FAI-11:15

EXPONENTIAL CONVERGENCE OF RECURSIVE LEAST SQUARES WITH EXPONENTIAL FORGETTING FACTOR*

Richard M. Johns tone l , C . Richard Johnson J r . 2 , Robert R . Bitmead3 and Brian D . O . Anderson1

'Department of Systems Engineer ing, Inst i tute of Advanced S t u d i e s ,

2Schoo l o f E lec t r i ca l Eng inee r ing , Corne l l Un ive r s i ty , I t haca , NY 14853, USA

3Depar tment o f E lec t r ica l and Elec t ronic Engineer ing , James Cook

Aus t r a l i an Na t iona l Un ive r s i ty , Canbe r ra , ACT 2600, Aus t ra l ia

University, Townsville, Qld 4811, Australia

*The work of Authors1 was suppor ted by t h e A u s t r a l i a n R e s e a r c h G r a n t s Commit tee , whi le that of Author2 was supported by NSF g r a n t ECS-8120998.

ABSTRACT

This paper demonst ra tes tha t , p rovided the sys tem i n p u t i s p e r s i s t e n t l y e x c i t i n g , t h e r e c u r s i v e l e a s t squa res e s t ima t ion a lgo r i thm wi th exponen t i a l fo r - g e t t i n g f a c t o r i s exponen t i a l ly conve rgen t . Fu r the r , i t i s shown t h a t t h e i n c o r p o r a t i o n o f t h e e x p o n e n t i a l f o r g e t t i n g f a c t o r is n e c e s s a r y t o a t t a i n t h i s c o n v e r - g e n c e a n d t h a t t h e p e r s i s t e n c e o f e x c i t a t i o n i s v i r - t u a l l y n e c e s s a r y . The r e s u l t h o l d s f o r s t a b l e f i n i t e - d imens iona l , l i nea r , t ime- inva r i an t sys t ems bu t has i ts c h i e f i m p l i c a t i o n s t o t h e r o b u s t n e s s o f t h e p a r a m e t e r e s t i m a t o r when t h e s e c o n d i t i o n s f a i l .

1. IhTRODUCTION

T h i s p a p e r d e a l s w i t h t h e e x p o n e n t i a l s t a b i l i t y o f a popu la r adap t ive e s t ima t ion scheme - t h e r e c u r s i v e least squa res (RLS) a l g o r i t h m . I n p r a c t i c e , t h e RLS a lgo r i thm i s m o d i f i e d t o i n c o r p o r a t e an exponen t i a l f o r g e t t i n g f a c t o r ( X ) between zero and one. The reason f o r i n c l u d i n g X has mos t o f ten been jus t i f ied by no t ing tha t such an inc lus ion d i scounts pas t measurements in f avour o f more r ecen t ones [ l ] and so o f f e r s some c a p a c i t y t o i d e n t i f y t i m e - v a r y i n g p l a n t s . W i t h o u t e x p o n e n t i a l f o r g e t t i n g ( X = l), and w i th r i ch i npu t s , t h e RLS a l g o r i t h m e f f e c t i v e l y t u r n s i t s e l f o f f , a f t e r some time. Th i s , o f cou r se , means t h a t t h e RLS a lgo - rithm p r o g r e s s i v e l y l o s e s t h e a b i l i t y t o i d e n t i f y t h e t ime-vary ing p lan t parameters . Converse ly , wi th A less than 1, o n e m a i n t a i n s t r a c k i n g c a p a b i l i t i e s a t t h e c o s t o f non-ze ro l imi t ing pa rame te r cova r i ance .

Our main r e s u l t i s t h a t , i f t h e s y s t e m i n p u t { u k l

i s p e r s i s t e n t l y e x c i t i n g t h e RLS algori thm with expo- n e n t i a l f o r g e t t i n g f a c t o r i s exponen t i a l ly conve rgen t , i n t h a t t h e p a r a m e t e r e r r o r v e c t o r ( t h e d i f f e r e n c e be tween the ac tua l p lan t and the es t imat ion scheme parameters ) approaches zero exponent ia l ly fas t . Fur the r i t i s shown t h a t w i t h o u t t h i s e x p o n e n t i a l f o r - g e t t i n g f a c t o r ( i e A = 1) exponent ia l convergence is l o s t . I t i s th i s p rope r ty o f exponen t i a l conve rgence which i s n e c e s s a r y t o e n s u r e r o b u s t p e r f o r m a n c e i n t h e presence o f measurement no ise , t ime-var ia t ions o f p lan t parameters , and under -model l ing o f p lan t o rder . The ease with which an exponential overbound on t h e p a r a - m e t e r e r r o r i s f o u n d c o n t r a s t s s h a r p l y w i t h t h e r e l a - t i v e d i f f i c u l t y [ 2 , 9 ] i n e s t a b l i s h i n g s u c h a conver- g e n c e r a t e f o r s t e e p e s t - d e s c e n t t y p e a l g o r i t h m s , T h i s r e f l e c t s t h e Newton-Raphson-like nature of RLS.

I n v i e w o f t h e r e c e n t r e s u l t s o f [ 2 ] t h e n e e d f o r p e r s i s t e n t e x c i t a t i o n o f { u k } s h o u l d n o t b e s u r p r i s i n g .

Indeed, i f we do n o t h a v e p e r s i s t e n t e x c i t a t i o n or' { u k l , t h e n it is t r i v i a l t o show t h a t we cannot en-

s u r e c o n v e r g e n c e o f t h e p a r a m e t e r e r r o r t o z e r o , a t a l l .

We a l s o show how an es t imat ion scheme due t o Landau [3] can be modified so t h a t an exponent ia l f o r g e t t i n g f a c t o r i s inco rpora t ed as i n [ 4 ] . An o u t - l i n e o f t h e p r o o f o f e x p o n e n t i a l c o n v e r g e n c e o f t h e modi f ied a lgor i thm i s g iven .

The approach o f t he p re sen t no te is l i m i t e d t o l i n e a r , t i m e - i n v a r i a n t , f i n i t e - d i m e n s i o n a l , s t a b l e p l a n t s which a r e s i n g l e - i n p u t , s i n g l e - o u t p u t (SISO) and dr iven by d e t e r m i n i s t i c i n p u t s . The SISO r e - s t r i c t i o n c a n d o u b t l e s s b e removed wi th an appropr i a t e pa rame te r i sa t ion scheme [SI. E x t e n s i o n s t o s t o c h a s t i c i n p u t s a l o n g t h e l i n e s o f [ 6 ] a l s o seem r e a d i l y p o s s i b l e , a l t h o u g h t e c h n i c a l l y i n v o l v e d . N o n l i n e a r , t ime-vary ing o r in f in i t e -d imens iona l p l an t s can on ly be accommodated t o t h e e x t e n t t h a t t h e r e s u l t s g i v e n h e r e a r e r o b u s t i n t h e p r e s e n c e o f some n o n l i n e a r i t y , some pa rame te r t ime-va r i a t ion and some modes which a r e neglected. See [7-91 f o r d i s c u s s i o n o f t h e e f f e c t s o f d e p a r t u r e f r o m i d e a l i t y i n a d a p t i v e s y s t e m p r o b l e m s . The s i g n i f i c a n c e o f t h i s p a p e r i s t h a t it demonst ra tes t h a t t h e c o n v e r g e n c e r a t e a d v a n t a g e s o f RLS over s t eepes t -descen t a lgo r i thms need no t be abandoned i n o r d e r t o g a i n a d a p t a b i l i t y ,

2 . THE RLS ALGORITHM WITH EXPONENTIAL FORGETTING

The RLS a lgor i thm can be descr ibed by the fo l lowing eaua t ions

and r

o r

(2.3)

where X E [ O , l ] , ?-l i s a p o s i t i v e d e f i n i t e m a t r i x ,

and 8 is a r b i t r a r y . The p a r a m e t e r e s t i m a t e v e c t o r

e can be shown [ l ] t o c o n v e r g e t o t h e v a l u e e which - 0

m i n i m i z e s t h e q u a d r a t i c c r i t e r i o n

where y i s t h e o u t p u t o f t h e p l a n t b e i n g i d e n t i f i e d and + i s the measurable information vector f rom which y i s presumed t o b e f o r m e d ,

I f we now d e f i n e a p a r a m e t e r e r r o r v e c t o r a s

e k = e - 'k ( 2 . 5 )

and r e s t r i c t o u r a t t e n t i o n t o t h e "homogeneous" case [6], where yk i s g iven exac t ly by & 0 , then (2 .3)

takes on the form

which i s a homogeneous, t ime-varying, l inear d i f f e r e n c e e q u a t i o n .

With A = 1 i n t h e above equat ions we have t he s t anda rd RLS a lgor i thm, whi le choos ing A < 1 can be s e e n v i a ( 2 . 4 ) t o b e e f f e c t i n g a n e x p o n e n t i a l w e i g h i n g on t h e p a s t d a t a . S i m i l a r l y , (2 .3) i s an asymptot i - c a l l y s t a b l e r e c u r s i o n o n l y f o r A c 1 s o t h a t i f $ i s p e r s i s t e n t l y e x c i t i n g P-' will remain bounded.

The q u e s t i o n o f t h e o r i g i n o f t h e i measurement vec to r will f o r t h e moment b e l e f t i n a b e y a n c e , t o be a d d r e s s e d l a t e r i n t h e d i s c u s s i o n o f t h e p e r s i s t e n c e o f e x c i t a t i o n c o n d i t i o n a n d i t s i n t e r p r e t a t i o n i n , for example, A R M model l ing .

3. MAIN RESULTS

As a p r e l i m i n a r y t o t h e main r e s u l t we d e f i n e p e r s i s t e n c e o f e x c i t a t i o n a n d t h e n p r o v e t h a t i f i+ 1 i s p e r s i s t e n t l y e x c i t i n g t h e n P-' i s uni formly

bounded above and un i fo rmly pos i t i ve de f in i t e ( i e bounded away f r o m z e r o ) . T h i s l a t t e r p r o p e r t y c a n b e seen to fo l low f rom (2 .3) because , wi th X < 1 , P i 1

i s t h e o u t p u t o f a n e x p o n e n t i a l l y s t a b l e s y s t e m d r i v e n by a p e r s i s t e n t i n p u t ,

D e f i n i t i o n

k

k -1

We s a y t h a t t h e v e c t o r s e q u e n c e I$ j i s p e r s i s -

t e n t l y e x c i t i n g i f f o r some c o n s t a n t i n t e g e r S and a l l j t h e r e e x i s t p o s i t i v e c o n s t a n t s a and E s u c h t h a t

k

j + S O < a I , < c ' j i d i , < B 1 < r n (3.1)

i= j

Lemma 1 : Suppose t h a t I $ 1 i s p e r s i s t e n t l y e x -

c i t i n g t h e n f o r a l l k > S k

e -1 a(T; - 1) 1

s+l I + L A 1 5 'k-1 2 k

s+l I > 0 (3 .2) 1 -A

Proof:

From (2.3) and (3.1) we o b t a i n t h a t

-1

-

Also, from (2.3) i t i s obv ious t ha t

P& 2

SO t h a t f o r j 3 0,

( - + - + . . . + 1) P: AS XS-1 1 1 1

a n d c o n s e q u e n t l y f o r a l l k 2

.l 5 a1 +s-1

: s

The upper bound follows from the solution of

Using (3.1) we conc lude t ha t

( 3 . 4 )

(3.5)

(3 .7)

+ AJ-s[P;J1 + . . . + p - 5 -1

We can now s t a t e our main r e s u l t .

Theorem 1: I f t h e measurement vector sequence I d 1 k i s p e r s i s t e n t l y e x c i t i n g , t h e n t h e RLS es t ima t ion scheme w i th exponen t i a l fo rge t t i ng f ac to r is exponen- t i a l l y s t a b l e .

We choose a Lyapunov func t ion cand ida te as Proof: Consider equat ions ( 2 . 2 ) , (2 .3) and ( 2 . 5 ) .

- T -1 - v k = e P k k-1 'k (3.10)

Then s t r a igh t fo rward a lgeb ra u s ing (2 .3 ) and (2 .1 ) y i e l d s

'k - "k-1 = 'kPk-1"k - 'k-1 k-2 k-1 -T -1 ; -T p-l 6

(3.11)

and t h e r e f o r e

F ina l ly , (2 .3 ) , ( 3 .10 ) and Lemma 1 ( p a r t i c u l a r l y the lower bound o f ( 3 . 3 ) ) i m p l i e s f o r k s

, s+l

A Q . E . D .

The main r e s u l t of Theorem 1 d e p e n d s c r i t i c a l l y o n t h e p e r s i s t e n t e x c i t a t i o n o f ill. 1 . Lemma 1 showed k *

995

t h a t Pk and Pi1 were bounded wi th pers i s ten t ly

e x c i t i n g I9 1 and i f P is unbounded i t i s not poss ib le

t o c o n c l u d e f r o m t h e f a c t t h a t BkPk-lSk +. 0 t h a t B k + 0 . k k - T -1

One c a n a l s o e a s i l y show t h a t if Jik i s bounded,

p e r s i s t e n c y b f e x c i t a t i o n of Jik i s necessa ry fo r P t o k - be bounded. Suppose that 0 < A < 1 1 Pk I / 2 B < m f o r

a l l k. Choose s s o t h a t A B<T. Then from ( 2 . 3 )

-1 S A

Applying the bounds shows that

o r

( 3 . 1 5 )

so t h a t qk i s p e r s i s t e n t l y e x c i t i n g .

ARMA Modelling

Given t h a t Theorem 1 guaran tees exponen t i a l convergence of RLS wi th A < 1 s u b j e c t t o s u f f i c i e n t exc i ta t ion o f the measurement vec tor Jik, we n o t e t h a t t h i s p rope r ty o f $k can be ach ieved i n t he impor t an t ARMA m o d e l l i n g s i t u a t i o n by r e q u i r i n g s t a b i l i t y of t h e p lan t , min imal i ty o f the parameter iza t ion , and pers i s tence o f exc i ta t ion o f the input sequence {ukl o n l y . I n p a r t i c u l a r , i f t h e p l a n t s a t i s f i e s

yk+alyk-l+...+ anyk-n = 8d~k-d+.. .+B m u k-m ( 3 . 1 6 )

and i s s t a b l e , and f u r t h e r

a ( z ) = z +alz +...+a n n-1 ( 3 . 1 7 )

8(z) = Bdz +8d+l~ +.. .+B z ( 3 . 1 8 ) n-d n-d-1 n-m

m

are r e l a t i v e l y p r i m e , t h e n Theorem 2 . 2 of [ 2 ] ensu res pe r s i s t ence o f exc i t a t ion o f $k = u ~ - ~ . . . u l T i f [uk-d.. .u 1 i s p e r s i s t e n t l y e x c i t i n g .

Nonexponential Convergence of RLS wi th X = 1

k-m+l k-m-n

To d e m o n s t r a t e t h a t RLS wi th X = 1 is n o t e x p o n e n t i a l l y c o n v e r g e n t i n s p i t e o f s u f f i c i e n t l y e x c i t i n g $ we cons ider the fo l lowing sca la r example .

Let e and 8 b e s c a l a r s a n d t h e p e r s i s t e n t l y e x c i t i n g scalar Jik = 1 f o r a l l k . Then choosing P =

c we have from ( 2 . 3 )

k A

-1

1 c+k+l Pk = -

and using (Z.l),

Consequently 8 decay a s - which cannot be over 1 K bounded by an exponential and s o does no t converge

e x p o n e n t i a l l y f a s t .

These problems with slow convergence ra te and non- p e r s i s t e n t l y e x c i t i n g i n p u t s are a l s o known t o o c c u r i n Kalman f i l t e r i n g w i t h o u t e x p o n e n t i a l d a t a w e i g h t i n g . A d i s c u s s i o n a l o n g t h e s e l i n e s c o n c e r n e d w i t h d a t a s a t u r a t i o n and f i l t e r d i v e r g e n c e is g i v e n i n [lo].

4 . THE "CLASS B" ALGORITHM OF LANDAU WITH EXPONENTIAL WEIGHTING

I n [ 3 ] Landau d e t a i l s a "Class B" algorithm which i s very much l i k e t h e RLS a lgo r i thm o f t h i s pape r . The equa t ions wh ich desc r ibe t h i s a lgo r i thm a re :

1 p k-1 JI k-1 J I T P k-1 k-1 P k = P - - k-1 X 1 T '+?k-lpk-1'k-l

and

T A n

i= 1 vo = k yk - "-lek-, + d iEk- i ( 4 . 3 )

'k = Yk - $k-Iek T A

( 4 . 4 )

where

= [Zk-l"zk-n Uk-l*''Uk-m] ( 4 . 5 )

and Zk i s t h e o u t p u t of t h e " p a r a l l e l " e s t i m a t i o n model ( s e e [ 3 ] , pp. 164-167 and [ 2 ] , S e c t i o n 3 ) .

Prov ided t ha t

1 + diz-'

1 + z aiz-i

h (z 1 = -1 i=l 1

2A - - ( 4 - 6 )

i= 1

is a s t r i c t l y p o s i t i v e r e a l t r a n s f e r f u n c t i o n , t h e a lo r i thm conve rges i n t he s ense t ha t t he pa rame te r e r r o r g o e s t o z e r o . The extra complexity of t h e a lgo r i thm i s o f f s e t by a tendency for measurement noise t o c a u s e less damage t h a n f o r a n o r d i n a r y least squa res a lgo r i thm.

The s i m i l a r i t y of ( 4 . 1 ) - ( 4 . 5 ) t o ( 2 . 1 ) - ( 2 . 3 ) is o b v i o u s . I n f a c t t h e o n l y s t r i k i n g d i f f e r e n c e s are t h a t t h e Landau a lgo r i thm assumes un i ty t ime-de lay i n t h e p l a n t ( i . e . d = 1); t h e r e i s a n a d d i t i o n a l t e r m a p p e a r i n g i n ( 4 . 3 ) which does not appear in ( 2 . 1 ) ; "pa ra l l e l " e s t ima t ion mode l ou tpu t s , z ~ - ~ , a p p e a r i n

( 4 . 3 ) whereas ARMA p l a n t o u t p u t s y a p p e a r i n t h e

d e f i n i t i o n o f $ i n S e c t i o n 3 of th i s pape r and ,

f i n a l l y , ( 4 . 2 ) does no t i nco rpora t e an exponen t i a l f o r g e t t i n g f a c t o r .

k-i '

k-d

Landau h a s p r o v e d t h e a s y m p t o t i c s t a b i l i t y o f t h e c l a s s B a l g o r i t h m u s i n g h y p e r s t a b i l i t y a r g u m e n t s . U n f o r t u n a t e l y o n e c a n n o t p r o v e e x p o n e n t i a l s t a b i l i t y o f t h e a l g o r i t h m a s i t s t ands . Th i s is e s s e n t i a l l y b e c a u s e the matr ix sequence {Pi ; l} cannot be guaranteed to be bounded. I n p a r t i c u l a r i f {$k-l} i s p e r s i s t e n t l y

e x c i t i n g t h e n use o f t h e m a t r i x I n v e r s i o n Lemma and ( l r . 2 ) shows tha t P- l+- . This p roblem may be c i rcum-

v e n t e d i f w e u s e a s l i g h t m o d i f i c a t i o n o f t h e class B a lgo r i thm [ 4 ] , which amounts to introduct ion of e x p o n e n t i a l f o r g e t t i n g .

k

Use m

P k = - 1 [I - - 1 pk-l'k-l'i-l X 1 T 'k-1

a?k-lPk-lJik-l < a <

( 4 . 7 )

996

i n p l a c e of (4 .2 ) . By u s i n g s imi l a r a rgumen t s t o ' those i n [2] ( see Sec t ion 3 of [2] ) i t i s then possible t o p r o v e t h e f o l l o w i n g r e s u l t . Due t o s p a c e l i m i t a t - ions on ly an ou t l ine o f the p roof is g iven .

Theorem 2 : Suppose that

( a ) t h e p l a n t is desc r ibed by (3.16) (with d = 1) and a(z-1) i s s t a b l e ,

( b ) a(2- l ) and B(2- l ) (wi th d = l), of (3 .17) , ( 3 . 1 8 ) , a r e r e l a t i v e l y p r i m e ,

( c ) {uk l i s a p e r s i s t e n t l y e x c i t i n g i n p u t sequence

-1 (d) h(z ) i n ( 4 . 6 ) i s s t r i c t l y p o s i t i v e r e a l .

t hen t he mod i f i ed c l a s s B a lgor i thm of ( 4 . 1 ) , (4 .7) and (4 .3)-(4.5) is e x p o n e n t i a l l y s t a b l e .

( a ) A Lyapunov f u n c t i o n c a n d i d a t e i s chosen as

where

(b) Fol lowing [2] we s e t - u p s t a t e v a r i a b l e equa t ions equ iva len t t o <4.1), (4 .7) and (4 .3 ) - (4 .4 ) .

( c ) Making use of t h e Kalman-Yakubovic l e m a equa t ions [ Z ] we o b t a i n an e x p r e s s i o n f o r V[ek+l ' 'k+11 - 'Ie k , 8,]. Th i s quan t i ty i s monotone dec reas ing un le s s ek = 0 ( and , i n ca se P i s bounded, -1

k-1 3k = 0 ) .

(d) We u s e a r g u m e n t s l i k e t h o s e i n t h e p r o o f of Lema 1 t o show t h a t i f { u 1 i s p e r s i s t e n t l y e x c i t i n g k then j : k-1 = [yk-l ' * ' yk-nuk-n * " ] i s p e r s i s t -

e n t l y e x c i t i n g .

T k-m

(e) From t h e a s y m p t o t i c s t a b i l i t y r e s u l t s of [4] we conc lude t ha t zk + yk. Lemma 3.2 of [2] then allows

u s t o conc lude t ha t {$k-l] is a l s o p e r s i s t e n t l y e x c i t i n g .

( f ) The Xa t r ix Inve r s ion Lema and (4 .7 ) g ives

lie conc lude , t hen , u s ing a rgumen t s a s i n t he proof of Lemmal tha t i f ;$,-,3 i s p e r s i s t e n t l y e x c i t i n g t h e n

{Pk I i s a bounded matrix sequence.

u s t o c l a im tha t t he mod i f i ed c l a s s B a lgor i thm of Landau i s e x p o n e n t i a l l y s t a b l e .

-1.

( 9 ) F i n a l l y , ( c ) , ( f ) and the form of ( 4 . 8 ) a l lows

5 . ~~o~cLus Ioss The main r e s u l t d e m o n s t r a t e s t h a t t h e RLS es t im-

a t i o n scheme w i t h e x p o n e n t i a l f o r g e t t i n g f a c t o r i s e x p o n e n t i a l l y s t a b l e p r o v i d e d t h a t + k i s p e r s i s t e n t l y e x c i t i n g . Wi th bounded input s igna ls the converse i s a l s o t r u e . T h i s a l g o r i t h m when used with fiK4 models i s sometimes called a " s e r i e s - p a r a l l e l ' ' a l g o r i t h m [ 3 ] ; we h a v e a l s o o u t l i n e d a n e x p o n e n t i a l s t a b i l i t y r e s u l t f o r t h e c o r r e s p o n d i n g " p a r a l l e l " RLS scheme, with e x p o n e n t i a l f o r g e t t i n g .

These r e s u l t s a r e p o t e n t i a l l y v e r y i m p o r t a n t i n the cases of t ime-varying systems or under-modelling [7-91 , and imply robus tness o f the a lgor i thms in the presence of measurement noise. A l s o , t h e r e s u l t s

complement those known fo r t he s tochas t i c app rox ima t io* a lgo r i thm [ Z ] ,

..

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