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uncertain z + w Outline Well-posedness and topological - - PowerPoint PPT Presentation
uncertain z + w Outline Well-posedness and topological - - PowerPoint PPT Presentation
Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation Dimitri PEAUCELLE & Didier HENRION & Denis ARZELIER all with LAAS-CNRS - Toulouse, FRANCE also with Czech Technical
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Topological separation
General framework
G (z, w)=0
z w
z z w w F (w, z)=0
Well-Posedness: Bounded ( ¯
w, ¯ z) ⇒ unique bounded (w, z) ∃θ topological separator: F(¯ z) = {(w, z) : F¯
z(w, z) = 0} ⊂ {(w, z) : θ(w, z) > −φ1(||¯
z||)} GI( ¯ w) = {(w, z) : G ¯
w(z, w) = 0} ⊂ {(w, z) : θ(w, z) ≤ φ2(|| ¯
w||)}
Related results :
➞ Stability (θ Lyapunov certificate), Passivity (θ storage function), IQC ... ➞ Robust analysis of Linear uncertain systems (Iwasaki)
2 GT MOSAR 9-10 juin 2005
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Well-posedness in the considered case
w + + z z w
Well-posedness: Null(E − A∇) is empty ∀∇ ∈ ∇
∇ ✪ Includes classical µ theory framework: I − A∇ non-singular for all (structured) norm-bounded ∇ ✪ Results of the paper extend to block diagonal uncertainties: ∇ = diag(δR
1 Ir1, . . . , δR NRIrNR, δC 1 Ic1, . . . , δC NCIcNC, ∆C 1, . . . , ∆C Nf) 3 GT MOSAR 9-10 juin 2005
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Well-posedness and descriptor systems
Stability of E ˙
x = Ax ⇔ Es − A full rank for all s ∈ C+ ⇔ W.P
. of
w = ∇z Ez = Aw
for all ∇ = s−1In ∈ C+
+ + x x
4 GT MOSAR 9-10 juin 2005
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Well-posedness and pole location
Let a region defined as a half plane or a disk
D = { s ∈ C : d1 + d2s + d∗
2s∗ + d3ss∗ ≤ 0 }
D-Stability of E ˙ x = Ax ⇔ Es − A full rank for all s ∈ D ⇔ W.P
. of
w = ∇z Ez = Aw
for all ∇ = ∇
∇
where ∇
∇ = { s−1In : d1s−1s−∗ + d2s−∗ + d∗
2s−1 + d3 ≥ 0 }. 5 GT MOSAR 9-10 juin 2005
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Well-posedness and polynomial descriptor systems
Stability of Adx(d) + Ad−1x(d−1) + · · · + A1 ˙
x + A0x = 0 ⇔ W.P
. of
w = ∇z Ez = Aw
for all ∇ = s−1Idn ∈ C+ where E =
Ad O · · · O O −I O
. . . ...
O O −I
A = −
Ad−1 · · · A1 A0 I O O
... . . .
O I O
6 GT MOSAR 9-10 juin 2005
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Robust stability of descriptor systems
Robust stability ∀∆ ∈ ∆
∆ of E(∆) ˙ x = A(∆)x
with
E(∆) = EA + (B∆ − EB)(ED − D∆)−1EC A(∆) = A + (B∆ − EB)(ED − D∆)−1C ⇔ W.P
. of
w = ∇z Ez = Aw
for all ∇ = s−1In
O O ∆
: s−1 ∈ C+ ∆ ∈ ∆ ∆
where E = EA
EB EC ED
A =
A
B C D
7 GT MOSAR 9-10 juin 2005
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Main result: Quadratic separation
W.P . of
w = ∇z Ez = Aw
for all ∇ = ∇
∇ ⇔ ∃Θ :
- I
E◦∗∇∗
- Θ
I ∇E◦
≤ O ,
∀∇ ∈ ∇ ∇
- EE◦
−A
⊥∗
Θ
- EE◦
−A
⊥
> O .
where the columns of E◦ form an orthogonal basis of E∗ and the columns of
- EE◦
−A
⊥ span the null-space of
- EE◦
−A
- .
8 GT MOSAR 9-10 juin 2005
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Application to descriptor systems
Stability of E ˙
x = Ax ⇔
W.P . of
w = ∇z Ez = Aw
for all ∇ = s−1In ∈ C+. Choice of separator:
- I
E◦∗s−∗
- Θ
-
O −E◦∗P −PE◦ O
I E◦s−1
= −2Re(s−1)E◦∗PE◦ LMI result ❶:
E◦∗PE◦ > O
- EE◦
−A
⊥∗
O E◦∗P PE◦ O
- EE◦
−A
⊥
< O
Equivalent to ❷:
E∗X∗ = XE ≥ O , A∗X∗ + XA < O
9 GT MOSAR 9-10 juin 2005
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Example of descriptor system
Scalar ’switch’ system: 1 ˙
x =
1
α
x If α = 0
- x(t) = 0 the system is stable
❶
- EE◦
−A
⊥
= [ ] →
LMI p > 0
❷ X =
- x1
x2
- →
LMI x1 ≥ 0 , 2x1 + 2x2α < 0 If α = 0
- ˙
x = x the system is unstable ❶
- EE◦
−A
⊥
=
1
1
→
LMI p > 0 , 2p < 0
❷ X =
- x1
x2
- →
LMI x1 ≥ 0 , 2x1 < 0
10 GT MOSAR 9-10 juin 2005
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Application to uncertain descriptor systems
Robust Stability of E(∆) ˙
x = A(∆)x
, ∆ = δI : |δ| ≤ ¯
δ ⇔
W.P . of
w = ∇z Ez = Aw
for all ∇ = s−1In
O O δIm
: s−1 ∈ C+ |δ| ≤ ¯ δ
. Choice of separator:
Θ =
E◦∗
2 Θ1E◦ 2
−E◦∗
1 P
E◦∗
2 Θ2
−PE◦
1
O O Θ2∗E◦
2
O Θ3
: ∇E◦ =
s−1E◦
1
∆E◦
2
constrained by the LMIs
E◦∗
1 PE◦ 1 ≥ O ,
E◦∗
2 Θ3E◦ 2 ≥ O
E◦∗
2 (Θ1 + ¯
δΘ2 + ¯ δΘ2∗ + ¯ δ2Θ3)E◦
2 ≤ O
E◦∗
2 (Θ1 − ¯
δΘ2 − ¯ δΘ2∗ + ¯ δ2Θ3)E◦
2 ≤ O 11 GT MOSAR 9-10 juin 2005
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Reducing conservatism ◆ Original system
EA
EB EC ED
˙
x z∆
= A
B C D
x
w∆
, ∇ = s−1In
O O δIm
❖ Augmented system
O I O O O EA EB O O EC ED O EA O O EB EC O O ED
¨ x ˙ x z∆ ˙ z∆
=
I O O O O A B O O C D O A O O B C O O D
˙ x x w∆ ˙ w∆
, ∇ =
s−1I2n
O O δI2m
12 GT MOSAR 9-10 juin 2005
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Reducing conservatism ◆ Quadratic separation on Original system ➞ Lyapunov stability with V (η, ∆) = η∗Pη ❖ Quadratic separation on Augmented system ➞ Lyapunov stability with V (η, ∆) = η∗P(∆)η
where in the case E = I (not descriptor)
P(∆) =
A(∆)
I
∗
P
A(∆)
I
13 GT MOSAR 9-10 juin 2005
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Robust stability analysis example
E =
8 4 8 10 9 10 3 5 3 1 3
, A =
3 −16 9 2 −8 −16 −19 −17 −20 1 4 1 2 −10 −6 3 7 −6
◆ Quadratic separation on Original system ➞ LMIs feasible up to ¯ δ = 0.22 - infeasible for ¯ δ = 0.23 ❖ Quadratic separation on Augmented system ➞ LMIs feasible up to ¯ δ = 0.45 - infeasible for ¯ δ = 0.46
14 GT MOSAR 9-10 juin 2005
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