title: Diagnostic des systèmes non linéaires : contribution aux méthodes de découplage
text: 6. d(t)( M'.. 2 7 6 / d(t)) ΣN L : ) ) 4 Q$KR( ) ) 4 2 6 /.2. 0 ) 5. /.2. 1 1 2 2 %) %') 1 0 1 QI$R' QILR' QI#R( 5'/ 2 ) 5 & 6 ). 6. / 1 2 QMJR QJIR 1 6' Q$$R' Q$LR Q$JR() 3  m   ẋ(t) = f0 (x(t)) + fi (x(t))ui (t) + Dx (x(t))d(t)   yx (t) = h(x(t)) i=1 I)$ ( x(t) ∈ Rn'u(t) ∈ Rm'd(t) ( {Dx }) = s < p, ∀x(t)) ∈ Rs.6. /.. h(x(t))..( )'/. {Dx } m k=0 )M(' 2 7 S I)$$( Dx [fk, S i ∩ E{dh}] 2 3 2 D(' 0. Dx Dx  = S S  i + i+1 [, ].  Dx   S0 = T ∈ Rp ∀i ∈ {0, · · ·, m}' fi ( {Dx }) 4 ' yx (t) S )I( dh 0 2 )I()!.= / 3 Dx Si.. 6 Dx Dx = Si+1 ⇒ S∗Dx = S i ( {Si Dx }) = n(' ).. I)$L(. S∗Dx...) 1''1 '. ()  Dx   {dh} Q0 = Θ ∩ m Dx  Lfk Qi +  Qi+1 = Θ ∩ 3 I)$J( {dh} k=0 & 5 '. QILR)'I)$$(). S∗Dx ','6 ) 5 & Ψ(x(t), u(t), yx (t))) Dx Si? ∩ E{dh}) /.2. 0 ) ) 4 0 9'. 0 @? yx (t)@'. !" #.2. (S∗Dx )⊥. 6 / S∗Dx,'' (S∗Dx )⊥ Φ(x(t))'. = {0}'.= / = {0}) 6 / 1 6'=. $()'I)$ ( 1 6 x̃(t) = 3  m   ̃ ̇  x̃ (t) = f (x̃ (t), x̃ (t)) + f ̃i,1 (x̃1 (t), x̃2 (t))ui (t) + D̃x (x̃1 (t), x̃2 (t))d(t) 1 0,1 1 2     i=1   m    ̃ x̃ ̇ 2 (t) = f ̃0,2 (x̃1 (t), x̃2 (t)) − Ψ̃0 (x̃2 (t), yx (t)) + fi,2 (x̃1 (t), x̃2 (t)) − Ψ̃i (x̃2 (t), yx (t)) ui (t)             + Ψ̃0 (x̃2 (t), yx (t)) + m i=1 Ψ̃i (x̃2 (t), yx (t))ui(t) i=1 yx (t) =h̃(x̃1 (t), x̃2 (t)) T Ψ̃0 (x̃2 (t), yx (t))+ m I)$M( Ψ̃i (x̃2 (t), yx (t))ui(t) = Ψ̃((x̃2 (t), u(t), yx (t))). 0 i=1'2 D D)M() >. =. 0 1.6 3 f ̃i,2 (x̃1 (t), x̃2 (t)) − Ψ̃i (x̃2 (t), yx (t)) = f ̆i,2 ( x̃2 (t)), ∀i ∈ {0, · · ·, m}. /./.. 0'2 7.'I)$#(.. / 2 2 x̃ ̇ 2 (t) x̃1 (t)) I)$M( 3  m   ̃ ̇  x̃1 (t) =f0,1 (x̃1 (t), x̃2 (t)) + f ̃i,1 (x̃1 (t), x̃2 (t))ui (t) + D̃x (x̃1 (t), x̃2 (t))d(t)     i=1 m m ̆ ̆ ̇  Ψ̃i (x̃2 (t), yx (t))ui (t) fi,2 (x̃2 (t))ui (t) + Ψ̃0 (x̃2 (t), yx (t)) + x̃2 (t) =f0,2 (x̃2 (t)) +    i=1 i=1    y (t) =h̃(x̃ (t), x̃ (t)) x 1 2 I)$K( 0 '. 0 6 / x̃(t) = Φ(x(t)) (S∗Dx )⊥ ) 0 2. 7 ),'∂φ(x(t)) ∂x(t) 7 Φ(x(t)).  2 7 ) ( 3 ∂φ(x(t)) ∂x(t)  ∂Φ(x(t))   =  −−−−−  ∂x(t) D ⊥ T (S∗ x ) T x̃2 (t) (S∗Dx )⊥ ). 1 / I)$ ( 2 6 6 '. ) ))) d(t) $.2. 7.. I /. ). / ) / 6, / 2 )!. 0 6. (0) (ρd −1) Ψ̃(x̃2 (t), u(t), u̇(t), yxd (t), · · ·, yxdm 6. n − ρdm ≤ ( 3 (t)) I)L( 3. / d(t)) ≤ n − 1 I)LI(, / -! 6... /.. 0 1 3 Ψ̃(x̃2 (t), u(t), yx(t)) 0≤ I)L (. / 3 ((S∗Dx )⊥ ) = (. / d(t)) ≤ n − 1 I)L$( T 1. 6 1. E{dh})'/ 0. /.. d(t) 0 I)L( I)L ((!" # ),'6.. / () ) 5 / : 6 I)LI( I)L$( )$) )$)$('. / ). 0 /... / / 6 )! 6 4 0 1 )'6 '.. 1 ) 4 2 S 2 6 ) &'3 7 QKR' 7 X /. / 2 QJ$R' QM#R' Q R' QMIR' QMMR) 6 /..... ) S 7 Q M' #R' Q$$' $JR' QILR' Q$IR' QLJR QLR' /. 6 ( S QL#R'. ) 1 6..) S Q KR' QM R) 4.6. 7 6 6. 6 )! 2.. ).2 7 ( '. 7'/ I)LL(' z̃2 (t) (S∗Dx )⊥ 7 I)$ () ΣF N L : z(t)    ż(t) =       f0 (z(t)) + m fi (z(t))ui (t) i=1 + {Ψ0 (z̃2 (t), yz (t)) − Ψ0 (z̃2 (t), yx (t))} m {Ψi (z̃2 (t), yz (t)) − Ψi (z̃2 (t), yx (t))} ui(t) + I)LL(        i=1   yz (t) = h(z(t)). '. 6 1 6 yz (t)( yx (t)( 3 r(t) = χ ◦ h(yx (t)) − χ ◦ h(yz (t)) 5 1'(r(t)) ≥ (w(t))( I)LJ( 1 ) & / 6'< 2 i( 6 ii( 3 i( 1 : 3 1 = 2 / 2' ii( 1 : = 3 1 ) 1.0 2 2 ) 7 6'2 / 6 3 S. 7' 6. 2 S 7 2 1 ('' 7 2 1 ) >.'7 Q R 6 QI R( I)$) ) 6 )... 7. 7 =. 1 ()! & 1'6 7 2 1..'/.. ( 6 -'+)&))-) 6 () 6 QLKR(), '.2 Q$JR QI#R() + &'1.0 Q#R' QL$R' QMLR( 2 2 r(t) 1 QJ R!$ XX XXX " XX XXX XXX / X 1 w1 (t) w2 (t) ··· wq (t) r1 (t) ··· r2 (t) ··· )) ) )) ) )) ) )) ) )) )) ) rq (t) ··· I)I3 > 6  r1 (t)  )  ))   r(t) =  ri (t)  )  ))  rq (t) q= 1  ⇐=     ⇐=     ⇐= ) 6 +)&))-) 3 1 w1 (t) 1 wi (t) 1 wq (t) I)LM( (w(t))).2. 7. 0/ I)LM( <. $' ) & ) I)I 6 >'6 ) +)&))-) I)LM() 1 7 =.2'QMR() /.'/ /. / './.. 1. ), 6' = 6 7. 2 1'QI$R(),'6?1 @ 5 1 )'' )))'/. 1)! 2 /. 7 +)&))-)' / 7 )'7. 7 ) " QJKR QJ R(''6.. ) $. 6 ) 6 6 6 7 ( (),.. 7 3 S 7 3 ΣF L  ż(t) = Az(t) + Bu(t) + L(y (t) − y (t)) z x : yz (t) = Cz(t) I)L#( S 7 3 ΣF N L : T L(•) 5    ż(t) =                 f0 (z(t)) + fi (z(t))ui (t) i=1 + {Ψ0 (z(t), yz (t)) − Ψ0 (z(t), yx (t))} m {Ψi (z(t), yz (t)) − Ψi (z(t), yx (t))} ui (t) + yz (t) = h(z(t))'0 ) /. 1'6..'6 < ) 7. ))) / &-! 1 = 0(. 1 0# 6 ) 3 5 9 3 7 7 ),'1'1. i / 1 ρw m'( ) Q$R' QMJR Q #R' Q$KR) & I)LK( i=1 Ψi (•) i = {0, · · ·, m}( 6 m 5)9) )' 76 I) (). 1'6. / 1 3. 1 ). ) 1 )../ 1!$ 1 u(t) w(t) 1 / w1 (t) 7 7 I) 0'3 ∀(λ, i) r1 (t) rq (t) 3 1 : A − λI =n Ci I)L ( T C = C1T · · · CiT · · · CpT ) ))) < / 1 3 yx (t). q / wq (t) 3 0# 7 9 1 u(t) )9) )' 76 I)$() w(t) 7 I)$3 /. yx (t) r(t) 7 3 ) 4 /.=./ 6 1 ) &-! 0 3 1 2 0# i : 3 - Y 9 u(t) 1 3 A − λI =n Ci -)9) )' 76 I)L() w(t) 1 / 7 3 q yx (t) r1 (t) rq (t) w1 (t) q / I)L3 I)J( 7 / 3 ∀(λ) 2 ) ))). : ) 5'1 / 7. wq (t). 6 '.2 / 1) &-! 3 1 :'. / 1... / 1) 0'1 / 7 ∀(λ, i) 1 : 3 2 2 3 A − λI =n C̄i I)JI(!% & T T T Ci+1 · · · CpT ) C̄i = C1T · · · Ci−1 5'2 1 6.. 2 5'/. 6 6 ) 2 O 6 / ) # $ 7 26. = 2' 1.'' ) 6 6 ) 6'1'+)&))-) ), '. 6,'1 6 I) ) )$(' 2 ) 1'2 ) /.. 6 ) 1 6 ) 5)9) )' )9) ) -)9) )( ) 2 / 2) 2 /,) -') 5 & 7 0 = 2 ) ) 4') 7 1. '.2 + 5 2 3   ẋ1 (t) = −x1 (t) + u1(t) + d(t)      ẋ2 (t) = x1 (t) − x2 (t)     ẋ (t) = x (t) − x (t) + x (t) + u (t) 3 2 3 4 2 ΣL :   ẋ4 (t) = −x3 (t)      yx1(t) = x3 (t)     yx2(t) = x4 (t) T Dx = 1 0 0 0 I)J ( T ) 7. 1 1 2 6 / ) )F)5) " 3 D 1 0 0 0 I)II((' C11 /. Dx1 = I)I ( ) 4. ). / /. 6 / L) 0 5 d(t) T 2 6 I) ) )$ 6 " CDx = 0) 6 I) ) )L() 3 2 I)IK(()'ρdm = ρdm1 = 3 7 I) ( /. 3    (0) x3 (t) yx1 (t)   y (1) (t)   x2 (t) − x3 (t) + x4 (t) + u2(t)     x1 = x̃(t) = Φd (x(t), u(t)) =     (2) (1)    yx1 (t) x1 (t) − 2x2 (t) − x4 (t) − u2 (t) + u2 (t) φ(x(t), u(t)) x4 (t)  I)J$(!% & 5 I)J ( . 3   x̃ ̇ 1 (t) = x̃2 (t)      x̃ ̇ 2 (t) = x̃3 (t)     x̃ ̇ (t) = −3x̃ (t) − 4x̃ (t) − 3x̃ (t) + x̃ (t) + u (t) + u (t) + 2u(1) (t) + u(2) (t) + d(t) 3 1 2 3 4 1 2 2 2 ΣL :   x̃ ̇ 4 (t) = −x̃1 (t)      yx1 (t) = x̃1 (t)     yx2 (t) = x̃4 (t) I)JL( Σ FL 7 3   z̃ ̇1 (t) = z̃2 (t)       z̃ ̇2 (t) = z̃3 (t)     z̃ ̇ (t) = −3z̃ (t) − 4z̃ (t) − 3z̃ (t) + z̃ (t) + u (t) + u (t) + 2u(1) (t) + u(2) (t) 3 1 2 3 4 1 2 2 2 :  z̃ ̇4 (t) = −yx1 (t)      yz1 (t) = z̃1 (t)     yz2 (t) = z̃4 (t) I)JJ(. 0 (0) (1) (2) Ψ̃(x̃4 (t), u(t), u(1) (t), yx1 (t), yx1 (t), yx1 (t)) = 0 0 0 −yx1 (t). z̃4 (t) / d(t)) T I)L((). / 6 / I. 6 / $) " 6 3 '. / 1 S∗Dx I)$L(( 7 3 S∗Dx       1  −1 1     0  1  −2       =  ,  ,    0  0   1       0 0 0 / 7'6 I)JM( I)$ ( 1 3  1 0 0  ∂Φ(x(t)) −1 1 0 = ∂x(t)  1 −2 1 0 0 0  0 0   0 1 I)J#( 5 I)J (. 3   x̃ ̇ 1 (t) = −x̃1 (t) + u1 (t) + d(t)       x̃ ̇ 2 (t) = x̃1 (t) − x̃2 (t) − u1 (t)     x̃ ̇ (t) = −x̃ (t) + x̃ (t) − x̃ (t) + x̃ (t) + u (t) + u (t) 3 1 2 3 4 1 2 ΣL :   x̃ ̇ 4 (t) = −x̃1 (t) − 2x̃2 (t) − x̃3 (t)      yx1 (t) = x̃1 (t) + 2x̃2 (t) + x̃3 (t)     yx2 (t) = x̃4 (t).. 0 Ψ̃(x̃4 (t), u(t), yx(t)) = 0 0 0 −yx1 (t) T I)L ((' 7 3 Σ FL   z̃ ̇1 (t) = −z̃1 (t) + u1 (t)       z̃ ̇2 (t) = z̃1 (t) − z̃2 (t) − u1 (t)     z̃ ̇ (t) = −z̃ (t) + z̃ (t) − z̃ (t) + z̃ (t) + u (t) + u (t) 3 1 2 3 4 1 2 :  z̃ ̇4 (t) = −yz1 (t)      yz1 (t) = z̃1 (t) + 2z̃2 (t) + z̃3 (t)     yz2 (t) = z̃4 (t) '. I)JK( z̃4 (t) I)J ( / ). / 6 / I. 6 / $) 0. 6 9' /..'6 ( / ) 7 ) & 3 3 (2) (1) (1) x1 (t) = y1 (t) + 2y1 (t) + 2y1 (t) − y2 (t) − u2 (t) − u2 (t) I)M(!% & 7 ΣF L : T 3 2 3 4 2   ż4 (t) = −z3 (t)      yz1 (t) = z3 (t)     yz2 (t) = z4 (t) (2) (1) (1) y1 (t) + 2y1 (t) + 2y1 (t) − y2 (t) − u2 (t) − u2 (t) 7.6. z3 (t) I)L ( 3   ż1 (t) = −z1 (t) + u1(t)     (2) (1) (1)   ż2 (t) = y1 (t) + 2y1 (t) + 2y1 (t) − y2 (t) − u2 (t) − u2 (t) − z2 (t)     ż (t) = z (t) − z (t) + z (t) + u (t) I)MI(. 
  
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  ''6 I) ) )L < 6.2 &. < / < 5') 4 ) 5 2 ) 2 %) %'< ) &. ),'< 6 I)J(). 62 I)$( / 1 2' I)J() '?1 @'. / 2 ) 5'62 '. 26 ) 5 '. 6 ) 6. = / ). 1 ) 5'6 1 / 1 ). 1 ) & '. 0 ∀(x(t), t))'5 ='' 6 ). / 1'7.2 ),'0. 2 < ).,. 6 ) 4')'' 7 I)L) '... 2... 2 ) 5 & z(t) 2 6. ) 7 = 6 7 6),'6. '.0 1. 2'0 ) 6.) &. / 0 1 6) 5 6'6' 6 1 2) 6.) 1' 1.. 1 /'6 2 ( ) 7' '.. < 1 2' ),7' )! 5'1 6 )I( /.. 7 ) (() ΣN L :  m   ẋ(t) = f0 (x(t)) + fi (x(t))ui (t) + Dx (x(t))d(t)   yx (t) = h(x(t)) i=1 )I( "'T x(t) ∈ Rn'u(t) ∈ Rm'd(t) ∈ Rs yx (t) ∈ Rp )  m   ż(t) = f0 (z(t) + fi (z(t))ui (t) + Ψ(•x (t), •z (t), •u (t)) ΣF N L : i=1   yz (t) = h(z(t)) z(t). ) /. x(t) Ψ(•x (t), •z (t), •u (t)). 0 •x (t)' •z (t) •u (t) '. 7 2 yx (t)(' ) ( ) (' () 2. 0 1.0 1 / 7 / =. ) 1 '. 6'7 ). =. 0 7 7. z̃2 (t)( z̃(t) = Φ(z(t))( / d(t) 3 0 1  m   ̃ ̇  z̃1 (t) =f0,1 (z̃1 (t), z̃2 (t)) + f ̃i,1 (z̃1 (t), z̃2 (t))ui (t) + Ψ̃1 (•x (t), •z (t), •u (t))     i=1 m Σ : F N L  z̃ ̇ (t) =f ̃ (z̃ (t), z̃ (t)) + f ̃i,2 (z̃1 (t), z̃2 (t))ui (t) + Ψ̃2 (•x (t), •z (t), •u (t)) 2 0,2 1 2    i=1    y (t) =h̃(z̃ (t), z̃ (t)) z 1 2 )$( 3 f ̃0,2 (z̃1 (t), z̃2 (t)) + m f ̃i,2 (z̃1 (t), z̃2 (t))ui (t) + Ψ̃2 (•x (t), •z (t), •u (t)) = x̃ ̇ 2 (t), ∀(d(t), u(t), t) i=1 )L( T x̃2 (t).,. 2 '. 0 Ψ(•x (t), •z (t), •u (t))... ) 2. 6 6.. = z̃1 (t)'1 / z̃2 (t) I)$$( 2 6..) > <. 6 ) ) / /.. 6 ) d(t) (. z̃1 (t) z̃2 (t) =. {0}) "!" (! ) " # 1 ) 5 & 6') 4 QI$R' QILR' QIMR QI#R / ) & % '.. 0 3 Ψ̃1 (•x (t), •z (t), •u (t)) = 0 )J( Ψ̃2 (•x (t), •z (t), •u (t)) = Ψ̃(z̃2 (t), u(t), yx(t)) − Ψ̃(z̃2 (t), u(t), yz (t)) 5 '.  Dx   S0 = 7 I)$$(( 3 {Dx } Dx Dx   Si+1 = S i + m )M( Dx [fk, S i ∩ E{dh}] k=0.= 3 Dx Si 52 Dx Dx = Si+1 ⇒ S∗Dx = S i 7 )#( S∗Dx ) 7. / d(t)' 2. 0 7 )J().2 (S∗Dx )⊥. z̃2 (t) /. )$('{0}) '. 6 &'(S∗Dx )⊥ 6. / ) /.. 2. / ) / "'()'ΣN L T Q JR( 3       x 0 0 (t)x (t) 1 4          x (t)(1 − x (t)) 0    0  u1 (t) + d1 (t) 4  3      ẋ(t) =    +      0   0 x1 (t) u2 (t) + d2 (t) :  1 0 0         x (t)    1   y (t) = x  x2 (t) x(t) ∈ R4'u(t) ∈ R2'yx (t) ∈ R2 d(t) ∈ R2 )K('' ) 6.. / 3 (Dx1 d1 (t)   0 0 0 0    Dx2 ) d(t) =   d(t) 0 x1 (t) 1 0'S∗Dx1 7 ) ( 3 1 S0Dx1 &.   0      0   = {Dx1 } =        0    1 I 0')I( S̄0Dx 1 = S0Dx1 ) S̄0Dx1 ∩ E{dh} = S̄0Dx1     0 0      0 0     E{dh} =  ,    1 0      0 1 )II( "!" (! )  S1Dx1            x1 (t)x4 (t) 0 0 0 0 0 x (t)(1 − x (t)) 0 0 0  0  0  3            4 = S̄0Dx1 +  ,   ,   +  ,   +               0 0 0 0 x1 (t) 0 0 1 0 1 1 1       −x1 (t)  0  0               x (t)  0 0  3      Dx1 = S̄0 +   +   +       0  0 0                0 0 0     −x 0 (t)   1    0  x (t)     3  =  ,    0  0       1 0 )I (     −x1 (t) 0 0  x (t)     3   ,   = 0  0  1 0   0      0      0      0 )I$( S̄1Dx1 = S1Dx1 ) 2 S̄1D x1 ∩ E{dh} = S̄0Dx1 & )IL('3 S2Dx1 = S1Dx1 ⇒ S∗Dx1 = S2Dx1 S∗Dx1 5.'S∗Dx2 S∗Dx2     0 −x1 (t)      0  x (t)     3  =  ,    0  0       1 0 )IJ( 7 3     0 0       0  −x (t)(1 − x (t))    1  4 =  ,   x1 (t)   x1 (t)x4 (t)      0 0. )IJ( )IM(( )I#(.'0 )IK(( )IM( 6 "'' / d1 (t) d2 (t)()     x3 (t)  0     0 x (t)    1  (S∗Dx1 )⊥ =  ,    1  0       0 0 )I#(     1 0      0 0     Dx2 ⊥ (S∗ ) =  ,   0 0      0 1, )I#(. z̃2 (t) )$() T (x̃T1 (t) x̃T2 (t))T = ΦT1 (x(t)) ΦT2 (x(t)) = T Dx1 ⊥ {dΦ1 } = S∗Dx1 {dΦT ) ) 5 2 } = (S∗ d1 (t) = 0' {dΦ1 } = S∗Dx2 z̃2 (t).2. 7 < )IK( )IK((' 2 7. z̃2 (t) = '. x̃2 (t) = {dΦ2 } = (S∗Dx2 )⊥ ) 7 x̃2 (t) d2 (t) = 0' )$( 6 ), '. 0
  
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