Time-Varying Stabilized Forgetting for Recursive Least Squares Identification

Copyright © IFAC Adaptive Systems in Control and Signal Processing, Budapest, Hungary, 1995

TIME-VARYING STABILIZED FORGETTING FOR RECURSIVE LEAST SQUARES IDENTIFICATION J. J. Milek and F. J. Kraus • • Automatic Control Laboratory. Swi.. F.d.ral In.titut. of T.chnology (ETH). B092-Zurich. Switz.rland. F=: (+.+1) 1 6321211. E-mail: mil.kOaut .••. ethz.ch.krau.§aut .••.•thz.ch

Abstract. Forgetting is a well established technique of enforcing adaptability of a recursive LS estimator, necessary for tracking parameters of time-varying systems. Such systems often appear in the adaptive control and signal processing context. The classical forgetting methods are unconditional in the sense that old data are forgotten even in the absence of appropriate new data. This causes undesired unbounded growth of the algorithm's gain (estimator blowup) and leads to unbounded noise sensitivity and to numerical difficulties. The blowup phenomenon can be avoided by using so called stabilized forgetting, that utilizes contractive covariance matrix update equations having stable finite time-invariant equilibrium for nonpersistently exciting signals. In this paper, the class of admissible stabilized forgetting functions is extended by time-varying functions with possibly time-varying equilibria. Key features of the resulting parameter estimation algorithms, like stability of the covariance matrix and the parameter convergence are discussed and shown to be direction-
1. INTRODUCTION

1988), (Parkum e' &I., 1990). (Kreiuelmeier, 1990), and (MiJek and KrauI, 1991) lor buic ideal, and (Parkum e' aI., 1992), (Kulhavy and Zarrop, 1993), (Gunnaruon, 1994), and (Ekvall and E«ard', 1994) for .pecific re.ult•. Simple example of the .tabilized forlettins method i. riven by

This pa.per addrenes .he problem of recuuive pa.ra.meter estimation of & linea.r, slowly time-varyins- .y.tem, deacribed by

(1)

(6)

(2) where '"'k E (0,1) and R nxn 3 Gk ~ O. Thil me' hod was pro· pOled in (Pr&ly, 1983) in Ihe Iime-invarian. lorm: '"'k Gk G.

= ,""

The re«reuion vector
=

Summa.rizinl the above di.cuuion and followinl the cited references we conclude that a de.ired recursive a.J«orithm ,uitable for param~ eter euimation of the posaibly time-varyinl .yatem (1)-(2) (a) is LS-bued, to provide recunive .tructure and clear .tochastic and determini.tic interpretation. and to enable euy in· corporation of a priori knowled,e; (b) include. for,ettinl, to enable parameter trackins: a.nd expo-nentia.l conversencej (c) iu cova.riance matrix i. upperbounded, to eliminate the blowup and iu conlequence'j (d) i. numeric&1ly .imple and well conditioned, to avoid opera.tions like eis:enva.lue decompo.itionj (e) ena.ble. euy a.daptation of the forlettin" to tra.de-off trackins: venu. noi.e rejection.

in~

(3) (b) estimate vector prediction error update

In order to en.ure requirement (e), one ha. to dea.l with on-line adju.tment of the forsettiDS parameters. Perhap. the .tronsest re.uh, concernins the parameter conversence of the a.lsorithma equipped wi.h lIabili.ed lor«eUin«, il «iven in (Parkum e' aI., 1992). The key atlumption therein i. that the for«ettins doe. Dot decrea.e the covariance matrix,

(4) (c) covariance matrix fOllenin, update

(5) (7) where Unfortuna.tely, condition (1) cannot be .ati.fied by a atabiliZed forlenin, like (6) when the forsettinl parameters #Ak and Ok are timevaryin,. Moreover, the condition fails alao for the time-invariant case, whenever the initial covariance matrix Po is larler than the equilibrium matrix p. ~G.

no forleninl exponential forlenins

=

linear forlenins

Thia paper analy.e. directional dability and conversence properties of recuraive LS-bued pau,meter estimation alsorithm6 wilh timevaryins 6tabilized forseuins. Freedom in choice of the forleuins parametera can further be exploited for improvins the trackins performance.

11 is well known tha.t the RLS alsorithm i. not appropriate for tr&ck~ ins pa.rameters of the time-varyins .y.tem (1)-(2), .ince it! covariance matrix Pk converse. to zero. The matm can be prevented from becomins .mall by u.inS .0 called forsettins, e.s., defined above exponen.i&l lor«euin« (EF) or linear lor«euin« (LF). However, unconditional forlettinl of the old information in the .ituation when there i. no appropriate new information lead. to unde.ired Irowth of the covariance matrix, called e.timator blow up.

2. TIME-VARYING STABILIZED LS ESTIMATORS The l0al of this .ectioD i.: (a) to demon.trate a parameter diver. ,ence, cau.ed by the .tabiliaed time-varyiDI for,ettins, and (b) to (ive eJl:ample. of well-behaved forlettiDI method., which avoid .uch a diverlence.

The e.timator blowup can be eliminated, e.s., by u.inS '0 called .tabilized (or, .elective) forsenins, havinl .table, finite equilibrium for non-perailtent excitation•. See (Janecki, 1988), (Sa.lsado et al.,

137

2.1. Diver«ence

EXAmpl~

3. STABILITY PROPERTIES

Consider 'he RLS "I~orilhm (3)-(5), equipped wilh the ,'"bilized for~ettin~ (6). The data. And the for«ettin« pArameters are

Po

=[

~.5

2 0

"'2k = [

~5

G 2k = [ 02 0.05

"'k

]

]

= 0.2

"'2k+1 = 0.05 1.925

]

,

Thi:5 section deacribu the evolution of the covariAnce matrix Pk' ~overned by (3), (5), "nd (8). The m"lrix is m"jori.. d by "n excitAtion-dependent, time-varying finite upper bound. The analy15i:5, partially omitted here, is bAaed on the followin« properties of the stabilized forgetting: (A) antimonotonicity with re:5pect to the excitation matrix 'Pk'P~, and (b) having for 'Plc == 0 exponentially 9 at able equilibria, lowerbounded by !!L1 I and upperbounded

G 2 k+1 =

[ [

=G

1 0.5 1.925 0.05

by <;i/';' ~/. 0.05 0.2

]

3.1. Di.cuuion of the Time-Invaria.nt Case The above choice of the for«ettin« para.meters ensures upperboundedneu of the cova.riance matrix for any excitation, cf. (Milek and Kr"u., 1991). A.. umin~ vk 0 "nd ek 0 in (1)-(2), "fler rudimentary trandormation6 we «et

=

92 k+2 -

e = P2k+2 P2~~1 P2k + 1 P2~1

'" [_~;~

For aimplicity, let us auume that the ttabilized linear forsetting (8) i. time inva.ria.nt, i.e., IJk = J.A and 9k = 1. It can be easily seen tha.t the matrix eqUAtion P = IJP+ J has a globally asymptotically .table equilibrium at P p. ~l. The forsettins cova.ria.nce

=

-~::~

(9 2 k

j(6

2k -

-

=

e)

e) Therefore the amount of the forgotten data depends on the difference between the equilibrium matrix and the cova.riance matrix. Bebavior of the cova.ria.nce matrix can be depicted graphically. By definition the covariance ma.trix is aymmetric and poaitive definite, and,

The last ma.trix h.as one ei«env&Jue A=:: 1.17, which for almost &11 initial utimatea 60 causes an unbounded ~rowth a.nd diversence of the parameter error Bit as It _ 00. As we see, the para.meter conversence cannot be just t ..ken for sranted.

for v E Rn the expresaion form.

2.2. "Well-Behaved" Time-Varyins Stabilized For~ettin~ Methods

The

tton

The set {v E Rn

Vi

p;l

tI

is a. p06itive definite quadratic

vi p;l v

=

I} is an ellipsoid in ~n,

called ellipl10id of concentration, (Fosel and Huang, 1982). hs main axe.s ~re the eigenvectors of matrix Pk with the length equal to doubled .quare rooU of the corre.pondin« eisenva.lue•. An exem· plary iteration of the forgeUins is .shown in Figs. 1-2. In Fig. 1

The previous example demonstra.tea that to suarantee the parameter conversence the directional freedom in choice of the forgetting parame.teu must ~.e reatricted. We pve now examples of numerIcally slmple stablhzed forsettin~ function6 tha.t are well behaved from the point of view of the pa.rameter convergence.

Stabilized Linear Forgetting (SLF).

=

upda.te equa.tion (8) can be rewritten as

EvoIuooll of rhe covariaoce maui.t

for~ellin~ func-

0.8 0.6

(8) 0.4 ope.r~tes dire~tly on the covaria.nce

POllttv:-definlt: matrix, 3

i:i, J! :

matrix; 1 is an arbitrary 1 > ~ > IJk > !!.. > 0 "I k,

0.2

"nd 3 9, 9' 9 > 9k > 9 > 0 '" k, The numeric"l burden rel"'ed to this m;thod is ne«lisible; the theoretical interpretation of the forseUin« is siven in (Milek, 1994).

0 -0.2 -0.4

St.abilized Exponential Forgetting. Thi, more complex "". blitzed for«etttns InvolVe! matrix inveuions and is siven by

-0.6 -0.8

(9)

·1 -I

where 3 r, 1:' 1 > T > Tk > 1: > 0 '" k "nd 3~, e, ~ > ek > .{ > 0 V k. The for«ettins Ca.n be implemented &s a si;ple operation actin« on the information matrix

+

=

1';;1'

-0.5

0.5

Fig. 1. One iteration of SLF for ?k+l -< pO; P*: solid line, ?k+l: dashed line, Pk + 1 = ~?k+l + J: dotted line

The time-va.ryin« cue

~k Tk ~onst i. an&1yzed in (Janecki, 1988), while the theoretical lnterpreta.tlon of the method is fiven in (Kulha.vy and Zarrop. 1993).

EvoIutioa of the oovariaDce matrix

The '''0 prevlously defined for«etting functions can be classified &s a mul~iplica.tive 6ta.bilizatio~ with a linear for«etting, and, corre6pondIn«ly, A! a.n exponential forseHing with an a.dditive stabiliza.tion. Then, by analo~y, ~n additive ttabilization with ~ linear forgeHin~ becomes

Additive Stabilization with Linear Forgetting.

0.8 0.6 0.4 0.2

(10) -0.2

with 9k And (k bounded as above. The method combines the Levenberg-Marqua.rdt regularization with the lineAr for~etting.

-0.4

A symmetric positive definite ma.trix 1 introduced in (Janecki, 1988) "nd u.ed in (8)-(10), i. c"lled he;e 'p"n m"'rix. U.in~ in (3)-( 5) a. .imple tra.nsformAtion of coordinates,

.;, = )0.5""

p,;,

)-0.5 p)-0.5,

9';' )-0.5 6, P ,;, )-0.5 P )-0.5,

the span matnx 1 16 transformed to the identity matnx )

J =

I

-0.6 -0.8 _I_L---""-O~.5:----~-,-,-""._.---:'0.5~-_-.J I

(11)

Fig. 2. One iteration of SLF for ~

?k+l

-<

p.

(1 E Rn x n denotes

Pk+l -< p. i. auumed. This is the usual operation mode, when enou«h information in &11 direction. i6 stored in the estimator. In this ca6e, the forgettin« operation increase. the covariance matrix, removing information from the ettima.tor. An interesting situation ari.es in the oppoaite case, cl. FiS· 2. Here Pk+1 --< p. doe6 not hold, a.nd there is not enough information in lome or in &11 directions of the parameter .pace. This i:5 typica.l in the initialization phase, where uau&11y Po ~}- p •. In thOle direction. v, for

For the sake of brevity, w~ reatrict the forthcomin« discussion to t~~ para.meter ettimation algorithm (3)-(5), equipped with the atab111zed linear forseHinS (8). However, a.s Ihown in (Milek, 1994), the prelented reluh. can be ealily extended for a whole class of the s~abili.zed for«eUing .functiona, includin« (9) and (10). Abo lome d.lrechon~ freedom 11 permitted, by ma.king the Ipan ma.trix lk hme-varYJnS·

which v'( p* -

138

Pk +1 )v :5

0, 'he ""bilized for~ellin~ decre".e, 'he

covaria.nce ma.trix a.nd reduces the utima.tor sa.in.

Auume now

zation of the excitation, like that the reKreuion vectors are bounded and that they excite all direction. in the parameter apace. Definition 2 The observation vectou 'Pk are .aid excitin« if

Evoluooa of the covvi&Dcc: mlUiI

be peuistently

\0

k+.

3 a, b E R+, , E

rv,

lI"'k 11

2

<

b

L

"

""i""~

>-

al.

(14)

.=k+1

U5inK thia Definition we «et: -Q.2

..

Theorem 2 (About Che Gap) Conaider the peui,tently excited RLS-SLF a.lsorithm. Lu {Pk } denote the aequence of the covariance matricea, and {P k }-the aequence of the modified cova.riance matrice6. Consider &.1150 the acalar bounding .equence {pZ}, defined in Definition 1. Then it holda

~

~.6 ~.I

·1 ·1

~..5

0

(15 )

0..5

where

Fig. 3. SLF avoids the blowup for any initial matrix Po (dotted), here Po -< p'

. 0.1.

(16)

The proof c&n be found in (Mile., 1994).

'0

Example of the Gap. In order illustrA'e 'he G&P Theorem, let U6 con.ider RLS-SLF with time-invariant forKettin« parametera (~ 0.9, gJ for the case of a perai.tently exciCin« aiKnal. As i\ can be .een in Fi«. 5, an upper bound atrictly less than p. = ~J = 1 can be found.

= to)

=

Evolutioa of die c::ovwiaDcc

marrix

0.1

Fig. 4. SLF avoids the blowup also for Po (dotted) such that the condition Po -< p' is not satisfied that there i6 no excitation, Le., 'Plc == O. It follow a from (3) that PJc+ 1 = Plc· (Thia would ca.uae blow up for the conventional forsettins achemea1ike EF a.nd LF.) Rewritinl (12) for thi6 ca.ae, we se'

.{ll

.1.LI----~~..5::-...:::::~~O--===----:-O~..5-----'

(13) Since 0 < JJ < I, mappins (13) ia contractive. It hai a Slobally a.symptotica.lly "able positive-definite equilibrium a.t P p •. Hence Plc - p. a.s le - 00. An example of the conversence to the equilibrium p. ii ahown in FiS6. 3-4. Two different initia.1 values of the cova.riance matrix Po (dotted line) are con.idered, na.mely Po -< p. in Fi~:. 3, a.nd -'PO -< p. in FiS. 4. In both caaes, the covaria.nce matrix Plc conver£e. exponentially to p •.

Fig. 5. Example of the Gap between the upper bound, p' == /-,P' + 9J = I, and the covariance matrices Pk for the persistently excited RLS-SLF

=

'0

Directional Propertie$ of the Gap. !l i. nsy e .. end the Ga.p Theorem for the ca.e when the excitation ia directiondependent, .0 that 01 in condicion (14) i6 replaced by a. poaitive (aemi)definite matrix A:

3.2. Re.uhs for the Time-Va.ryin£ Ca..e The ma.in difference with respect to the previously discuued timeinva.riant ca.ae ii that now the upper bound becomes J and ii time-varyin£. The bound .til1 repre.enu the a..ymptotic beha.vior of the cov&ria.nce ma.uix when there ia no excitation:

Pk

k+.

3 0

<

a E R, , E

rv

L

""i"": >-A

"k E

rv.

.=k+1 Definition 1 A .c&lar .equence {pZ}r=O' z:ener&ted recuuively as pZ+ 1 = h(pZ) wi.h Po = Amaz( J-O.5 Po J- O.5 ), i. c&Ued & 6ca.la.r boundin~ aequence. (Notice that IIc(') denotes here a iCa.la.r func.ion, defined &S hP) = I'k A 9k')

In auch a case, (16) in the Gap Theorem ahould be repla.ced by

+

·A.

Theorem 1 (About Stability) The .ca.1a.r boundin£ .equence {pZ }~O h&. 'he followins 'wo properties: (&) i' i. upperbounded by and (b) it upperbounda tbe .equence of the lar£est ei«enva.!uell of the tranaformed covaria.nce ma.uix J-O.5 Plc J-O.5.

Fisure 6 depicu what happen, to the cova.riance matrix of RLSSLF (I' = 0.9, gJ = when 'he exci,&,ion i. no' uniform in apace. Thi6 should be compared to Fil. 5.

w,

To)

For 'he proof See (Mile., 1994).

Propertie$ for the Excitation Limited to a Hyper-

space. We con.ider now the time evolution of the ei«enva.luea a.nd eiKenvectou of the covariance matrix if there exiac aome not excited direction.. Aa it could be already auppo.ed by inapectinK Fig. 6, the covariance matrix Pie "rota.tu" in a particular way. The

Hence, tbe blow up phenomenon ia avoided. In order to obtAin ti£hter bound. on the cova.riance ma.uice. (Le., a sap between the bound J and PIt;), one need6 to ha.ve a. more a.ccura.te ch&racteri-

Pk

139

Evolutioe of the oovariaDce maail

1bc anaJk:ac ei

10' ~

~

.5-

awlue of

Ill"

10"

10" 10"

0 ti.meiuWlf: Aa lie betweea ei

10'

..

!

• GJ l·mI

t

aYeaOl'" IDd UDC1c:ite:d directioa 10'

10"

u

~ 10"

.1.LI-----o~..l:-....:::::=-~--===----=O..l7--~

Ifr'

Fig. 6. Directional properties of the Gap described by the Gap Theorem

0

10

13

33

Fig. 8. The largest eigenvalue of the covariance matrix converges to pZ; the corresponding eigenvector rotates towards the unexcited direction [1 0]'

cilenvectou of the matrix ca.n be a..ymptotic~y pa.nitioned into two lroup•. The fiut lroup .pAns entirely the excited hyperapa.ce ~, And the .ecoDd onc cODt&ins vectors ortholoDal to the excited hypenpace ~.

direction. Let the sequence of the relreuion vectors V'k E Rn conatitute a. not peuistent excitation, .ueh thAt for any Jc > 0 .pan(\PO, \Pl'" ',\pk)

= "',

and dim("')

=

Tn

<

n,

4

(17)

Condition (17) meAn. tbAt there is no excitAtion ouuide the hyper. spa.ce 4'.

The «0&1 of this .ection i. to .tudy chosen pa.rameter conver«encc properties of the sta.bilized &1«orithmlS with time-va.rying for,euinl para.meteu. The analysis utilizes the directiona.l .ta.bility results of tbe previou••ection. Let the .ystem be de.cribed by (1)-(2). The time evolution of the parameter error, Sic :: 91c - 9Jc, is de.cribed by the followinl Lemma.

Theorem 3 Let the RLS-SLF allorithm be excited by a not penis. tent excitation (17). Then: (A) n - m eisenva.lues of the transformed cova.riADce matrix, J-O.5 PJc )-0.5, converle exponentially to (defined in

Ph

Definition 1); the converlence rate is at lea.t

DETERMINISTIC PARAMETER CONVERGENCE PROPERTIES

Lemma 1 The para.meter error Sic in &1«orithm (3)-(5) with An ar· bilrary for«ellin« funclion Pk+l = fk(P k + l ) .ali.fies the foUowin~ reCUUlon

~;

(b) the eigenvectou, correipondinl to the afore.a.id eilenvaJuei, a.symptotica.Jly spa.n the ortholonaJ complement ~.l. of the exciled hype .. pace '" (Rn '9'_ E ",.l,

11_11 =

1

= '" 6l lim k-oo

(19)

",.l):

_'(pZp;;l -

J)_ = 0;

(18) This is a standard result for RLS-like &1«orithms, which follows di· rectly from the &llorithrn's definition, (CAneui and Espana, 1989). For the determini.tic conver«ence properties, we ta.ke elc 0 and -k 0 in (19), «ellin«

(c) the rema-ining m eigenvectora of the cova.ria.nce matrix asymp· totica.l1y .pan entirely the excited hyperspace 4>.

=

=

The proof can be found in (Milek, 1994).

(20) As a.n exa.mple, con.ider the time evolution of the cova.riance matrix in the RLS-SLF a1«orilhm wilh I' = 0,9 and gJ = Lel lhe

nrI,

excitation be persistent in a hyperspace

+,

with 4>.1..

=

[1 0]'.

Fi,·

Equation (20) de.cribes an autonomous dynAmical .y.tern, hAvins an equilibrium at BIc = O.

EvoIutiol of the oovm..ce 1DIUi1

4.1. Suitable Lyapunov Functions and the Scalar Re«ula.rity Con· dition The liability propertie. of the sy.tem (20) determine the para-meter conver«ence of the e.timation a.1«orithm, and can be investi«ated usin« tbe Lya.punov method. However, the usual functions, like e~ 91c , are not .uitable here, since the forleuinl is allowed to

p;1

decrea.e the covariance matrix Plc even in the time-inVAriant case, when the initia.1 matrix Po is lar«er tha.n the equilibrium matrix p. (u .bown in Fig. 4). Such a decreue of Plc may cause an increa..e of 8~p;-181c. In order to compen.ate it, the function can be premuhiplied by the sca.la.r boundin« .equence (.ee Definition 1). It is .hown in (Milek, 1994) tha.t the followinl four function'

(21)

(22) a.re appropriate a. Lya.puDov functions. The following condition

Fig. i, The covariance matrix grows in the unexcited directions and asymptotically achieves the upper bound p. = I

"Y

ure 1 .hows the time evolution of Plc. which indeed .a.tisfiu (18). Fi,ure 8 depicu the .malle.t ei,envaJue of PJc I, and the Anlle between the corre.pondinl ei,envector and the unexcited direction (1 0)'. The a.ngle conver«es to zero, .0 indeed onc of the eilenvec. tou of the covariance matrix Plc "roU,tea" towArdlS the unexcited

Pk

=0<),$1;, inf k

d7k1>:> d)'

> 0,

(23)

in (Milelr:, 1994) called .ca.la.r resula.rity, plAYS the crucia.l role in the pa.rAmeter converlence, enaurinl that the pArticula.r LYApunov function. (21)-(22) never increue. The condition (23) is sa.ti,fied by the propo.ed .. "bili.ed for«ellin~ funclion. (8), (9). and (10),

140

The Transition Matrix and Its Directional Properties Depending on the Span Matrix. Uain~ Ibe coordin"le Irana·

4.2. The Transition Manix Auumin~ J = I in 'he tequel, let. us consider 'be ala.'e 'ranai,ion matrix for 'be weigh'ed parameter error, Pkl8k:

Definit.ion 3 The ma'rix T k == p;;l Ma'rix.

Pk

forma.t.ion (11) it. i~ pouible to ~eneralize .ligh,ly t.he Transi,ion Mat.rix approach. Auumin~ in SLF (8) an arbitrary positive definite apan manix J and con"ant forseUin« parame'eu J.!k ~ and 9k 9, 'he Transi,ion Ma'rix become.:

This manix, for the first time uaed in (Janecki, 1988), makes t.he convergence analyais par'icularly aimple. h is related to the Lyapunov func,iona (22) and appeara in the convergence problem,

where

= TkP;19k.

e.~., one C&n wrile

Hence,

8=

The Transition Matrix has for RLS-SLF the

followin~

'0

(a) T k i. aymmetric, its eigenvec,ou are identical to tho.e of P k

Pk ;

"nd

4.3. Conver~ence Properties Without Auump'ion.5 Abou' the Ex-

(b) if the exci'ation is peraistent, t.hen

o

~ Tk ~

~ p; (I

ci'ation

>

- ~..,6 f . 1) for k

a;

In the sequel nothing is a.asumed abou' the exci'ation. We con.ider a theorem, in its orisinal form due '0 (Goodwin et aI., 1980), that for over a deca.de has been playing an imporca.nt role in 'he conver~ence analysis of a dass of adap,ive controllers, incorpora'in~ various RLS-based iden,ifica,ion algorit.hms. So, the theorem is a quality mark for a recuuive pa.rameter e.tima,ion algori'hm. (The re.ulu in (Bill&nli et "I., 1990) form" new improved w"y for 'he convergence ana,lyiJili; however, this method i. not applicable for RLS-SLF.)

k

( c)

o

~

Tk

P:-I

~ - - a- I

Pk

. .

for any exclu.uon;

(d) if the exci'ation is peraiatent, then conver~ence ill exponent.ial.

n;:C=l Tk

o

"nd Ihe

The conICant. -r is specified by condit.ion (23), fJ f is pven by the Gap Theorem.

Theorem 5 The RLS-SLF algori'hm hai, in a noise-free environment, the followin~ properties:

h is interestin« to compare how the Tranaition Manix looka like in Ihe c"ae of RLS, RLS-EF, "nd RLS-SLF (wilh con.l"nl for~ellin~ parameters ):

•

For the conventional RLS

al~orithm

(withou'

(a) 'he norma.lized predic'ion error is square aummable, 00

L...J

wei~hled p"r"melererror, p;1 9k .

I

+ 'P k 'Pk

00;

(b) the parameter e.timates are bounded,

evolves in JRn as a sequence of points alon« st.raight. lines, converging exponentia.lly towarda the oripn. Thus the RLSEF al~orithm ia exponentially conver~ent., provided p;;l is

3E

<

co

(c) the cha.nse in the e.tima'es is .quare aummable,

lowerbounded. The influence of the exponential for«ettin~ on is uniform in the weighted parameter apace and both

co

L

TfF TfF

and the conver«ence ra.'e do not depend on the excita,ion (actually, this is not a very a.ppea.lin~ propeny). Fi~ure 9 shows convergence of the weighted parameter error to the equilibrium [0 DJ'.

1I 0k - 0k+11I

2

<

co.

k=O The proof C&n be found in (Mile., 1994).

4.4. TIme el'Olutioa of the weigtled

<

I

k=O

For EF, TfF = Ak1. The

2

,.

' " ' (Yk - 'PklJk)

for~euin«),

TfLS = 1. •

h is now of interest to analyze those direc-

(24), and, in view of J- 0 . 5 8, they have no direct impac' on 'he conver«ence rat.e. However, J influencea al60 the tra.nsformed exciCation, JjJ = .fJ.5 "p. ThiIJ haiJ an obvious directional impact, first on the Ga.p Theorem, and then on each Transition Matrix t k , defined in (25) and appea.rin« in (24). As a reauh, the conver~ence rate in the eisendirec'iona corre.ponding the large.t ei«envalueiJ of t.he span matrix J is increased.

(24) Theorem 4 propeuies:

p. = ~I.

,ional propercies of the matrix T k which influence 'he paramuer conver~ence «overned by (24). Firs', not.ice 'hat the factou J±0.5, which appear in (25), cancel in all hut t.wo factors of the product

aince (20) i. equivalenl 10 P;;19k+1 = Tk+1 p;1 9k "nd ,,1.0 10

P;;19k+1

=

=

will be called the Tranait.ion

Conver~ence Properties

for PeuiiJtent Excitation

C&ti~e

Equuion (24:) and Theorem 4: immedia'ely imply the following reauh:

0.1 0.08

Theorem 6 Consider the peuistently excited RLS-SLF algorithm, auumin~ noi.e free environment. Then, the eatimation error 8k conver«es exponentially faat to zero.

0.06 0.04

~ 8E 8 '0

.... 0

0.02

The above Theorem is of ba.sic importance for adap'ive control applica'ions, because, as ahown in (Andeuon and Johns'one, 1983), t.he exponentiaJ conversence of an el&imator impliu capabili'y of t.racking .lowly t.ime.varyin« .ys'em6. To«uher with Theorem 5, t.he addreued propercy is a quali'y mark for a recursive parameter el&imation a.lgorithm .

0 '().02 .().04

Parameter Convergence Rates Depend on the Excitation. The direc,ional dependence between the convergence rates

.().06

and 'he excitUion lS demons'ra'ed in the following example. The RLS-SLF &1~orilhm wilh I' = 0.999, gJ = 0.011, "nd = p. = 0.11 i. used identify a. .econd order FIR dynamical sy.'em. In 'he first experiment, 'he sys'em input. ili uk = nk 0.2, where nk E N(O, I) &nd E{n;n)} 6;)' Thi. inpul produce. "lmoat uni·

.().08 .().I .().I

'().llS

0

0.1lS

=

1S( ax:opoDeDt

For SLF using conS&an' forgettin~ parameters J.!k

= gl, one obl"in.

=

#-!

=

=

and

Tt LF = (I + g(p;1 _ (p.)-I ))-1, where p. = ~1. A .. umin~ Po ~ p., we ~el Tt LF ~ 1. From Ihe G"p Theorem il followa Ih"l Tt LF dependa on Ihe Gk

+

form spatial distribution of the excit.ation V'k (uk Uk_1]'. As .hown in Fig. 10, 'he pa.rame'er conver~ence is a.lao ra.ther uniform. In the .econd experimen', the system input is uk 0.2nk + 1Such an input ca.uses 'he direction [1 1]' to be exci'ed more in'ensively tha.n the one orcho!onal to it, [1 - 1]'. This reauha in non-uniform convergence, al ahown in Fi~. 11. In the leas exci'ed direction the convergence is a160 exponen'ial, but the correspondin~ time conUants are much Ia.r~er.

Fig. 9. Evolution of the weighted parameter error, Pk- I Ok, for RLS-EF, where T k = AkI •

Po

'0

0.1

4.5. Convergence Properciea for Peuistent Excitation in a. Hyperspace

excit.a'ion, bein~ amaller for more exci'ed directioniJ. This property cau.ea exis'ence of different conver«ence raCes for differen' direc'ions, which Can be useful for tradin~-off 'he trackin~ abili'y veuue 'he noise rejection.

Theorem 7 Con.ider the RLS-SLF algorithm in a noise free environment, persisten&1y exched in a hyperspace ~ (we omit here the

141

T.". 0Y0luti0n of lho _igIItod panlmat.' .timet. inv(P{k))·he~lho"'(k))

Traj8C1ories of the parameter estimate hat(theta(k))

2 1.5

.. /

./ -0.5

0.1

-1

3 2

o

.0.1

0.1

2nd component

1 et oomponent

-1

-0.5 1st component

Fig. 10. In the uniformly excited RLS-SLF, the weighted parameter error Pk-10 k converges for various directions with similar convergence rates

Fig. 12. Parameter convergence depends on the persistence of the excitation unbiased estimates for moving avera«e systema, while in the case of autoregreuive systems .ome estimation biai appeare. In order to tra.de-off the parameter tracking ability vereuIJ the noi.e rejection, mechanisms for adaptive tuning of the forgeuing are proposed. The6e mechanisma include estimation of the auociated hypermodel or the filter bank approa.ch.

6. CONCLUSIONS This paper contributes by giving an in.ight into the directional stability and convergence properties of recuraive LS-ba.ed pa.rameter estimation algorithms with time-varying stabilized forgetting. Demonatra.ted propertiei and freedom in choice of the forgetting can be further exploited for improving the estimator's tra.cking performance.

c

.

j

-E

0.1 0.05

7. REFERENCES

Andeuon, B. D. O. &nd R. M. John.lone (1983). Ad&plive .yllem. and time. varying plants. Int. J. Control 37, 367-377. Bitta.nti, S., B. Bolzern and M. Campi (1990). Recuuive lea.t.quares identification with incomplete excitation: convergence analy.i. and application to adaptive control. IEEE Tran. on Au10m&l. CODlr. 85(12), 1311-1373. CaneHi, R. M. and M. D. Espana (1989). Convergence analy.is of the least-.quares identification a.lgorithm with a variable forgetting factor for time-varying linear .ystems. Automatica 25, 609612. Ekvall, J. and B. Egardt (1994). Identification for adaptive control. algorithm design from a frequency domain perspective. In: 10th IFAC Symposium on Identification and Sy.tern Parameter Estim&lion. Vol. 2. Copenh&sen. pp. 509-514. Fogel, E. and Y. Huan« (1982). On the value of information in system identification-bounded .noi.e case. Automatica 18. 229-238. Goodwin, G. C., P. J. R&m&dse &nd P. S. C&ine. (1980). Di.creletime muhiva.riable a.daptive control. IEEE TraDe on Automat. Conlr. 25(3), 449-456. Gunna.ruon, S. (1994). On covariance modifica.tion and re«ulariza.tion in recuuive least squares identification. In: 10th IFAC Symposium on Identification and System P&ra.meter Estimation. Vol. 2. Copenh&sen. pp. 561-666. Janecki, D. (1988). New recuuive para.meter estimation al«orithrn wilh nryins bUI bounded s&in m&lrix. 1nl. J. Conlrol 47(1), 1584. Kreiuelmeier, G (1990). Stabilized lea.st .• quares adaptive type idenlifiers. IEEE Tan. on AUIOm&l. Conlr. 85(3), 306-310. Kulhavy, R. and M. Zarrop (1993). On & Kener&l concept of for«etlinS· 1nl. J. Conlrol 58(4), 905-924. Milek, J. J. (1994). Stabilized Adaptive For«eHinK in tbe Recuuive Pa.rameter Estimation. PhD thesii. Swiss Federal Institute of Technolosy (ETH). Zurich, SwilZerl&nd. No. 10893. In prinl. Milek. J. J. a.nd F. J. Kraus (1991). Stabilized least. squarea e.tima.tou for time variant proceues. In: Fiut IFAC Sympo6ium on De.ign Methods of Control System •. Vol. 1. Zurich, Switzerland. pp. 430-435. P&rkum, J. E., N. K. Poul.en &nd J. Hol.. (1990). Seleclive for«eUin~ in adaptive procedures. In: 1Uh IFAC World Con«reu. T&1lin, Elloni&. pp. 180-185. P&rkum, J. E., N. K. Poul.en &nd J. Hol.. (1992). Recursive forsellinS &1sorilhm •. 1nl. J. Conlrol 55(1),109-128. Pra.ly, L. (1983). Robustness of model-reference adaptive control. In: 3rd Y&le Work.hop. New H&ven, CT. pp. 224-226. S&IS&do, M. E., G. C. Goodwin &nd R. H. Middlelon (1988). Modified least squares algorithm incorporatin« exponential reseUing &nd forsellins. 1nl. J. Conlrol 47(2), 477-491.

2nd oomponent 0.1

.0.1

1 at component

Fig. 11. In the non-uniformly excited RLS-SLF, the weighted parameter error p;;IOk converges for various directions with different convergence rates definition). Let us decompose the e.timatioD error al where

e~

E 4> &nd

et E 4> J..

Bit = e~ + et

I

Then, Ihe followins holds:

(a) the projection of the pa.ra.meter error on the excita.tion bypenpace converges to zero:

lim k-oo

1I0~1I =

0;

(b) iu ouh0s-onal complement remAin. bounded: 3E

<

00

lIotll
"'kEN.

The proof c&n be found in (Milek, 1994).

Consider 'he followin& example. For a. persistent excitat.ion "Pit; a.nd any init.ia.l estima.te 8~, the estima.te

ek • • hawn

the loUd linea, converges to the true value

0, =

e = [1

in Fig.

12 using

1 1)', ma.rked as

o. The initial enimate. 8 i 1,2,3,., a.re located at. the cornere of the da.shed squa.re. If the excita.tion i. not persistent (only tbe direction cJl = .pan([o a al') is excited), the vector of parameter estimates, .hown there u.in~ dots, does not conYer~e to the true value but "lands" on a hypeupace ~.L that cODu.in. the true value of parameter vector 6 and is ortho~onal to •.

°

5. EXTENSIONS (Milek. 1994) di.cuuea .everal exten.ion. of the pre.ented results, concernin~ parameter estimation &.1«orithms equipped with timevaryin« (or adaptive) atabilized for«ettin«. For example, the al«orithm. are shown to be exponential1y .table with respect to sin!le .tate perturba.tion,. In the .tocha'tic ca.se, the a.l«orithms deliver

142