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i f p r e m i s e t h e n c o n s e q u e n c e , su c h a s t h e c o n s e q u e n c eof a ru l e i s an a f f i ne l oca l model ; t he g loba l model i so b t a i n e d b y t h e s u m o f t h e l o c a l m o d e l s w e i g h t e d b ya c t i v a t io n f u n c t i o n s a s s o c i a t e d t o e a c h o f t h e m .F o r s t a t e e s t i m a t i o n , t h e s u g g e s t e d t e c h n i q u e , c o n s i s t s i na s s o c i a t i n g t o e a c h l o c a l m o d e l a l o c a l o b s e r v e r . T h eg l o b a l o b s e r v e r ( m u l t i p l e o b s e r v e r ) , i s t h e s u m o f t h el o c a l o b s e r v e r s w e i g h t e d b y t h e i r a c t i v a t i o n f u n c t i o n s ,w h i c h a r e t h e s a m e t h a n t h o s e a s s o c i a t e d w i t h t h e l o c a lmodel s (Pa t t on , 1998) . Our con t r i bu t i on l i es i n t he des igno f th i s g l o b a l o b s e r v e r b y e l i m i n a t i n g t h e u n k n o w n i n p u t sf r o m t h e s y s t e m . T h e s t a b i l i z a t i o n o f t h e m u l t i p l eo b s e r v e r i s p e r f o r m e d b y t h e s e a r c h o f s u i t a b l e L y a p u n o vm a t r i c e s a n d t h e i m p r o v e m e n t o f t h e p e r f o r m a n c e s o f t h em u l t i p l e o b s e r v e r b y p o l e a s s i g n m e n t i s f o r m u l a t e d i n aL M I c o n t e x t .

2 . M U L T I P L E O B S E R V E R O F A S Y S T E M W I T HU N K N O W N I N P U T S

T h i s s e c t i o n c l a r i f i e s t h e c o n s t r u c t i o n o f t h e o b s e r v e r .T h i s l a s t h a s a n a n a l y t i c a l f o r m r e s u l t i n g f r o m t h ea g g r e g a t i o n o f l o c a l o b s e r v e r s a n d t h i s f o r m i sp a r t i c u l a r l y s u i t a b l e f o r s t a b i l i t y a n d c o n v e r g e n c e s t u d yo f th e e s t i m a t i o n e r ro r . T h e n u m e r i c a l a s p e c t s c o n c e r n i n gt h e d e t e r m i n a t i o n o f t h e g a i n s o f t h e o b s e r v e r s w i l l b ea l so t r ea t ed .2 . 1 P r i n c i p l e o f t h e r e c o n s t r u c t i o nL e t u s c o n s i d e r a s y s t e m i n m u l t i p l e m o d e l ( w i t h r l o c a lm o d e l s ) f o r m a n d d e p e n d e n t o n u n k n o w n i n p u t s :

~ ; t ) = .(~(t))(Aix(t)+Biu(t)+F ff(t)+Di) 1 )Ly( t ) =w h e r e x ( t ) 6 R n i s t he s t a t e vec to r , u ( t ) E R l i s t he i npu tv e c t o r , f i ( t ) 6 R q t h e v e c t o r o f u n k n o w n i n p u t s a n dy ( t ) ~ R m t h e v e c t o r o f m e a s u r a b l e o u t p u t . F o r t h e i hm o d e l A i E R nxn i s t he s t a t e mat r i x , / ~ ~ R n x l i s themat r i x o f i npu t , F ~ R n x q i s t h e m a t r i x o f i n f l u e n c e o f t h eu n k n o w n i n p u t s a n d D ~ R n x l i s a m a t r i x d e p e n d i n g o nthe opera t i ng po in t . F ina l l y , C ~ R m x n i s t he mat r i x o fo u t p u t a n d ~ ( t ) r e p r e s e n t s t h e v e c t o r o f d e c i s io nd e p e n d i n g o n t h e i n p u t a n d / o r t h e m e a s u r a b l e s t a t ev a r i a b l e s : t h e v a l u e o f ~ ( t ) a l l o w s t o d e t e r m i n e w h a t a r et h e a c t i v e l o c a l m o d e l s a t t i m e t . T h e p r o c e d u r e t h a ta l l ows t o ob t a in t h i s s t ruc tu re and t o es t imat e i t sp a r a m e t e r s i s n o t d e v e l o p e d h e r e . L e t u s s t a t e t h a t o n ec a n e i t h e r u s e s t e c h n i q u e s o f p a r a m e t r i c e s t i m a t i o n( G a s s o , 2 0 0 1 ) o r t e c h n i q u e s o f l i n e a r i z a t i o n ( J o h a n s e n ,2000) .L e t u s c o n s i d e r t h e g l o b a l f u n c t i o n a l s t a t e m u l t i p l eo b s e r v e r , J ( t ) , d e s c r i b e d a s f o l lo w s :

r~( t )=Zkt i (~( t ) ) (Uiz ( t )+Gi lU( t )+Gi2 +LiY( t ) ) (2a)

i=1~ c (t )= z ( t ) - E y ( t ) (2b)

N i E R n x n , G i l E R n x l , L i ~ R n x m i s t h e g a i n o f t h el o c a l o b s e r v e r , Gi2 E R nxl i s a cons t an t vec to r and E am a t r i x o f t r a n s f o r m a t i o n . A l l t h e s e m a t r i c e s o r v e c t o r sh a v e t o b e d e f i n e d s o t h a t t h e r e c o n s t r u c t e d s t a t ea s y m p t o t i c a l l y c o n v e r g e s t o th e a c t u a l s t a t e x ( t ) .T h e r e c o n s t r u c t i o n e r r o r o f t h e s t a t e is g i v e n b y :

. ~ ( t) = x ( t ) - ~ c ( t ) 3 )t ha t i s wh i l e u s ing (2b ) :

~ ( t ) = ( l+ E C ) x ( t ) - z ( t )I t s t ime var i a t i on i s :

~c ( t ) 2 i ( ~ ( ) ] (P( a i x (t )+ Bi u ( t )+ Ff f ( t) + Di ) - ) 4 )= , - ,t , ,~U iz ( t)_Gi lU( t )_Gi2_L iY( t )i=1wi th

P = I + E C (5)T h e e x p r e s s i o n ( 4 ) c a n b e r e w r i t t e n

r t ) +N i ~ ( t ) + ( P A i - N i P - t i f x~c t ) = Z ~ i ~ t ) ) / e n i - a i l ) u t ) +

i=1 [ , (P D - G i 2 ) + P U f f( t )(6)

I f t h e c o n d i t i o n s ( 7 ) a r e s a t i s f ie d ( M a q u i n , 2 0 0 0 ) a n d( G a d d o u n a , 1 9 9 5 ) :

P = I + E CL i C = P A i - N i PG i l = P B iG i 2 = P D iP F = O

r

i = 1.. r (7)

Z k t i ( ~ ( t ) ) N i s t ab l ei=lt hen , t he r econs t ruc t i on e r ro r o f t he g loba l s t a t e t ends

a s y m p t o t i c a l l y t o w a r d s z e r o , a n d ( 4 ) is r e d u c e d t o :

. ~ t ) = ~ i ( ~ ( t ) ) N i . ~ ( t )i=1

8 )

T h e n u m e r i c a l s o l u t i o n o f t h e s y s t e m ( 7 ) r i s e s f r o m t h eu s e o f t h e g e n e r a l i z e d i n v e r s e ( C F ) - o f ( C F ) , t hee x i s t e n c e c o n d i t i o n s b e i n g s p e c i f i e d i n ( K u d v a , 1 9 8 0 ):

E = - F ( C F ) -P = I - F ( C F ) - CG i l = P B i (9)G i 2 = P D iN i = P A i - K i CL i - K i - N i E

I t i s impor t an t t o no t e t ha t t he s t ab i l i t y o f mat r i cesN i , V i 6{1 . .. . r } does no t gua ran t ee t he s t ab i l i t y o f ther

m a t r i x E ~ t i ( ~ ( t ) ) N i . T h i s i t e m i s d i s c u s s e d i n t h e n e x ti=1

p a r a g r a p h .

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2.2 G l o b a l c o n v e r g e n c e o f th e m u l t i p le o b s e r v e rIn this part , we wil l develop the sufficient condi t ions ofthe asympto t i c g lobal convergence o f the s t a te es t imat ionerror.The dynamic equat ion (8 ) i s g lobal ly asympto t i ca l lystable i f there exis ts a posi t ive defini te symmetricalmatrix X, such that (Bo yd, 1994):

N f X + X N i< O , v ie{ 1 .... r} (10)The search o f a mat r ix X which i s common to a l lmatrices i in the preceding equat ions can be a ratherconservat ive s tep. In what fol lows, another method,based on the exis tence of various local matrices X i > O,will be used.The orem 1 (Chadl i , 2001 ): If there exis ts symm etricaland defini te posi t ive matrices X i such as:

N i h X i + X i N i h < O , Vi6{1 . . . . r} , N i h = 2 N i + N f ) 1 1 )where ih i s the hermit ian part associated to the matrixN i , then the mult iple observer (2) is global lyasympto t i ca l ly convergen t .P r o o fTo solve (11), one uses N i = P A i - K i C from (9) and theineq uality (11) b eco m es, g i6 {1, . .. r} :

e A i - K i C ) + P A i - - K i c ) T ) x i +(12)S i P A i - g i G ) + P A i - g i f ) T ) < o

I t i s noted unfortunately that the preceding inequal i t iespresen t the d i sadvan tage o f be ing non l inear wi th respec tto the variables K/ and X i (more precisely bi l inear) . Anumer ica l p rocedure o f reso lu t ion by l inear i za t ion i spresented in the fol lowing sect ion.2.3 M e t h o d o f r e s o l u t i o nMethods o f reso lu t ion were p roposed to so lve non l inearand in part icular bi l inear inequal i t ies (see (Chadl i , 2001)and inc luded references ) . The method tha t we adop ted i sknown as local , because i t i s based on the l inearizat ion,with respect to variables K/ and X i , wel l chosen aroundthe initial values Koi and X o i . One poses :

K i = K o i + O K i a n d X i = X o i - F O X (13)The inequal i ty (12) then becomes:

+ O X i +P A i - K o i + a K i ) C )

P A i - g o i + O K i )C ) + IX o i + O X i ) ~ P A i _ K o i + O K i ) c ) T ~ < OX o i + ~ X i ) > O

(14)

By neg lec t ing the second o rder t e rms o f the inequal i ty(14), one obtains:e A i - K o i C ) + e A i - g o i f ) r ) o x i +

~9Xi P~. - KoiC) + PAi - KoiC )r) -O K i C X o i _ C X o i ) TO KF _ 1 5 )c r a X T X o , - X o , a < c +

P A i - K o i C + P A i - KoiC)r )X o i +X o i P A i - K o i C ) + P A i - K o i C ) T ) < 0The system (15) is then of LMI type and i ts resolut ion iss tandard (Boyd, 1994). Let us note that the choice ofinitial values Koi and Xoi remains the majord i sadvan tage o f the method and moreover convergencetowards a solut ion is not always guaranteed.Unfor tunate ly , f rom a p rac t i ca l po in t o f v iew, one can beled to tes t various choices of ini t ial values in order toobtain a solut ion.R e m a r kThe L M I sys tem (15) i s va l id on ly in the v ic in i ty o f K o iand X0i ; this enc ourag ed us , to impro ve the resolut ion,to propose the fol lowing addi t ional constraints ( in orderto l imit the variat ions of matrices K and X ):{ 1 1 1 w i t h 0 < e ol rl K 0 / [ [ l n n ) O g i 1 (17)[L o f t E l l g o il ll m x m ) > 0

Indeed, i f the LMI system (15) and (17) is real izable,then the mult iple observer (2) global ly asymptot ical lyest imates the s tate of the mult iple model (1) .

3. P O L E A S S I G N E M E N T

In th i s par t , we examine how to improve theper fo rmances o f the mul t ip le observer in par t i cu lar wi thregard to the ra t e o f convergence towards zero o f thestate error es t imat ion. For bet ter es t imat ing the s tatevar i ab les o f the mul t ip le model , the dynamics o f themult iple observer is selected in a manner which isappreciably fas ter than that of the mult iple model .Def in i t ion : The mul t ip le observer wi th unknown inpu t s(2) is known as local ly observable i f the pairs P A i , C )are observable, V i~{1 . .. . r} (Pat ton, 1998).To ensure a cer t a in dynamics o f convergence o f the s t a t eest imat ion error, one defines , in the complex plane, anarea S (cz, 13) built by the inters ectio n b etw een a circle, o fcenter (0 , 0) and o f radius 13, and the left half planel imited by a vert ical l ine of X-coordinate equal to (-cz)with cz being a posi t ive constant . The LMI formulat ion isp roposed by the fo l lowing coro l l a ry .

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Corol lary: The eigenvalues of the m a t r i x L ~ t i ( ~ ( t ) ) N ii=1

belong to the area S(a,13 ) , i f there ex is ts m atricesO X and ~ K i such that:

- ~ X o i + ~ X i ) N o T . X i - ~ K i C ) T X o i ] x3 > x2 ; the processmodel i s g iven by the equat ion (19).Indeed, taking into account the fundamental laws ofconservat ion of the f luid , one can describe the operat ingmode of each tank; one then obtains a nonl inear modelexpressed by the fol lowing s tate equat ions (Zolghadri ,1996):

a dX ( t)=-CZlSn(2g(xl( t) -x3( t) ) )V2+Ql( t)+Qfl f f ( t)d ta dx2 (t) =o~3Sn 2g(x 3 t)_x2 (t)))v 2 -o~2Sn(2gx2 (t)) 1/2d t

+Qz (t)+Qfz-ff(t) (19)a dx3 (t)=(ZlSn(Zg(x, ( t) -x3 ( t) ) ) t/2d t

0 ~ 3 S n 2 g x 3 t ) - - X 2 t ) ) ) v 2 + ~ 3 ~ t )w h e r e Zl, x 2 and a 3 are constants , ~ ( t ) i s regardedas an unkno wn input. Qf/ f i ( t ) , ie{1 . .. .. 3} denote theaddi t ional mass f lows into the tanks caused by leaks andg is the gravity constant.The mul t iple model (1) , wi th ~ ( t ) = u ( t ) , whichapproximates the nonl inear system (19), i s described by:

42 t ) = ~ . , g ~ u t ) ) a ~ x t ) + B ~ u t ) + f - ~ t ) + O ~ ) 2O)i=1y O=Cx O

~ Q 1 , Q 2

i i iii iiiii iiiiiii ii:i: :i: : ',: i : i : i : : : : : : : :: : : : : : : :: : : : : : : i : i : i ~ : : ii i ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i ii i ii ii ii i i i i . : .: . : . : . : . : . : : i i i i i i i i .: . : . : . : : : : : : : . ~ : ~ : ; : I : ~ : ~ : ~ : : : . : . : . : . ~:i: i : i : i : i : i : : i : i ~ i i ~ SHV V VQ f ~ a t ) Q f 3 a t ) Q f ~ a t )

Figure 1 . Three tank system

The matrices ~ . , B and D~ are calculated by l inearizingthe ini t ial sys tem (19) around four points chosen in theoperat ion range of the system. Four local models havebeen selected in an heuris t ic way. That numberguarantees a good approximat ion of the s tate of the realsystem by the mul t iple model . The fol lowing numericalvalues were obtained:

-0 .01 09 0 0 .0109 ] [ -2 .861A I = 0 - 0 . 0 2 0 6 0 . 01 0 6 D = 1 0 - 3 | - 0 . 3 8 [

0.0109 0.0106 -0 .0 215 L 0.11 JI 0 0 0 0 0 0 t

A = -0 .0205 0.01044 D = 10-3 -0.34L O . 0 1 1 0 . 0 1 0 4 4 - 0 . 0 2 1 5 L O .O 3 8 j-0 .0084 0 0 . 0 0 8 4 ] [ 3 . 7 0 ]

A3= 0 -0 .0206 0.009 5~ D3=10-31-0.14 /0 .0084 0 .0 09 5 -O.O18J L 0 .69 JF-0.0085 0 0.0095~. 0 0 8 5 ] , -3.671

Z 4 = -0.0205 O n 1 0 ~ [ 0 . 1 8 [L 0 .0085 0 .0 09 5 -0 .0183 L 0 .62 ]

B i= C = 1 0 - 1

In the following, the functions Qfl, Qf2 and Qf3 areconstant , E ( t ) i s a random sequence and the numericalappl icat ion are performed with:

Q f/= 10 -4, v i E {1 .... 4} and t e [0 ,x,[~1 = 0.78, ~2 = 0.78 and ~3 = 0.75g=9 .8 , Sn = 5 x l O -s and A=0.0154D e t e r m i n a t i o n o f t h e m u l t i p le o b s e r v e rThe s t ructure of the mul t iple observer is defined in (4) .The matrix P is obtained by solving (9)

1 - 1 2P = - 3 - 1 - 1The observabi l i ty of the pairs ( P A i ,C) i s checked. Toobtain the matrices K0i and X0 / , one can proce ed in thefol lowing way.

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i I I I i i t10.8

0 6f J l l i0 2130 4 0 0 6 0 0 8 0 0 I (B 0 1 2 0 0 1 4 0 0 1 0 3 0 1 8 0 0 3 0 0 0

i i i i i i i

0.4

0 . 1 1 ' 9 t , i i i t i i0 2 00 4 0 0 0 3 0 8 0 0 I (B 0 1 2 0 0 1 4 0 0 1 6 0 0

I

1800 2O300 . 8 i i

0.6 . ~ - . . . _ ~ . ~ ~ ' ~ ' ~ ' ~ - ' ~ ' ~x a ~ c t a n x a ~ r n ~

0.4 i0 200 400 600 800 100 0 120 0 1400 160 0 1800 2000Figure 4 . Non l inear sys tem 19)and mul t ip le observ er 2 )

0.4

0.2

00

i i i i i i is t a t e e s t i rr ~ i o n e r r t r f o r x l

330 4(30 600 8(13 1600 1200 14(I) 1600 1800i i i i i i i i

s t a t e e s l i r r ~ i o n e r r o r f o r < 20 . 5 i

0

-0.5 J0 2000 . 0 2 ,o . O

-0.01 i0 330

I I I I I I I I400 600 8130 1(300 12130 1400 1600 1860 20 03i i i i i

~ t a t e e s t i n ~ i e n e r r o r f o r x 3

I I I I4 6 O 6 0 0 8 3 O I 3 6 0 1 2 6 0 1 4 6 0 1 6 1 3 O

Figure 5 s ta tes es t imat ion er ro r1803 2660

5 . C O N C L U S I O NUs i n g a m u l t i p l e m o d e l r e p re s e n t a t i o n , we s h o we d h o wto des ign a mul t ip le observer us ing the p r inc ip le o f thein te rpo la t ion o f loca l observers . More over , on econs idered the case where some inpu ts o f the sys temwere unknown. The ca lcu la t ion o f the ga in o f the g lobalobserver reduces to the ca lcu la t ion o f the ga ins o f theloca l observers ; the s tab i l i ty o f the who le requ i res t ak ingin to accoun t the coup l ing cons t ra in t s be tween the loca lobservers , wh ich leads to the reso lu t ion o f a p rob lem ofthe B MI Bi l inear M at r ix Inequal i t i es ) type . Thereso lu t ion o f these BMI cons t ra in t s i s car r ied ou t byl inear iza t ion and the essen t ia l numer ica l d i sadvan tage o fth i s method on ly res ides in the cho ice o f the in i t i a lvar iab les o f mat r ices K i and X i .

The d i rec t app l ica t ion o f th i s observer cou ld be , thanks totak ing in to accoun t the unknown inpu ts , the base fo r thedes ign o f a de tec t ion p rocedure and loca l iza t ion o f fau l t so f ac tua to rs .

6 . R E F E R E N C E SBoyd S., L. Elghaoui, E. Feron, V. Balakrishnan, 1994 ).

Li ne ar m a t r i x i ne qua l i t y i n s y s t e m and c on t r o l t he or y ,Siam.Chadli M., D. Ma quin, J. Ragot, 2001). On the stabilityanalysis of multiple model systems, Eur ope an C on t r o lConference , Portugal 4-7 september.Gaddoun a B. 1995). Contribution au diagnostic des syst~meslin6aires inva riants b. entr6es in connu es, App lication b. unprocessus hydrauliqu e, Th?~se de docto rat IN PL, 3 avril.Gasso K. 2001). Identification in multiple modelrepresentation: e limination an d merging of local models,

C onf e r e nc e on D e c i s i on an d C on t r o l, C D C 2001 , Orlando,USA, D ecember.Gua n Y., M. Saif, 1991). A novel approach to the design ofunknown input observers, I EE E Tr ans . on Au t om at i c

C o n t r o l 36 5), pp. 632-635.Johanse n T. A., R. Shorten, R. Mu rray-Sm ith, 2000).On theinterpretation and identification dynamic Takagi-Sugenofuzzy models, I EEE Tr ans . on Fuz z y Sy s t e m s , 8 3), pp.297-313.Johnson C. D., 1975). Obse rvers for linear systems withunknown and inaccessible inputs, Int . J. Control , 21, pp.825-831.Kobayashi N., T. Nak amiz o, 1982). An observer design forlinear systems with unknown inputs, Int . J. Control , 35, pp.605-619.Kudva P., N. Viswanadham, A. Ramakrishna, 1980). Observerfor linear systems with unknown inputs, I EEE Tr ans . onAu t om at i c C on t r o l , 25 1), pp. 113-115.Lyubc hik L. M., Y. T Kostenko, 1993). The output control ofmultivariable systems with unmeasurable arbitarydistrubances - The inverse model approach, E u r o p e a n

C on t r o l C on f e r e nc e , G r on i nge n , The Netherlands, pp.1160-1165, june 28-july 1.Ma quin D., J. Ragot, 2000). Diagnos t ic des sys tkmes l indaires ,

C o l l e c ti on p~dagog i que d au t om at i que . Ed i t i on H e r m k sScience .Med itch J.S., G. H. Hostetter, 1974). Observe rs for systemwith unknown and inccessible inputs, Int . J. Control , 19,pp. 473-480.

Patton R. J., J. Chen, C. J. L o p e z - Toribio, 1998) . Fuzzyobserver for nonlinear dynamic systems, I EEE C on f e r e nc eon D e c i s i on and C on t r o l, C D C 98 , Tampa Florida,December 16-18.

Takagi K., M. Sugeno, 1985). Fuzzy identification of systemsand its application to modeling and control, I EEE Tr ans . onSy s te m s , M a n an d C y be r ne t i c s , 15, pp. 116-132.Wa ng S. H., E. J. Daviso n, P. Dorato, 1975). Observing the

states of systems with unmeasurable disturbances, I E E ETr ans. on Au t om at i c C on t r o l , 20, pp. 716-717.Zolghad ri A., D. Henry, M. Monsion, 1996). Design ofnonlinear observers for fault diagnosis: a case study,

C ont r o l Eng i ne e r i ng Pr ac t i c e , 4 11), pp. 1535-1544.

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