1232019

Supervision dynamique d’un canal d’irrigation
Nadia Bedjaoui
To cite this version:
Nadia Bedjaoui. Supervision dynamique d’un canal d’irrigation. Automatique / Robotique. Institut
National Polytechnique de Grenoble - INPG, 2006. Français. �tel-00140232�
HAL Id: tel-00140232
https://tel.archives-ouvertes.fr/tel-00140232
Submitted on 5 Apr 2007
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INSTITUT NATIONAL POLYTECHNIQUE DE GRENOBLE
N° attribué par la bibliothèque
|__|__|__|__|__|__|__|__|__|__|
THESE
pour obtenir le grade de
DOCTEUR DE L’INP Grenoble
Spécialité : « Automatique-Productique »
préparée au laboratoire Cemagref de Montpellier
dans le cadre de l’Ecole Doctorale
« Electronique, Electrotechnique, Automatique, Traitement du Signal »
présentée et soutenue publiquement
par
Nadia Bedjaoui
Le 06 Décembre 2006
Titre :
Supervision dynamique d’un canal d’irrigation
DIRECTEUR DE THESE :
Damien Koenig
CO-DIRECTEUR DE THESE
Xavier Litrico
JURY
M.
M.
M.
M.
M.
M.
Didier Georges
Jacques Sau
Dominique Sauter
Vincent Kulesza
Damien Koenig
Xavier Litrico
,
,
,
,
,
,
Président
Rapporteur
Rapporteur
Examinateur
Directeur de thèse
Co-encadrant
Remerciements
Ce travail a été préparé au Cemagref de Montpellier au sein de l’équipe
TRANSCAN dont le thème de recherche est : Modélisation et Régulation
Des Canaux d’Irrigation.
Je tiens à remercier Mon directeur de thèse Damien pour tout le soutien
et la confiance qu’il m’a témoignés tout au long de ces années de travail ensemble, pour sa disponiblité aux réunions tenues à Montpellier, pour m’avoir
permis et offert la possibilité d’enseigner dans son établissement.
Je voudrais également remercier Didier Georges d’avoir accepté d’être
mon directeur de thèse pour une période, pour ses recommandations, conseils
et pour avoir accepté d’être président de jury.
Je voudrais exprimer mon respect à Dominique Sauter pour l’intérêt qu’il
a porté à mon travail et d’avoir accepté de lire et de critiquer mon manuscrit.
Je souhaiterais témoigner ici de ma grande sympathie à Jacques Sau pour
tous les conseils et les connaissances dont il m’a fait part et pour le temps
qu’il a consacré à lire mon manuscrit.
Je voudrais exprimer ma reconnaissance à Xavier Litrico pour sa confiance,
son soutien moral, ses encouragements, ses précieux conseils, sa forte sympathie et ses critiques constructives.
A TRANSCANette, je dis : Je remercie Dieu de m’avoir donné la chance
de travailler parmi des gens aussi compétents et sympas à la fois.
A Caroline, je dis merci pour toute ton aide professionnelle, morale et
affective.
Je voudrais également remercier tout le personnel du Cemagref pour la
sympathie et la gentillesse dont ils m’ont fait part.
Un grand merci à toutes les personnes qui ont contribué de près ou de
loin à l’achèvement de ce travail notamment ma famille et mes amis.
Résumé
L'intérêt principal des canaux d'irrigation est d'assurer une bonne distribution de l'eau aux usagers tout en préservant la ressource en eau. L'automatisation des ouvrages de régulation des canaux a largement contribué à
l'amélioration de la gestion de l'eau. La commande automatique d'un canal
consiste à agir sur l'ouverture et la fermeture des vannes suivant les données
issues des diérents instruments de mesures placés sur le canal. Le bon déroulement du processus de commande exige que ces mesures soient ables.
En pratique, ces mesures sont souvent entachées de bruits ou encore d'erreurs. Ces erreurs sont généralement l'eet de défauts dans les capteurs ou
actionneurs. Il est par conséquent impératif de munir le canal d'un système
de supervision permettant de détecter et de localiser ces défauts.
Nous développons dès lors une méthodologie de détection et de localisation des défauts dans les canaux d'irrigation. Cette méthodologie s'appuie sur
les techniques de réconciliation de données et de diagnostic à base d'observateurs. La première partie est consacrée à l'application de la réconciliation de
données sur un site réel du Canal de Gignac. Les canaux d'irrigation étant
des systèmes à retards, la deuxième partie de ce travail porte sur l'extension
des méthodes de diagnostic à base d'observateurs pour systèmes à retards.
Canal d'irrigation, diagnostic de défauts, observateur, réconciliation de données, systèmes à retards, détection et localisation.
Mots clès :
Abstract
The main objective of irrigation canals is to satisfy the users' demands on water while preserving water resource. Manual for a long time, canals' control is
now automatic which improves water management. Canal automatic control
consists on manipulating structures according to measurements of sensors.
The good unfolding of control process imposes correct measurements. However in practice, measurements are never correct and are usually subject to
random noise and errors. These errors are generally instruments faulty effect. Consequently, it seems important to supply a monitoring system which
allows fault detection and isolation.
The work consists in developing a fault detection and isolation method for
irrigation canals. This method is based on data reconciliation and diagnosis
based observer. The rst section concerns the application of data reconciliation on real data of the canal de Gignac. Irrigation canals are time-delay
systems. The second section is about the extension of the diagnosis based
observer to time-delay systems.
Keywords : Irrigation Canal, Observer, Data Reconciliation, TimeDelay Systems, Fault Detection and Isolation.
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s
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p22 (s) =
q0 (s)
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ì d ; "‡Qî
λ2 eλ1 X −λ1 eλ2 X
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yi (s) =
ai −τi s
qi (s)
se
−
ai
s (qi+1 (s)
+ pi (s))
ì d ; €›:î
4/$L8+-8Z:*€—
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qi+1 (s) = bi yi (s) + ki+1 ui+1 (s) + b̄i ȳi (s)
ì d ; "ºQî
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yi
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yif = yi + δyi
qi+1f (s) = bi yif (s) + ki+1 ui+1 (s) + b̄i ȳi (s)
qi+1f (s) = bi (yi (s) + δyi (s)) + ki+1 ui+1 (s) + b̄i ȳi (s)
qi+1f (s) = qi+1 (s) + bi δyi
qi+1f (s) = qi+1 (s) + δqy (s)
ì d ; "Qî
?`ž ?baO*181'S+->&"*2 1'8(>$#%ZQ"*1R*Z:*+šCE*BK1', ''4/$?,"! ?& '#%
ȳif =
ȳi + δ ȳi
qi+1f (s) = bi yi (s) + ki+1 ui+1 (s) + b̄i ȳif (s)
qi+1f (s) = bi yi (s) + ki+1 ui+1 (s) + b̄i (ȳi (s) + δ ȳi (s))
qi+1f (s) = qi+1 (s) + b̄i δ ȳi (s)
qi+1f (s) = qi+1 (s) + δqȳ (s)
?`ž ?baO*171',+A*YZ:*$$WšCE*Y2\4:#%#-4/$¯ $¯(?baO*17)(4/$$? 2*'
ì d ; d:‡Qî
ui+1f =
ui+1 + δui
qi+1f (s) = bi yi (s) + ki+1 ui+1f (s) + b̄i ȳi (s)
qi+1f (s) = bi yi (s) + ki+1 (ui+1 (s) + δu(s)) + b̄i ȳi (s)
qi+1f (s) = qi+1 (s) + ki+1 δu(s)
qi+1f (s) = qi+1 (s) + δqu (s)
ì d ; d0€î
?`ƒ '? +- ZQ K $LM4/1D*22\4/'H(N(? _#%,šCJN(? _#%Sa¸4/1'$#E*Z:*$LH+A*)2'#-N"! ?& '#%
q
@+-N(? _#%8*2'@+A*)2'#->"! ?& '#%
qi+1f (s) = qi+1 (s) + pi (s)
qi+1f (s) = qi+1 (s) + δqp (s)
ì d ; d:dQî
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Œ:‡
È ã=È ãI‚
åæj`•éL+—WcŒIŽ†c“4ŒL‡†c†`é
ž *$S+O! 4/_05& #Ša(]! *ZQ4/#%'8(S'?1+% *@9Q? $? '#-[L1"I$4/[email protected]'#%K4/[email protected][email protected](4/$0—
$?(01 K4F( +-84/1a¸4/'K8*/(0#%K $#-4/$$ +%+-M[L1J! 4/$ ' 2'? $LN*ZQ&H+O! bc02\4: *$L
∗;
* î ¢ ] |ñF¤§D|m¥òͧFE€ E§F¤¬|m}èm§¶µ
+-)' *'(R+-)2+%1M9:' *$(U ∗
I
I
I
τm
s = τm s yr = ur = ai τm Q0 yi∗ = yy
I
I
q∗ = q ;CJ?[L1*#-4/$Wì d ; €›:î Rì d ; "ºQî
ȳi∗ = yȳ u∗i = uu τi∗ = ττ
i
Q
"! ?& '#%ZQ $L8(4/$&š
°F4/#%
i
i
ri
i
ri
i
0
i
m
1 −τi∗ s∗ ∗ ∗
qi (s )
s∗ e
yi∗ (s∗ ) =
i
ri
i
−
ì d ; d:ŒQî
1 ∗
∗
s∗ qi+1 (s )
_ î ¢ ]|ñF¤§D|m¥òͧFE€ E§F¤¬|§R¤WE§
• $.*22+%#-[L1*$L<+-m$4/ *#-4/$<2'?&?( $L¬TH+O! ?[L1*#-4/$(+A*HZ:*$$:I/4/$74/_#- $LHš
ì d ; d’Fî
q ∗ (s∗ ) = b∗ y ∗ (s∗ ) + k ∗ u∗ (s∗ ) + b̄∗ ȳ ∗ (s∗ )
44™
i+1
i i
yr
yr
yr
b∗i = bi Q
, b̄∗i = b̄i Q
, ki∗ = ki Q
0
0
0
;
i+1 i+1
i i
& î ¢ ]|ñF¤§D|m¥òͧFE€ E§F¤¬|m}~QE¤
$ 2\4: *$L ∗ t > $U&4/K,_#%$*$LVì d ; d:ŒQî .ì d ; d’Fî I]4/$Í4/_#- $L>+A* ' +A*#-4/$
• U
(0^F$*K7#-[L1>t $L=
'>τ[email protected] $L'?
S+-[email protected]/'#- 2\4/1'H&*/[L1N_#-ba;
m
yi∗ (s∗ ) =
1 −τi∗ s∗ ∗
∗
(bi−1 yi−1
s∗ e
(ui , ȳi−1 )
∗ )−
+ ki∗ u∗i + b̄∗i−1 ȳi−1
yi
1
∗ ∗
s∗ (bi yi
∗ u∗
∗ ∗
+ ki+1
i+1 + b̄i ȳi )
ì d ; d:”Qî
$• *22+%#-[L1*$L8+A*7' *$˜a¸4/'K?)(,CE*2+A*/&,#%$LZQ 'BT7+A*7' +A*#-4/$ýì d ; d:”Qî I 4/$
U
4/_#- $LH+O! ?[L1*#-4/$D(0#]? ' $L#- +%+-)(01V2' K7#- '84/'(0'>1#%Z:*$Lš
∗ (t∗ − τ ∗ ) + k ∗ u∗ (t∗ − τ ∗ ) + b̄∗ ȳ ∗ (t∗ − τ ∗ )
ẏi∗ (t∗ ) = b∗i−1 yi−1
i
i i
i
i−1 i−1
i
∗ u∗ (t∗ ) − b̄∗ ȳ ∗ (t∗ )
−b∗i yi∗ (t∗ ) − ki+1
i+1
i i
ì d ;d · î
584/1©122\4:4/$ (*$ $4/'W&"*/ ]( ! ? 1(W[L1D+A*ÍZ:*'#A*#-4/$ (01`$#%ZQ"*1 +-
2+%1V*K4/$LV $L1+%+- y∗ = 0 ;8
5 4/1 122\4:4/$©? 9L*+- K $LV2\4/1'©122'#%K '
+O! #%$L? ' */& #-4/$3(01W_#-ba 3 *0ZQ&B+-8_#-ba¸ 1 2 [L1,+-)$#%ZQ"*1W+-)2+%1N*Z:*+ Ȳ &4/$ *$LHì¥(Z:*'#A*#-4/$B$L1+%+- ȳ∗ = 0 î m[L1+A*@Z:*$$S*Z:*+#%K7K4/_#%+- u∗ 2= 0 ;
3
CJH?[L1*#-4/[email protected]/$L8*+-4/'@'?(012#%8*10cVbc02'#-4/[email protected]#%Z:*$L,š
ẏ1∗ (t∗ ) = k1∗ u∗1 (t∗ − τ1∗ ) + b̄∗0 ȳ0∗ (t∗ − τ1∗ ) − b∗1 y1∗ (t∗ ) − k2∗ u∗2 (t∗ ) − b̄∗1 ȳ1∗ (t∗ )
ẏ2∗ (t∗ ) = b∗1 y1∗ (t∗ − τ2∗ ) + k2∗ u∗2 (t∗ − τ2∗ ) + b̄∗1 ȳ1∗ (t∗ − τ2∗ ) − b∗2 y2∗ (t∗ )
ì d ; dQ›:î
ì d ; d:ºQî
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Œ0
( î ¢ ]|ñF¤§R|¬«%£Q"0.|m}~QE¤
CJ8K4F( +-M(]! ? *S&4/''2\4/$(*$LH1$©K4F( +-NT)? * S $L'?S' *'(?S(
+A*,a¸4/'Kš
P2
P2
P2
∗ ∗ ∗
∗
∗ ∗ ∗
∗
∗ ∗ ∗
∗
ẋ∗ (t∗ ) =
i=0 Ai x (t − τi ) +
i=0 Bi u (t − τi ) +
i=0 B̄i ȳ (t − τi )
∗
∗
∗
∗
∗
y (t ) = C x (t )
ˆ ™
ì d ; d:Qî
I
T
J+-mZQ& 1'E(]! ? *"I ∗
u = u∗1 u∗2
¬+-HZQ& 1'S&4/K7K.*$(:I ȳ∗ = ȳ∗ ȳ∗ T ∈ <2 ¬+-HZQ& 1'S( $0—
∈ <2
0
'?K1'? ∗
1+-HZQ& 1'S(4/'#-K1'?";
∗
∗ T
2
B+A*RK.*'#-&Yy (]! =? *"I y1 ∗ y2B+A*R∈K.<*'#-&Y(]! $L'?:; ∗ 4/$L7+-BK.*'#-&
A∗0
(]! ? *8' *'(?M ∗ 4/$L8B+-0HK.*'#-&M(]! $L'?M' *'(A?i "I
;
Bi
i = 1, 2 C ∗
+A*)K.*'#-&,(N4/'#-:;
τ0∗ = 0 x∗ =
T
∈ <2
−b∗1 0
0
∗
,
A
=
1
0
−b∗2
0
∗
0 −k2∗
k1
∗
∗
B0 =
, B1 =
0 0
0
∗
0 −b̄∗1
b̄0
∗
∗
B̄0 =
, B̄1 =
0
0 0
1 0 ;
C∗ =
0 1
A∗0 =
y1∗ y2∗
0
0
, A∗2 =
0 0
b∗1 0
0
0
, B2∗
=
0 0
0 k2∗
0
0
, B̄2∗
=
0 0
0 b̄∗1
,
î ¢ ]|ñF¤§R|m¥òͧFE€ E§F¤¬|§L7|£¶µâ}J"
584/1)*ZQ4/$)K4/$L'? [L1.[L1 +[L1.4/#%,+-â^F2\( (?baO*1)4F&& 1'' $LB(*$
1$D&"*$*+OI#%+J2\ 18*ZQ4/#%'8+-NK\ K,&] H[L1J! 1$R(?baO*18*/& #-4/$$ 1'M H[L1N+O! ? 1(
(B& +%1#Š—â& #<2\ 'K >()9Q? $? ' *+%#- 'N+O! ? 1(T.4/1N+-N*1'Nâ^F2\N(B(?baO*1";
584/17&4/$#-(? '4/$.(7+-4/'(*$7$4/'Y^0 K©(]! ? 1(V1$#-[L1 K $L.+-(?b—
aO*1M*/& #-4/$$ 1'M[L1J! 4/$RK4F(? +%#- ' *.&4/K7K)1#%Bš u∗ (t) = u∗ (t) + f ∗(t) ; ƒ *'
&4/$?[L1 $L"I€+A*@' 2'? $L *#-4/$7(]! ? *<(01,K4F( +-(01B^0f K¬ $)(?baO*1m"! ?& '#%š
ẋ∗ (t∗ ) =
P2
∗ ∗ ∗
i=0 Ai x (t
+E ∗ f ∗ (t∗ )
y ∗ (t∗ ) = Cx∗ (t∗ )
ˆ ™
E ∗ = B0∗
È ã=È ã=È
f ∗ = δu∗
− τi∗ ) +
P2
∗ ∗
i=0 Bi u
(t∗ − τi∗ ) +
P2
i=0
;
B̄i∗ ȳ ∗ (t∗ − τi∗ )
ì d ; Œ:‡Qî
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7
ž *$N&B[L1#<1#%"I\$4/1>*+%+-4/$NZ:*+%#-( 'M+-BK4F( +-Vì d ; d:Qî .ì d ; Œ:‡Qî ;]CE* +-4/$0—
9:1 1' (D&*/[L1D_#-ba>(01`&"*$*+M&4/$#-(? '?D ( Œ:‡:‡:‡ KY;¬CJ.' *'( 4/$L
#-( $L#-[L1D122\4:?V&4/$$L1D Y&4/$ *$L¯ì τ̇ = τ̇ = 0 î ;@CJ 1'YZ:*+- 1'Y
1
2
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Œ:d
;<CJ*1'72*' *K '(01ýK4F( +-©4/$L72\4/1'7Z:*+- 1'
τ1 = τ2 = 647s
;ECJB&4F&à.& #- $LB(.+%#%$?"*'#- *#-4/$¯(,Z:*$$B4/$L)+-,Z:*a+- 11=
'
−5
a2 = 3.21.10
1#%Z:*$L b̄
I
I
2
2
0 = b̄1 = b̄2 = −29.36m /s b0 = b1 = b2 = 29.05m /s k0 = k1 =
;CJ,(? _#%H$4/K7#%$*+m
; ƒ *'M&4/$?[L1 $L"I+-
2
3
k2 = 18.11m /s
Qnom = 14m /s
K.*'#-&H(01YK4F( +->4/$LH(4/[email protected]*'>š
−0.4084 0
=
−0.2787
0
0
0
∗
−3
A2 = 10
−0.4675 0 0 −0.3714
B1∗ = 10−3
0 0
1 0
C∗ =
0 1
A∗0
10−3
0 −0.2432
=
0 0
0.55 0
∗
−3
B0 = 10
−0.42
0
0
0
∗
−3
B2 = 10
0.6320 0
A∗1
10−3
ì d ; Œ0€î
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Z
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Ì ZQ&
P2
P2
ẋ(t) =
i=0 Ai x (t − τi (t)) +
i=0 Bi u (t − τi (t)) + Ef (t)
y (t) = Cx(t)
WEX
š c XeZ
0 ≤ τi (t) ≤ hi , 0 ≤ τ˙i (t) ≤ di ≤ 1
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> Œ ; ?,½¸HVY2A Q`OP K N2RSQFGDLK DLB4O9QSC
∆V
6%
Ž‘
2\ $(*$L"I4/$D$N' *Z:*#%+%+->2*/H(0#%'& K $LN*ZQ&>[email protected]$#%ZQ"*10cY(]! "*1JIK.*#-
*ZQ&M+-S&4/U5N6sÎI0*#%$#J4/$©#- $LH? 9L*+- K $LH&4/K72N(S#%$& '#%1(@1'+-
*1 1'H(01V' */(0#- '8 [email protected](?&"*+A*9QH $L'>+-p"? '4:H(M+O! ?& +%+-, @+-M' */(0#- '
C! #%$& '#%1(,&4/K7#->1'
Z = Y + 0echelle
Z
H*+-4/'
∆Z = ∆Y + ∆echelle = 0.02m
;
∆Z = 0.02m
ô ¤¥}E0€ ¦|§R¤W/E~Q§D|§L.òͧL€}E§L
ƒ 4/1'8&"*+-& 1+- 'H+A*)Z:*'#A*$&,(SK1'"I$4/18*ZQ4/$84/2?N2\4/1'@+A*)K? 4F(
1#%Z:*$L8_*/?M1'+-HaO*#%[L1:IF2\4/1'1$©$#%ZQ"*1V(M&4/$0= *$&N( :” IF+O! #%$& '#Š—
1(>S+-N(4/1_+->(N+O! ?&"*'˜—Oâ^F2\ 0 · 9d 1 š
1
σ 2 = ( ∆)2
2
ˆ $Y4/_#- $LH+-SZ:*'#A*$&H1#%Z:*$L,š
”:‡«8ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
σZ
7 × 10−3
σU
5 × 10−3
σV
3 × 10−2 V %
> Œ ; d?,½¸HVY2A Q`O9Qj^&HJFLK HVN2RSQ
Ž‘
½ §FòÍ2âJ}E§ y“’a”,•—– ˜1™š›™6œfŸžfžaœ ¡£¢{žAV¤E¢{ž¥3¦§žAŸžfŸ¨© ¤=¦EªN¥až ” ¡£¤E«¬¥¤­žaœ¦N¥{™V”q¥¡£”®¨L¯
”{– ™V¤CŸžŸž9V¤ °±¨ ¢{ž •¦Eœ9³²Ÿ´ ”®žaœµ
ä ã=ä ã=ä
ñ“4Š¸Ë¶Œ `‹ŒL‡†Ì`ÅéL— ˍ`‡†c`—W†cŠ¸[—W†`—WéG” ‹ŒIê ’`
ƒ +%1#- 1'>K4F( +-> *#-[L1N4/$L)T.$4/'(0#-2\4:#%#-4/$U2\ 'K *$L,(7&"*+Š—
& 1+- @
' +A*BZ:*+- 1'@(01D(? _#%,š
? ¢ ]|ñF¤§‘á ðJ|}¤¥=âJ}E§Y|§R¤RE§ /
Qv = Cd LU
p p
2g H1 − H2
44 ™ C H+-,&4F&à.& #- $LM(,(? _#%"I L +A*7+A*'9Q 1'M(01R&"*$*+m g +O! */&&? +-? ' *€—
#-4/$W(d >+A*72\ *$L 1'"; U ' 2'? $L)+O! 4/1ZQ '1'B(>Z:*$$:I H H +-
2
*1 1'H(]! "*1R*K4/$L8 H*Z:*+TB+A*)Z:*$$> Q +-N(? _#%"; 1
v
I
;
Cd = 0.68 L = 1.53m
§ / CE*&4/1'_\,(N *' *9Q,(4/$$>1$
? ¢ ]|ñF¤§R|§W¤Í~Q }Eè§D|§D"­+
;E8! N1$
' +A*#-4/$Í1$#%ZQ4F[L17 $L'+-7(? _#% N+A* *1 1',(]! "*1
Q
H2
' +A*#-4/$`(? 'K7#%$?Rbc02\? '#%K $L *+- K $LY*1`(? _1 (. *#-4/$72'#%$L *€—
$#- '+-4/'[L18+-M&"*$*+]K7#- $V"*1Y*2'@*ZQ4/#%'@? ?8$ 4"^Q?N(84/1
+-N*+%9:1N >*1'N(? r2 /M N(?& ";\C! 4/2\? ' *#-4/$3>aO*#%82*'M1$ 5*10—
9Q"*9QVTD(0# ]? ' $L7Z:*+- 1'B(01ý(? _#%B $¯1$“2\4/#%$L7(4/$$?V(01¯&"*$*+O;mCJ
5*19Q"*9QH' 2\4:81''4/#-(4/$$?NšQ+A*>ZF#%@K4"^Q $$N(]! ?&4/1+- K $L V I
+A*N*1 1'(]! "*1©(*$+-H&"*$*+ H IL[L1#OIF*ZQ&[email protected]'4=+ ' *$ZQ ' *+\(01.+%#%"I
(? +%#%K7#%M+A*)& #-4/$©K4/1#%+%+-? S ;FCJM2 (? _#% Q S&"*+-& 1+-?M $©K,1+%#%2+%#A*$LH+A*
& #-4/$YK4/1#%+%+-?N2*'@+A*BZF#% Q = V S ;
EC *,&$#-[L1M' */(0#%#-4/$$ +%+-N(M(? 'K7#%$*#-4/$Y(01V(? _#%S&4/$#-NT>2+-4/$0—
9Q ' (*$.+-Y&4/1' (]! "*1JIaO*/&D*1`&4/1' *$L"I1$K4/1+%#%$ ? +%#-&4V¶O(*+N4/1
$&4/')1$U&4/1' *$L4/K '? +-& '4/K.*9:$? #-[L1DìZQ4/#%'8=9:1' Œ ; "dQî ; ˆ $R'b—
+- ZQD+- ZF#% (01 &4/1' *$LV $ (0# ]? ' $L©2\4/#%$L©#%1?©1'.2+%1#- 1'
ZQ '#-&"*+-H(N+A*B& #-4/$YK4/1#%+%+-?)*€=$Y(>(?(01#%'N+A*BZF#%8K4"^Q $$ V ;
5*19Q"*[email protected]& +A*/#-[L1$S2\ 1` '2' *#-[[email protected][L12\4/$& 1 +%+- K $L"; Ì 2'
(@$4/K,_' 10c>5*19Q"*9Q"IL4/$ ? *_+%#%+A*>&4/1'_\H(@ *' *9Q:; • +%+-Hbc02'#%KH+A*
&4/''2\4/$(*$&, $L'[email protected](? _#%H @+-S*1 1'H(]! "*1D(*$S+->&"*$*+O;
Qct = a(H2 − H0 )b
44™ I 4/[email protected](@&4/$ *$LH +A*)*1 1'H(]! "*1Y2\4/1'+A*/[L1 +%+-N+-
(? _ a#%HbS$L1+OI H ' 2'? $L>+A*)H0*1 1'8(]! "*1R*Z:*+ETB+A*)Z:*$$> Q +-
2
ct
î_ î_VÂ`ÄJÉ%ÂHÁÉ%qÂ'"ÒÂ8Â8ÄÍÆMÅu2É32.8Â
”0
:.;=<.> Œ ; "d?A«¡2MVDLM$O9BRSM&B9FIHVN%DLM&T$ÀDGFLQ ¢SA QSRDGFLM&TUHVí&N2¢DLK „%B2QjB9DLK A K CG¢j´–M&B9F‡A QSC›·GHVB2í&QXHVí&QSC
(? _#%";
W d œœQÓ àŠÚ Öؙà Ѐ×Y×/ÛQÙ Ÿ ܙ๸ÛQÔ³º
ß
a = 1.95 × 10(−4+2b) b = 1.8126
ÔØ
H0 = 1.4 ×
.š&4/1'_\72\ 'K ,((?(01#%'+-NZ:*+- 1',(013(? _#%>2\4/1',(0#]? ' $L
10−2 m
Z
*1 1'8(]! "*1Y' +- ZQ?";
:.;=<.> Œ ; "Œ?,½¶FIHVRS¢cO9Q A H.RSM&B9FLY–QjO9Q DIHJFIHVí&Q
Qct = a(H2 − H0 )b
? ¢ ]|ñF¤§¯|§¯¤ý~Q|§“|§ÍEA"§L""§/ ž '$#- ' K $L"IE+A*©K1'72'?& #-
&4/$L#%$L1M(@+A*NZF#%@K4"^Q $$:IF $ & ' *#%$ *#-4/$"IL¬1$.2'4/9:'
&4/$#-(? ' *_+-:; • +%[email protected]¬4/_ [email protected]*'<+A*MK? 4F(@(<1+%' *€—â4/$' *ZQ ' *$L
+- &4/1'B(]! "*1J;mCJ.K4F( +- ( &4/'( (.ZF#%2\ 'K "ImTY2*'#%'B(.+A*
K1'D(©+A*WZF#%V $+%#%9:$ V (4/$$?Y2*'+-©K7#%$#-4/$#-&:I¬(01#%' *$L
(]! "*1 H (*$+-N&"*$*+J @(01V' *22\L4/' K = V $L'N+A*)ZF#%HK4"^Q $$
(,+O! ?&4/1+- K $L V M(>+A*ZF#%, $D+%#%9:$ VV I (,(? 'K7#%$ 'M+-)(? _#%
L
L
”:d«8ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
2*'@+A*,a¸4/'K,1+-.š
Q
Qcv = Lc HVL K
44 ™
N+A*©+A*'9Q 1')(013&"*$*+OI
' 2'? $L+A*V1'˜aO*/&7K4/1#%+%+-?.(01
Lc H
&"*$L*+Oc;<CJ©' *22\4/'
\
2
1
\
'
V*22'4F&?V2*'B1$ý2\4/+%^F$«/KV(01
K = VV
( 10c0#- KB4/'(0'
(4/$LN+-8&4F&à.& #- $L I K = (c1 + c2 H + c3 H 2 )
4/$L8(? 'K7#%$?Hbc02\? '#%K $L *+- K $L";\CJ>K4F( +-M=$*+E(N+A*7&c4/1 '(c,2 (
c3
ZF#%M1#%+%#-?N,š
L
Qcv = Lc VL H(c1 + c2 H + c3 H 2 )
d œœQÓ àŠÚ Öؙà Ѐ×R×/ÛQÙ Ÿ ܙ๸ÛQÔ L = 1.33m ß c = 1.8445 ß c = −1.9594 ÔØ
c
1
2
Z
0.9383 š
CE*ZF#%>K4"^Q $$ M(? 'K7#%$?)2*'8+O! ?[L1*#-4/$
44™
V
Q = VS
W
Lc H
ä ã=ä ã÷þ
S+A*BZ:*+- 1'@(N+A*)2*#%'
(Q, H)
c3 =
S#%'?>(N+A*B&4/1'_\>(N *' *9QS:; =
åæj`•éL“èéIŒI†`˜«—r˶ŒI“2˜“L$‡’`ËñéL— ˘Š¸‡†cŠÂŒIéIŒL—6‹ŒL‡†
JC SK4F( +-1#%+%#-?S4/$LS$4/$V+%#%$?"*#%'"; ® +-4/$LH*+-4/'S+%#%$?"*'#[email protected]*14/1'H(
+- 1'@2\4/#%[email protected](Ma¸4/$& #-4/$$ K $L";
? ¢ ]|ñF¤§|§¯E§º/ q = k z + k z + k u Im*ZQ& k I k k (
&4/$ *$L"; z I z I u qv ' 2'1?1 $L $L2+-2Z:*'u#A*#-4/$(¬1 $#%ZQ2"*10c.u(]! "*1
*K4/$LH H*Z:*1+OI(28+O! 4/1ZQ 'v1',(8Z:*$$N @(01Y(? _#%H*14/1'H(01©2\4/#%$LH(
+%#%$?"*'#- *#-4/$J;
*ZQ&
1$&4/$ *$L:;
§ /
? ¢ ]|ñF¤§B|§7¤.~Q }Eè§)|§)"­#
' 2'? $L $L<+-Z:*'#A*#-4/$<(q¬ct+A*@=*k1cth 12 '<(]! "*k1ct*Z:*+(¬+A*@Z:*$$
h2 qct
@(01D(? _#%8*14/1'H(01Y2\4/#%$LH(M+%#%$?"*'#- *#-4/$J;
*ZQ&
(
§ /
? ¢ ]|ñF¤§D|§D¤U~Q|§Y|§VEA"§L""b
&4/$ *$L"; h I v q ' 2'? $L $LN+-qcv8Z:=*'k#A*hh#-4/+$Nkv(v,+A*.*k1h 1'Nk(]v ! "*1
*13$#%ZQ"*1¯(01“&"*2 1cv')(ZF#%:IE(+A*©ZF#%. ,(01“(? _#%7*14/1'B(01
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$ 5Ns
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2
q =2 k z + k v
q =k z
ct
Ú!l
ct 2
cv
h 2
v
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ZF' *9QVìZ:*$$"I(? ZQ '4/#%' î 4/1¯(01“K4F( +- (.+A*D&4/1'_\ (. *' *9Q:; ® +bc0#-
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Ì =$B(&4/$#-(? ' '<+-m( 10cB&"*/"I$4/1m&4/K7K $«34/$¬(*$1$B2' K7#- '< K72
2*' 'm+A*@'?&4/$& #%+%#A*#-4/$.((4/$$?< $B1#%+%#- *$Lm1$#-[L1 K $L<+-mK4F( +-m(
Z:*$$S (S&4/1'_\S( *' *9Q:; ƒ 1#-"I:(*$<1$( 10c0#- KS K72"I/$4/1<' *#% '4/$
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ä ã÷þ`ãI‚
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$4/'.&"*/,(B=9:1':IJ+-7ZQ& 1')(]! ? *
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z 1m
1'
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$4/$3K1'?,um
;
Xnm = q
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z1
 z2  I]+-ZQ&
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X =
 u 
q
1')(7Kb—
1'8(MK1'H @+-NZQ& 1'H(@Z:*'#A*_+-
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B+A*R&4/Y*Z:*+S( +A*DZ:*$$V u 1B+O! 4/1ZQ '1'Y( +A*DZ:*$$©&4/K7K2
#%$(0#-[L1?H1'S+A*,=9:1' Œ ; ’ ;
• *$L,(4/$$?B[L1)+-M&"*2 1'>4/$LM#%$(? 2\ $(*$L"I]+-)ZQ& 1' X 2'?b—
m
Yam
Yav
U 6- Q
:.;=<.> Œ ; ’?#iRI¡2¢STUHjFLQS´9FL¢SCGQSN%DIHJDLK ZO9Q A H^&HVN2N2QjO9Q FL¢Sí&B2A HJDLK M&NgO9B†ÂHJFGDLK DLQSB9F
$LN+A*)K.*'#-&,(MZ:*'#A*$&b—â&4"Z:*'#A*$&)1#%Z:*$Lš
σz21
V = 0
0
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0
σz22
0

0
0 
σu2
d ; å £Q"§L/ò3¥E§L ¤§òÍ]|ñF¤§ / CJVK4F( +-V(0#%2\4/$#%_+-D4/$LY& 10c (W+A*
Z:*$$> @(N+A*B&4/1'_\>(M *' *9Q.š
qv = k 1 z1 + k 2 z2 + k u u
qct = kct z2
ì Œ ; d:‡Qî
ì Œ ; d0€î
”’68ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
ˆ $RN'4/1ZQ>#-& #E(*$H+->&"*/H44™Y+A*7K1'>(01R(? _#%HK.*$[L1:;«584/',&"*/
7#%K72+-V B$ $?&#%©2*/B1$ K.*'#-&V( 2'45& #-4/$
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#%$&4/$$L1:6; 584/[email protected]/_ $4/[email protected]$N$4/1ZQ +%+->?[L1*#-4/$3š
ì Œ ; d:dQî
k1 z1 + (k2 − kct )z2 + ku u = 0
ƒ 4/1'©& D*22+%#-&"*#-4/$JI¬$4/1 *+%+-4/$K.*$#%21+- ' -+ Z:*+- 1'_'1.(
K 1' I *1W+%#- 1U()+- 1'8Z:*'#A*#-4/$ I *14/1'>(01
z 1 z2
u
2\4/#%$L TZ1 Z2 U
(4/[email protected]*'>š
X0 = (Z10 , Z20 , U0 )
z 1 = Z 1 − Z 10 , z 2 = Z 2 − Z 20 , u = U − U 0
• $©' K72+A*43"*$L z I z u 2*'+- 1'Sbc02'#-4/$(*$+O! ?[L1*#-4/$3ì Œ ; d:dQî I
$4/1H4/_ $4/[email protected]+-M1 K4F2( +-š
ˆ ™
MX = R
M = ( k1 k2 − kct ku )
R = M X0
Œ ; û §L"§L¯¤¦¤¥|0€ |§LUòͧL€}E§L 584/1B*ZQ4/$7*22+%#-[L1?.+A*Y'?&4/$#Š—
+%#A*#-4/$ *#-[L1U1'©1$W2\? '#-4F(W( "‡ 54/1'Y*+%+A*$LD(01 ‡:º ß ‡:Œ ß ‡:” *1
"º ß ‡:Œ ß ‡:” ; Ì =$R(, '8+A*Z:*+%#-(*#-4/$U(HK1'"I $4/18&4/K7K $«34/$
2*'@*22+%#-[L1 '+- *#-#-[L1"_; 584/1(0#-2\4:4/$(]! 1$V 1+ '?#-(01V(
_#%+A*$JIJ+-7N9:+-4/_*+¬$J! ,(4/$&72*/N1#%+%#-:IJ#%+¸1 à,(]! *22+%#-[L1 ',+-7
$4F(*+E1'S+-N'?#-(01Y(N_#%+A*$Y$4/'K.*+%#-?:;
8
a584F(*+
⇔ Si |rni | > 1.96
*+-4/'
H0
'™5 ?
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? '#- 1'S(@+A*­p"4/$8(@$4/$©(? & #-4/$V&H[L1# #%9:#%$#Š=8[[email protected]+-¬K1'4/$L
&4/? ' $L8 H[L1J! 4/$V$>(? & ,*1& 1$>2'? $&>(N(?baO*1";
résidu normalisé
3
2
1
Seuil
0
−1
−2
−3
0
2
4
jours
6
8
:.;=<.> Œ ; "”?fʇ¢SCGK O9BsO9QjY2K A HVN
10
î_ ý¸<Â`ÄJÉ%ÂHÁÉ%¯ÆMÅDÄJ“Ëaï$'əÄJəÂHÁÉ%“ÆMÅDÆ\ïÅÇ>Ç]ÁÂHÁÉ%¥t"Ŭ”:”
courbe du niveau amont
70.8
Z amont
Z amont estimé
70.7
courbe du niveau aval
70.5
Z aval
Z aval estimé
70.4
70.6
70.3
70.5
70.2
70.4
70.1
70.3
70.2
0
2
4
jours
6
8
70
0
10
:.;=<.> Œ ; · ?LB6MVDLQjHVT$M&N%DWQSCEDLK T$¢SQQD¤T$QË
CGB9FL¢SQ
2
4
jours
6
8
:.;=<.> Œ ; €›g?LB6MVDLQcHX^&HVAQSCEDLK T$¢SQ QD‡T$QSCGBeË
FL¢SQ
courbe de l’ouverture de vanne
0.4
U
U estimé
0.35
0.3
0.25
0.2
0
2
4
jours
6
8
10
:.;=<.> Œ ; "º?LÌB2^£QFGDLB9FLQ O9Q ^&HVN2N2Q QSCEDLK T$¢SQjQD¤T$QSCGB9FL¢SQ
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[[email protected](4/[email protected]'?&4/$& #%+%#-?84/[email protected]'S2'4F&H(SK1'";
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10
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courbe du débit
1
débit (ct)
débit estimé
0.9
m3/s
0.8
0.7
0.6
0.5
0
2
4
jours
6
8
10
:.;=<.> Œ ; "?fˆ ¢SY2K D¤QSCEDLK T$¢jQD¤T$QSCGB9FL¢
·0; û L§ "§LW¤“|£Q"§L~:€ X|§L©èm‘/ ® & #OI$4/1>*ZQ4/$M' *€54/1?B& B? *2\
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1$Y_#A*#[email protected]( Œ & Ki1'S+-N$#%ZQ"*1R*Z:*+J2\ $(*[email protected]$D& ' *#%$Y K72";
GH?1+% *.š]CJ79:' *2(>'?#-(01>(7_#%+A*$`ìA=9; Œ ; d:‡Qî K4/$L'.[L1J! #%+^U*
(? & #-4/$R(>&M_#A*#-"I+-M'?#-(01D(? 2*/M+-> 1#%+J[L1N+O! 4/$Y"! =c?:;
résidu normalisé
3
2
1
Seuil
0
−1
−2
−3
−4
−5
0
1
2
3
4
5
jours
6
7
8
9
:.;=<.> Œ ; d:‡?fˆ ¢DLQSRDLK M&NgOP B2NY2K HVK C¤CGB9F‡A QSC¤T$QSCGB9FLQSC
CJ,Z:*+- 1'>'?&4/$& #%+%#-?)4/$L,#%+%+%1'?)1'>+-N=9:1' ì Œ ; d0€î Iì Œ ; d:dQî ì Œ ; d:ŒQî ;
2\ $(*$LM(¸·R*1YK.*$[L1>(]! #%$0a¸4/'K.*#-4/$Ya¸4/1'$#J2*'@[email protected]( +-"I4/$Y$
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>'? 2\ '& 1)1'@+-M*1'8 84/$D$>2\ 18(4/$&>2*/MT7+O! *#-(,(H'?#-(01
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î_ ý¸<Â`ÄJÉ%ÂHÁÉ%¯ÆMÅDÄJ“Ëaï$'əÄJəÂHÁÉ%“ÆMÅDÆ\ïÅÇ>Ç]ÁÂHÁÉ%¥t"Ŭ”Q›
courbe du niveau amont
70.8
Z amont
Z amont estimé
70.7
courbe du niveau aval
70.5
Z aval
Z aval estimé
70.4
70.6
70.3
70.5
70.2
70.4
70.1
70.3
70.2
0
2
4
jours
6
8
70
0
10
:.;=<.> Œ ; d0U?LB6MVDLQjHVT$M&N%DWQSCEDLK T$¢SQQD¤T$QË
CGB9FL¢SQ
2
4
jours
6
:.;=<.> Œ ; d:d!?LB6MVDLQcHX^&HVAQSCEDLK T$¢SQ QD‡T$QSCGBeË
FL¢SQ
courbe de l’ouverture de vanne
0.4
U
U estimé
0.35
0.3
0.25
0.2
0
2
4
jours
6
8
10
:.;=<.> Œ ; d:Œ?LÌB2^£QFGDLB9FLQ O9Q ^&HVN2N2Q QSCEDLK T$¢SQjQD¤T$QSCGB9FL¢SQ
ä ã÷þ`ã=È
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4/17&4/$L' *#%$LR(B'4/#-7K4F( +-[L1#@4/$L.TR$4/'Y(0#-2\4:#%#-4/$Xšm+-©K4F( +-
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8
10
”:º«8ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
; å £Q"§L/ò3¥E§L©¤§L.òͧL€}E§L#/




z1
z
1
m
 z2 

 [email protected]& 1'S(HK1'
 z 2m 
CJHZQ& 1'S(]! ? *Mš



X=
u
X
=
m


 um 
 v 
vm
>+-BZQ& 1',(MZ:*'#A*_+->q$4/$UK1'? X = q ;\CJ,&"*2 1'>4/$L
#%$(? 2\ $(*$L"I/+A*@K.*'#-&(Z:*' *#%$&b—â&4"Z:*'#A*nm
$&S(01BZQ& 1'
"! ?& '#%Sš
Xm
 2

σ z1 0
0 0
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0 0 
2

V =
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0 σu2 0 
0
0
0 σv2
d ; å Q£ "§L/ò3¥E§L“¤§`òÍ]|ñF¤§/ 584/17(0#-2\4:4/$B(B'4/#-BK4F( +-Yì Œ ; d:ŒQî I
ì Œ ; d’Fî ,ì Œ ; d:”Qî [L1#J&"*+-& 1+- $L8+-MK\ K>(? _#% q ;
ì Œ ; d:ŒQî
q = k 1 z1 + k 2 z2 + k u u
q = kct z2
q = k h z2 + k v v
ƒ 4/1'8? +%#%K7#%$ 'H+-N(? _#%H[L1#]$J! @2*/SK1'?:I#%+J1¸àH(M'?"*+%#- ',š
ì Œ ; d’Fî
ì Œ ; d:”Qî
(3.24) = (3.23)
(3.24) = (3.25)
CJNK4F( +->4/_ [email protected](01Yâ^F2\
M=
MX = R
ˆ ™
k1 k2 − kct ku 0
0 ky − kct 0 kv
Œ ; û §L"§LV¤W¤¥|0€ ¦|§L.òͧL€}E§L
?Òû §L.­ ¤èU¤ / CE*7[L1*$L#%?
R = M X0
φ = r T Vr−1 r ∼ χ2 (2)
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(?baO*1,ìZQ4/#%'S=9:1':; Œ ; d’Fî φ;
;
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Chi2 test
14
12
10
8
Seuil
6
4
2
0
0
2
4
jours
6
8
10
:.;=<.> Œ ; d’?,½_QSCED‡í&A M&Y4HVA
?Òû §LBE]|¤/ CJH'?#-(01 (H_#%+A*$ 2'? $L8( 10c &4/K72\4: *$L
(4/[email protected]*'>š
r1 = M 1 X − R 1
*ZQ&
k1 k2 − kct ku 0
M2 =
0 ky − kct 0 kv
r2
R1 = M 1 X0
R2 = M 2 X0
/
4
1
*
+
/
4
'
'™5 ? )
8 S$4F(*+
⇔ (Si |rn | > 1.96
|rn | > 1.96
H0
CE*B=9:1':; Œ ; d:” K4/$L')[L1>+->@Z:*+%#-(>+O! L^F2\4/
21#-[L1N+-
H0
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r2 = M 2 X − R 2
*ZQ&
M1 =
r1
1
2
Résidus de bilan
4
r1
r2
3
2
1
0
−1
−2
0
2
4
jours
6
:.;=<.> Œ ; d:”?#½_QSCED¤N2MiO2HVA
8
10
· ‡«8ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
’ ; ½ £L~Q E~L¥¤¥§Lͤ§LRòͧL€}E§L / °L1'>+->=9:1'"; Œ ; d ·€— Œ ; d: IE4/$L,#%+%+%1'?
+-NK1'>'?&4/$& #%+%#-?, N+-NK1'>_'1";ýCJ>9:' *2NK4/$L' $L
courbe du niveau amont
70.8
courbe du niveau aval
70.5
Z aval mesuré
Z aval estimé
Z amont mesuré
70.7
Z amont estimé
70.4
70.6
70.3
70.5
70.2
70.4
70.1
70.3
70.2
0
2
4
jours
6
8
10
70
0
:.;=<.> Œ ; d · ?|B6MVDLQ`HVT$M&N%D
4
6
jours
8
10
:.;=<.> Œ ; Qd ›!?|B6MVDLQ`HX^&HVA
courbe de l’ouverture de vanne
0.4
2
courbe de la vitesse
0.8
U mesuré
U estimé
vitesse mesurée
V estimé
0.75
0.35
0.7
V[m/s]
0.3
0.25
0.65
0.6
0.55
0.2
0.5
0
2
4
jours
6
:.;=<.> Œ ; d:º?L¸M&CGK DLK M&Ng^&HVN2N2Q
8
10
0.45
0
2
4
jours
:.;=<.> Π; d:?f[K DLQSCGCGQ
[L1N+-SZ:*+- 1'S'?&4/$& #%+%#-?84/[email protected]'S2'4F&H(SK1'";
6
8
10
î_ ý¸<Â`ÄJÉ%ÂHÁÉ%¯ÆMÅDÄJ“Ëaï$'əÄJəÂHÁÉ%“ÆMÅDÆ\ïÅÇ>Ç]ÁÂHÁÉ%¥t"Å ·
” ; ½ §L~Q E"/}§X¤x| E£L§ ò͇âJ}E "§ / íH$Va¸4/#-.+A*U'?&4/$& #%+%#A*#-4/$
'K7#%$?:I+A*BK1'NK.*$[L1*$L S'?#%K?>2*'@[email protected](4/[email protected]'?&4/$0—
& #%+%#-?";CE*)=9:1':; Œ ; Œ:‡ K4/$L'>+-Nq(? _#%H#%K?:;
courbe du débit estimé
1
0.9
Q [m3/s]
0.8
0.7
0.6
0.5
0.4
0
2
4
jours
6
8
10
:.;=<.> Œ ; Œ:‡!?Aˆ ¢SY2K D¤QSCEDLK T$¢
·0; û §L"§L,¤.|£Q"§L~:€ U|§L8èm/ ƒ 4/1'#%K,1+- '<+A*@2'? $&S(]! 1$7(?baO*1"I
$4/1N*ZQ4/$8' *€54/1?,1$R_#A*#-M( ” & K²1'H+O! 4/1ZQ '1'B(,Z:*$$,2\ $(*$L
1$N2\? '#-4F(N(4/$$?:;
?Òû §Lý­ ¤è¤ CJD 9:+-4/_*+8K4/$L'W[L1Y+-D'?#-(01 ©TU+O! #%$L? '#- 1'
(V+AH
* p"4/$D(Y$4/$`(? & #-4/$ $¦+O! *_ $&R(D(?baO*1. $`( 4/'
(Y& 1
p"4/$Y $¦2'? $&D(Y(?baO*1RìZQ4/#%'=9:1':;8ì Œ ; Œ0€îî ;CJV
9:+-4/_*+82\ 'K ©+A*3(? & #-4/$ K.*#- $D2\ 'K ©2*/.+A*Í+-4F&"*+%#- *#-4/$ (01
(?baO*1";EC! #%$0a¸4/'K.*#-4/$“#-1(+A*VK1'.(7ZF#%:I]$J! *22\4/'.'#- $
2*'S' *22\4/'8TB+A*)+-4F&"*+%#- *#-4/$D $Y1#%+- *[email protected]$#-[L1 K $LH+-NS9:+-4/_*+O;
Chi2 test
15
10
Seuil
5
0
0
2
4
jours
6
8
10
:.;=<.> Œ ; Œ0?LB6MVDLQ`HX^&HVA¶QSCEDLK T$¢SQjQD¤T$QSCGB9FL¢SQ
?Òû §L ]|¤./ °L1'+A*N=9:1' Œ ; Œ:d I04/$©&4/$ *M[L1J! 1$©_#A*#-(? & ?
1$#-[L1 K $L,1'N+-)'?#-(01 r ;E& #N(¸3
· *1WaO*#%>[L1B+O! 4/1ZQ '1'(
+A*)Z:*$$
u
#%$L 'ZF#- $LH1$#-[L11 K $L8(*[email protected]+-M'?#-(01
r1
;
· d«8ÂÉÁËHÅñî_mËaï$'əÄJəÂHÁÉ%¯ÆMÅDÆ\ïÅÇMÆ10 "ºÂ8Â8Ä.Æ10 əËHËHÉ32ÂHÁÉ%
ž *$B+-©&"*/744™ý4/$¦*D2*/B1$©&4/$$*#- *$&Y*D2'#-4/'#@( +O! 4/'#%9:#%$Y(01
_#A*#-"I4/$V2\ 1H4/12r34/$$ 'H[L1M+-N(?baO*[email protected]' 4/#%@1' ;
u
z1
Résidus
4
r1
r2
3
2
1
0
−1
−2
0
2
4
jours
6
8
10
:.;=<.> Œ ; Œ:d?fˆ ¢DLQSRDLK M&NgOP B2NY2K HVK C¤CGB9F‡A QÏÎSQF¤FL¢SCGK O9BsB2N2K „%B2QST$QSN%D
• $.&] "I/+-S'?#-(01 r aO*#%<#%$L 'ZQ $#%'¬+-Z:*'#A*_+- z I z u ;QCJ'?#-(01
aO*#%#%$L 'ZQ $#%' z 1 v ;QCE*>(? & #-4/$©1' r 1$#-[L11 K2 $L#%9:$#Š=H[L1
r2
1
K,1$)*ZQ& r ; ž *$N+-
+-7(?baO*1>N1'M+-2NK1'N( r $4/$Í&4/K7
1
2
&"*/"I4/$Y2\ 1S' $&4/$L' '[email protected](0# ]? ' $LH#%1*#-4/$,š
? °F 1+ r (? 2*/,+-, 1#%+m()(? & #-4/$ýš ž ?baO*181'8+A*K1')(,+A*
1
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4/$L z 4/1D1'S+O! 4/1ZQ '1',(MZ:*$$ u ;
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?
r2
ZF#%:;
? r r (? 2*/ $LM+-B 1#%+m()(? & #-4/$š • '' 1'N1'8+A*.K1'B(
+A*B1 &4/,2*Z:*+ ; ˆ $R*)+A*) *_+->(>(?& #-#-4/$D1#%Z:*$Lš
z2
r1 = 0
r2 = 0
r2 = 1
v
r1 = 1
z1 ∪ u
z2
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1
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CJ.Z:*+- 1' '?&4/$& #%+%#-?V4/$L #%+%+%1'?©1' +-=9:1'"; Œ ; Œ:Œ — Œ ; Œ ·0;
584/1,2\4/1ZQ4/$)' K.*'[L1 'B[L1.K\ K #¬+- (?baO*1B,1$#-[L1 K $L
1' u I0#%+EM'? 2\ '& 1>1'S4/[email protected][email protected](4/[email protected]'?&4/$& #%+%#-?";
î_ ý¸<Â`ÄJÉ%ÂHÁÉ%¯ÆMÅDÄJ“Ëaï$'əÄJəÂHÁÉ%“ÆMÅDÆ\ïÅÇ>Ç]ÁÂHÁÉ%¥t"Å · Œ
courbe du niveau amont
70.8
courbe du niveau aval
70.5
Z amont mesuré
Z amont estimé
70.7
Z aval mesuré
Z aval estimé
70.4
70.6
70.3
70.5
70.2
70.4
70.1
70.3
70.2
0
2
4
jours
6
8
10
70
0
:.;=<.> Œ ; Œ:Œ?LB6MVDLQ`HVT$M&N%D
4
jours
6
8
10
:.;=<.> Œ ; Œ’?LB6MVDLQcHX^&HVA
courbe de l’ouverture de vanne
0.4
2
courbe de la vitesse
0.8
U mesuré
U estimé
vitesse mesurée
V estimé
0.75
0.35
0.7
V[m/s]
0.3
0.25
0.65
0.6
0.55
0.2
0
0.5
2
4
jours
6
:.;=<.> Œ ; Œ:”?A¸M&CGK DLK M&Ng^&HVN2N2Q
ä ã÷þ`ã=ä
8
10
0.45
0
2
4
jours
6
:.;=<.> Œ ; Œ · ?A[K DLQSCGCGQ
ñ˜«Ë¶ŒGö‡+`ÅŽj`•éL“HÏñ˜«Ë¶ŒGö‡[`[éL—øŠ¸‡’`Ëÿ$Å`+‹–—r˗¤
CJ4/'[L1>$4/1M*ZQ4/$M&] & 1?>+->H2'?&?( $LM1'H1$>+-4/$9:1,2\? '#-4F(,(
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y (s) = p (s)q (s) + p (s)q (s)
1
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1
22
2
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y0
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y1
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q1
q2 -
6u2
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p22
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xk+1 = Axk + Buk + Bu
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15 × 10−4
0
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4
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u
2
15
f2
f1
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10
5
0
seuil
−5
−10
0
ey
n
20
40
t
:.;=<.> Œ ; Œ:? • '' 1'H(]! #%K.*#-4/$
60
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4/$Lm'? 1$#[email protected]+A*@a¸4/#-m(m4/1#%+-<(¬'? 9:1+A*#-4/$JIQ(¬K1' <(12\ 'ZF#-#-4/$J;/íH$
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Ì ^L*$L1$ý 1+S'?#-(01ý(? 2\ $(*$L.(©4/1B+-BK1'"Im4/17+-BK1'
4/$L84/1r2 34/$$?M? '4/$$?"; ž *$8&>â^F2\>(,&"*$*10c]I1$>'?&4/$& #%+%#A*#-4/$U *€—
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ž *$¬1$ ( 10c0#- KH K72"IQ+-H&"*/¬2*'#-& 1+%#- '(@$4/'8#%H[L1# (0#-2\4:@&4/$L' *#Š—
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r1 = 0
r1 = 1
z 1 ∪ z2 ∪ u
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y2
î
v
r1 = 0
r2 = 0
r2 = 1
v
r1 = 1
z1 ∪ u
z2
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[[email protected]+O! 1$8(H&¬K1'";FíH$.K4F( [email protected]\4:#%_+-H(8&4/K72*' '+O! 4/1ZQ '1'M(
Z:*$$>9* à.&? *ZQ&>& +%+->bc0#%9Q?N2*'S+O! *14/K.* ; ž *[email protected]&N&"*/S+AT0I# $Y(?baO*1"I+-M'u4/#-#- K>'?#-(01 r = u − u @+-M'?u#-a(01 r ' *#- $LH$4/$Y$Lu1+-";
3
r1 r2 = 00
r3 = 0
r3 = 1
×
r1 r2 = 01
v
×
a
r1 r2 = 10
z1
u
2
r1 r2 = 11
z2
(u ⊗ v) ∪ (u ⊗ y2 )
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[email protected] $L'? *#%$#J[L1N+A*B4/'#- 4/[email protected]/12r34/$$?H $D(?baO*1";
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Si fs 6= 0
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ris 6= 0, i = 1, m
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ris 6= 0
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ì ’ ; "dQî
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∀u, x, d
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ì’ ; · î
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f
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∀u, x, d, fj
ì ’ ; d:dQî
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y(t) = Cx(t)
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ì ’ ; d:ŒQî
ė(t) = F e(t) + T Ei fi (t) + T Ej fj (t)
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ì ’ ; d’Fî
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Singular Values
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−30
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ì ’ ; dQ›:î
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º:‡
ì ’ ; Œ:‡Qî
ì ’ ; 0Œ €î
T A − F T + GC = 0
ì ’ ; :Œ dQî
kTf r (jω)k∞ ≤ γ
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T = In − N C
i
T
N
F
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T N F G
T N F G = ΨΘ+ − K(I2n+2p − ΘΘ+ )
In
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 0
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*+-4/'
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4/1S+A*,a¸4/'Kš
ˆ ™
T = χT − KβT , N = χN − KβN , F = χ0 − Kβ0
χT = ΨΘ+ ϕT , βT = (I2n+2p − ΘΘ+ )ϕT
χ0 = ΨΘ+ ϕ0 , β0 = (I2n+2p − ΘΘ+ )ϕ0

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A
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0
C
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r(t) = Ce(t)
ì ’ ; Œ’Fî
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ì ’ ; Œ:”Qî
V̇ (t) + r T (t)r(t) − γ 2 fiT (t)fi (t) < 0
⇔
ζ
T
ì ’ ;Œ · î
eT [P χ0 + χT0 P − P Kβ0 − (P Kβ0 )T + C T C]e − γ 2 fiT fi
+eT P (χT − KβT )Ei fi + [eT P (χT − KβT )Ei fi ]T < 0
⇔
P χ0 + χT0 P − P Kβ0 − (P Kβ0 )T + C T C P (χT − KβT )Ei
(P (χT − KβT )Ei )T
−γ 2 If
ζ<0
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U = PK
P χ0 + χT0 P − U β0 − (U β0 )T + C T C (P χT − U βT )Ei
((P χT − U βT )Ei )T
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à
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ẋ(t) = A(∆)x(t) + B(∆)u(t)
y(t) = C(∆)x(t)
ì ’ ; ŒQ›:î
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ì ’ ; Œ:ºQî
ẋ(t)
x(t)
= M (∆)
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y(t)
u(t)
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N
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^FK? ='#-A(∆)x(t)
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(*$L8*10cY4/K7K >š
AT
(j)
P + P A(j) < 0, ∀j ∈ {1, ..., N }
? ï è"§L€0"§F}E¦èm}E"§ E çFE¯|¬«-¥E~Q§L€€A€}E|§Lý ¤Að]"m=âJ}E§L ® +
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2[x (t)N1 + x (t − τ (t))N2 + ẋ(t)N3 ] × [x(t) − x(t − τ (t)) −
Z
t
ẋ(s)ds] = 0
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ì ” ; "”Qî
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
X11 X12 X13
T
τm η T (t)Xη(t) −
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X22 X23  ≥ 0
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ì” ; · î
V̇ = xT (t)(P A0 + AT0 P )x(t) + 2xT (t)P A1 x(t − τ (t)) + xT (t)Sx(t)
Rt
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ì ” ; €›:î
V̇ ≤ xT (t)(P A0 + AT0 P )x(t) + 2xT (t)P A1 x(t − τ (t)) + xT (t)Sx(t)
Rt
−(1 − d)xT (t − τ (t))Sx(t − τ (t)) + τm ẋT (t)Z ẋ(t) − t−τ ẋT (s)Z ẋ(s)ds
+(5.14) + (5.16)
⇒ V̇ ≤ ζ T (t)Γζ(t) −
Z
t
t−τ
ζ T (t, s)Ψζ(t, s)
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ẋ(t)
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
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13
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Γ11
Γ12
Γ13
Γ22
Γ23
Γ33
Γ12
Γ22
Γ23
τm N2T
Γ13
Γ23
Γ33
τm N3T
I

τm N1
τm N2 
<0
τm N3 
−τm Z
ì ” ; "ºQî
= Q + N1 + N1T − AT0 T1T − T1 A0
I
= N2T − N1 − AT0 T2T − T1 A1
= P + N3T + T1 − AT0 T3T
= −(1 − d)Q − N2 − N2T − T2 A1 − AT1 T2T
= −N3T + T2 − AT1 T3T
= τm Z + T3 + T3T
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S
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x(s)
ds
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ẋT (s)
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R0 Rt
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ẋ(s)
2[xT (t)N1 + xT (t − τ (t))N2 + ẋ(t)N3 + ẋ(t − τ (t))N4 ]
Rt
×[x(t) − x(t − τ (t)) − t−τ ẋ(s)ds] = 0
2[xT (t)T1 + xT (t − τ (t))T2 + ẋ(t)T3 + ẋ(t − τ (t))T4 +
×[ẋ(t) − A0 x(t) − A1 x(t − τ (t))] = 0
Rt
t−τ
ds
ì ” ; "Qî
ì ” ; d:‡Qî
x(s)dsT5 ]
ì ” ; d0€î
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V̇ ≤ η T (t)Γη(t) −
Z
t
[η T (t)N̄ + ζ T (s)Z̄]Z̄ −1 [N̄ T η(t) + Z̄ T ζ(s)]ds
t−τ (t)
 T T

A0 T5
x(t)
ˆ ™ ζ =  x(t − τ (t))  I N̄ =  AT1 T5T
 −T5T
 ẋ(t)

ẋ(t − τ (t))
0

Ω11 Ω12
 ∗
I
Ω22
Γ = Ω + τm N̄ Z̄ −1 N̄ T Ω = 
 ∗
∗
∗
∗

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
N1
N2 
 I Z̄ = S
N3 
WT
N4

Ω13 Ω14
Ω23 Ω24 

Ω33 Ω34 
∗
Ω44
W
Z
I
ì ” ; d:dQî
Z̄ > 0
Ω
N̄ T
N̄
−τm Z̄
<0
ì ” ; d:ŒQî
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:”
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4/$LH(@^0 [email protected]? *@ H $L'[email protected]' *'(?H(N+A*,a¸4/'Kš
ˆ ™
P τ
P τ
ẋ(t) = ni=0
Ai x(t − τi (t)) + ni=0
Bi u(t − τi (t)) + Ef (t)
y(t) = Cx(t)
ì ” ; d’Fî
+O! ? *@(01Y^0 K:I
+O! $L'?:I
+A*B4/'#-N u ∈ <p
y ∈ <m
*$L+-^0 K:;CJEK.*'#-&
n×n n×p
4/$L'2\& #%ZQ K $LS+-K.*'#-&S(]! ? *@ (]! $L'?:; A0 ∈ <n×n B0 ∈ <n×p
Bi ∈ <
2\4/1' i = 1, n 4/$LM'2\& #%ZQ K $L>+-8K.*'#-&N(]! ? A *iN∈' < *'(?N N
(]! $L'?
' *'(?"; τ m×n >+A* K.*'#-&.(4/'#-7 >
A
+
*
K.*'#-&
n×n
(8(?baO*1"; C ∈ <
4/$L+-K,1+%#%2+-S' *'(Z:E*'∈#A*$L<S(*$+-H K72 +-
τi , i = 1, nτ
[L17š
x ∈ <n
+-(?baO*1<*9]&
f ∈ < nf
f
τi (t) = 0
pour i = 0
0 ≤ τi (t) ≤ hi et
0 ≤ τ̇i (t) ≤ di < 1 pour i = 1, nτ
ì ” ; d:”Qî
ƒ 4/1'& 8& +A*/@(8^0 K$4/1(? ZQ +-4/22\4/$(4/_ 'Z:* 1'(@â^F2\
C]1 L$ _\ '9Q '8(N'1& 1', K,_+A*_\,TB& +%+->(01D^0 K7š

P τ
P τ
P nτ
Fi z(t − τi (t)) + ni=0
T Bi u(t − τi (t)) + ni=0
Gi y(t − τi (t))
 ż(t) = i=0
x̂(t) = z(t) + N y(t)

ŷ(t) = C x̂(t)
ì ” ;d · î
.(©+A*
ˆ ™
n m 4/$L7'2\& #%ZQ K $L +O! #%K?Y(©+O! ? *
4/'#-x̂ y∈I z< = T xŷ ∈∈<<n 8+O! ? *N()+O! 4/_ 'Z:* 1'"I u y 4/$LM+-Mx $L'?N(
+O! 4/_ 'Z:* 1'"; CJHK.*'#-&
n×m
T ∈ <n×n , N ∈ <n×m , Fi ∈ <n×n
4/$LB+-)2*' *K '( +O! 4/_ 'Z:* 1'.TR(? 'K7#%$ '(.aO*434/$TR *Gi#-˜∈aO*#%<'.+-
4/_05& #Ša¸@(?#%'?";
+A*©(0#%"I\2\4/1'>#%K72+%#Š= '>+A*©^F$L7(>4/_ 'Z:* 1'"I\$4/1>&4/$#-(? '4/$
+-,^0 KBTY( 10cÍ' *'()1$#-[L1 K $LYì n = 2 î */&*$LB[L1+A*VK? 4F(
τ
2\ [email protected]? $? ' *+%#- 'MT n ' *'(";
τ
¨, E|mA€ EB|§D|£Q"§L~:"èm¥¤¥A"£
CJ>^0 Kì ” ; d’Fî @(? & *_+-,#š
rang
sIn −
C
P2
i=0
Ai E
0
= n + nf
ì ” ; dQ›:î
8ÂÉÁËHÅúÑ_bÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ#t"ËqÇÒ¬Ç]ÁÊÈRÅÇ|âxËHÅ8ÁÂ8ËHÆ>Ç
·
gl
ønUt|x wÔ6uJnU ÖtxjvÒn$Ñvx p6y~.Ô6xjoprv
j
Ð
½-
ž *$H& >2*'#-:I$4/1H(? ZQ +-4/22\4/$H1$R4/_ 'Z:* 1'MT7 $L'?H#%$&4/$$L1
( B+A*.a¸4/'KYì ” ; d:ºQî (4/$L>+-N2*' *K ',(4/#%ZQ $L ')=c?N()aO*434/$3T ' $(0'
+A*W(0^F$*K7#-[L1Y(©+O! '' 1'.(]! #%K.*#-4/$
*/^FK724/#-[L1 K $L.$L1+%+-
e = x − x̂
[L1 +-[L1>4/#- [email protected][email protected] $L'? [email protected]&4/$(0#%#-4/[email protected]#%$#%#A*+- @[email protected](?baO*1 ;
u
x
f
CE*7(0^F$*K7#-[L1>(N+O! '' 1'H(]! #%K.*#-4/[email protected](4/$$?N2*',š

P
P
= 2i=0 Fi e(t − τi (t)) + 2i=0 (T − (In − N C))Bi u(t − τi (t))+
 ė(t)
P2
i=0 (Fi T − T Ai + Gi C)x(t − τi (t)) − T Ef (t)

r(t) = Ce(t)
ì ” ; d:ºQî
4/K7K2'? $L?S(*$+-&*2#%'S2'?&?( $L"I:&& 0# 2\ 1<¬'?"*+%#- '#04/$B#%K72\4:
[email protected]&4/$(0#%#-4/$81#%Z:*$L>š
;
ì ” ; d:Qî
T = I − NC
n
d;
ì ” ; Œ:‡Qî
Fi = T Ai − Ḡi C
Œ;
ì ” ; Œ0€î
Ḡi = Gi − Fi N
’;
ì ” ; Œ:dQî
TE = 0
ƒ *'8&4/$?[L1 $L"I+A*7(0^F$*K7#-[L1>(M+O! '' 1'8M'?(01#%8T
ė(t) =
2
X
i=0
ì ” ; Œ:ŒQî
Fi e(t − τi (t))
8
+->2'4/_+- K)' ZF#- $LNT'4/1ZQ '8+-HK.*'#-& F +%+-M[L1>+-)^0 K ì ” ; Œ:ŒQî
i
4/#%H*/^FK724/#-[L1 K $LM *_+-:;
Ì =$¦(]! ^¯2*'ZQ $#%'"I¬$4/1 */(4/24/$ ( 10c? *2\";CE*W2' K7#- 'R&4/$#-DT
'?4/1(0'+->&4/$(0#%#-4/$B*+%9Q? _'#-[L1,2*'>+A*©K? 4F(.(7+A*©2 1(4/#%$LZQ '9Q?b—
$? ' *+%#-?>[L1#]$4/[email protected] \'N(S( 9:'?H(M+%#%_\ '?M$4/1S2\ 'K *$L8(M2*/ 'HT)+A*
&4/$(>? *2\>[L1#J&4/$#->T7 *#-˜aO*#%'M+A*B *_#%+%#%?)*/^FK724/#-[L1:;
D
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GH??& '#%ZQ4/[email protected][email protected]&4/$(0#%#-4/$Bì ” ; ’L”Qî ,ì ” ; ’F›:î 4/1a¸4/'KNK.*'#-& #- +%+-.š
T
N
F0 Ḡ0 F1 Ḡ1 F2 Ḡ2
Θ1 = Ψ 1
ì ” ; Œ’Fî
_ ý¸ $'Å`8ÁÉ%qÆMÅÇbÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ["É%
Q›
Ñ
Ì ZQ&






Θ1 = 





In
C
0
0
0
0
0
0
E
0
0
0
0
0
0
0
Ψ1 =
A0
0
−In
−C
0
0
0
0
A1
0
0
0
−In
−C
0
0
A2
0
0
0
0
0
−In
−C
In 0 0 0 0
(§Fò3òͧ yú’a”‰žV¤^œ¦§ž¡£•žPœ3²V” ¡ü¦=¤C²V•






 ∈ <(4n+4p)×(4n+q)





∈ <n×(4n+q) .
œ3™V«
F0 Ḡ0 F1 Ḡ1 F2 Ḡ2
¦=¤^œ3™V¤^œ3™V•¦Ÿµ!"Hœ ¤ž¦9œž £
¡ ”®ž fž •¦9œ ¤P”q™ƒ¥{²V•:V¤=¦=¤C²V•“ŸžtŸ¨¥{²V¡š:”q™!Ÿ
# žŸžaœHž •¦=¢{¨{žaœ
¤E•:¥{²V•À•À¡Àžaœ%$
T
N
ì ” ; Œ:”Qî
rang(CE) = rang(E)
žaœ¦'&V¨ ¢ ¤ (Ϩ{žµ*)gž¦C¦§ž1¥{²V•:V¤=¦=¤C²V• žaœ¦P¥až ”E”®ž,+¡Àž|”{– ²V•5¦=¢²V¡-&Vž.#Ÿ¨ •¨ ¢™V”®ž fž •¦Ä™V•›œA”q™
œ ˜V•¦/›´aœž9Ÿžaœ10—’32úœ¦ ™V•:Ä™V¢6œ546879
¹ §F}J § y
T
¦=¢ ¤C¥až
:™œ3²V” ¡ü¦=¤C²V•
Ÿž,Ÿµ!"Ÿž šÀž •:HV¡¢™V•-#HŸž³”q™Ô³™V«
N F0 Ḡ0 F1 Ḡ1 F2 Ḡ2
0/•†
ž œ3²V” ¡ü¦=¤C²V•>žV¤^œ¦§žPœ ¤ž¦gœž ¡£”®ž fž •¦œ ¤;478<9
Θ1 .
rang
Θ1
Ψ1
ì ” ;Œ · î
= rangΘ1
= •¡ü¦=¤E” ¤^œ3™V•¦Ï¥až¦C¦§ž9¢{ž ”q™T¦=¤C²V•>Ÿµ-7?"Až¦o”q™LŸ¨(g•À¤=¦=¤C²V•HŸžP”q™f³™T¦=¢ ¤C¥až
²Ÿ©¦=¤ ž •¦
rang
In E
C 0
@ ¨(g•À¤^œaœ3²V•›œ†”q™³³™T¦=¢ ¤C¥až9•:²V•1œ ¤E•-#V¡£” ¤ ´ ¢{ž
V0 =
In
0
C −Ip
,
™V”q²V¢aœ
In E
(5.37) ⇔ rangV0
= n + rankE
C 0
⇔ rangCE = rankE
)gžA+¡£¤o¥{²VNš:”®´¦§ž9”q™†š:¢{ž ¡-&Vž
ž¦
Ψ1
ªÏ²V•
ì ” ; ŒQ›:î
= n + rangE
Θ1
8ÂÉÁËHÅúÑ_bÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ#t"ËqÇÒ¬Ç]ÁÊÈRÅÇ|âxËHÅ8ÁÂ8ËHÆ>Ç
:º
ž 0 · ‡21 I H4/1,ì ” ; Œ:”Qî I+A*74/+%1#-4/$Y9Q? $? ' *+->(.ì ” ; Œ’Fî >š
T
N
F0 Ḡ0 F1 Ḡ1 F2 Ḡ2
+
= Ψ 1 Θ+
1 −K(I2n+2p −Θ1 Θ1 )
ì ” ; Œ:ºQî
ˆ ™ + J+A*2 1(4/#%$LZQ '( 2(n+p)xn J1$<K.*'#-&*'_#%' *#%'
L[ 1J! 4/Θ$.12\4/1'' *N=c '*€=$©(H *#-˜ΘaO*1#%'@+-K¬∈&4/<$(0#%#-4/$$4/$©*+%9Q? _'#-[L1Nì ” ; Œ:ŒQî ; ž +A*SK\ KK.*$#- '[L1¬(*$J+-¬&*2#%'¬2'?&?( $L"I+-E2*' *K '(+A*@4/+%1#-4/$
2\ 1ZQ $LH"! ?& '#%'>4/1S+A*)a¸4/'K7š
Fi
Fi = χi − Kβi
ˆ ™
Ì ZQ&
χi = Ψ1 Θ+ ϕi , βi = (I2n+2p − Θ1 Θ+ )ϕi






A0
A1
A2
 0
 0
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
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




 0

 0

 0







 −C  I
 0
 0
 
 ϕ1 = 
 ϕ2 = 

ϕ0 = 
 0
 0
 0









 0

 −C 
 0







 0

 0

 0

0
0
−C
• $Y' K72+A*43"*$L8+A*B4/+%1#-4/$“ì ” ; Œ:ºQî (*[email protected]+A*B&4/$(0#%#-4/$ ”Qî (01V?4/' K €î I

F0 F1 F2
=
T
N
F0 Ḡ0 F1 Ḡ1 F2
ƒ *'8&4/$?[L1 [email protected]+A*7(0^F$*K7#-[L1>(M+O! '' 1'8( ZF#- $L,š
D
ã÷þ`ã=È





Ḡ2 





P
ė(t) = 2i=0 (χi − Kβi )e(t − τi (t))
r(t) = Ce(t)
A0
0
0
−C
0
0
0
0
A1
0
0
0
0
−C
0
0
A2
0
0
0
0
0
0
−C
ì ” ; Œ:Qî
Éʘ“2‡éI’j‹ŒL‡†Ì`“¨Š¸‡†ccŒG‹ŒL‡†c“†`‡†{—Wé ¤˜«ÿc˶ŒIê ’`“
JC 2'4/_+- KS' ZF#- $LTH'4/1ZQ '<+A*8K.*'#-& K +%+-S[L1+-S^0 KNì ” ; ’LQî */^FK724/#-[L1 K $L7 *_+-:; ƒ 4/1'B& +A*V_*/?.1'>+->' *Z:*10c“()Î'#-(0K.*$¯(*$
0 dQ›91 I0$4/[email protected]? $4/$«34/$H+-M?4/' K,1#%Z:*$L,š
û áE£LñFòͧ yCB²V¡Ÿœ”q™
Ä²V•À•¨{žaœPŸž
ª
i i = 1, 2
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ªg¤E”žV¤^œ¦§ž†¡£•ž†³™T¦=¢ ¤C¥až
¦§ž ”E”®žE+¡Àž­”{– ²Ÿ©aœž ¢&6™T¦§ž ¡£¢ŸµGFH"
K
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:
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až œ¦o™6œ ˜VNš¦ ²T¦=¤I+¡Àž fž •¦œ¦ ™Ÿ© ”®žÏš›²V¡£¢9¦ ²V¡ü¦§žJ3²V•:¥3¦=¤C²V•HV¤ °±¨ ¢{ž •¦=¤C™Ÿ© ”®ž
œ3™T¦=¤^œ3™V¤E«
τi (t)
œ3™V•¦:š›²V¡£¢P¦ ²V¡ü¦
ª
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t ≥ 0 0<τi (t)≤hi , τ̇i (t) ≤ di < 1, i = 1, 2
n×n
³™T¦=¢ ¤C¥ažaœ
ª
Q1 = QT1 > 0, Q2 , Q3 , Z̄i1 , Z̄i2 , Z̄i3 S̄i = S̄iT > 0, R̄i = R̄iT > 0
š›²V¡£¢
ž¦
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U ∈<
i = 1, 2








ᾱ11
∗
∗
∗
∗
∗
ᾱ12
ᾱ22
∗
∗
∗
∗
0
ᾱ23
ᾱ33
∗
∗
∗

R̄i 0
 ∗ Z̄i1
∗
∗
2ON
ᾱ11
ᾱ12
ᾱ22
ᾱ23
ᾱ33
ᾱ24
ᾱ44
:™³³™T¦=¢ ¤C¥až
0
ᾱ24
0
ᾱ44
∗
∗
QT2
QT3
0
0
−h−1
1 R̄1
∗

i R̄i Ai
Z̄i2  ≥ 0
Z̄i3
QT2
QT3
0
0
0
−h−1
2 R̄2
;




<0



ì ” ; ’L‡Qî
ì ” ; ’€î
i = 1, 2
P
P
= Q2 + QT2 + 2i=1 hi Z̄i1 + 2i=1 S̄i
P
P
= Q3 − QT2 + 2i=0 i QT1 χi − U βi + 2i=1 hi Z̄i2
P
2
= −Q3 − QT3 +
i=1 hi Z̄i3
= χT1 − AT1 1 Q1 − β1T U T
= − (1 − d1 ) S̄1
= χT2 − AT2 2 Q1 − β2T U T
= − (1 − d2 ) S̄2
K
žaœ¦ÏÄ²V•À•¨{žÏš›™V¢
ì ” ; ’LdQî
K = Q−1
1 U
¹ §F}J § ¡QP›²V¤E¢KRN•À•žŸž'R9µ
¤­/A9áòͧ yS)‰™6œf’T"DB²V¡ŸœUŸµ"ž¦.ŸµGF
VW">¥/ü²V¤^œ ¤E¢
ª
ª
i ∈ <n×n i = 1 2
ž¦P¢{¨aœ3²V¡›V¢{ž”®žaœX:ML|’{œYŸµ Z<?"Fž¦[Ÿµ Z6?"™&Vž¥
Q1 = QT1 > 0, Q2 , Q3 , S̄i =
ž¦
µ @ – ™aš:¢{´aœ,Ÿµ \F]"eªEŸµ8H"
T
T
S̄i > 0, Z̄i1 , Z̄i2 , Z̄i3 , U
Ri = Ri > 0, i = 1, 2
ž¦o”q™|¥{²V•:V¤=¦=¤C²V•1"|V¡¦/›¨²V¢{´ fž*FTªŸ¨V¡£¤E¢{ž”®žaœ³™T¦=¢ ¤C¥ažaœ
ª
K, T N, Fi , Ḡi , Gi ,
¢{žaœ šÀž¥3¦=¤&Vž fž •¦
i = 0, 1, 2
.
O
D
(
ã÷þ`ã=ä
jŠ¶ç`˜Ž—`[cŒL—‡†`‡“9‹ŒIŠ
4
Ì =$U(B(? & '> M+-4F&"*+%#- 'M+- p (?baO*1
4/_ 'Z:* 1'@9Q? $? ' *$L8&*/& 1$D1$Y'?#-(01
n
f = δu
f
rj = y − C x̂j , j = 1, 2, ..., p
I]4/$U&4/$'1#%M1$U_*$&
8ÂÉÁËHÅúÑ_bÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ#t"ËqÇÒ¬Ç]ÁÊÈRÅÇ|âxËHÅ8ÁÂ8ËHÆ>Ç
"‡:‡
ˆ ™7+-S'?#-(01 r <#%$ $#%_+-H*1(?baO*1 f $#%_+-H*10c
;CE*B *_+->(j>#%9:$*1',*/4F& #-?NH*+-4/j'H(4/$$?N2*',š
fj̄
®a
r1
f1 6= 0
f; 2 6= 0
;;
C! ? *
x̂j
S+A*74/'#-N(01
fp 6= 0
j me
‡
;
;;
r2
‡
;;
···
···
;
p−1
*1'¬(?baO*1
rp
;;
;
.
‡
···
4/_ 'Z:* 1'
P
P
ż (t) = 2i=0 Fi z (t − τi (t)) + 2i=0 T Bi u (t − τi (t))
P2
+ i=0 Gi y (t − τi (t))
x̂j (t) = z (t) + N y (t)
[L1#[email protected]&4/$«3 1D2\4/1'S+-
ˆ ™
j ime
(?baO*1,ì ” ; ”:dQî ;
P
P
ẋ (t) = 2i=0 Ai x (t − τi (t)) + 2i=0 Bi u (t − τi (t)) + Ej fj (t) + Ej̄ fj̄
y (t) = Cx (t)
I
I
ì ” ; ’LŒQî
I
Ej = B0j Ej̄ = ... B0j−1 B0j+1 ... fj (t) = δuj
T +A* j ime &4/+-4/$$B( B . °F4/1 f = 0,
fj̄ = ... δuj−1 δuj+1 ...
B 0j
+-.K.*'#-& F , G , i = 0, 1, 2, T N 4/$L©(? 'K7#%$?© 0+%+- [L1 j̄x̂ (t)
&4/$LZQ '9QV*/^FK7i 24/i #-[L1 K $L©ZQ '
$¦+O! *_ $&R(D(?baO*1
.2\j4/1'
x(t)
fj̄
$J! #%K72\4/' ϕ (t) , ϕ (t) , ϕ (t) , y, u, f ; 8 *$L,[L1 E M+%#%$?"*#%' K $L,#%$(?b—
2\ $(*$L,(M1+A*BK.*2'#-&>(>3 (0#-'#%_1#-4/$Rj (@ $L'[email protected]#%$j̄ &4/$$L1 E *+-4/'
j
P
ė (t) = 2i=0 Fi e (t − τi (t)) + T Ej̄ fj̄
rj (t) = Ce (t)
2\4/1'
; • $R1#%+%#- *$L8+-
2\ 1 raj (t)
'N6=&04/$'1#%f j̄ 6= 0
p
'?#-(01"I+A*
= r1 ∩ ... ∩ rj−1 ∩ rj+1 ∩ .. ∩ rm
aj
=
=
#
1 δuj 6= 0
#%$4/$
0
j ime
*+A*'K)1#%Z:*$L
, j = 1, 2, ..., m
Ì $4/ 'N[L1,2\4/1'N*19:K $L 'N+O! #%$ $#%_#%?,aO*/&7*10cR $L'?M#%$&4/$$L1"I\+-
&"*/((?baO*1$4/$#%K,1+% *$? 71$#-[L1 K $L &4/$#-(? '?:; • $2' *#-[L1:I<+A*
2'4/_*_#%+%#%?,[L1M+-H(?baO*[email protected]'ZF#- $$ $L8#%K,1+% *$? K $LMS#%$0=$#%?#%K.*+-:;
D
ã÷þ`ã÷þ
éIŒIŠ¸—6‹ŒL‡†—W’‰¼{Š¸—W†`—W’‰¼{~7ŒLË˶Œ ¤—6‹ŒL‡†
t
ž *$ & Y2*'#-:I¬$4/1 b c02\4:4/$+-.'?1+% * /4 _ $L1 +-4/' (Y+O! *22+%#Š—
&"*#-4/$“(B+O! 4/_ 'Z:* 1')(? ZQ +-/4 22\?.(*$,&7&*2#%' 1'N+-^0 K7(&"*$*+
_ ý¸ $'Å`8ÁÉ%qÆMÅÇbÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ["É%
"‡0
Ñ
2'? $L?V(*$B+-©&*2#%'  4F(? +%#- *#-4/$J; • $ _*/ *$L1'B+- K4F( +-©(?& '#%
2*'M+-MK.*'#-&.ì d ; Œ0€î I]+A* &4/$(0#%#-4/$3(B(?&4/12+A*9QDì ” ; Œ:”Qî N *#-˜aO*#%:;\íH$
4/_ 'Z:* 1'<í ®ˆ (+A*@a¸4/'KNì ” ; d:ºQî 2\ 1<(4/$&. 'S&4/$«3 1B2\4/1'm$4/'^0 K:;
ž *[email protected]&>&"*/"I+-N(0#A*9:$4:#-&>2\ 1H8aO*#%'> $Y&4/$'1#- *$LH1$Y_*$&>(>( 10c
4/_ 'Z:* 1')44 ™3+O! 4/_ 'Z:* 1' >#%$ $#%_+-©*1“(?baO*1 f (t) ,+O! 4/_ 'ZQ '
2
S#%$ $#%_+-,T f (t) ;CE*B *_+-1>(N#%9:$*1',*/4F& #-?NH(14/$$?N2*',š
2
°L#
r1
f1 =
6 0
f2 =
6 0
‡
r2
‡
C! *22'4F&7(? 2\ $(*$LB(8' *'(MN*22+%#-[L1?,2\4/1'
d = d = 0;
1
D
I
h1 = τ 1 h2 = τ 2
2
ã÷þ`ãED
Éʘ“4’céG‹–—6‹“`ÅéL—ø“9”`†‡‹ç`•“2
−0.6724
F01 =
−0.0024
0.0000
G01 =
0.0000
−0.0000
T1 =
0.0000
0.0022
0.0518
, F11 =
−0.6724
0.0003
−0.0014
−0.0000
, G11 =
0.0323 −0.0000
−0.0057
1.0000
, N1 =
0.4866
−0.0000
−0.0002
−0.0222
, F21 =
0.0519
−0.0000
0.0004
−0.0035
, G21 =
−0.0252
0.2949
0.0057
0.5134
$ 1#%+%#- *[email protected]+O! *+%9Q4/'#%K I$4/1H*ZQ4/$H4/_ $L1D+-S'?1+% *H1#%Z:*$L,š
• Y
* îˆ _ 'Z:* 1' #%$ $#%_+-NT IL+-H2*' *K ' 4/_ $L1V(M& 4/_ '˜—
f1
T1
Z:* 1'SZQ? '#Š=.š 1
*
+
/
4
'
@
L
[
1
−0.4597
−0.0022
−16
T E = 10
≈0
T E =
1
1
1
+0.4162
_ îSˆ _ 'Z:* 1' S#%$ $#%_+-,T
−0.3747
−0.3053
0.16
H4/$V2*' *K '
f2
*+-4/'@[L1
≈0
Diagostic à base du modèle ID
r1
r2
f1
f2
0.12
0.1
0.08
0.06
0.04
0.02
0
0
20
−0.1838
ZQ? '#Š=š
T2
0.0941
T 2 E1 =
0.1061
0.14
Résidus
T2 E2 =
10−15
2
2
40
t
60
80
:.;=<.> ” ; ?#iK TB2A HJDLK M&NCGB9F‡A QjT$MiO9ÀSA Q @Eˆ
0.0000
−0.0225 −0.0001
0.0109
"‡:d
8ÂÉÁËHÅúÑ_bÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ#t"ËqÇÒ¬Ç]ÁÊÈRÅÇ|âxËHÅ8ÁÂ8ËHÆ>Ç
Diagnostic à base du modèle SIC
0.16
r1
r2
f1
f2
0.14
Résidus
0.12
0.1
0.08
0.06
0.04
0.02
0
0
20
40
t
60
80
:.;=<.> ” ; d?#iK TB2A HJDLK M&[email protected]
Ì =$B(]! #%+%+%1' 'm+-&? K.*8(01B(0#A*9:$4:#-&:I/4/$B9Q? $ '$4/$B#%K,1+% *$? K $L¬(
_#A*#-,( Œ & K &*/& 1$“1'>1$Z:*$$(7'? 9:1+A*#-4/$J;584/1>2\4/1ZQ4/$,4/_ 'ZQ '
1'S+-=9:1' ” ; ” ; d [L17š
? Ì Z:*$L Œ JI¬+A*Í2\? '#-4F(R[L1#HK.*'[L1R+O! *_ $&R(©(?baO*1"I¬+- '?#-(01
$( $LHZQ '
p"? '4&>[L1#J#%9:$#Š=N1$N#%K.*#-4/$R&4/''& >(N+O! ? *";
? ÌxŒ JIF+-N(?baO*1 f 1'ZF#- [email protected]'+A*)Z:*$$ u ;0C! 4/_ 'Z:* 1' ^F$L? #-?
2\4/1s
' ')#%$ $#%_1+-BT
K.*#%$L#- $LN+-,'?#-(011
T p" '4 2\ $1(*$LN[L1,+-
³
f1
r1
'?#-(01 r 9Q? $? '?N2*'@+O! 4/_ 'Z:* 1' $#%_+-,T f @(0# ]? ' $L8(†
p"? '4;
2
2
2
K K,2? $4/K $B*''#%ZQ,+-4/'[L1>+-,(?baO*1 f 1'ZF#- $LM1'H+A*BZ:*$$
? CJ>\
'
$(*$LH+-M'?#-(01 r (0# ]? ' $L8(†
p"? '4 r $L21+O;
u2
2
CE*)=9:1' ” ; K4/$L'>+-S'4/#[email protected]&"*/@&4/$#-(? '?,š
°L#%K,1+A*#-4/$
°L#%K,1+A*#-4/$ d f1 6= 0 f2 = 0
°L#%K,1+A*#-4/$ Œ f1 = 0 f2 6= 0
f1 = 0 f 2 = 0
Ì 10c[L1 +-7&4/''2\4/$( $L.+-B'?1+% *1#%Z:*$L[L1#S4/$L $&4/$&4/'(*$&
*ZQ&>+A*) *_+->(>#%9:$*1'š
GH?1+% *
GH?1+% * d r1 = 0 r2 6= 0
GH?1+% * Πr1 6= 0 r2 = 0
r =0 r =0
1
gl
ÐÒÑvx p6y~.Ô6xjoprv
2
H∞
í $D(?&4/12+A*9Q>2*'˜aO*#%H(@ $L'[email protected]#%$&4/$$L1H$J! @2*/S4/1/54/1'@2\4:#%_+-:;
H
584/1H2'4/2\4:4/$8(H+-4/'@1$B*? $L1*#-4/$U(>' *$˜a¸ '8(8 $L'?8#%$&4/$$L1
1'@+-N'?#-(01R $R*22+%#-[L1*$L81$>^F$L H ;CJH&4/$(0#%#-4/$MT7ZQ? '#Š= 'H2\4/1'
[L1M+O! 4/_ 'Z:* 1'Bì ” ; d:ºQî 4/#%S1$D4/_ 'Z:* 1' ∞H 4/$L,š
∞
;
ì ” ; ’:’Fî
T = In − N C
d;
ì ” ; ’L”Qî
F = T A − Ḡ C
i
i
i
_ _ Ã@ÇFÅËU¦0ÂHÁÅ!"ËMÇ
Ñ Ñ
Œ;
"‡:Œ
H∞
ì ” ;’ · î
Ḡi = Gi − Fi N
’;
ì ” ; ’F›:î
kTf r (s)k∞ < γ
ã ãI‚
Éʘ“2‡éI’j‹ŒL‡†‰`“¨Š¸‡†ccŒG‹ŒL‡†c“#—Wé ¤˜«ÿc˶ŒIê ’`“
D ED
GH??& '#%ZQ4/[email protected][email protected]&4/$(0#%#-4/$)ì ” ; ’L”Qî )ì ” ; ’F›:î 4/1a¸4/'KMK.*'#-& #- +%+-.š
In
C
0
Ì ZQ&
0
0
0
0
°L#]+A*7&4/$(0#%#-4/$ 0






Θ2 = 





T
N
F0 Ḡ0 F1 Ḡ1 F2 Ḡ2 Θ2 = Ψ2

A1
A2

0
0


0
0

 0
0
 Ψ2 = In 0 0 0

−In 0


−C 0

0
−In 
ì ” ; ’LºQî
A0
0
−In
−C
0
0
0
0
0
−C
Θ2
rang
= rang(Θ2 )
Ψ2
*+-4/'S+O! ?[L1*#-4/$¯ì ” ; ’LºQî 2'? $LN+A*74/+%1#-4/$Y9Q? $? ' *+-š
T
N
F0 Ḡ0 F1 Ḡ1 F2 Ḡ2
+
= Ψ 2 Θ+
2 − K(I2n+2p − Θ2 Θ2 )
ˆ ™ + 8+A*72 1(4/#%$LZQ 'B(
2(n+p)xn 1$,K.*'#-&7*'_#%' *#%'
L[ 1J! 4/Θ$.22\4/1'' *M=c '*€=$ (H *#-˜aO*Θ#%'[email protected]+-K&4/$∈(0<#%#-4/$$4/$©*+%9Q? _'#-[L1Mì ” ; Œ:ŒQî ; ž +A*K\ K¬K.*$#- '[L1(*$E+-¬&*2#%'¬2'?&?( $L"I+-J2*' *K 'm(+A*@4/+%1#-4/$
T 2\ 1ZQ [email protected]"! ?& '#%'M4/1+A*Na¸4/'KBš F = χ − Kβ T = χ − K⠈ ™
F
i
i
ƒ
i
T
T
+
χi = Ψ 2 Θ+
2 ϕi , βi = (I2n+2p − Θ2 Θ2 )ϕi
Ì
i
+
χT = Ψ 2 Θ+
2 ϕT , βT = (I2n+2p − Θ2 Θ2 )ϕT







A0
A1
A2
 0

 0

 0









 0

 0

 0















ZQ& ϕ =  −C  I ϕ =  0  I ϕ =  0  ϕ = 
0
T
 0
 1  0
 2  0









 0

 −C 
 0









 0

 0

 0


*'H&4/$?[L1 $L0H+A*B(0^F$*K7#-[L1>0(N+O! '' 1'H( ZF#-−C
$L)š
P
ė(t) = 2i=0 (χi − Kβi )e(t − τi (t)) + (χT − KβT )f (t)
r(t) = Ce(t)
In
0
0
0
0
0
0
0





;






ì ” ; ’LQî
"‡’
ã ã=È
D ED
8ÂÉÁËHÅúÑ_bÃ@ÇFÅËU¦0ÂHÁÅ!"ËMÇ#t"ËqÇÒ¬Ç]ÁÊÈRÅÇ|âxËHÅ8ÁÂ8ËHÆ>Ç
Éʘ“2‡éI’j‹ŒL‡†Ì`“¨Š¸‡†ccŒG‹ŒL‡†c“†`‡†{—Wé ¤˜«ÿc˶ŒIê ’`“
JC 72'4/_+- K.' ZF#- $L)T©'4/1ZQ ',+A*©K.*'#-&
+%+-[L17+-^0 KYì ” ; ’LQî
K
4/#%8*/^FK724/#-[L1 K $LM *_+-:;«584/[email protected]#%+%#-4/$H(?4/'K.*#[email protected][email protected]'?1+% *H( 0 ’£1
2\4/1'Mbc02'#%K '8+-M&4/$(0#%#-4/$M() *_#%+%#%?)(01W^0 K ì ” ; ’LQî ;r584/18? $4/$«34/$
*+-4/'@+-M?4/' K,1#%Z:*$L,š
û áE£LñFòͧ ^
¡ :9– ž ¢ ¢{ž ¡£¢UŸµ `_"t¤^œ1™6œ ˜VNš¦ ²T¦=¤I+¡Àž fž •¦­œ¦ ™Ÿ© ”®ž1¦ ²V¡ü¦ž •5œ3™T¦=¤^œ3™V¤E«
œ3™V•¦aŸµ bVW"Fœ ¤š›²V¡£¢
ª
ª
ª
(MŸ¨aœaª¤E”9žV¤^œ¦§ž>Ÿžaœt³™T¦=¢ ¤C¥ažaœ
i i = 1, 6 ¯i i = 1, 2
ª
ª
ª
ª
ª
ª
ª
ª
ª
ž¦
Zi > 0 Si > 0 Ri > 0 Qi > 0 Ui Wi i = 1, 2 Hi i = 1, 7 U
P >0
¦§ž ”E”®ža%
œ +¡Àž9”®žaK
œ :ML|’{œ†œ ¡£¤&6™V•¦§žaœ­œ3²V•¦,œ3™T¦=¤^œ3™V¤=¦§žaœ%$
Qi
UiT
U1
Ri
ì ” ; ”:‡Qî
≥ 0, i = 1, 2


Φ h1 H̄1
h2 H̄2
 ∗ −h1 Z̄1
<0
0
∗
∗
−h2 Z̄2
2ON
Z̄i =


2ON




Φ=








H̄i = 




Φ11
∗
∗
∗
∗
∗
∗
Si
WiT
W1
Zi
, i = 1, 2
−¯
i (P χ0 − U β0 )T
−¯
i (P χ1 − U β1 )T
−¯
i (P χ2 − U β2 )T
¯i P
0
0
−¯
i (P χf − U βf )T
Φ12
Φ22
∗
∗
∗
∗
∗
Φ13
Φ23
Φ33
∗
∗
∗
∗
Φ14
Φ24
Φ34
Φ44
∗
∗
∗
ì ” ; ”0€î
Φ15
Φ25
Φ35
Φ45
Φ55
∗
∗
H1
H2
H3
H4
H5
H6
H7
Φ16
Φ26
Φ36
Φ46
Φ56
Φ66
∗










Φ17
Φ27
Φ37
Φ47
Φ57
Φ67
Φ77










P
Φ11 = 2i=1 (hi Si + Qi ) + 1 Sym(P χ0 − U β0 ) + 2Sym(H1 )
+C T C
Φ12 = 1 (P χ1 − U β1 ) + 2 (P χ0 − U β0 )T − H1 + 2H2T
Φ13 = 3 (P χ0 − U β0 )T + 1 (P χ2 − U β2 ) + 2H3T − H1
P
Φ14 = P + 2i=1 (Ui + hi Wi ) + 2H4T + 4 (P χ0 − U β0 )T − 1 P
Φ15 = 2H5T + 5 (P χ0 − U β0 )T
Φ16 = 2H6T + 6 (P χ0 − U β0 )T
Φ17 = 2H7T + 1 (P χf − U βf )
_ _ Ã@ÇFÅËU¦0ÂHÁÅ!"ËMÇ
Ñ Ñ
H∞
"‡:”
Φ22 = −(1 − d1 )Q1 − Sym(H2 ) + 2 Sym(P χ1 − U β1 )
Φ23 = −H3T − H2 + 2 (P χ2 − U β2 ) + 3 (P χ1 − U β1 )T
Φ24 = −H4T + 4 (P χ1 − U β1 )T − 2 P
Φ25 = −(1 − d1 )U1 − H5T + 5 (P χ1 − U β1 )T
Φ26 = −H6T + 6 (P χ1 − U β1 )T
Φ27 = −H7T + 2 (P χf − U βf )
Φ33 = −(1 − d2 )Q2 − Sym(H3 ) + 3 Sym(P χ2 − U β2 )T
Φ34 = −H4T + 4 (P χ2 − U β2 )T
Φ35 = −H5T + 5 (P χ2 − U β2 )T
Φ36 = −(1 − d2 )U2 − H6T + 6 (P χ2 − U β2 )T
Φ37 = −H7T + 3 (P χf − U βf )
P
Φ44 = 2i=1 (Ri + hi Zi ) + 4 P
Φ45 = 5 P
Φ46 = 6 P
Φ47 = 4 (P χf − U βf )
Φ55 = −(1 − d1 )R1
Φ56 = 0
Φ57 = 5 (P χf − U βf )
Φ66 = −(1 − d2 )R2
Φ67 = 6 (χf − U βf )
Φ = −γ 2 If
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H5T
H6T
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;
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ì ’Fî
ì ”Qî
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ġ(e, f, s) ≤ ΓT1 (Φ + h1 X̄1 + h2 X̄2 )Γ1
P Rt
X̄i H̄i
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Γ2 (t, s)ds
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h2 H̄2T
H44™
Φ11
Φ12
Φ13
Φ14
Φ15
Φ16
Φ17
Φ22
Φ23
Φ24
Φ25
Φ26
Φ27
Φ33
Φ34
Φ35
Φ36
Φ37
Φ44
Φ45
Φ46
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Φ55
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Φ66
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≥0
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h1 H̄1
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0
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P
= 2i=1 (hi Si + Qi ) + 1 Sym(P F0 ) + 2Sym(H1 ) + C T C
= 1 P F1 + 2 (P F0 )T − H1 + 2H2T
= 3 (P F0 )T + 1 P F2 + 2H3T − H1
P
= P + 2i=1 (Ui + hi Wi ) + 2H4T + 4 (P F0 )T − 1 P
= 2H5T + 5 (P F0 )T
= 2H6T + 6 (P F0 )T
= 2H7T + 1 P T E
= −(1 − d1 )Q1 − Sym(H2 ) + 2 Sym(P F1 )
= −H3T − H2 + 2 P F2 + 3 (P F1 )T
= −H4T + 4 (P F1 )T − 2 P
= −(1 − d1 )U1 − H5T + 5 (P F1 )T
= −H6T + 6 (P F1 )T
= −H7T + 2 P T E
= −(1 − d2 )Q2 − Sym(H3 ) + 3 (P F2 )T
= −H4T + 4 (P F2 )T
= −H5T + 5 (P F2 )T
= −(1 − d2 )U2 − H6T + 6 (P F2 )T
= −H7T + 3 P T E
P
= 2i=1 (Ri + hi Zi ) + 4 P
= 5 P
= 6 P
= 4 P T E
= −(1 − d1 )R1
=0
= 5 P T E
= −(1 − d2 )R2
= 6 P T E
= −γ 2 If
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T E = χf − Kβf
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