(58),

E ^ Æ(t)we T (t) = t 1 X =t m E ^ Æ(t)w T (+1) (63) = t 1 X =t m (I n Q) t  1 E ^ Æ(+1)w T (+1) = 2 t 1 X =t m (I n Q) t  R w =O( 2 ) (64)

Atlast,fortheexpectationtermin (61)which turnsouttobeofthemain

orderO()weobtain(remindingthat weconsider t=km)

E ^ Æ(t) e Æ T (t) = 1 m t 1 X =t m E ^ Æ(t) ^ Æ() t 1 X s= w(s+1) ! T (65) = 1 m t 1 X =t m E ^ Æ(t) ^ Æ T () 1 m t 1 X =t m t 1 X s= E ^ Æ(t)w T (s+1) (66)

Here,foreach ;s=t m;:::;t 1,weobtainfrom (48)and(51)

E ^ Æ(t) ^ Æ T ()=(I n Q) t  E ^ Æ() ^ Æ T () ! k !1  2  2 e I n +o( ) and(similarlyto(63){(64)) kE ^ Æ(t)w T (s+1)k ! k !1 O( 2 )

Hence,from (65){(66)follow

E ^ Æ(t) e Æ T (t) ! k !1  2  2 e I n +o( ) (67)

(57)intoaccount,wearriveat thefollowingequation ^ R m =(I m Q) m ^ R m +   2  2 e I n +o( )  S T +O( 2 ): (68) Since (I m Q) m =I m mQ+O( 2 ) as!0;

theasymptoticsolutiontotheequation(68)isasfollows

^ R m = 2m  2 e Q 1 S T +o( ) (69)

Thus, combining(43),(56),(57)and(69)wearriveat(44). 

Acknowledgement ThisworkwassupportedbytheSwedishResearch

Coun-cilunderthecontractonSystemModeling. Theauthorswouldalsoliketothank

Prof. J.P.LeBlanc (Lulea)forhiscreativeacronyms.

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