title: Diagnostic des systèmes non linéaires : contribution aux méthodes de découplage
text: 0 7 ) 5 /.'76.. 6. ) 5 2 6 7
  
  )! ΣN L :, 3  m   ẋ(t) = f0 (x(t)) + fi (x(t))ui (t) + P (x(t))w(t)   yx (t) = h(x(t))... 1 7 $)I( i=1 w(t) /. x(t)' 3     Φ1 (x(t)) x̃1 (t)     x̃(t) = Φ(x(t)) =  − − −  = − − − Φ2 (x(t)) x̃2 (t) $) ( $ * T {dx̃T 1} = (dx̃2 )⊥ T ) /. 1 ΣN L : $)I(. 3  m   ̃ ̇ x̃1 (t) = f0,1 (x̃1 (t), x̃2 (t)) + f ̃i,1 (x̃1 (t), x̃2 (t))ui (t) + P̃11 (x̃1 (t), x̃2 (t))w1 (t)    i=1      +P̃12 (x̃1 (t), x̃2 (t))w2 (t)  − − − − − − − − − − − − − − − − − − − − − − − − − − − − −−   m    ̃ ̇  f ̃i,2 (x̃1 (t), x̃2 (t))ui (t) + P̃2 (x̃1 (t), x̃2 (t))w2 (t) x̃2 (t) = f0,2 (x̃1 (t), x̃2 (t)) +     i=1   y (t) = h̃(x̃ (t), x̃ (t)) x 1 2 $)$( T w1 (t) 1 w(t) / w2 (t). $)$(. U. 6 /.2 x̃1 (t)' 1. 6 w(t)) O. 6 1 : w1 (t).( ), '.. x̃2 (t)) &.. x̃1 (t) x̃2 (t)) 1.. '. x̃2 (t) 6. w1 (t)).. 0 0 1.2 x̃2 (t))'/. $)$( x̃1 (t) 7 3  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    − − − − − − − − − − − − − − − − − − − − − − − − − − − − −− Σ : m F NL   ̃ ̇  f ̃i,2 (z̃1 (t), z̃2 (t))ui(t) + Ψ̃2 (•x (t), •z (t), •u (t)) z̃2 (t) = f0,2 (z̃1 (t), z̃2 (t)) +     i=1   yz (t) = h̃(z̃1 (t), z̃2 (t)) $)L( T Ψ̃(•x (t), •z (t), •u (t)). 0.2 ) () 0 f ̃0,2 (z̃1 (t), z̃2 (t)) + m 1.6 3 f ̃i,2 (z̃1 (t), z̃2 (t))ui (t) + Ψ̃2 (•x (t), •z (t), •u (t)) = x̃ ̇ 2 (t), ∀(w(t), u(t), t) i=1 $)J(.2 1 '. 0 1'7 6 6 ).. 0. 6 7 6) ) $!" ( + #& 5 6'. 0. Ψ̃(z̃2 (t), ∆) T ∆ 62 ), '. ) 5 & 62. 0 ) 5.2. ) 4'' 2 (. 0 1) $,. 7 $)L('. 6 3 Ψ̃1 (•x (t), •z (t), •u (t)) = 0 $)M( Ψ̃2 (•x (t), •z (t), •u (t)) = Ψ̃(z̃2 (t), u(t), yx(t)) − Ψ̃(z̃2 (t), u(t), yz (t)).0. 7 /. 1 ). 6'2. 0. 6 7 7) 6 T 7 /. . 1 T 7.0 1. 1 ) 5 6'. 7 )M ( )#( 1 )'1. 1 6 / 7' q ))) $. I)LM(() 5 7 '. 2.. 1 6 & q'1). / / 1 ) 08. P S∗ j 9 1 '. j = {1, · · ·, q}' 7 q 3   j   j Φ1 (x(t)) x̃1 (t)     x̃j (t) = Φj (x(t)) =  − − −  = − − − Φj2 (x(t)) x̃j2 (t)  P S∗ j T ∂Φj (x(t))   = −−−  ∂x(t) P (S∗ j )⊥     T $)#( $ * ΣF N Lj    z̃ ̇1j (t)        j ̇ : z̃2 (t)          yz (t) T. 7. 7 3 j = f ̃0,1 (z̃1j (t), z̃2j (t)) + m j (z̃1j (t), z̃2j (t))ui (t) f ̃i,1 i=1 j = f ̃0,2 (z̃1j (t), z̃2j (t)) + m j (z̃1j (t), z̃2j (t))ui (t) f ̃i,2 $)K( i=1 +Ψ̃j (z̃2j (t), u(t), yx (t)) − Ψ̃j (z̃2j (t), u(t), yz (t)) = h̃j (z̃1j (t), z̃2j (t)) / z̃1 (t). j (w1 (t) · · · wj−1(t) wj+1(t) · · · wq (t))T. 0 $)M(()'j. / z̃2 (t) 6. 1 wj (t)) '. 0 Ψ̃j (• 
  
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  1.6 3 z̃ ̇2j (t) = x̃ ̇ 2 (t), ∀wi (t).2 j (z̃ j (t)) + f ̃0,2 m i = {1, · · ·, j − 1, j + 1, · · ·, q} $) (. 0 3 j j (z̃ j (t))ui (t) = f ̆0,2 (z̃2j (t)) + f ̃i,2 i=1 m j (z̃2j (t))ui (t) + Ψ̃j (z̃2j (t), u(t), yz (t))(t) f ̆i,2 i=1 j Ψ̃j (z̃2j (t), u(t), yz (t))(t) = f ̃0,2 (z̃ j (t)) + m ⇓ j (z̃ j (t))ui − f ̃i,2 j f ̆0,2 (z̃2j (t)) + i=1 m j (z̃2j (t))ui (t) f ̆i,2 i=1 j = {1, · · ·, q} $)I( > 7.. $)II( 3 ΣF N Lj  m  j j j j j  ̃ ̇ z̃ (t) = f (z̃ (t), z̃ (t)) + (z̃1j (t), z̃2j (t))ui (t) f ̃i,1  1 0,1 1 2    i=1 m : ̇j j j f ̆i,2 (z̃2j (t)) + (z̃2j (t))ui (t) + Ψ̃j (z̃2j (t), u(t), yx(t)) z̃2 (t) = f ̆0,2     i=1   yz (t) = h̃j (z̃1j (t), z̃2j (t)) &'$)I('. 3. 7. )$)J)I) j f ̃i,1 j f ̆i,2 τ (x(t)) ⊂ 2 D() $)II(. 0 1 {dz̃1T } / /, τ (x(t)) ⊆ {dz̃1 T }, ∀i ∈ {0,
  
  · · ·, m}. 0 '.2 $)I ( 6 $!" ( + + ΣN L : & )J#(( 3   ẋ1 (t) = x2 (t)x3 (t) + w(t)      2   ẋ2 (t) = −x1 (t)x2 (t) + x3 (t)u(t) ẋ (t) = x2 (t) − x3 (t)  3    yx1 (t) = x2 (t)     yx2 (t) = x3 (t) 1'/. 6' 1 w(t). 7).. / 7 1 ΣF N L : $)I$( 0 / x3 (t)) $)K( 3   ż1 (t) = z2 (t)z3 (t) + Ψ1 (z(t), yx (t)) − Ψ1 (z(t), yz (t))      2   ż2 (t) = −z1 (t)z2 (t) + z3 (t)u(t) + Ψ2 (z(t), yx (t)) − Ψ2 (z(t), yz (t)) ż3 (t) = z2 (t) − z3 (t) + Ψ3 (z(t), yx (t)) − Ψ3 (z(t), yz (t))     yz1(t) = z2 (t)     yz2(t) = z3 (t), 0. 6 )M$(' 1 $)IL( 7. 3        (t) (t)z (t) ż z  1 2 3  =   z̃ ̇1 (t) =    2  ż2 (t) −z1 (t)z2 (t) + z3 (t)u(t)  Σ : ̇ F N L z̃2 (t) = ż3 (t) = z2 (t) − z3 (t) + Ψ̃(z̃(t), yx (t)) − Ψ̃(z̃(t), yz (t))     yz1 (t) = z2 (t)     yz2 (t) = z3 (t) 6 z̃2 (t). 0 1).0 1.2 ) & z2 (t) / 6 $)K(('. z̃ ̇2 (t) $)IJ( / z̃1 (t) 1 /.. 0 1 Ψ̃(z̃(t), yz (t)) = z2 (t) = yz1 (t) $ *.T 7 3      z2 (t)z3 (t)     z̃ ̇1 (t) =    2  −z1 (t)z2 (t) + z3 (t)u(t)  Σ : ̇ F N L z̃2 (t) = −z3 (t) + Ψ̃(z̃(t), yx (t))     yz1 (t) = z2 (t)     yz2 (t) = z3 (t) $)IM( Ψ̃(z̃(t), yx (t)) = x2 (t) = yx1(t)) / 7 '.6 < 6 x3 (t) z3 (t)) '. ),'6 ) 5. 2 )M ( )#(' rj (t) y '. /. 2 rj (t) 3 rj (t) = χPj ◦ h(x(t)) − χPj ◦ h(z(t)) $)I#( = χPj ◦ h(x̃2 (t)) − χPj ◦ h(z̃2 (t)) wj (t)(' q 7 q 1 $)I#('.. 1 w(t)) 7'' / 6 /.. 1 2 ). '. 7 / 1 ))) $ 08 7 Pj S∗ φPi,···,Pj P · · ·'S∗ j ) /. 1 ) φPi,···,Pj S∗Pi')'Pj. S∗ 7 7 P (i) φ ⊆ S∗Pi ∩ · · · ∩ S∗ j = {0} (ii) φPi,···,Pj ∩ S∗Pk = {0} k = {i, · · · Pi,···,Pj 6 j = {1, · · ·, q}( ) 1 /.. 7' 1 / 1 9 1'6 QLR(), '. w(t)). 3 i = j ', j}' $!" ( + φPi,···,Pj. 7 / Pj {0}() & 7 φPj. S∗ S∗Pk T k = {1, · · ·, j −1, j +1, · · ·, q}) / 7'Pj. S∗ 3 S∗ j = φPj ⊕ φPj,Pi ⊕ · · · ⊕ φ···,Pj,··· P / {0} φPj $)IK( / 7 ) ) w(t) 0 1, #AB #C7 /! wj (t) j = {1, · · ·, q} φPj! wj (t) 1 ) 0 #AB #C7 : P 5 {Pj } ⊆ (S∗ j )⊥! P P S∗ j ) {Pj } ⊆ S∗ j P S∗Pi ⊆ S∗ j ) i = {1, · · ·, j −1, j + 1, · · ·, q} 3 q! 0 ) ),,( S∗Pi ⊆ P S∗ j 0! ) {Pj } ⊆ P S∗ j ⊥ i=1,i=j q ⊥ 0 ⊆ S∗Pi i=1,i=j 1 {Pj } = {0}! φPj = {0} 4. /.. / 76 $)I T 7 1 ) 7'1 76 ) 6 S∗P1 &'{Pj }) 9. 3 q S∗P2'P S∗ j 2 S∗P3 S∗P4 $)I {Pj } 7 {0}. / 3 {P1 } ⊆ S∗P1 ) ( 6 / φPj ) x̃(t) = Φ(x(t))( ∂Φ(x(t)) = φP1 · · · φPq φPk1,Pk2 · · · φPk3,···,Pk4 γ ∂x(t) T $)I ( $ * X {P1 } S∗P1 S∗P1 S∗P4 S∗P2 S∗P3 $)I3.. $!" ( + T γ /. 6 γ ⊂ (φ• )⊥'∀•() φ P S∗ j j = {1, · · ·, q} k1, k2, k3, · · ·, k4 ∈ {1, · · ·, q}) 2 & ∂Φ(x(t)) 6 ∂x(t) • T x̃(t) = x̃1 (t) · · · x̃q (t) x̃k1,k2 (t) · · · x̃k3,···,k4 (t) x̃γ (t) 5 x̃• (t)'.2 0 5 = φPi 2 φ Pi φPi,Pj'/.. 1 wi(t)) 6 / k = i) φ Pi,Pj, =. x̃i,j (t). φ ···,Pk,··· wi (t) wj (t)' k = i, j ) j = {1, · · ·, q} j = {1, · · ·, q}() 3 S∗ j = φPj ⊕ φPj,Pi ⊕ · · · ⊕ φ···,Pj,··· P T.,7 Pj. S∗ 6 '..2 1 = 6 / & x̃i (t) 1W' 6 = 1 ) 2 /.. φ···,Pk,···'7. = Φ(x(t))) φ• $) ( 6 ) {dxT γ } 1 w(t) 0 )'2. 0. 7 2 6 +. ) / 1.. $) (' ) 5'T.. <. $ 1) 5 2 1 )'1 $ * 2 S∗P1 3     0 0      0 1     =  ,    0 1      1 0 φP1 = S∗P1'φP2 = S∗P2 φP1,P2 S∗P2     1 0      0 1     =  ,    0 0      0 0. 7 S∗P1 ∩ S∗P2 = {0}) $) I(. $) ( / 3 S∗P1 = φP1 φP1 1 )'7 Φ(x(t)) φP2. 6 / = S∗P2 = φP2 $).. 6 2.)'6 6 φ P1 φ P2 $) $((' 7 3  0 0 0 1 φP1      − − − − −  − − − − −  1 0 0 0  P T  ∂Φ(x(t))      = φ 2  =  − − − − − ∂x(t)      − − − − −  0 1 0 0  T γ 0 0 1 0  ( γ ⊆ (φP1 )⊥ 9 T   γ ⊆ (φP2 )⊥ ). = ),'1 w(t) (S∗P1 )⊥
  
  (S∗P2 )⊥ ) '. 6. .. 1 6 γ. γ( 0 1 0 0, 2 1. S∗P2 ) ). / $) $( w1 (t) 6 T w2 (t)) 1 w2 (t) =. 0 0 1 0 = 0 6 w1 (t) w2 (t)... 7) $ 2 S∗P1 3     0 0      0 1     =  ,    0 0      1 0 S∗P2     1 0      0 1     =  ,    0 0      0 0 $) L( T $!" ( +, φ P1 φ P2'5 '. S∗P1 = φP1 ⊕ φP1,P2 S∗P2 = φP2 ⊕ φP1,P2 φ P1,P2 = 0 1 0 0 T ) 1 3 φP1 ⊆ (φP2 )⊥ φP2 ⊆ (φP1 )⊥ φP1 ⊆ (φP1,P2 )⊥ φP2 ⊆ (φP1,P2 )⊥ $) J( $)I (. 3  T   0 0 0 1 φP1     − − − − −  − − − − −    T    1 0 0 0   φP2     ∂Φ(x(t))     = − − − − − =  − − − − −    ∂x(t)    φP1,P2 T  0 1 0 0          − − − − −  − − − − −  (γ)T 0 0 1 0 . 0 $) M(. ). /. 1 ΣN L : $) ( $)I (. 1'6 $)I( 3  m   P −1 1 1 ̃ ̇ x̃1 (t) = (φ.ẋ(t)) ◦ (Φ) (x̃(t)) = f0 (x̃(t)) + f ̃i1 (x̃(t))ui (t) + P̃ 1 (x̃(t))w1 (t)     i=0   ))   )    m    q Pq −1  ̃ ̇  f ̃iq (x̃(t))ui (t) + P̃ q (x̃(t))wq (t) x̃q (t) = (φ.ẋ(t)) ◦ (Φ) (x̃(t)) = f0 (x̃(t)) +    i=0   m    x̃ ̇ k1,k2 (t) = (φPk1,Pk2.ẋ(t)) ◦ (Φ)−1 (x̃(t)) = f ̃0k1,k2(x̃(t)) + f ̃ik1,k2(x̃(t))ui (t) i=0  ))    )    m   k3,···,k4  Pk3,···,Pk4 −1 ̃ ̇ .ẋ(t)) ◦ (Φ) (x̃(t)) = f0 (x̃(t)) + f ̃ik3,···,k4(x̃(t))ui(t) x̃k3,···,k4(t) = (φ     i=0  m    ̇  x̃γ (t) = (γ.ẋ(t)) ◦ (Φ)−1 (x̃(t)) = f ̃0γ (x̃(t)) + f ̃iγ (x̃(t))ui (t)     i=0    y (t) = h ◦ (Φ)−1 (x̃(t)) = h̃(x̃(t)) x $) #( T x̃(t) = x̃1 (t) · · · x̃q (t) x̃k1,k2 (t) · · · x̃k3,···,k4 (t) x̃γ (t) = Φ(x(t)) (x̃• (t)) = (φ• )) wj (t) :. 6 j. x̃j (t) j = {1, · · ·, q}) $ * $)I) '. 0 7 7 3  m   1 ̃ ̇  z̃ (t) = f (z̃(t)) + f ̃i1 (z̃(t))ui (t) + Ψ̃1 (z̃(t), u(t), yx (t)) − Ψ̃1 (z̃(t), u(t), yz (t))  1 0    i=0   ))    )    m   q  ̃ ̇  f ̃iq (z̃(t))ui (t) + Ψ̃q (z̃(t), u(t), yx(t)) − Ψ̃q (z̃(t), u(t), yz (t)) (t) = f (z̃(t)) + z̃ q  0    i=0  m    k1,k2  ̃ ̇ z̃k1,k2 (t) = f0 (z̃(t)) + f ̃ik1,k2(z̃(t))ui (t)     i=0  k1,k2 (z̃(t), u(t), yx (t)) − Ψ̃k1,k2(z̃(t), u(t), yz (t)) + Ψ̃ ΣF : NL  ))    )    m   k3,···,k4  ̃ ̇  f ̃ik3,···,k4 (z̃(t))ui(t) (t) = f (z̃(t)) + z̃ k3,···,k4  0    i=0    k3,···,k4  +Ψ̃ (z̃(t), u(t), yx(t)) − Ψ̃k3,···,k4(z̃(t), u(t), yz (t))    m  γ    ̇γ (t) = f ̃0γ (z̃(t)) + f ̃i (z̃(t))ui (t) + Ψ̃γ (z̃(t), u(t), yx (t)) − Ψ̃γ (z̃(t), u(t), yz (t)) z̃     i=0    yz (t) = h̃(z̃(t)) $) K(. ) 5 & ) 4 )M ( )#((.. $) (' 7 Σ : F NL 3  m   1 1 ̆ ̇  z̃1 (t) = f0 (z̆∗ (t)) + f ̆i1 (z̆∗1 (t))ui (t) + Ψ̃1 (z̆∗1 (t), u(t), yx (t))     i=0   ))   )     m   q q  ̆ ̇  f ̆iq (z̆∗q (t))ui (t) + Ψ̃q (z̆∗q (t), u(t), yx(t)) z̃q (t) = f0 (z̆∗ (t)) +     i=0  m    k1,k2 k1,k2 ̆  ̇ z̃ f ̆ik1,k2(z̆∗k1,k2 (t))ui (t) + Ψ̃k1,k2 (z̆∗k1,k2 (t), u(t), yx(t)) (t) = f (z̆ (t)) +  0 ∗  k1,k2 ))  ) i=0   m   k3,···,k4 k3,···,k4  ̆ ̇  (t) = f (z̆ (t)) + f ̆ik3,···,k4(z̆∗k3,···,k4(t))ui ( t) z̃ k3,···,k4  0 ∗    i=0    k3,···,k4 k3,···,k4  (z̆∗ (t), u(t), yx(t)) +Ψ̃    m    ̇  f ̆iγ (z̆∗γ (t))ui (t) + Ψ̃γ (z̆∗γ (t), u(t), yx (t)) z̃γ (t) = f ̆0γ (z̆∗γ (t)) +     i=0    yz (t) = h̃(z̃(t)) $) ( $!" ( + z̆∗1 (t) · · · z̆∗q (t) z̆∗k1,k2 (t) · · · z̆ k3,···,k4 (t) z̆∗γ (t) = &'&'&'z̃1 (t), z̃γ (t) · · · z̃q (t), z̃γ (t) z̃k1 (t), z̃k2 (t), z̃k1,k2(t), z̃γ (t) &'&'· · · z̃k3 (t), · · ·, z̃k4 (t), z̃k3,···,k4 (t), z̃γ (t) z̃γ (t) $)$(. 0 0 1.2. 6 f ̆0• (z̆∗• (t)) + 7 $)$(),'3 m m f ̆i• (z̆∗• (t))ui (t) + Ψ̃• (z̆∗• (t), u(t), yz (t)) = f ̃0• (z̃(t)) + i=0 f ̃i• (z̃(t))ui (t) i=0 Ψ̃• (z̆∗• (t), u(t), yz (t)) = f ̃0• (z̃(t)) + m ⇓ f ̃i• (z̃(t))ui (t) − f ̆0• (z̆∗• (t)) + m i=0 f ̆i• (z̆∗• (t))ui(t) i=0 $)$I( • = {[1], · · ·, [q], [k1, k2], · · ·, [k3, · · ·, k4], [γ]}). $)$I( 7 2 1. y. 0. rj (t) ∃rj (t) T x̆j∗ / φPj'j = {1, · · ·, q}. 6.2. / / 1 / w(t)). )M (' wj (t) q ) 3 χPj ◦ h̃(x̃(t)) = χPj (x̆j∗ ) $)$ ( z̆∗j ). 7 7 $). 1 ) 7 ( $)$ (. 1 6 +)&))-)()'1.. 1 / 6 ) 5 6. 0 / ) )))! 6. . = 1 0 0. 1 )I ) +)&))-) QLIR ) 5'$ * 7 ). /'1 0 7'). /. $) ( 3 S∗ j = φPj ⊕ φPj,Pi ⊕ · · · ⊕ φ···,Pj,··· P,') 5 & $)$$( $)I.. ) 4 ),'φPj'j = {1, · · ·, q} / = ) / 1 ΣN L : x̃(t) = Φ(x(t))' $)I(. 3  m   k1 ̃ ̇  x̃ (t) = f (x̃(t)) + f ̃ik1(x̃(t))ui(t) + P̃ k1(x̃(t))wk1 (t)  k1 0    i=0   ))    )    m    k2 ̃ ̇  f ̃ik2(x̃(t))ui(t) + P̃ k2(x̃(t))wk2 (t) (t) = f (x̃(t)) + x̃ k2  0    i=0  m    k3,k4  ̃ ̇ x̃k3,k4 (t) = f0 (x̃(t)) + P̃lk3,k4(x̃(t))wl (t) f ̃ik3,k4(x̃(t))ui (t) + i=0 l={k3,k4}  ))    )    m   k5,···,k6  ̃k5,···,k6 (x̃(t))ui (t) + ̃ ̇  f (x̃(t)) + x̃ P̃lk5,···,k6 (x̃(t))wl (t) k5,···,k6 (t) = f0  i    i=0 l={k5,···,k6}   m     f ̃iγ (x̃(t))ui (t) x̃ ̇ γ (t) = f ̃0γ (x̃(t)) +     i=0    yx (t) = (h) ◦ (Φ)−1 (x̃(t)) = h̃(x̃(t)) $)$L('x̃(t) = x̃k1 (t) · · · x̃k2 (t) x̃k3,k4 (t) · · · x̃k5,···,k6(t) x̃γ (t) ∂Φ(x(t)) ∂x(t) & = φPk1 · · · φPk2 φPk3,Pk4 · · · φPk5,...,Pk6 γ {Pj }.. / {P1 }', ) φ k1, · · ·, k6 ∈ {1, · · ·, q}) wj (t) wj (t)'1 ) 76 $) ), 76 $)I $) : 6. S∗P3 ) 2. 0'/. Pj Pj. S∗ )'x̃j (t) x̃j,i(t) · · · x̃···,j,···(t) S∗P2 T T / 2 ). 0 $!" ( + X {P1 } S∗P1 S∗P4 S∗P2 S∗P3 $) 3.. $ * + ΣN L : 3   ẋ1 (t) = x1 (t) + u1 (t) + w1 (t)      2   ẋ2 (t) = x3 (t)x4 (t) + x1 (t)u2 (t) + w2 (t) ẋ3 (t) = x2 (t)x3 (t) + x4 (t)    2   ẋ4 (t) = x4 (t) + u3 (t) + w3 (t)   yx (t) = [x21 (t) x2 (t)x3 (t) x3 (t)]T x(t) ∈ R4'u(t) ∈ R3'yx (t) ∈ R3'1'x1 (t)' x2 (t) x3 (t) P1 w(t) ∈ R3 $)$J( < )   1 0 0 0 T   P2 P3 = 0 1 0 0) 0 0 0 1, T E{dh} = ) 0 0 0 1'5. 3 P C∗ 1 = P S∗ 1 =     1   0       0   0 = P   0     1       0   0 P C∗ 2 = S∗ 2 =.  0    0   ,   x3 (t)     0   0    0,   1 0   0      0,   0     1 = Am P C∗ 3 = P S∗ 3 =   1     0       0   0     0   0       0   1    0       1        0       0 P (S∗ 2 )⊥  0    0   ,   x3 (t)     0     0   1       0   0    1       0        0       0 P (S∗ 1 )⊥   0  1  ,   0 0   0    0,   1 0   0      0,   0     1   0     0       0   1 P (S∗ 3 )⊥ =   0    2x (t)x (t)    4, 3    1    2x4 (t)   0    2x (t)x (t)    4, 3    1    2x4 (t)  0  2x (t)  , 3   0  2  0  2x (t)  , 3   0  2     1       0          0      0 7. )#K((. 3   1 0 0   Am = 0 1 1 0 0 1 6'=φ p1 = S∗p2 =φ p2,p3 )I T 1 0 0 0, = 0 1 0 0 φp 3 = j = {1, 2, 3} 6 1 3 S∗p3 = φp3 ⊕ φp2,p3 $)$M( 7) Pj S∗, S∗p1 1 T, 0 0 1 0 T, 0 0 0 1 T $)$#( $!" ( + 7 3 x̃(t) = Φ(x(t)) 5 2 φp2,p3 7 γ ∂Φ(x(t)) = φp1 φp3 φp2,p3 ∂x(t). 7... T $)$K( φp 1'φp 3 ) 5'7. 3    z̃ ̇1 (t) = z̃1 (t) + u1 (t) + Ψ1 (z̆∗1 (t), u(t), yx (t)) − Ψ1 (z̆∗1 (t), u(t), yz (t))          3 3 ̇   z̃ ̇3 (t) = z̃31 (t) = z̃2,3 (t)z̃31 (t) + z̃32 (t) +Ψ (z̆∗ (t), u(t), yx (t)) 2 Σ : (t) + u3 (t) z̃ ̇32 (t) z̃32 −Ψ3 (z̆∗3 (t), u(t), yz (t)) F NL    2   (t) + z̃1 (t)u2 (t) + Ψ2,3 (z̆∗2,3 (t), u(t), yx (t)) − Ψ2,3 (z̆∗2,3 (t), u(t), yz (t)) z̃ ̇2,3 (t) = z̃31 (t)z̃32     yz (t) = [z̃ 2 (t) z̃2,3 (t)z̃31 (t) z̃31 (t)]T 1 $)$ ( (z̃3 (t)) = (φ ) = 2 $)$( 7 z̆∗1 (t) p3 T 1. 3 z̆∗3 (t) z̆∗2,3 (t) = & z̃1 (t)'& z̃3 (t)'& z̃3 (t), z̃2,3 (t)'$)L( z̆∗1 (t) = z̃1 (t) ; z̆∗3 (t) = 3 z̆∗1 (t) 3 z̆∗2 (t) = z̃31 (t) z̃32 (t)   2,3   z̆∗1 (t) z̃31 (t)   2,3   ; z̆∗2,3 (t) = z̆∗2 (t) =  z̃32 (t)  2,3 (t) z̆∗3 z̃2,3 (t) $)LI(. 0 7 7
  
  1 2. 
  
  ) 6
  
  7 $)$I( = 1 3    z̆ 1 (t) + u1 (t) + Ψ1 (z̆∗1 (t), u(t), yz (t)) = z̃1 (t) + u1 (t)   ∗      3 (t) (t)z̃ (t) + z̃ (t) z̆∗2 z̃ 31 32   + Ψ3 (z̆∗3 (t), u(t), yz (t)) =  2,3  3 2 2   (z̆∗2 ) (t) + u3 (t) z̃32 (t) + u3 (t)     z̆ 2,3 (t)(z̆ 2,3 )2 (t) + Ψ2,3 (z̆ 2,3 (t), u(t), yz (t)) = z̃31 (t)z̃ 2 (t) + z̃1 (t)u2 (t) ∗1 ∗2 3 ∗ $)L ( 32    Ψ1 (z̆∗1 (t), u(t), yz (t)) = 0       yz2 (t)  Ψ3 (z̆∗3 (t), u(t), yz (t)) =    0    1  Ψ2,3 (z̆ 2,3 (t), u(t), yz (t)) = yz1 (t)u2 (t) ∗ $)L$( $ * 7. ΣF N L /   z ̇1 (t) = z1 (t) + u1 (t)    1   2  z ̇ yx1 (t)u2 (t) 2 (t) = z3 (t)z4 (t) +  : z ̇3 (t) = yx2 (t) + z4 (t)     z ̇4 (t) = z42 (t) + u3 (t)     yz (t) = (z12 (t) z2 (t)z3 (t) z3 (t))T'x1 (t) ∀w2 (t)' w3 (t) 6 / S. (z3 (t) z4 (t))! 1 x2 (t) ∀w1 (t) 3'S.. z2 (t) $)LL( / z1 (t) / 1 S. . 6 /. 6 3 / z2 (t)'7'. = w2 (t) w3 (t) = 0) / 1 w1 (t)) z1 (t) w1 (t) = 0) w2 (t) = 0. / (z3 (t) z4 (t)) / 6 / (x3 (t) x4 (t)) ∀w1 (t)' w2 (t)'''1'w3 (t)) w3 (t)) w3 (t) = 0) 6 7 7.). ( 2. 1 w(t)' y ) D rj (t) 7 {drjT }. {drjT } ⊆ φPj'P φ j ⊕ φPj,Pi ⊕ · · · ⊕ φ···,Pj,··· ), '. 6 )I'2 q. 1. 2 / 1 ) $, 4 0 6 7 ) 1 /.. 0 7. = 6 ),'q. '
  
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