[PPT] - Ellipsoidal Sets for Static Output-Feedback by Dimitri Peaucelle PowerPoint Presentation

SLIDE 1

IFAC World Congress July 21-26 2002, Barcelona

Ellipsoidal Sets for Static Output-Feedback

by Dimitri Peaucelle & Denis Arzelier & Regis Bertrand

Laboratoire d’Analyse et d’Architecture des Syst` emes du C.N.R.S. Toulouse, FRANCE

SLIDE 2

Motivation

1

Σ ∆

Θ :

h
∆
i

Θ

∆
∆

∆

h

Σ

jω
i

Θ

Σ jω
ω
R

Topological separation for robust analysis

Ellipsoidal Sets for Static Output-Feedback

SLIDE 3

Motivation

1-a

for synthesis

:

Σ

K

Θ :

h

Σ

jω
i

Θ

Σ jω
ω
R

h

K
i

Θ

K
non-empty set

Ellipsoidal Sets for Static Output-Feedback

SLIDE 4

Matrix ellipsoids

2

Ellipsoids of the vector space

Rn

Centre xo

Rn, radius r and geometry Z
Rnn s.t. Z
,

kZ k 1.

x
Rn

:

x xo

Z

x xo

r
Matrix ellipsoids of

Rmp

Centre Ko

Rmp, radius R
Rpp and geometry Z
Rmm s.t. Z
,

kZ k 1.

K
Rmp

:

K Ko

Z

K Ko

R
fX, Y, Z

g-ellipsoid : definition

K
Rmp

:

h

K
i
X

Y Y

Z
K
Z
Ellipsoidal Sets for Static Output-Feedback

SLIDE 5

Matrix ellipsoids

3 ✪ Algebraic rules: Ko

Z

1Y , R Ko ZKo X.

✪ Non-emptiness condition : R

X

YZ 1Y

✪ LMI description :
X

YK K Y

K

Z

ZK

Z

✪

fX, Y, Z g-ellipsoid : a compact convex set.

✪ VOL

fX, Y, Z

g-ellipsoid

q

det

Rm

det

Z p VOL

f,

, g-ellipsoid.

Ellipsoidal Sets for Static Output-Feedback

SLIDE 6

Static output-feedback

4

Notations

Σ

K : Σ

K :

˙

x

t

Axt
But
yt
Cxt
Dut
ut
Kyt
Σ is stabilisable via static output-feedback

iff

K s.t.

Σ

K is stable.

iff there exist a Lyapunov matrix P and a non-empty

fX, Y, Z g-ellipsoid s.t.:

A

B

P

P

A

B

C

D

X

Y Y

Z
C

D

Ellipsoidal Sets for Static Output-Feedback

SLIDE 7

Remarks

5

A set of control laws:

V

x x Px proves Σ K stability for any gain K in the fX, Y, Z g-ellipsoid.

The non-convex constraint

SOF stabilisability

LMIs

a non-linear constraint (X YZ 1Y ).

Example for K

R11 : z

1 et x y2

y x

Ellipsoidal Sets for Static Output-Feedback

SLIDE 8

Fragility and resilience

6

Definition : Let Ko be a stabilising gain and ∆

K an additive uncertainty.

✪ Fragile :

∆K ∆ K s.t. Σ

Ko

∆K is unstable.

✪ Resilient :

∆K ∆ K

: Σ

Ko

∆K is stable.

✪ Quadratically resilient : A unique quadratic Lyapunov function (V

x x Px) proves the resiliency.

Ellipsoidal Sets for Static Output-Feedback

SLIDE 9

Fragility and resilience

7

Corollaries

✪ ∆

K :

ellipsoidal matrix set centred at the origin. LMI constraint

non-linear constraint X YZ 1Y

centre Ko of the

fX, Y, Z g-ellipsoid is quadratically resilient to ∆

KZ∆K

R.

✪ ∆

K : norm-bounded uncertainty.

LMI constraint

Z

non-linear constraint
ρ
YY
X
centre Ko of the

fX, Y, Z g-ellipsoid is quadratically resilient to ∆

K∆K

ρ .

✪ ∆

K : multiplicative uncertainty with radius ¯

δ LMI constraint

non-linear constraint X

1

¯

δ2

YZ 1Y

centre Ko of the

fX, Y, Z g-ellipsoid is quadratically resilient to ∆

K

δKo with jδ j ¯

δ.

Ellipsoidal Sets for Static Output-Feedback

SLIDE 10

Bounded or dissipative specifications on K

8

Definition

Design a stabilising control law K that belongs to a given

fX

K, YK, ZK

g-ellipsoid.

Example 1 : Find a control law with bounded gain (K

K ρK )

Exemple 2 : Find a passive control law (K

K
)

LMI constraint

ν

ν

XK

YK YK

ZK
X

Y Y

Z
Ellipsoidal Sets for Static Output-Feedback

SLIDE 11

Pole location

9

fXR, YR, ZR g-stability

The poles of Σ

K belong to an ellipsoidal region of the complex plane: f s

C : XR

sYR s YR

ss

ZR

g

Examples: half-planes, discs, sectors, parabolas...

Static output-feedback

fXR, YR, ZR g-stabilisability h

i
XR

P

YR

P

YR

P

ZR

P

A
B
h
i
X
Y
Y
Z
C
D
non-linear constraint

Ellipsoidal Sets for Static Output-Feedback

SLIDE 12

Extensions to multi-objective synthesis

10 ∋ {X ,Y ,Z }−ellip.

K K K

K <γ <γ <γ

2 3 1

∋ ∆ ∆ Λ

R R R

{X ,Y ,Z }−stable Σ

✪ An LMI constraint for each specification. ✪ A Lyapunov function for each specification. ✪ A unique non-linear constraint (X

YZ 1Y )

Ellipsoidal Sets for Static Output-Feedback

SLIDE 13

Algorithms to solve non-linear matrix inequalities

11 ✪ SOF design has no convex formulation in the general case. ✪ Elimination based approach

LMIs (XY

)

proved to be NP-hard [Fu & Luo 1997]. ✪ Heuristic algorithms such as coordinated-descent iterative resolutions of BMIs. ✪ Efficient sub-optimal first order algorithms: ✫ Cone complementarity algorithm [El Ghaoui & al 1997]. ✫ Alternation projection algorithm [Grigoriadis & Skelton 1996].

Ellipsoidal Sets for Static Output-Feedback

SLIDE 14

Algorithms to solve non-linear matrix inequalities

12

Numerical experiments with cone complementarity algorithm

✪ Considered non-linear constraint: X

1

¯

δ2

YZ 1Y .

✪ Linear relaxation: X

1

¯

δ2

ˆ

X YZ

1Y

ˆ

X ✪ Algorithm designed for: ˆ X

YZ

1Y .

✪ Stopping criterium: X

1

¯

δ2

YZ 1Y .

Ellipsoidal Sets for Static Output-Feedback

SLIDE 15

Algorithms to solve non-linear matrix inequalities

13

SOF design such that K

h

k1 k2

i

R21.

Stability Poles location (Repˆ

les
015)

and resiliency bounded K (radius 10, center

h

10 10

i

).

and resiliency

−0.55 −0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

2

k1 k

0.4 0.6 0.8 1 1.2 −0.5 −0.4 −0.3 −0.2

δ = 0.25 δ = 0 δ = 0.5

k1 k2

0.5 1 1.5 2 2.5 3 3.5 2 4 6 8 10 12 14 16 18

1 2 3 4 8 12 16

δ = 0 δ=0.5 δ = 0.25

Ellipsoidal Sets for Static Output-Feedback

SLIDE 16

Conclusions and prospectives

14 ✪ New static output-feedback design based on the topological separation theory. ✪ No conservatism when compared to LMI analysis techniques. ✪ Encouraging numerical results. ✫ Develop new adapted algorithms. ✫ Extensions to other design problems.