Observateurs non linéaires de multimodèles. Application à la synchronisation et aux communications

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Submitted on 5 Dec 2006

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Observateurs non linéaires de multimodèles. Application à la synchronisation et aux communications

Estelle Cherrier, Mohamed Boutayeb, José Ragot

To cite this version:

Estelle Cherrier, Mohamed Boutayeb, José Ragot. Observateurs non linéaires de multimodèles. Ap-

plication à la synchronisation et aux communications. May 2006, pp.CDROM. �hal-00118548�

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-)(,M;2:/B*

y

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}\>J&N/BG)&)(.6J*+>L=GOQ-)>P2S&)>%=2=>R@UOJ&)8:1F4=>R4'>U-)ES2698=G)2:(+-/0&(,1;2pt:/0H

-)OP>f1;t:->QG)]0/B&)>P%:GcL1;%=GW*./ƒ6P*+/;-)-)>f4'>R-E'-&)bP@U>-c4'OJl2=(,>L:/BGRVˆ§SY^

~*a>Q-&A13GM;/32=(+-)Oh6P13@U@U>p-%=(,&Q^³"/?L:/3G&(,>‚~~>Q-&<6J132-)/;6JGOP>hg

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698=G132=(.-/0&)(+132K$-1;%:-r13G@R>‚4'>p"´v~[-PK\>P2“%'&)(+*,(.-/B2S&A*+/C&)8=OQ13G(,>

4'>T"ES/BL=%=2:10];H TG9/3-)10]F-F(,(i^}\>Q-n6J1;2:4'(,&)(+132:-e-)132S&€&13%'&a4!#/Btx13G94

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p

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4=/32:-e*+/L:/BG)&)(+>m~)~~e-%=Ge*i#>PoF>Q@UL=*,>T4!#%=2q@%=*,&)(+@U1F4=bP*+>j6P13@UL1S-O

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4'%p6JGEFL'&/3M3>04'O6JGESL=&/BM;>W4!#%=2p-(+M32/B*-)132=1;G)>;^

J F! I +!":$#%F":­!

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&)(+@U1'4'bP*+>?6P132:-&)G%=(,&hgŸL:/3G&(,Gq4'>C4'>Q%'o-E'-&)bP@U>-ƒ698:/31B&)(.dS%=>Q-

6J1;@UL1;G&9/B2S&f%=2©G)>P&/3G4!KL=%=(.-f*./p6J1;2:-[&G)%6&)(+132©4!#%=2Ÿ13t->QG)]0/0H

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-)%+<-)/32;&>TM;/BG9/B2S&(+--)/32S&e*./U-EF2:698=G132:(+-/0&(,1;2h4'>N*i#1;t:-)>PG]/B&)>Q%=GQK

-)13%:-‚*./œr13G@R>Ÿ4!#%=2=>Ÿ"´p~^j}\>J&)&)>­4'OQ@A/BG9698=>I-)E'-[&OP@A/0&(+dS%=>

>Q-&ql:2:/B*+>P@U>Q2;&‚OJ&>P2:4=%=>Ÿ/B%76Q/3-qM3OQ2=OPG9/B*j4!#%:25@%=*s&(,@U1'4'bQ*,>

-Q#/3L=L=%=ES/B2S&c-%=G

p > 0

-)E'-[&bP@U>Q-a698/B1B&(+dS%=>-eG)>P&/BG94'O-P^

,.-./

'‰iŒ10UŽ %2P,…9Ž0WŠ=Ž0¤43R 3…U ;…4%5<¤

¤9‰20U…J¤

’m2C6J1;2:-(.4'bQG)>4=>P%'ov-)EF-&)bQ@U>Q-j698:/B13&)(.d;%:>Q-QKx4'>R4'EF2:/B@U(.dS%=>Q-

G>Q-)L>6&)(+]3>-T

˙

x = A

¹

x + f

¹

(x) + g

¹

(x

τ

)

^Vˆ®SY

˙

x = A

2

x + f

2

(x) + g

2

(x

τ

)

^Vi¯3Y

’m2C4'OPl:2=(,&T/3*,1;G-c*,>R@%=*,&)(+@U1'4'bP*+>-)%=(+]/32S&QK!g<L:/3G&(,Gj4'>hVˆ®SYW>J&

VX¯;Y







˙

x = µ(y)(A

1

x + f

1

(x) + g

1

(x

τ

)) +(1 − µ(y))(A

2

x + f

2

(x) + g

2

(x

τ

)) y = Cx ∈ R

VX°;Y

"/Wr1326&)(+132

µ

4'1;(s&`]3OQG)(,l:>QG

µ ∈ [0, 1]

^`’m2R/m698=1;(+-)(

ξ(t) = y(t)

4=/32:-a*i#>PoFL:G)>-)-)(,1;2?Vi¥3YK6P/BG

y

>Q-&c*,>f-)>P%:*-(+M32/B*4'(.-Lx132:(,t=*+>f/B%

2=(+]3>/B%‚4'%‚G)O6J>PL=&)>P%:GQ^

6c13%-q/3*,*+132-ƒOP&/Bt:*,(+Gh*+>Q-‚6P132:4'(,&)(+132-hMS/BG9/B2S&)(.--)/32;&<*./“-EF2'H

698=G132=(.-/0&)(+132qgUt:/3-)>N4!#13t:-)>PG]0/0&)>Q%=Ga4'%p@%:*s&(,@U1'4'bP*+><Vˆ°SY^e’m2

698=>QG698=>Ugp6J1;2:-&)G%=(,G>R%=2©13t:-)>PG]0/0&)>Q%=GN4'%Ÿ-E'-&)bQ@R>pVX°;YKZ4'132S&

*./R4=ES2/B@U(+dS%=>T>-[&c*./U-%=(+]0/B2S&)>A

˙ˆ

x = µ(y) (A

¹

ˆ x + f

1

(ˆ x) + g

1

(ˆ x

τ

) − K

1

(y − Cˆ x)) +(1 − µ(y)) (A

2

x ˆ + f

2

(ˆ x) + g

2

(ˆ x

τ

) − K

2

(y − Cˆ x))

V±BY

6c1;%:-U/B*+*,1;2:-R(,24'(+dS%=>QGU*+/?L=G1F6POQ4'%:G)>qL1;%=GR&)G13%=];>PGR*,>-RM;/B(+2:-

K

1

>P&

K

2

^`">a];>Q6&>P%=G$4#>QG)G>P%=G$4=>c-EF2:698=G132:(+-/0&(,1;2>Q-&€4'OPl:2=(

L:/3G

e = x − x ˆ

^`_k/BGc4=OPG(,]0/0&(,1;2K'(+*]F(,>Q2;&f

˙

e = µ(y) ((A

1

− K

1

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1

(x) − f

1

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1

(x

τ

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−g

¹

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τ

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²

− K

2

C)e +f

²

(x) − f

2

(ˆ x) + g

2

(x

τ

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2

(ˆ x

τ

))

VX²;Y

_`13%=Gc/3*,*+OPM;>PGn*+>Q-n2=13&/0&(,1;2:-QK'132qL1S->A

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1

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1

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2

− K

2

C)

^VX—;Y

>J&

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i

= f

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τ

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(ˆ x

τ

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V[•˜;Y

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-)ES2698=G)1;2=(.-)/B&)(+132³4'>v*i#1;t:->QG)]0/B&)>P%:GvVi±BY</]3>Q6‚*+>v@%=*s&(,@U1'4'bQ*,>

Vˆ°SYj

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; (*+>Q-a6P132:4=(s&(,1;2:-a-%:(,]0/B2S&>Q-a-)132S&a]3OPG(,l:OP>-m

•3^a">Q-Ur1;2:6&(,1;2:-

f

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6J1;2:-&/B2S&>Q-n4'>f"(+L:-698=(,&)zTG)>-Lx>Q6J&)(+]3>-

k

f i

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k

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i = 1, 2

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kf

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, ∀x, y ∈ R

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P

^K

4'>Q%'oR@A/B&)G(+6P>Q-

K

1

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K

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ξ > 0

-)13*+%'&)(+132A4'>-$"´p~[-\-%:(,]0/B2S&>Q-

V

i = 1, 2

^YW

(Ai−KiC)^TP+P(Ai−KiC)+(ξ+k_{f i})I P

P − ¹

k_{f i}I

!

<0 ^<MÌQ4=

>J&

(k

gi

− ξ )I P P −

_k¹_gi

I

< 0

^V[•§;Y

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ˆ

x(t) → x(t)

^dS%:/B2:4

t → ∞

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>e¡…4'¢0…?: _`1;%=G /B2/B*+EF-)>PG *./ 6J1;2F]3>PGM3>Q2:6J> 4'>

*i#1;t:->QG)]0/B&)>P%:GQK«132 (+2S&)G1F4=%=(s&™*+/„r1;2:6&(,1;2 4=> "ES/BL=%=2:10];H

TG9/3-)10]F-F(,(Z

V = V (e(t), e

τ

(t)) = e

^T

P e+ξ Z

⁰

−τ

e(t +θ)

^T

e(t+θ)dθ

^V•P®FY

P ξ > 0

"$#>PGG>P%=GŸ4'>-)EF2:698=G132=(.-)/B&)(+132 6J132F];>PGM3>œ/;-EF@UL'&1B&)(.dS%=>P@U>Q2;&

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V(†Y

V (e, e

τ

) ≥ 0

V(+(†Y

V ˙ (e, e

τ

) ≤ 0

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P > 0

>J&

ξ > 0

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V˙ = e^T`

M^TP+PM+ξI´

e+ 2e^TP“

µf˜₁+ (1−µ) ˜f₂” +2e^TP(µ˜g₁+ (1−µ)˜g₂)−ξe^T_τeτ

<

M4=

; (c132³/BL:L=*,(.dS%=>p-%6P6J>-)-)(+]3>P@U>Q2;&U*+>Q-U(+2=OPMS/B*+(s&OQ-A4'>?}e/3%:698FE;H

; 698n/BGzm>P&W4'>$13%=2=MK=132q13t=&)(+>P2S&f

2µ(y)e^TPf˜₁≤µ(y)(kf1e^TP P e+k_f1e^Te)

2(1−µ(y))e^TPf˜₂≤(1−µ(y))(kf2e^TP P e+kf2e^Te) ^<

M4=

2µ(y)e^TP˜g₁≤µ(y)(kg1e^T_τP P eτ+k_g1e^T_τeτ)

2(1−µ(y))e^TPf˜₂≤(1−µ(y))(kg2e^T_τP P eτ+kg2e^T_τeτ) ^<

M=

6J>NdS%=((+@RL:*,(.dS%=>A

V˙ ≤ e^T“

M^TP+PM+ (µ(y)k_f1+ (1−µ(y))kf2)P² +(µ(y)kf1+ (1−µ(y))kf2+ξ)I”

e +e^Tτ

“

(µ(y)kg1+ (1−µ(y))kg2)P² +(µ(y)k_g1+ (1−µ(y))k_g2−ξ)I” eτ

<

M 4=

_`/3Gƒ6J1;2:-)OQdS%=>P2S&K\>Q2“%=&)(+*,(.-)/32S&CVˆ—;YR>P&CVX¥;YKa-(W*,>-AOQdS%:/B&)(+132:-

-)%=(,]0/32;&>Q-QK=4'>T&[EFL>f{c(+6Q6P/0&(XK=-)132S&a]3OQG)(,l:OQ>Q-j

(Ai−KiC)^TP+P(Ai−KiC) +kf iP²+ (ξ+kf i)I <0, i= 1,2 ^<

M4=

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k

gi

P

²

+ (k

gi

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^Vi¥B˜SY

132‚/BGG(,];>jgq

V ˙ ≤ 0

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;

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&)(+]3>Q@U>P2S&cV[•¥;Y>P&cV[•Q§SYK36P>edS%=(=6P13@UL=*+bJ&>\*./T4'OP@U132-[&G/B&)(+132<*,>

]3>6&>P%=Gn4!#>PGG)>Q%=G\4'>T-)ES2698=G)1;2=(.-)/B&)(+132

e

6J1;2S];>PGM3>at=(+>P2h]3>QG-

0

^K

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W…0U‹B¡4:… :

">-ƒ"´p~[-vV[•¥3YJKaL1;%=G

i = 1, 2

KnLx>P%=];>P2S&ƒuJ&G)>pGOQ-)13*+%=>-A2F%'H

@UOPG(+dS%=>Q@U>P2S&Q^ ; (W1;2³2=1B&>

L

i

= P K

i

VX6J>CdS%=(mOQdS%=(+]0/B%'&hg

K

i

= P

⁻¹

L

i

L=%=(.-)dS%=>

P

>Q-&m(+2F]3>QG-)(,t:*,>YK132v1;t'&)(+>P2S&N4!#/BL=GbQ- V[•—;Ym

(A

i

− K

i

C)

^T

P + P (A

i

− K

i

C) + (ξ + k

f i

)I

= A

^T_i

P + P A

i

− C

^T

L

^T_i

− L

i

C + (ξ + k

f i

)I

Vi¥3¥3Y

">m@U>P@t:G)>W4=>m4'G13(,&)>c>-[&\*,(+2=OQ/3(,G>c>Q2

P

^>J&

L

i

K'4'1;2:6a*+>Q-€"´p~[-

V[•¥3Yc>J&RV•Q§SY9K

i = 1, 2

KLx>P%=];>P2S&muP&)G>Nrˆ/36P(,*+>P@U>Q2;&jG)O-1;*,%:>Q-aL:/3G

%=2p/B*+M31;G)(,&)8:@R>T-&/B24=/BG94h4#1;L'&)(+@U(+-/0&(,1;2q6P132F]3>PoF>;^

-

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p

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4!#%=2Ÿ@%:*s&(,@U1'4'bP*+>h6P13@ULx1;-)O<4'>

p > 0

-)E'-[&bP@U>Q-698:/B13&)(.dS%=>Q- G>J&/3G4=OQ-QKF4=OJl:2=(4'>N@A/B2=(+bPG>N/B2:/3*,1;M3%=>Tg‚Vˆ°;Ym

x ˙ = P

p

i=1

µ

i

(y) (A

i

x + f

i

(x) + g

i

(x

τ

)) y = Cx

Vi¥B§;Y

p

X

i=1

µ

i

(y) = 1, 0 ≤ µ

i

≤ 1

^VX¥B®SY

¨m>*./A@UuP@U>f@A/B2=(+bPG>dS%#gU*X#OQdS%:/B&)(+132­V±BYKx*./<4'EF2:/B@U(.dS%=>4'>

*i#1;t:->QG)]0/B&)>P%:Gn4=><VX¥B§SY\>Q-&a698:13(.-(+>N4'>N*+/r1;G)@U>U

˙ˆ x =

p

X

i=1

µ

i

(y) (A

i

x ˆ + f

i

(ˆ x) + g

i

(ˆ x

τ

) − K

i

(y − Cˆ x))

^VX¥3¯;Y

">j&8=OP1;G)bQ@R>U•NLx>P%'&W/3%:--(!uP&)G>jM;OP2=OQG/3*,(.-)O

7839Ž0¡20U… :

; (*+>Q-a6P132:4=(s&(,1;2:-a-%:(,]0/B2S&>Q-a-)132S&aG)>Q@RL:*,(+>Q-T

•3^a">Q-er1326&)(+132-

f

i

K:>J&

g

i

K

i = 1, p

-1;2;&aG>Q-)L>6&(,];>P@U>P2S&

k

f i

H

>J&

k

gi

H*+(,L:-698=(,&)zQ(,>Q2=2=>Q-NVˆ]31;(,GfV•3•YY

¥'^n~*>Po'(+-&)>

ξ > 0

K=%=2=>N@A/0&G)(.6J>N-)EF@ROP&)G(+dS%=>;K=4'OJl2=(,>L1S-(,&)(+]3>

P

^>P&

p

@U/B&)G(+6P>Q-

K

i

K

i = 1, p

-)13*+%'&)(+132:-a4=>Q-

2p

"´p~[-T

(Ai−KiC)^TP+P(Ai−KiC)+kiI P

P −¹

k_iI

!

<0 ^<Q4=

>J&

(k

gi

− ξ )I P P −

_k¹_gi

I

< 0

^VX¥S±BY

/B*+13G9-c4=/32:-W6P>6Q/3-QKVX¥;¯3Yn>-[&j%=2v1;t:->QG)]0/B&)>P%:GW4'%v@%=*s&(,@U1'4'bQ*,>

VX¥3§;YJ

ˆ

x(t) → x(t)

^dS%:/32:4

t → ∞

^{^}

>e¡…4'¢0…?: "/Ÿ4=OP@U132:-&)G9/0&(,1;2³>Q-&ƒ/B2:/3*,1;M3%=>pgŸ6J>Q*,*+>C4'%

n8=OQ13GbP@U>U•3^ ; (1;2h(+2S&)G1F4=%=(s&c*./R2=1B&9/0&(,1;2

M =

p

X

i=1

µ

i

(y) (A

i

− K

i

C)

^VX¥3²;Y

*+>Q-OQdS%:/0&(,1;2:-4=><&[EFL>‚{a(.6P6Q/0&)(TV[•—;Y->hM3OP2:OPG9/B*+(+-)>P2S&-)13%:-*+/

r13G@U>U

M

^T

P + P M +

p

X

i=1

µ

i

(y)k

f i

P

²

+ X

^p

i=1

µ

i

(y)k

gi

+ ξ I < 0

VX¥3—;Y

JJ R!I!=4J ) &

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4'(,žxOQG)>Q2S&-Q^

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



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˙

x

1

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¹

+ αx

2

+ f (x

¹

)

˙

x

2

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1

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2

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3

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x

3

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Vˆ§;˜;Y

13¶R*+>Q-`2=1;2=*,(+2=O/BG(s&OQ-

f

^>P&

g

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f (x

¹

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¹

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¹

(t)) = ε sin(σx

¹

(t − τ ))

Vˆ§:•Y

f g

4'>A"(+L:-698=(,&)z3KL1;%=GN4'>Q-N6P132:-&/32;&>Q-jOPMS/B*+>Q-mG)>-Lx>Q6J&)(+]3>Q@R>Q2S&Ng

|αδ|

^>P&

|εσ|

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1

^

α

1

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¹

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¹

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¹

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1

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1

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1

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VX§;¥3Y

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2

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2

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VX§3§;Y

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0 0.5

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−0.2 0 0.2

y x z

M

! "#

1

0.05 0.1

0.15 0.2

0.25 0.3

0.1 0.2 0.3 0.4 0.5 0.6

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y x z

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µ(.)

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698=1;(+-)(,>N4'>T*./Rr13G@U>

µ(y) = 1 + tanh(ηy) 2

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η

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µ(.)

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0

^>J&

1

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µ(.)

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W…0U‹B¡4:…%$:

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k

f i

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k

gi

K

i = 1, 2

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4=/32:-NVˆ°SYj

C = 1 ζ 0

^VX§;¯3Y

ζ 0 < ζ < 1

Vˆ/];>Q6

h = f

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g

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kh(x

1

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1

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2

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2

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≤ k|ζ|kx

2

− x ˆ

2

k

Vˆ§;°;Y

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&)1;%'&)>-f@%=*s&(,L:*,(+OP>-NL:/BG

ζ

^‚~[6J($1;2­/‚L=G(+-

ζ = 0.01

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k

f1

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ζ

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4'%‚@%:*,(,&)@U1'4'bP*+>3^

−0.5 0

0.5 1 1.5

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y x z

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K

1

=

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−22.73009924194811 41.15127226834878 13.21359829809531

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

Vˆ§F±BY

K

2

=

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−8.89211050110927 23.15660169589151 8.79844323851249



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Vˆ§;²;Y

*./@A/B&)G(+6P>

P

>Q-&c4'132:2=OP>N>Q2IVX§3—;YJK'>J&

ξ = 2

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0.1 0.1 0.1

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0 0.01 −0.1

^{T V®•Y}

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µ(y(t))

>Q-&€&)G9/36POP>alM3%=G>c®:KF-/3698:/32;&ed;%:>

η = 0.7

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*./lM3%=G>f¯'^

P =





0.66711349131098 −0.24680949689941 0.12692206278854

−0.24680949689941 0.72345874659580 −0.21169486677705 0.12692206278854 −0.21169486677705 0.29023507485856





Vˆ§;—;Y

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

/ ! "+ !

µ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.1 0 0.1 0.2 0.3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.2

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.5 0 0.5 1

"

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y

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4'>

y

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y

2

(t)

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x

3

(t)

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Lx>P2:4=/32S&W4'%‚@R>-)-/BM;>

u(t)

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u ∈ [0, 1]

^Ym

y

2

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3

(t − T

u

u(t))

^V®F¥3Y

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u

^Km1;2

-)%=L=Lx1;-)>fd;%:>

0 < T

u

< T

e

K:13¶

T

e

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; ('1;2R/3L=L=*+(+dS%=>e*+/mr1;G)@%=*,>n4'> Z/ES*+13G)H"/BM;G/32=M3>kgj*i#>PoFL:G)>-)-)(,1;2

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x

3

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2

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3

(t)T

u

u(t)

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˙

x

3

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u(t) = x

3

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2

(t) T

u

x ˙

3

(t)

V®;®SY

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ˆ

u(t) = x ˆ

3

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2

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u

x ˙ˆ

3

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K

i

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i,1

κ

i,2

κ

i,3

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u(t) ∈ [0, 1], ∀t > 0

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