Détection, localisation et estimation de défauts : application véhicule

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Détection, localisation et estimation de défauts : application véhicule

Ahmad Farhat

To cite this version:

Ahmad Farhat. Détection, localisation et estimation de défauts : application véhicule. Automatique

/ Robotique. Université Grenoble Alpes, 2016. Français. �NNT : 2016GREAT056�. �tel-01372957v2�

THÈSE

Pour obtenir le grade de

DOCTEUR DE LA COMMUNAUTE UNIVERSITE GRENOBLE ALPES

Spécialité : Automatique - Productique

Arrêté ministériel : 7 août 2006

Présentée par

Ahmad FARHAT

Thèse dirigée par Damien KOENIG

préparée au sein du GIPSA-Lab dans l'École Doctorale EEATS

Détection, localisation et estimation de défauts : Application véhicule

Thèse soutenue publiquement le 22/09/2016 devant le jury composé de :

M. Luc DUGARD

Directeur de recherche, CNRS Grenoble, Président

M. Saïd MAMMAR

Professeur, Université d'Evry Val-d'Essonne, Rapporteur

M. Didier THEILLIOL

Professeur, Université de Lorraine, Rapporteur

M. Michel BASSET

Professeur, Université de Haute Alsace, Examinateur

M. Damien KOENIG

Professeur associé, HDR, Institut Polytechnique de Grenoble, Directeur

N R C R e(.)

S (M ) M

M ^T M

M ^∗ M

M ⁺ M

M ≻ ( )0 M

M ≺ ( )0 M

(⋆) L ∞

L 2

RH ∞

k . k 2 2

k . k ∞ ∞

k . k − −

H − /H ∞

H ₋

H _∞ H _∞

H _∞

H _∞

H ₋

r(t)

z s z ˙ s

H ₋ /H _∞

H _∞

H _∞ H ∞

H _∞

H _∞

H ₋ /H _∞ H ₋ /H _∞

H ₋ /H _∞

H _∞

H ∞

H _∞

ρ ₁

ρ ₂

•

•

•

•

•

•

◦

+2, 4%

•

•

•

•

1 : 5

r(t) r(t)

f (t) f (t)

r(t) 6 = 0 f (t) 6 = 0

| r(t) | > γ f(t) 6 = 0

γ

Actionneurs Syst`eme Capteurs ^y(t)

u(t)

G´en´erateur de r´esidu

Logique de d´esicion

nature, valeur du d´efaut

H ₂ /H _∞ H ₋ /H _∞ H _∞ /H _∞

H − /H ∞

H _∞ H ₋ H _∞

H ₋ /H _∞

H _∞

H ₋

H ₋

H − /H ∞

H ₋ /H _∞

H ∞

H ∞

H ∞

H ∞

H − /H ∞ H ∞

f : R ⁿ × R ^m 7→ R ⁿ g : R ⁿ × R ^m 7→ R ^l ( x(t) = ˙ f (t, x(t), u(t))

y(t) = g(t, x(t), u(t))

x(t) X ⊂ R ⁿ u(t)

U ⊂ R ^m y(t) Y ⊂ R ^l

f (.) g(.)

( x(t) = ˙ Ax(t) + Bu(t) y(t) = Cx(t) + Du(t)

( x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) + Du(k)

G(.) = ^y(.) _u(.)

G(s) = D + C(sI − A) ⁻¹ B , G(z) = D + C(zI − A) ⁻¹ B ,

A, B, C, D G(s) G(z)

G(s) ,

"

A B C D

#

(x e , u e ) δx(t) δu(t)

x(t) = x e (t) + δx(t) u(t) = u e (t) + δu(t)

(δx, δu) dδx

dt =

∂f(x, u)

∂x ∗

δx +

∂f(x, u)

∂u ∗

δu

( ^∗ ) (x e , u e ) A B

C D

A = ^∂f _∂x ^(x,u) x = x e

u = u e

, B = ^∂f _∂u ^(x,u) x = x e

u = u e

,

C = ^∂g(x,u) _∂x x = x e

u = u e

, D = ^∂g(x,u) _∂u x = x e

u = u e

.

T d

A d = e ^{A c kT d}

B _d = A ⁻¹ _c (e ^{A c kT d} − I )B c T _d C d = C c

D d = D c

e ^{A c kT d} = I n + T _d A c

T _d

A _d = I n + T _d A c

D _d = D _c

A ∈ R ^n×m B ∈ R ^p×q

A ⊗ B = h A ij B i

ij =







a ₁₁ B . . . a _1m B a _n1 B . . . a nm B







M M = A + BCD A C

[A + BCD] ⁻¹ = A ⁻¹ − A ⁻¹ B

DA ⁻¹ B + C ⁻¹ −1

DA ⁻¹

n × n ∀ u ∈ R ⁿ

u ^T F u ≻ ( ≺ )0

G(s) G

ω | G(jω) |

G(jω)

σ i (G(jω)) := p

λ i (G(jω)G(jω) ^∗ )

i=1.. (m,l)

= q

λ i (G(jω)G( − jω) ^T )

i=1.. (m,l)

σ(G(jω)) σ(G(jω))

σ(G(jω)) ≥ σ ₁ (G(jω)) ≥ σ ₂ (G(jω)) ≥ · · · ≥ σ(G(jω)) l = m = 1

σ(G(jω)) = σ(G(jω)) = | G(jω) |

u(t) y(t)

σ(G(jω)) ≤ k y(jω) k 2

k u(jω) k ₂ ≤ σ(G(jω)) L _∞

L ∞ C C 7→ C

f(jω) ∀ ω ∈ R sup

ω∈ R

σ[f(jω)]

H ₂

H 2 e(t) L 2

H 2

k e(t) k ₂ = R ∞

0 e(τ ) ^T e(τ )dτ

H _∞ RH _∞

H _∞ L ∞

f (jω) ∀ ω ∈ R R e(jω) > 0 H _∞

RH ∞

H ∞ RH ∞

L 2 L 2

w(t) z(t) H ∞

k G(s) k _∞ = sup

ω∈ R

σ(G(jω))

= sup

ω∈H 2

kz(s)k ₂ kw(s)k ₂

= max

ω∈L 2

kzk ₂ kwk ₂

H −

H ₋

[ω, ω]

[0, ∞ ]

H ₋ G(s)

k G(s) k ^[ω,ω] − = inf

ω∈[ω,ω] σ(G(jω))

ω ω ∞

x ∈ R ⁿ F (x) = F (0) +

m

X

i=1

F _i x _i ≻ ( ≺ )0 F i = F _i ^T ∀ i ∈ { 1, .., m }

˙

x(t) = Ax(t) V (t) = x ^T (t)P x(t) V (t) > 0 V ˙ (t) < 0

x ^T P x > 0 x ^T (A ^T P + P A)x < 0

F (P ) =

"

− P 0 0 A ^T P + P A

#

≺ 0

P = P ^T F (P) ≺ 0 P

F (x) ≺ 0 x

x opt

P Q M

"

P M

M ^T Q

#

≺ 0

⇔

( Q ≺ 0

P − M Q ⁻¹ M ^T ≺ 0

⇔

( P ≺ 0

Q − M ^T P ⁻¹ M ≺ 0

P − M Q ⁻¹ M ^T Q

Ψ N θ M

Φ + M ^T θ ^T N + N ^T θM < 0

W X X

θ

W _M ^T ΨW _M < 0 W _N ^T ΨW _N < 0

X Y ∆ ∆ ^T ∆ < I

α > 0

X ^T ∆Y + Y ^T ∆X ≤ αX ^T X + 1

α Y ^T Y

X ₂ , Y ₂ ∈ R ^n×m X ₃ , Y ₃ ∈ R ^m×m

"

X X 2

X ₂ ^T X ₃

#

≺ 0

"

X X 2

X ₂ ^T X ₃

# −1

=

"

Y Y 2

Y ₂ ^T Y ₃

#

"

X I I Y

#

≻ 0

"

X I I Y

#

≤ n + m

"

X I I Y

#

≻ 0 (I − XY ) ≤ m

A

A P > 0

A ^T P + P A < 0

A ^T P A − P < 0

t i

x i t i t _f t i u(t)

[t i , t f ] x i ∀ t 1

C (B,A) = h

B AB A ² B . . . A ⁿ⁻¹ B i C (B,A) ) = n _C (B, A)

n _C

n _C = n (B, A)

(B, A)

t i

x i t i t f t i y(t)

u(t) [t i , t f ] x i

O (A,C) =















 C CA CA ² CA ⁿ⁻¹

















P P AP ⁻¹ =

"

A ₁₁ 0 A 21 0

#

, CP ⁻¹ = h C ₁ 0 i

,

A 11 n _O × n _O n _O < n C 1 l × n _O A 21 (n − n _O ) × (n − n _O )

n _O

n _O = n

C(sI − A) ⁻¹ B = 0, ∀ s ∈ C B

"

sI − A C

#

= n, ∀ s ∈ C .

(A, C) (A, C )

0 < γ < ∞ A k G(s) k _∞ < γ

P = P ^T > 0

"

P A + A ^T P + C ^T C P B + C ^T D

⋆ D ^T D − γ ² I

#

< 0 P







P A + A ^T P P B C ^T

⋆ − γI D ^T

⋆ ⋆ − γI





 < 0

S S = P ⁻¹







AS + SA ^T B SC ^T

⋆ − γI D ^T

⋆ ⋆ − γI





 < 0

V = x ^T P x V > 0 V < ˙ 0 H _∞ y ^T y < γ ² u ^T u

x ^T P x > 0

˙

x ^T P x + x ^T P x ˙ + y ^T y − γ ² u ^T u < 0

P S γ

P γ _∞ H _∞

G(s)

P, γ γ ∈ R ⁺

=

P, γ γ ∈ R ⁺

=

S, γ γ ∈ R ⁺

= k G(s) k _∞ = γ ∞

P S γ







A ^T P A − P B ^T P A C ^T

⋆ B ^T P B − γI D ^T

⋆ ⋆ − γI





 < 0

D

D = { z ∈ C : L + zM + ¯ zM ^T < 0 }

L M L ^T = L

f _D = L + zM + ¯ zM ^T

D D ^R

D ^R = { z ∈ C : R ₁₁ + R ₁₂ z + R ^T ₁₂ z ¯ + R ₂₂ z¯ z < 0 }

R ₂₂

D

A P

R ₁₁ ⊗ P + R ₁₂ (P A) + R ^T ₁₂ ⊗ (A ^T P ) + R ₂₂ ⊗ (A ^T P A) < 0

• R ₁₁ = 0 R ₁₂ = 1 R ₂₂ = 0

• R ₁₁ = − 1 R ₁₂ = 0 R ₂₂ = 1

• R e(z) ≤ − δ

R ₁₁ = 2δ R ₁₂ = 1 R ₂₂ = 0

• 2φ

ξ = (φ) R ₁₁ = 0 _2×2 R ₁₂ =

"

(φ) (φ)

− (φ) (φ)

#

R ₂₂ = 0 _2×2

(l < n)

t x(t)

x(t)

Σ _{LT I} :

( x(t) = ˙ Ax(t) + Bu(t) y(t) = Cx(t)

y(t) ← y(t) − Du(t)

x(t) OBSV

OBSV :

( z(t) ˙ = F z (t) + Ju(t) + Ly(t) w(t) = M z(t) + N y(t) x(t) ∈ R ⁿ u(t) ∈ R ^m y(t) R ^l

z(t) ∈ R ^q T x(t) w(t)

e(t) = T x(t) − z(t) 0

t→∞ lim e(t) = T x(t) − z(t) = 0

x(t) z(t) = ˆ x(t) x(t) ˆ x(t)

˙ˆ

x(t) = A x(t) + ˆ Bu(t) + L(y(t) − y(t)) ˆ ˆ

y(t) = C x(t) ˆ

L(y(t) − y(t)) ˆ L

˙ˆ

x(t) = (A − LC)ˆ x(t) + Bu(t) + Ly(t)

˜

x(t) = x(t) − x(t) ˆ

˙˜

x(t) = (A − LC)˜ x(t)

˜

x(0) = x(0) − x(0) ˆ

˜

x(t) = e ^(A−LC)t x(0) ˜

0 t t = 0 ˜ x(0) 6= 0

x(0) = ˆ x(0)

˜

x(t) −→ 0 t −→ ∞ L

A − LC

(A, C)

A − LC (A, C )

L A − LC

L

L

n − l

C x(t)

C = h

C ₁ C ₂ i , x =

"

x ₁ x ₂

# , (C ₁ ) = l C ₁ ∈ R ^l×l x ₁ ∈ R ^l×l

C ₁ ) = l x(t) = ¯ P x(t)

P =

"

C ₁ C ₂ 0 _n−l,l I _n−l,l

# ,

Σ LT I :

( x(t) = ˙¯ A ¯ x(t) + ¯ ¯ B u(t) ¯

¯

y(t) = C ¯ x(t) ¯ A ¯ = P AP ⁻¹ B ¯ = P B C ¯ = CP ⁻¹

C

x(t)

A =

"

A 11 A 12

A ₂₁ A ₂₂

#

, B =

"

B 1

B ₂

# , C = h

I l 0 i

, x(t) =

"

x 1

x ₂

# .

˙

x 1 (t) = A 11 x 1 (t) + A 12 x 2 (t) + B 1 u(t)

˙

x ₂ (t) = A ₂₁ x ₁ (t) + A ₂₂ x ₂ (t) + B ₂ u(t) y(t) = x ₁ (t)

l (y(t) = x ₁ (t)) x ₁ (t)

x ₁ (t)

ξ(t) x 2 (t)

u(t)

ξ(t) = ˙ x 1 (t) − A 11 x 1 (t) = A 12 x 2 (t) + B 1 u(t) x ₂ (t) v(t)

˙

v(t) = A 21 x 1 (t) + A 22 v(t) + B 2 u(t) + K[ξ(t) − ξ(t)] ˆ x ₁ (t) = y(t)

ξ(t) = A 12 v(t) + B 1 u(t)

ξ(t) y(t)

z(t)

z(t) = v(t) − Ly(t) z(t)

˙

z(t) = v(t) ˙ − L x ˙ 1 (t)

= A ₂₁ x ₁ (t) + A ₂₂ v(t) + B ₂ u(t) − L[A ₁₁ x ₁ (t) + A ₁₂ v(t) + B ₁ u(t)]

z(t) = v(t) − Ly(t) y(t) = x 1 (t)

˙

z(t) =F z(t) + Ju(t) + L 1 y

F =A 22 − L 1 A 12

J =B ₂ − L ₁ B ₁

L ₁ =A ₂₁ + A ₂₂ L − LA ₁₁ − LA ₁₂ L

v(t) =z(t) + Ly

e(t) e(t) = x ₂ (t) − v(t) e(t)

˙

e(t) = F e(t) F = A 22 − LA 12

F

(A ₂₂ , A ₁₂ ) F

L (A, C) (A ₂₂ , A ₁₂ )

(A, C) (A 22 , A 12 ) (A, C)

∀s ∈ C ,

sI − A C

=





sI − A 11 −A 12

−A 21 sI − A 22

I l 0



 = n

∀s ∈ C , =

A 12

sI − A 22

= n − l (A 22 , A 12 )

˙

z(t) =F z(t) + Ju(t) + L ₁ y

F J L x ˆ ₁ x ˆ ₂

x ₁ x ₂

ˆ

x 1 (t) = C ₁ ⁻¹ [(I − C 2 L)y(t) − C 2 z(t)]

ˆ

x ₂ (t) = z(t) + Ly(t)

H − /H ∞

˙

x(t) = Ax(t) + Bu(t) + E _d d(t) + E _f f(t)

y(t) = Cx(t) + Du(t) + F _d d(t) + F _f f (t)

Syst`eme

Observateur ^{y ˆ}

u y

d f

r G´en´erateur de r´esidu

A ∈ R ^n×n B ∈ R ^n×m E _d ∈ R ^{n×n d} E _f ∈ R ^{n×n f} C ∈ R ^l×n D ∈ R ^l×m F _d ∈ R ^{l×n d} F ∈ R ^{l×n f} d ∈ R ^{n d} f ∈ R ^{n f}

"

sI − A E f

C D f

#

= n + n _f , ∀s ∈ C .

d(t) f (t)

r(t)



 

 

˙ˆ

x = Aˆ x + Bu + L(y − y) ˆ ˆ

y = C x ˆ + Du r = y − y ˆ

kr d k ₂ < γ kdk ₂

kr f k ₂ > β kf k ₂

˜

x = x − x ˆ

˙˜

x = Ax + B u + E d d + E f f − Aˆ x − Bu − LC x ˜ − LF d d − LF f f

= (A − LC)˜ x + (E d − LF d )d + (E f − LF f )f

r = C x ˜ + F d d + F f f

A ^∗ = A − LC E _d ^∗ = E d − LF d E ^∗ _f = E f − LF f

T rd T rf

T _rf (s) = C(sI − A ^∗ ) ⁻¹ E _f ^∗ + F _f T rd (s) = C(sI − A ^∗ ) ⁻¹ E _d ^∗ + F d

H _∞ H ₋

kT rd k _∞ < γ kT rf k ₋ > β

H − /H ∞ L γ

β

(A, C)

H _∞ γ ∈ R ⁺

kr _d k ₂ < γ kdk ₂ γ ∈ R ⁺

kT _rd (s)k _∞ < γ

γ ∈ R ⁺ L P > 0

"

P (A − LC) + (A − LC) ^T P + C ^T C P (E _d − LF _d ) + C ^T F _d

⋆ F _d ^T F d − γ ² I

#

< 0

γ ∈ R ⁺ U P > 0

"

P A + A ^T P + U C + C ^T U + C ^T C P E d + U F d + C ^T F d

⋆ F _d ^T F d − γ ² I

#

< 0

L = −P ⁻¹ U

γ ∈ R ⁺ U P > 0







P A + A ^T P + U C + C ^T U P E _d + U F _d C ^T

⋆ −γ ² I F _d ^T

⋆ ⋆ −I





 < 0 L = −P ⁻¹ U

H ∞

kr d k ₂ < γ kdk ₂ ⇔ kr d k ₂ kdk ₂ < γ, kT rd (s)k _∞ = sup

d ∈ R ^nd

kr d k ₂ kdk ₂ < γ

V > 0 V < ˙ 0 V | f=0 = ˜ x ^T P x P > ˜ 0 V < ˙ 0

r _d ^T r d − γ ² d ^T d < 0

V ˙ + r ^T _d r d − γ ² d ^T d < 0

⇔ x ˙˜ ^T P x ˜ + ˜ x ^T P x ˙˜ + r ^T _d r d − γ ² d ^T d < 0

x ˜ d

^T

P A ^∗ + A ^∗T P + C ^T C P E _d ^∗ + C ^T F d

⋆ F _d ^T F d − γ ² I

˜ x d

< 0

∀ x ˜

d

6= 0

P A ^∗ + A ^∗T P + C ^T C P E _d ^∗ + C ^T F d

⋆ F _d ^T F d − γ ² I

< 0

U = −LP

H − F f

β ∈ R ⁺

kr _f k ₂ > β kfk ₂

kT _rf (s)k ₋ > β

β ∈ R ⁺ L P

"

P (A − LC) + (A − LC) ^T P − C ^T C P (E f − LF f ) − C ^T F f

⋆ β ² I − F _f ^T F _f

#

< 0 L = −P ⁻¹ U

β ∈ R ⁺ U P

"

−P A − U C + C ^T C − A ^T P − C ^T U −P E _f − U F _f + C ^T F _f

⋆ F _f ^T F f − β ² I

#

> 0 L = −P ⁻¹ U

V ˙ | d=0 − r ^T _f r f + β ² f ^T f < 0

P H ₋

H − /H ∞ H ∞ P

H ₋

ω→∞ lim G(jω) = lim

ω→∞ (C(jωI − A) ⁻¹ B + D) = D D = 0 H −

L

H −

β

D ^T D ≥ β ² I L D ^T D −β ² I

Syst`eme

Observateur ^{ˆ y}

u y

d f

r G´en´erateur de r´esidu

W f

˜

r r ¯

D _add

H ₋ /H _∞

f = 0 f << ∞

H ₋ /H _∞

H ∞

σ(G(jω)) = 0 σ(G(jω)) L

β [0, ω] H −

H −

F _f = 0 T _rf (s)

β = 0

|G(jω)|

D add

σ(G(jω)+D _add ) > 0 β = D _add H ₋

W f

W f (s) =

s/ω ₁ + 1 s/ω ₂ + 1

m

ω 1 < ω 2 m

10 ⁻² 10 ⁻¹ 10 ⁰ 10 ¹ 10 ² 10 ³ 10 ⁴ 10 ⁵

−70

−60

−50

−40

−30

−20

−10 0 10 20

Fr´equence (rad/s)

G a in (d B )

T rf (s)

T rf (s) + D add

(T rf (s) + D add ) W f (s)

H ₋

D add W f (s) T _{¯ rf} (s) = _f ^{¯ r(s)} _(s)

T rf (s) = _f ^r(s) _(s)

D add << 1 W f (jω) | ω<ω ₁ = 1

T _{¯ rf} (s) | _{ω<ω 1} = T _rf (s) | _{ω<ω 1} = C(sI − A + LC) ⁻¹ E _f + F _f

W f (s) :=

"

A _F B _F C F D F

#

W f

˙ˆ

x = A x ˆ + Bu + L(y − y) ˆ ˆ

y = C x ˆ + Du

˜

r = (y − y) + ˆ D add f

˙

x F = A F x h + B F ˜ r

¯

r = C F x h + D F r ˜

˜ x

"

˙˜

x

˙ x F

#

=

"

A − LC 0 B F C A F

# "

˜ x x h

# +

"

E _d − LF _d B F F d

# d +

"

E _f − LF _f B F (F f + D add )

# f

¯

r = h

D _F C C _F i

"

˜ x x F

# + h

D _F F _d i d + h

D _F (F _f + D _add ) i f

x ^a = h

˜

x ^T x ^T _F i T

˙

x ^a = A ^a x ^a + E _d ^a d + E _f ^a f

¯

r = C ^a x ^a + F _d ^a d + F _f ^a f

A ^a =

"

A − LC 0 B F C A F

#

, E ^a _f =

"

E f − LF f

B F (F f + D add )

#

, E _d ^a =

"

E d − LF d

B F F d

#

C ^a = h

D F C C F

i

, F _f ^a = D F (F _f + D _add ) F _d ^a = D F F _d

F _f =

"

F f 1

0

#

D add =

"

0 εI

#

H ₋ H _∞ T rf ¯ :=

"

A ^a E _f ^a C ^a F _f ^a

#

, T rd ¯ :=

"

A ^a E _d ^a C ^a F _d ^a

#

H ₋ / H _∞

H − /H ∞

γ β U ^a

P ^a > 0









P ^a A ₀ + A ^T ₀ P ^a + U ^a C ₀ +C ₀ ^T U ^aT + C ^aT C ^a

P ^a E _d0 + U ^a F d

+C ^aT D ^a _d

⋆ D ^aT _d D _d ^a − γ ² I









< 0









− P ^a A ₀ − A ^T ₀ P ^a − U ^a C ₀

− C ₀ ^T U ^aT + C ^aT C ^a

− P ^a E _f0 − U ^a F f

+C ^aT D ^a _f

⋆ D ^aT _f D _f ^a − β ² I









> 0

A 0 , B _f0 , B _d0 , C 0 I 0

A ₀ =

"

A 0

B F C A F

#

, C ₀ = h C 0 i

, I ₀ =

"

− I 0

#

E _d0 =

"

E d

B F F d

#

E _{f 0} =

"

E f

B F (F f + D add )

#

L = I ₀ ^T (P ^a ) ⁻¹ U ^a

T ¯ rd







P ^a A ^a + C ^aT C ^a P ^a E _d ^a +A ^aT P ^a +C ^aT D _d ^a

⋆ D _d ^aT D _d ^a − γ ² I





 < 0

A ^a E _d ^a

P ^a A ^a = P ^a

A 0 B F C A F

+ P ^a

− I 0

L

C 0

P ^a E _d ^a = P ^a E d

B F F d

+ P ^a

− I 0

LF d













P ^a (A 0 + I 0 LC 0 ) +(A 0 + I 0 LC 0 ) ^T P ^a

+C ^aT C ^a

P ^a (B 0

+I 0 LF d ) +C ^aT D _d ^a

⋆ D ^aT _d D _d ^a − γ ² I













< 0

U ^a = P ^a I 0 L

H _∞

H _∞

H _∞

H _∞

H ∞

r u y

d u d y

η ε

G(s) K(s)

u(s) y(s)

u(s) = (1 + G(s)K(s)) ⁻¹ (K(s)r(s) − K(s)d _y (s)

− K(s)η(s) + K(s)G(s)d u (s)) y(s) = (1 + G(s)K(s)) ⁻¹ (G(s)K(s)r(s) + d y (s)

− G(s)K(s)η(s) + G(s)d u (s))

S (s) , (1 + G(s)K(s)) ⁻¹

T (s) , 1 − S (s) = (G(s)K(s)(1 + G(s)K(s)) ⁻¹ KS (s) , K (s)(1 + G(s)K(s)) ⁻¹

SG (s) , (1 + G(s)K(s)) ⁻¹ G(s)

u(s) = KS (s)r(s) − KS (s)d y (s) − KS (s)η(s) − T (s)d u (s)) y(s) = T (s)r(s) + S (s)d _y (s) − T (s)η(s) + T (s) G (s)d _u (s))

T (s) = y(s)

r(s) = − y(s)

η(s) = u(s) d u (s) S (s) = y(s)

d y (s) = 1 − T (s) = ε(s) r(s) KS (s) = u(s)

r(s) = − u(s)

d y (s) = − u(s) η(s) SG (s) = y(s)

d _u (s)

u

d u d y

ε y

r Contrˆoleur K ∞ Syst`eme Σ ^t

η

u

d u d y

ε y

r Contrˆoleur K ∞ Syst`eme Σ ^t

η W ^e e ε

H ∞

S (s) W e

e ε

S (s) W ^e (s)

|S (s) | ≤

1 W e (s)

∀ s

kW ^e (s) S (s) k ∞ ≤ 1

W ^e (s) S (s) W ^T (s) T (s) W u (s) KS (s) W SG (s) SG (s)

∞

≤ 1

S (s) KS (s)

u

d u d y

ε y

r Contrˆoleur K ∞ Syst`eme Σ ^t

η W ^e e ε

W ^u e u

u

W ^e (s) S (s) W u (s) KS (s)

∞

≤ 1

H ∞

H _∞ P (s)

w

u e

y

w ^T = h

r d ^T _u d ^T _y i T

z ^T = h ε u ^T

i T

y s = ε u s = u W I W O

W ^I W ^O

W I = I W ^O = ( W ^e , W ^u )

P P ^′

P (s) :=



 

 

˙

x(t) = Ax(t) + B w w(t) + B u u(t) z(t) = C z x(t) + D wz w(t) + D uz u(t) y(t) = C y x(t) + D wy w(t) + Duyu(t) K

K(s) :=

( x ˙ _K (t) = A _K x _K (t) + B _K y _s (t) u s (t) = C K x K (t) + D K y s (t) x ∈ R ⁿ w ∈ R ^{n w} u s ∈ R ^{n u} z ∈ R ^{n z} y s ∈ R ^{n y} x K ∈ R ^{n K}

H _∞ w

z H ∞

T zw (s)

k T zw (s) k _∞ < γ _∞

P (s)

K (s)

u s

y s

z w

H ∞

P (s)

K(s)

u s

y s

¯

z ^{w ¯} W ^I (s) ^w

W ^O (s)

z

P’

H _∞ T zw (s) L l (P(s), K(s))

P (s), K(s) F (P(s), K(s)) H _∞

K γ _∞

H _∞ H _∞

H ∞

F (P (s), K(s)) (A, B u )

(A, C _y )

y s u s D yu = 0

L 2 w z γ k z k 2 < γ k w k 2

γ > 0 P cl (n + n K ) ×

(n + n _K )







P cl A cl + A ^T _cl P cl B _cl ^T P cl C _cl ^T

⋆ − γI D _cl ^T

⋆ ⋆ − γI





 < 0 P _cl > 0

A cl =

"

A + B u D K C y B u C K

B K C y A K

#

, B cl =

"

B w + B u D K D yw

B K D yw

# , C _cl = h

C z + D zu D K C y D zu C K

i D _cl = h

D zw + D zu D K D yw

i .

F (P (s), K(s)) P K











˙ x(t)

˙ x K (t)

z(t) y s (t)











=











A 0 B w B u

B K C y A K B K D yw 0 C z 0 D zw D zu

C y 0 D yw D yu



















 x(t) x K (t)

w(t) u s (t)











w z







˙ x(t)

˙ x K (t)

z(t)





 =







A + B u D K C y B u C K B w + B u D K D yw

B K C y A K B K D yw

C z + D zu D K C y D zu C K D zw + D zu D K D yw











 x(t) x K (t)

w(t)







L ^l (P (s), K (s)) =

"

A cl B cl

C cl D cl

#

P cl A K B K C K D K

H _∞

H ∞

γ > 0 X Y A ˜ B ˜ C ˜ D ˜











M ₁₁ ⋆ ⋆ ⋆ M ₂₁ M ₂₂ ⋆ ⋆ M ₃₁ M ₃₂ M ₃₃ ⋆ M ₄₁ M ₄₂ M ₄₃ M ₄₄











< 0

"

R I n

I n S

#

> 0

M ₁₁ = AR + RA ^T + B u C ˜ + ˜ C ^T B _u ^T M ₂₁ = ˜ A + A ^T + C y D ˜ ^T B ^T _u

M ₃₂ = B _w ^T S + D _yw ^T B ˜ ^T

M ₃₃ = − γI n u , M ₄₄ = − γI n y

M ₄₁ = C z R + D zu C ˜

M ₄₂ = C z + D zu DC ˜ y

M ₄₃ = D zw + D zu DD ˜ yw

D K = ˜ D

C _K = ( ˜ C − D _K C y R)M ⁻¹ B K = N ⁻¹ ( ˜ B − SB u D K )

A K = N ⁻¹ ( ˜ A − SAR − SB u D K C y R

− N B u C k R − SB u C K M ^T )M ⁻¹

M N

M N ^T = I − RS

P cl

P cl =

S N N ^T U

, P _cl ^{− 1} =

R M M ^T V

P cl P _cl ⁻¹ = I P cl

R M ^T

, =

I 0

P cl Π 1 = Π 2

Π 1 =

R I M ^T 0

Π 2 = I S

0 N ^T

Π 1

Π ^T ₁ , I, I

D ˜ = D K

C ˜ = C K M ^T + D K C z R B ˜ = N B K + SB w D K

A ˜ = N A K M ^T + N B K C y R

+ SB u C K M ^T + S(A + B u D K C y )R

H ∞

γ >

0 R S

"

N R 0 0 I n w

#







AR + RA ^T RC _z ^T B _w

⋆ − γI n z D zw

⋆ ⋆ − γI n w







"

N R 0 0 I n w

#

< 0

"

N S 0 0 I n z

# T 





A ^T S + SA ^T B w S C _z ^T

⋆ − γI n w D _zw ^T

⋆ ⋆ − γI n z







"

N S 0 0 I n z

#

< 0

"

R I n

I n S

#

> 0

N ^R N ^S h

B _u ^T D ^T _zu i h

C y D yw

i n k < n

(I − RS) ≤ n _k

θ =

D ^K _i C _i ^K B _i ^K A ^K _i

A cl B cl C cl D cl

L l (P (s), K (s)) =

"

A cl B cl

C cl D cl

#

=

"

A 0 + B θ C B 0 + B θ D yw

C 0 + D ^zu θ C D 0 + D ^zu θ D ^yw

#

A 0 =

A 0 0 0 n k

; B 0 = B w

0

; C 0 = C z

0

; B =

0 B u

I n k 0

;

C =

0 I n k

C y 0

; D ^zu = 0 D zu

D ^yw = 0

D yw

;

Ψ P cl + Q ^T θ ^T X P cl + X _P ^T _cl θQ < 0

X P cl =

B ^T P cl 0 D ^T zu

; Q =

C D ^yw 0

;

Ψ P cl =





P cl A 0 + A ^T ₀ P cl B ₀ ^T P cl C ₀ ^T

⋆ − γI D _zw ^T

⋆ ⋆ − γI





P =

B 0 D ^zu

X P cl =





P cl 0 0 0 I 0 0 0 I



 P X P cl P

Q W ^X W ^P W ^Q W ^X =





P _cl ^{− 1} 0 0 0 I 0 0 0 I



 W ^P P cl θ

W P ^T Φ P cl W P < 0 W Q ^T Ψ P cl W Q < 0

Φ P cl =





A 0 P _cl ^{− 1} + A ^T ₀ P _cl ^{− 1} B ₀ ^T P _cl ^{− 1} C ₀ ^T

⋆ − γI D _zw ^T

⋆ ⋆ − γI





W X ^T Ψ P cl W ^X < 0

W P ^T





P _cl ⁻¹ 0 0 0 I 0 0 0 I





T

Ψ P cl





P _cl ⁻¹ 0 0 0 I 0 0 0 I



 W P < 0

P _cl ⁻¹ P _cl ⁻¹

W P ^T























A 0 0 0

R M

M ^T V

+

R M ^T

M V

A i 0 0 0

B w

0

R M ^T

M V

C _z ^T 0

B _w ^T 0

− γI D ^T _zw C z

0

R M

M ^T V

D zw − γI























W P < 0

⇔ W P ^T









AR + RA ^T AM B w RC _z ^T M ^T A 0 0 M C _z ^T B _w ^T 0 − γI D ^T _zw C z R C z M D zw − γI







 W ^P < 0

P =

B 0 D zu

=

0 I n k 0 n k × n w 0 B _u ^T 0 0 n u × n w D ^T _zu

,

W 1

W 4

B ^T _u D _zu ^T

W P P

W P











W 1 0

0 0

0 I n w

W 4 0











W P





W 1 0 W 4 0 0 I n w





T 



AR + RA ^T RC _z ^T B w

B _w ^T − γI D zw

C z R D ^T _zw − γI









W 1 0 W 4 0 0 I n w



 < 0

W Q ^T























A 0 0 0

S N

N ^T U

+

S N N ^T U

A i 0 0 0

S N N ^T U

B w

0

C _z ^T 0

B ^T _w 0

S N

N ^T U

− γI D _zw ^T C z 0

D zw − γI























W ^Q < 0

⇔ W Q ^T









A ^T S + SA A ^T N SB w C _z ^T N ^T A 0 N ^T B w 0

B _w ^T S B _w ^T N − γI D ^T _zw C z 0 D zw − γI







 W ^Q < 0

Q =

C D ^yw 0

=

0 I n k 0 0 n k × n z

C y 0 D yw 0 n y ×n z

W 2

W 3

C y

D yw

W Q Q

W Q =











W 2 0

0 0

W 3 0 0 I n z













 W 2 0 W 3 0 0 I n z





T 



A ^T S + SA SB w C _z ^T B _w ^T S − γI D ^T _zw

C z D zw − γI







 W 2 0 W 3 0 0 I n z



 < 0

N R = W 1

W 4

N S = W 2

W 3

P cl

Syst`eme K _{f ∞}

d

f y f ˆ z

H _∞

n _k = n

K

R S M N

I − RS

U U = N ^T RM ^+T

M ⁺ M

P cl

K

H ∞

H _∞

˙

x = Ax + E d d + E f f y = Cx + F _d d + F _f f

K _f∞ (s)

z = ˆ f − f w ^T = h

d ^T f ^T i

d

P (s)

K _∞ (s)

u = ˆ f y

¯

z ^{w ¯} W ^I (s) ^w

W ^O (s)

z d

f

H ∞

K f ∞ (s)

˙

x ^F = A ^F x ^F + B ^F y f ˆ = C ^F x ^F + D ^F y

H _∞ z = ˆ f − f

w = h d ^T f ^T

i T

y s = y u = ˆ f

P (s) :=



 

 

˙

x(t) = Ax(t) + B w w(t) + B u u(t) z(t) = C _z x(t) + D _wz w(t) + D _uz u(t) y(t) = C y x(t) + D wy w(t) + D uy u(t)

A = A, B w = h

B E _d E _f i

, B u = 0

C z = 0, Dzw = h

0 0 − 1 i

, D zu = h

1 i

C y = C, D yw = h

B F d F f

i

, D yu = h 0 i

D yu = 0 H ∞

W I W O

H _∞

α(t) : t ∈ R ⁺ ( Z ⁺ ) 7→ I := { 1, 2, . . . , N }

( h ◦ x(t) = f _α(t) (t, x(t), u(t)) y(t) = g _α(t) (t, x(t), u(t)) x ∈ R ⁿ u ∈ R ^m y ∈ R ^l

f i : R ⁿ × R ^m 7→ R ⁿ g i : R ⁿ × R ^m 7→ R ^l ∀ i ∈ I h ◦ x(.)

h ◦ x(t) = ˙ x(t) h ◦ x(t) = x(t + 1)

h ◦ x(t) = A _α(t) x(t)

A _α(t) ∈ A = A ₁ , A ₂ , . . . , A _N

˙

x(t) = A _α(t) x(t)

P > 0 ∀ i ∈ I

P > 0 A ^T _i P + P A i < 0

P > 0 A ^T _i P A i − P < 0

V (x) = x ^T (t)P x(t) V (t) > 0 V ˙ (t) < 0 V (t + 1) − V (t) < 0

V (k, x(k), α(k)) = x ^T (k)P _α(k) x(k)

∀ α ∈ I P i ∀ i ∈ I

"

P j A ^T _i P i

⋆ P _i

#

> 0 ∀ (i, j) ∈ I × I P _i G _i ∀ i ∈ I

"

G i + G ^T _i − S i G ^T _i A ^T _i

⋆ S j

#

> 0 ∀ (i, j) ∈ I × I

V (k, x(k), α(k)) = x ^T (k)P α(k) x(k) ∀ α(k) ∈ I V (k, x(k), α(k)) > 0

V (k + 1, x(k + 1), α(k + 1)) − V (k, x(k), α(k)) < 0

∀ k ∈ Z ⁺

x ^T (k + 1)P α(k+1) x(k + 1) − x ^T (k)P α(k) x(k) < 0

∀ x(k) ∈ R ⁿ

A ^T _α(k) P α(k+1) A α(k) − P α(k) < 0

⇔ A ^T _α(k) P _α(k+1) ^T P _α(k+1) ^{− 1} P α(k+1) A α(k) − P α(k) < 0

"

− P α(k+1) − A ^T _α(k) P _α(k+1) ^T

⋆ − P α(k)

#

< 0, ∀ α(k) ∈ I

α(k) = i i ^me

i = α(k) 6 = α(k + 1) = j − P j − A ^T _i P i

⋆ − P i

< 0, ∀ (i, j) ∈ I × I

S i = P _i ^{− 1} S j = P _j ^{− 1} − S _j ⁻¹ − A ^T _i S ⁻¹ _i

⋆ − S _i ⁻¹

< 0, ∀ (i, j) ∈ I × I

I, S i

S j − A ^T _i S i A i = T ij > 0 G i = S i + g i I g i g i

g ⁻² _i (S i + 2g i I) > A ^T _i T ij A i ∀ (i, j) ∈ I × I

S i + 2g i I − g i A ^T _i

⋆ T ij

∀ (i, j) ∈ I × I

G i T ij

P i = P ∀ i ∈ I N = 2

{ A ₁ , A ₂ } (N = 2)

δ ∈ [0, 1]

A _eq = δA ₁ + (1 − δ)A ₂

α(t) ∈ { 1, 2 }

A eq = δA ₁ + (1 − δ)A ₂ δ ∈ [0, 1] P

Q

δ(A ^T ₁ P + P A 1 ) + (1 − δ)(A ^T ₂ P + P A 2 ) = − Q

δx ^T (A ^T ₁ P + P A ₁ )x + (1 − δ)x ^T (A ^T ₂ P + P A ₂ )x = − xQx < 0, ∀ x ∈ R ⁿ { 0 }

x ^T (A ^T ₁ P + P A 1 )x x ^T (A ^T ₂ P + P A 2 )x ∀ x ∈ R ⁿ { 0 }

V (x) = x ^T P x

A

a ij ≥ ∀ i 6 = j

M c M d

Π ∈ R ^N×N

Π ∈ M c :

N

X

i=1

π _ij = 0

Π ∈ M d :

N

X

i=1

π ij = 1

∀ j = 1, .., N

A ^T _i P i + P i A i +

N

X

j=1

π cji P j + Q i < 0

A ^T _i





N

X

j=1

π _dji P j



 A i − P j + Q i < 0

τ D τ D ∈ N P i > 0 ∀ i ∈ I

A ^T _i P i A i − P i < 0 ∀ i ∈ I

(A ^τ _i ^d ) ^T P j A ^τ _i ^d − P i < 0 ∀ (i, j) ∈ I × I , i 6 = j

τ D

τ D = 1

τ moy

τ _moy

τ moy

N α (T, t) [t, T ]

α(t) N ₀ τ moy

N α (T, t) ≤ N ₀ + T − t τ moy

N ₀

τ D τ moy

H ₋ /H _∞ H ∞

H _∞

t s us

i = { f, r } j = { l, r }

X, Y, Z (x, y, z)

θ

m x ¨ = F tx f + F tx r − F aero − R tx f − R tx r − mg (θ)

F tx f F tx r

F aero

R tx f R tx r

m g

ρ C d A F V x V vent

ρ C _d A _F

F aero = c aero ( ˙ x + V vent ) ²

R tx f + R tx r = f(F zf + F zr ) f

F zf = − F aero h aero + m¨ xh + mgh (θ) − mgl r (θ) l f + l r

F zr = F aero h aero + m xh ¨ + mgh (θ) + mgl _f (θ) l f + l r

l r l f h

h aero

F tx f F tx r

F t

F t

θ z

x

X Z

m ω I t

˙

x r ef f ω i

β x i = ρ ₁ (r _{ef f i} ω i − x) ˙

ω i i ^me r _{ef f s}

ρ ₁

1 ρ ₁ =

( x ˙ , ( ˙ x < 0)

r ef f ω i , ( ˙ x > 0)

F tx ij = D x

(C x ) B x β x ij − E x (B x β x ij − (B x β x ij )

F tx i = c xi β xi

c xi

Acc´el´eration

D´ec´el´eration

coefficient de glissement

F or ce d u p n eu [N]

3000

2000

1000

0

−1000

− 2000

− 3000

−4000

− 0.6 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 Non Lin´eaire

Lin´earis´ee pour de petits glissements

I t f ω ˙ t f = T t f − r ef f f F x f

I t r ω ˙ t r = T t r − r _{ef f r} F x r

I t T t f

T t r

m

m¨ x = c x (r ef f ω − x)ρ ˙ ₁ − c aero ( ˙ x + V vent ) ² − f mg (θ) − mg (θ) I t ω ˙ = − r ef f c x (r ef f ω − x)ρ ˙ ₁ + T i

θ = 0

ρ ₂ = ˙ x

"

¨ x

˙ ω

#

=

"

− ^β _m ^x ρ ₁ − ^{c aero} _m ρ ₂ ^{r ef f} _m ^{β x} ρ ₁

r ef f β x

I t ρ ₁ − ^r

2 ef f β x

I t ρ ₁

# "

˙ x ω

# +

"

0

1 I t

# T _i

c trans

y =

"

1 0

0 _c ¹

trans

# "

˙ x ω

#

l f l r L = l r + l f

δ

X Y ψ X, Y

ψ V ~ = X ~ ˙ + Y ~ ˙ β

δ r 6 = δ _l

≈ R

F _l = mV ²

R

δ F tx f l

F ty f l

l r

l f

F ty rl

F tx rl

y

x V ~ = X ~ ˙ + Y ~ ˙

t r

β

X ψ

t f

A

B

CoG



 

 



 

 



X ˙ = V (ψ + β) Y ˙ = V (ψ + β) ψ ˙ = ^V _l ^(β)

r +l f (δ) β = ⁻¹ _l

f +l r (δ) l r +l f

y ψ v

y ma _y (t) = F _yf + F _yr

= F tx _f (δ) + F ty _f (δ) + F ty r + F _dy

a y = ¨ y + v ψ ˙

¨ y = v β ˙

I z ψ ¨ = l f F yf − l r F yr

= l _f (F _{tx f} (δ) + F _{ty f} (δ)) − l _r F _{ty r} + M _dz (δ) ∼ = 0 (δ) ∼ = 1

mv( ˙ β + ˙ ψ) = F ty f + F ty r + F dy

I z ψ ¨ = l _f F ty _f − l r F ty r + M _dz

F ty f F ty r β

F ty f = c yf β yf

F _{ty r} = c _yr β _yr

β yf = δ − β + l _f ψ ˙ v β yr = − β + l r ψ ˙

v







mv( ˙ β + ˙ ψ) = c yf

δ − β + ^{l f} _v ^{ψ ˙} + c yr

− β + ^{l r} _v ^{ψ ˙} + F dy

I z ψ ¨ = l _f c _yf

δ − β + ^{l f} _v ^{ψ ˙}

− l r c yr

− β + ^{l r} _v ^{ψ ˙} + M _dz

F d l d

"

β(t) ˙ ψ(t) ¨

#

=





− ^{c yr} _mv(t) ^{+c yf c yr} _mv ^{l r −c} 2 (t) ^{yf l f} − 1

c yr l r −c yf l _f

I z − ^{c yr} _I ^{l 2 r} _z ^+c _v(t) ^{yf l 2 f}





"

β(t) ψ(t) ˙

# +

" _c

yf

c mv yr l _f I z

# δ(t) +

"

1 mv l _d

I z

#

F _d (t)

a y = v( ˙ β + ˙ ψ) = v

( − c _yr + c _yf

mv )β + ( c _yr l _r − c _yf l _f

mv ² − 1) ˙ ψ + c _yf mv δ

+ v ψ ˙

y =

"

ψ ˙ a _y

#

=

"

0 1

− ^{c yr +c} _m ^{yf c yr l r} _mv ^{−c yf l f}

# "

β ψ ˙

# +

"

0

c _yf mv

# δ

• m s

z s

• m us

z us

z r

k t c t

k s

( m s z ¨ s = − (F sz + F dz ) m _us z ¨ _us = F _sz − F _tz

F _dz F tz

F tz = k t (z us − z r ) + c t ( ˙ z us − z ˙ r ) F sz

F sz = F _k (.) + F _d (.)

F _k (.) F _d (.)

• F _k (.)

z def = z s − z us

F k = k s z def

k s

• F d (.)

F d (.) = F d ( ˙ z s − z ˙ us )

F _d (.) = F _d ( ˙ z _s − z ˙ _us , u) u

F d (.)