Robust pole placement by static output feedback

Jérôme Bosche, Olivier Bachelier and Driss Mehdi

Abstract— This paper tackles the problem of pole placement
by static output feedback. The considered systems are LTI
and subject to both polytopic and norm-bounded uncertainties
i.e. the uncertain closed-loop state matrix can be written

A

o

= A + BK

o

C + J EL. Matrices A, B, C, J and L belong

to polytopes of matrices whereas

E is unknown. K

o

is the

matrix associated with the static output feedback. It is aimed
to compute

K

o

so as to place the spectrum of

A

o

in an

EEMI-

region, denoted by

D

U

, while maximizing the acceptable 2-

norm of

E ensuring the D

U

-stabity of

A

o

. Nevertheless, the

pole placement by static output feedback presents two major
difficulties :
- the strict pole placement by static output feedback is usually
possible only if Kimura’s condition holds.
- the non strict pole placement methods through Lyapunov
based approaches leads to non convex problems such as

BMI

(Bilinear Matrix Inequality) rather than

LMI (Linear Matrix

Inequality).
This paper proposes a heuristic technique to compute

K

o

. It

aims at circumventing the difficulties mentionned herebefore.
This technique is based on genetic algorithms (not detailed
here) and on the resolution of

LMI.

I. INTRODUCTION

Pole placement arouses the interest of many authors

working on linear state representations [3], [6], [12].
Indeed, transient performances are strongly influenced by
the location of the closed-loop state matrix eigenvalues. It
can then be desirable that all those eigenvalues belong to
a region

D of the complex plane. This property, known

as

D-stability, is an essential concept in this article. The

regions considered in this work are referred to as

EEMI-

regions [2] (

Extended Ellipsoidal Matrix Inequality)

and are generically denoted by

D

U

. We then talk about

D

U

-stability. This class of regions enables to handle unions

of several possible disjoint and non symmetric subregions.
State-space LTI models are considered. They are affected
by polytopic and additive norm-bounded uncertainties.
The objective of this work is to compute some output
feedback matrix

K

o

so as to guarantee the

D

U

-stability

of the closed-loop state matrix, while maximizing the size
of the uncertainty domain. In other words, it is aimed
at robustly

D

U

-stabilizing the closed-loop system by a

static output feedback.

LMI (Linear Matrix Inequality)

conditions for robust

D

U

-stabilization exist [4] and can

involve Parameter-Dependent Lyapunov Matrix (PDLM)
[10], [17].
The paper is organized as follows : after this introduction,

This work was not supported by any organization
J.

Bosche,

O.

Bachelier

and

D.

Mehdi

are

with

the

Laboratoire

d’Automatique

et

d’Informatique

Industrielle

-

Ecole

Supérieure

d’Ingénieurs

de Poitiers, Bâtiment

de Mécanique,

40

Avenue

du

Recteur

Pineau,

86022

POITIERS

C

EDEX

,

France

[e-mail]

the second section recalls formulations of

EEMI-regions

and of the polytopic and norm-bounded uncertainties in
order to state the precise problem. It is solved in the third
part from the nominal point of view and in the fourth one
from the robust point of view. An illustration is proposed
in the fifth section before to conclude.

Notations : We denote by

M

, the transpose conjugate

of

M , by H(M ) the Hermitian expression M + M

. The

Kronecker product is denoted by

⊗. ||M||

2

is the 2-norm

of matrix

M induced by the Euclidean vector-norm, i.e. the

maximal singular value of

M . II

n

is the identity matrix

of order

n, O is a null matrix of suitable dimension.

Matrix inequalities are considered in the sense of Löewner
i.e. “

≺ 0” (“ 0”) means negative (semi-)definite and

”

0” (“ 0”) positive (semi-)definite. HPD stands for

Hermitian Positive Definite. Small letters are used for scalar
numbers and vectors while capital letters denote matrices
or sets. In the Hermitian matrices, notation (

•)’ prevents us

from repeating the symmetric blocks. At last,

i denotes the

imaginary unit.

II. P

ROBLEM

S

TATEMENT

In this part are introduced all the preliminary useful

concepts. First, the uncertain state matrix is presented. The-
reafter, the description of the considered clustering regions
is specified. Once these preliminaries established, a nominal
D

U

-stabilization condition is recalled before to formulate

our precise purpose.

A. The uncertain matrix

Consider a complex uncertain matrix

A ∈ lC

n×n

defined

by :

A = A + E with E = J EL .

(1)

Matrices

A and E are both uncertain. E is an additve

uncertainty in which

E ∈ lC

q×r

is unknown.

E is assumed

to belong to

B(ρ), the ball of (q × r) complex matrices

checking

||E||

2

≤ ρ. Define matrix M by :

M =

A J

L O

=

A(θ) J(θ)

L(θ)

O

.

(2)

Matrix

M is assumed to belong to a polytope of matrices :

M =

˘

M = M(θ) ∈ lC

(n+r)×(n+q)

|

M(θ) =

N

X

i=1

(θ

i

M

i

) ; θ ∈ Θ

)

(3)

where

Θ is the set of all barycentric coordinates :

43rd IEEE Conference on Decision and Control
December 14-17, 2004
Atlantis, Paradise Island, Bahamas

0-7803-8682-5/04/$20.00 ©2004 IEEE

TuB12.3

869

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.

Θ =

⎧

⎪

⎨
⎪

⎩

θ =

⎡
⎢

⎣

θ

1

..

.

θ

N

⎤
⎥

⎦ ∈ {IR

+

}

N

|

N

i=1

θ

i

= 1

⎫

⎪

⎬
⎪

⎭

. (4)

Extreme matrices

M

i

,

i = 1, ..., N are the vertices of M :

M

i

=

A

i

J

i

L

i

O

.

(5)

Therefore matrix

A can read A = A(θ) + J(θ)EL(θ). It

is well known that this formulation encompasses the case
where

M linearly depends on some parameter deflections

around a nominal value. If no polytopic dependence is
assumed (only one vertex for

M) then A is only subject to

a classical norm-bounded uncertainty [18].

B.

EEMI-regions

Definition 1: [2] Let

R be a set of ¯k Hermitian matrices

R

k

defined by :

⎧

⎪

⎪

⎨
⎪

⎪

⎩

R

k

= R

k

=

R

k

00

R

k

10

R

k

10

R

k

11

∈ lC

2d×2d

R

k

11

0, R

k

11

∈ lC

d×d

∀k ∈ {1, ..., ¯k} .

(6)

The set of points

D

U

defined by :

D

U

=

8

>

<
>

:

z ∈ lC | ∃ ω =

2
6

4

ω

1

..

.

ω

¯

k

3
7

5 ∈ {IR

+

}

¯

k

| f

D

U

(ω, z)

=

¯

k

X

k=1

(ω

k

ˆ

II

d

˜zII

d

˜

R

k

»

II

d

zII

d

–

) ≺ 0

9

=
;

(7)

is called an open

EEMI-region of degree d.

When ¯

k = 1, this description reduces to EMI formu-

lation (

Ellipsoidal Matrix Inequality) proposed in [17].

An

EEMI-region is possibly nonconnected and can result

from the union of possibly complex and disjoint

EMI-

subregions. Such

EMI-subregions are convex and can be,

for instance, shifted and nonsymmetric half planes, classical
or hyperbolic sectors, vertical or horizontal strips, discs or
insides of ellipses... Here,

R

k

is allowed to be complex, i.e.

EMI-subregions can be nonsymmetric (with respect to the
real axis).
Some necessary and sufficient conditions (NSC) for

D

U

-

stability of real or complex matrices exist [2]. Such a NSC
is now recalled.

C. Nominal

D

U

-stability of a complex matrix

A reformulation of the conditions suggested in [2] is now

proposed :

Theorem 1: [2] Let

D

U

be an

EEMI-region as introduced

in definition 1. A matrix

A ∈ lC

n×n

is

D

U

-stable if and

only if there exists a set

P of ¯k HPD matrices P

k

∈ lC

n×n

,

k = 1, ..., ¯k, such that :

U(A, P

k

) =

II

dn

II

d

⊗ A

U(P

k

)

II

dn

II

d

⊗ A

≺ 0

(8)

with :

U(P

k

) =

¯k

k=1

(R

k

⊗ P

k

) .

(9)

Proof : see [2].

Remark 1: In the previous theorem, the matrices are consi-
dered as complex. Thus, to specify

R

k

as complex enables

to consider nonsymmetric subregions which is an advantage
in term of variety of regions. The possible complexity of
A and thus E is however pratically useless in Automatic.
Nevertheless, it turns out that the tools and theorems pre-
sented in this paper are valid for complex matrices (and
thus a fortiori for real matrices). In practice, if

A

i

,

J

i

,

L

i

can be considered as real, it is impossible to impose the
reality of

E which is unknown. The non consideration of

this realness may induce some conservatism.

D. The precise purpose

In this paper, we consider the closed-loop state matrix of

a multivariable system that can be written.

A

o

= A

o

(θ, E) = A(θ) + B(θ)K

o

C(θ) + J(θ)EL(θ) .

(10)

We will adopt formulation (10) afterwards. Formulation
(10) allows, for example, to consider fragility analysis.
Indeed, as it is impossible in practice to precisely implant a
control law, we can suppose that the feedback matrix equals
to

K

o

nom

+ ∆

K

o

where

K

o

nom

is a nominal term and

∆

K

o

an uncertain one. The uncertain closed-loop state matrix is
then

A

o

(θ, E) = A(θ) + B(θ)K

o

nom

C(θ) + B(θ)∆

K

o

C(θ)

(11)

which is a particular writing of (10).
The present purpose is to compute an output feedback
matrix

K

o

which assigns the eigenvalues of the uncertain

closed-loop state matrix

A

o

in a

EEMI-region D

U

defined

as in (7).

K

o

is derived with calculating a bound on

||E||

2

, denoted by

ρ

3

, as large as possible, such that

D

U

-

stability of is guaranteed, for any radius

ρ ≤ ρ

and for

any

θ ∈ Θ. The obtained bound is known as a robust

D

U

-stability bound. This pole placement technique must

implicitly involve PDLM in order to take the polytopic
structure of the uncertainty into account.

III. NOMINAL POLE PLACEMENT

In this part, its first explained that the strict pole placement

by static output feedback is generically difficult if Kimura’s
condition is not satisfied. Then, a

D

U

-stabilization condition

in the nominal case is presented.

870

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.

A. Eigenstructure assignment and Kimura’s condition

Eigenstructure assignment by output feedback is a major

topic in multivariable systems [7], [15], [22] and many
results have been presented since the middle of the seventies
[8], [23]. These techniques use the degrees of freedom (dof)
offered by eigenvectors. Nevertheless, the

m×p dof offered

by the output feedback matrix are not always sufficient to
place the all desired spectrum. Several conditions linking
m, n and p allowing a complete pole placement by static
output feedback were proposed [20], [24]. It seems that
if the problem is restricted to real matrices, the condition
is

mp > n. However, when it is aimed to apply effective

methods of eigenstructure assignment, Kimura’s condition
[11], presented as generically sufficient to place

n poles,

also proves generically necessary for methods [5] and [22]
to be applied. This is why we retain Kimura’s condition
which is now presented.

Consider a multivariable realization, reachable and obser-
vable, with

n states, m imputs and p outputs, if the condition

m + p > n

(12)

is satisfied, then it is possible to place any of the spectrum
for the closed-loop output state matrix

A

o

= A + BK

o

C

as long as a slight modification of the required spectrum is
tolerable.
According to this condition, two classes of systems are
considered :

1) systems of class

S

+

: their models check Kimura’s

condition (i.e.

m + p >

n). It is then possible

to place any spectrum by static output feedback.
Several approaches, not detailed in this paper, can be
exploited [5], [19], [22].

2) systems of class

S

−

: their models do not check

Kimura’s condition (i.e.

m + p ≤ n). Concerning

these systems, finding a strict placement law by
static output feedback becomes a much more delicate
task and we will propose a conservative method of
nonstrict placement in an

EEMI-region (which can

also be used for systems of class

S

+

).

The next part proposes a technique that may enable to
compute an output feedback matrix (if Kimura’s condition
is satisfi ed or not).

B. Nominal

D

U

-stabilization

This part proposes a method to compute a robust control
law by static output feedback for any nominal multivariable
linear systems. This method is based on the resolution of
tractable

LMI conditions and relies on an idea originally

proposed in [1], [16] and then reformulated in [13]. This
idea is here extended to the concept of

D

U

-stability and

consists in finding a state feedback and an output feedback
checking the same

D

U

-stability property through the exis-

tence of the same set of Lyapunov matrices. It amounts
to solving the following

LMI system (13) ∀ k ∈ {1, .., ¯k}

where ¯

k is the number of subregions and U(•) is defined

in (8) :

U(A

s

, P

k

) ≺ 0

U(A

o

, P

k

) ≺ 0

(13)

A

s

and

A

o

are state matrices defined by :

A

s

= A + BK

s

and

A

o

= A + BK

o

C .

(14)

K

s

and

K

o

are then respectively associated with state and

output feedback. Theorem 2 proposes a NSC for

LMI

system (13) to hold.

Theorem 2: Let a LTI system be modelled by the triple
of matrices (

A; B; C) et D

U

, be an

EEMI-region defined

as in (7). Let also

K

s

be a matrix associated with a static

state feedback such that

A

s

= A + BK

s

is

D

U

-stable.

The

LMI system (13) is satisfied if and only if there

exist a set

P, made up by ¯k HPD matrices P

k

∈ IR

n×n

,

k ∈ {1, .., ¯k}, a nonsingular matrix G ∈ IR

m×m

as well as

H ∈ IR

m×p

such that the following

LMI holds :

H

U

=

»

II

dn

II

d

⊗ A

s

O

II

d

⊗ B

–

U(P

k

)

»

II

dn

O

II

d

⊗ A

s

II

d

⊗ B

–

+

H

j»

O

II

dm

–

(II

d

⊗ G)

ˆ

−II

d

⊗ K

s

−II

dm

˜ff

+

H

j»

O

II

dm

–

(II

d

⊗ H)

ˆ

II

d

⊗ C O

˜ff

≺ 0

(15)

with

U(P

k

) defined in (9).

An output feedback matrix is then given by :

K

o

= G

−1

H .

(16)

Proof : According to (16),

H = GK

o

. Once developed,

inequality (15) becomes :

Ψ

k

+

O

II

d

⊗ S

G

(•)

−II

d

⊗ (G + G

)

≺ 0

(17)

with

Ψ

k

=

2
6

6

6

6

4

U(A

s

, P

k

)

¯

k

X

k=1

(R

k

10

⊗ P

k

B + R

k

11

⊗ A

s

P

k

B)

(•)

¯

k

X

k=1

(R

k

11

⊗ B

P

k

B)

3
7

7

7

7

5

and

S = K

o

C − K

s

.

Define matrices

N

u

=

II

dn

O

and

N

v

=

II

dn

II

d

⊗ S

(18)

respectively the left and the right orthogonal comple-
ments of

U =

O

II

dm

and

V =

II

d

⊗ S −II

dm

.

(19)

871

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.

First, notice that

Ψ

k

+ H(U(II

d

⊗ G)V ) ≺ 0

(20)

corresponds to condition (15).
Then, a simple algebraic manipulation shows that

N

u

Ψ

k

N

u

= U(A

s

, P

k

) and N

v

Ψ

k

N

v

= U(A

o

, P

k

) .

(21)

It becomes clear that, in virtue of the elimination lemma
[21],

LMI system (13) is equivalent to condition (15). 2

Here, it is important to emphasize that matrices

A

s

and

A

o

« share » the same set of Lyapunov matrices

P.

It is the idea suggested in [1] and [16]. Although this
constraint is conservative, it allows, by checking the same
property of

D

U

-stability for

A

s

and

A

o

, to circumvent a

BMI problem. Indeed, if the state feedback K

s

which

D

U

-stabilizes the system is not initially calculated, the

condition (15) corresponds to a

BMI. Matrix K

s

can thus

be considered as an initialization of the technique presented
in this part, enabling the computation of a control law by
static output feedback. This « initialization step » is very
important for this technique because among the infinity of
matrices

K

s

which

D

U

-stabilize the model, some may not

lead to a static output feedback checking (13) and (15). In
the following section a heuristic resolution enabling to use
the

D

U

-stabilization condition of theorem 2 is proposed

for the more general case of the robust placement.

IV. ROBUST POLE PLACEMENT

The model is now affected by a polytopic and norm-

bounded uncertainty and it aims at computing a robust
control law by static output feedback.
A robust

D

U

-stabilization condition by static output feed-

back is presented in the first part whereas in the second
part, a heuristic of robust placement is proposed.

A. Robust

D

U

-stabilization

The uncertain closed-loop state matrix is the one defined

in (10). The objective is to compute the matrix

K

o

which

leads to the largest size of

B(ρ), i.e. which leads to the

largest bound

ρ

.

To do that, the idea presented in the previous section
consists in finding a state feedback and an output feedback
checking the same property by the same (or the same ones)
set of Lyapunov matrices, is still exploited. This will be
checked if there exists a solution to the following

LMI

system :

U(A

s

, P

k

(θ)) ≺ 0

U(A

o

, P

k

(θ)) ≺ 0

∀θ ∈ Θ .

(22)

with

U(A

•

, P

k

(θ)) =

ˆ

II

dn

II

d

⊗ A

•

˜

¯

k

X

k=1

(R

k

⊗ P

k

(θ))

»

II

dn

II

d

⊗ A

•

–

≺ 0

(23)

A

s

and

A

o

are respectively the uncertain state matrices of

the systems controlled by static state and output feedbacks.
A sufficient condition for the existence of a solution to
system (22) is now proposed :

Theorem 3: The

LMI system (22) is satisfied if there

exist

N sets P

i

, each one made up by ¯

k HPD matrices

P

k,i

∈ lC

n×n

,

k ∈ {1, .., ¯k} , a nonsingular matrix

G ∈ IR

m×m

as well as

F ∈ IR

d(2n+m)×dn

and

H ∈ IR

m×p

such that the

LMI (24) is satisfied ∀ i ∈ {1, ..., N}

for a given state feedback associated with the matrix

K

s

D

U

-stabilizing the pair

(A(θ), B(θ)).

H

U

i

=

2
6

6

6

4

U(P

i

)

O

II

d

⊗ L

i

O

O

O

O

O −II

dq

O

II

d

⊗ L

i

O

O

−γII

dr

3
7

7

7

5

+ H

8

<
:

2
4

F

O

O

3
5 ˆ II

d

⊗ A

s

i

−II

dn

II

d

⊗ B

i

II

d

⊗ J

i

O

˜

9

=
;

+ H

8

>

>

>

<
>

>

>

:

2
6

6

6

4

O

O

II

dm

O

O

3
7

7

7

5

II

d

⊗ G

ˆ

−II

d

⊗ K

s

O −II

dm

O O

˜

9

>

>

>

=
>

>

>

;

+ H

8

>

>

>

<
>

>

>

:

2
6

6

6

4

O

O

II

dm

O

O

3
7

7

7

5

II

d

⊗ H

ˆ

II

d

⊗ C

i

O O O O

˜

9

>

>

>

=
>

>

>

;

≺ 0 .

(24)

with

U(P

i

) =

¯k

k=1

(R

k

⊗ P

k

i

) et γ = ρ

−2

.

(25)

An output feedback matrix is then given by :

K

o

= G

−1

H

(26)

Proof : For any vector of barycentric coordinates

θ ∈ Θ,

let us consider the convex combination

H

U

=

N

i=1

(θ

i

H

U

i

) .

Since all

θ

i

are positive, it comes :

H

U

≺ 0 .

(27)

In the sequel, we will adopt the following notations :
A

s

(θ) = A

s

,

B(θ) = B, C(θ) = C, J(θ) = J and

L(θ) = L.

Let us define the matrix

S = K

o

C − K

s

, the condition (27)

becomes with (26) :

H

U

=

2
6

6

6

6

4

U(P(θ))

II

d

⊗ S

G

O

O

O

II

d

⊗ L

O

(•)

−II

dq

⊗ (G + G

)

O

O

−II

dq

O

(•)

(•)

−γII

dr

3
7

7

7

7

5

+ H

8

<
:

2
4

F

O

O

3
5 ˆ II

d

⊗ A

s

−II

dn

II

d

⊗ B

II

d

⊗ J

O

˜

9

=
;

≺ 0 .

(28)

872

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.

Since

γ 0, applying Schur’s lemma to (28) gives :

H

U

=

2
4 U(P(θ))

II

d

⊗ S

G

O

(•)

−II

d

⊗ (G + G

)

3
5

+ H

˘

F

ˆ

II

d

⊗ A

s

−II

dn

II

d

⊗ B

˜¯

+

»

II

d

⊗ ρL

O

– ˆ

II

d

⊗ ρL O

˜

+ F (II

d

⊗ JJ

)F

≺ 0 .

(29)

By virtue of the lemma proposed in [25], it exists a matrix

˜

E checking ˜

E

˜

E ≤ II

dr

such that :

H

U

=

2
4 U(P(θ))

II

d

⊗ S

G

O

(•)

−II

d

⊗ (G + G

)

3
5

+ H

˘

F

ˆ

II

d

⊗ (A

s

+ Jρ ˜

EL) −II

dn

II

d

⊗ B

˜¯

≺ 0 .

(30)

Let consider

E = ρ ˜

E, it comes :

H

U

=

⎡
⎣ U(P(θ))

II

d

⊗ S

G

O

(•)

−II

d

⊗ (G + G

)

⎤
⎦

+ H

F

II

d

⊗ A

s

−II

dn

II

d

⊗ B

≺ 0 . (31)

At this stage, the reasoning is the same as the one used in
the proof of theorem 2 but with considering equation (17)
with

A

s

= A

s

,

B = B, C = C and P

k

= P

k

(θ). It then

leads to

LMI system (22).

2

B. Heuristic resolution

As we already noticed, the technique presented in this

paper requires an « initialization step » which consists in
computing a state feedback matrix

K

s

. We can thus speak

about « good initializations » and « bad initializations »
with respect to the system robustness conferred by

K

s

.

Indeed, according to the value of the matrix

K

s

, the

approach leads or not to a more or less robust control
law by static output feedback. In other words, each
matrix

K

s

can lead to a bound

ρ

, significant of the

system robustness. Nevertheless, for the time being, there
is no effective method enabling to determine a « good
initialization » with respect to the bound, even less « the
best one » .

The resolution method that we propose is as follows :

Step 1 : Several state feedbacks

K

s

, each one that

D

U

-

stabilizes the pair (

A(θ), B(θ)), are computed by using

a beforehand specified eigenstructure assignment technique
and exploiting the dof on the eigenvalues and eigenvectors.

The technique used for the numerical example is that
proposed in [14].
Step 2 : A robust control law by static output feedback is
computed, if possible, for each matrix

K

s

thanks to the tool

presented by theorem 3. In other words, when a matrix

K

s

enables to compute a robust control law by static output
feedback, we obtain a matrix

K

o

and its corresponding

robustness bound

ρ

. When

K

s

does not enable to satisfy

LMI (24), the robustness bound is imposed zero (ρ

3

= 0).

Test : Then we can define a convergence criterion which,
if it is satisfied, a priori picks up the « best »

K

o

(denoted

by

K

3

o

) in the sense of the criterion, and stops the process.

If the convergence criterion is not satisfied, an evolution
process based on genetic algorithms (not detailed in this
paper) is used to improve the robustness of the control law,
iteration after iteration, until the convergence criterion is
finally satisfied.

V. NUMERICAL ILLUSTRATION

The model considered for this example, inspired from [9],
is that of a satellite. The state, input and output matrices of
the system are respectively given hereafter :

A(Γ, δ) =

2
6

4

0

1

0

0

−10Γ −10δ 10Γ 10δ

0

0

0

1

Γ

δ

−Γ

−δ

3
7

5 ; B =

2
6

4

0

0

0

1

3
7

5 ;

C =

»

−2

−5 −7

−2

−2, 5 3, 3 8, 5 −3, 3

–

.

Then, the considered system belongs to the class

S

−

, i.e.

Kimura’s condition is not satisfied (

m + p < n).

The torque is affected by a polytopic uncertainty such that
Γ = Γ

0

+ ∆

Γ

where the nominal term is

Γ

0

= 0, 35 and

the uncertain term is such that

|∆

Γ

| ≤ 0, 03. In the same

way, the viscous damping

δ is also affected by such an

uncertainty and it is assumed to read

δ = δ

0

+ ∆

δ

with

δ

0

= 0, 028 and |∆

δ

| ≤ 0, 001.

The uncertain open-loop state matrix

A can then be written :

A = A(Γ

0

, δ

0

) +

2
6

4

0

1

0

0

−10∆

Γ

−10∆

δ

10∆

Γ

10∆

δ

0

0

0

1

∆

Γ

∆

δ

−∆

Γ

−∆

δ

3
7

5 =

A(Γ

0

, δ

0

)+

2
6

4

0

0

0

0

−10

0

10

0

0

0

0

0

1

0

−1

0

3
7

5 ∆

Γ

+

2
6

4

0

0

0

0

0

−10

0

10

0

0

0

0

0

1

0

−1

3
7

5 ∆

δ

.

and the describes polytope with four vertices.

The clustering region

D

U

considered in this example results

from the union of three subregions :

D

R

1

,

D

R

2

and

D

R

3

.

D

R

1

and

D

R

2

are two discs of radius

1, centered around

−2 ± 2i whereas D

R

3

is a vertical half-plane defined by

x < −4. The three subregions are then disjoint and D

U

is

symmetric. The optimization problem to be solved is :

max

P

i

,F,G,H

ρ

under the

LMI constraint (24).

The evolution process based on the genetic algorithms

873

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.

enables to improve the robustness of the closed-loop system
and eventually leads to

K

3

o

=

74, 2 −20, 9

On figure 1, the roots of the uncertain closed-loop state
matrix (

A

o

) are plotted for many random values of

∆

K

oinit

and

∆

K

o

such that

||∆

K

oinit

||

2

≤ ρ

(on the left) and

||∆

K

o

||

2

≤ ρ

(on the right). This figure shows the

improvement of the robustness during the optimization
process. This improvement is possible owing to the genetic
algorithms. The conservatism induced by this approach
appears « relatively » weak according to this figure (on the
right-hand side) and this, in spite of the relative complexity
of the model.

−8

−7

−6

−5

−4

−3

−2

−1

0

1

−4

−3

−2

−1

0

1

2

3

−7

−6

−5

−4

−3

−2

−1

0

−3

−2

−1

0

1

2

3

Fig. 1.

Spectrum of

A

o

VI. CONCLUSIONS

In this paper, a pole placement technique by static output
feedback is proposed. It enables to place the poles of a
reachable and observable LTI system, in a region of the
complex plane reading on

EEMI formulation. It enables

the system to reach some transient performances level.
This technique uses recent results linking the state feedback
and the output feedback. It allows to circumvent well known
difficulties concerning the pole placement by static output
feedback (

BMI) by using heuristic resolution based on

genetic algorithms.
When the system is affected by uncertainties belonging to
some domain of which we do not wish to fix the size a
priori, the technique presented here, also allows a pole
placement while maximizing the uncertainty domain. The
considered uncertainties are polytopic and norm-bounded.
It is thus possible to reduce the conservatism of the crite-
rion by considering PDLM. Owing to this, the technique
presented in this paper enables to compute a robust

D

U

-

stabilizing static output feedback.
In a very traditional way, this technique could be easily
extended to the dynamic output feedback. It would also be
interesting to improve the resolution process by using more
efficient algorithms so as to obtain even more robust control
laws.

R

EFERENCES

[1] D. Arzelier, D. Peaucelle, and S. Salhi.

Robust static output

feedback stabilization for polytopic uncertain systems : improving
the guaranteed performance bound. In Proceedings of ROCOND,
Milan-Italy, Juin 2003.

[2] O. Bachelier and D. Mehdi. Robust

D

U

-stability analysis. Inter-

national Journal of Robust and Non-linear Control, 13(6) :533–558,
2003.

[3] J. Bernussou, J. C. Geromel, and P. L. D. Peres.

A linear pro-

gramming oriented procedure for quadratic stabilization of uncertain
systems. Systems and Control Letters, 13 :65–72, Juillet 1989.

[4] J. Bosche, O. Bachelier, and D. Mehdi. Robustness bounds for matrix

D

u

-stability. Europeen Control Conference, Septembre 2003.

[5] C. Champetier and J. F. Magni. On eigenstructure assignment by

gain output feedback. SIAM Journal of Control and Optimisation,
29(4) :848–865, 1991.

[6] M. Chilali and P. Gahinet.

H

∞

Design with Pole Placement

Constraints : An LMI Approach. IEEE Transactions on Automatic
Control, 41(3) :358–367, mars 1996.

[7] M. Fahmy and J. O’Reilly. Parametric eigenstructure assignment

by output feedback control : the case of multiple eigenvalues.
International Journal of Control, 48(4) :1519–1535, 1988.

[8] L. R. Fletcher and J. F. Magni. Exact pole assignment by output

feedback part 1. International Journal of Control, 45(6) :1995–2007,
1987.

[9] G. J. Franklin, J. D. Powell, and A. Emami-Naeini. Feedback Control

of Dynamic Systems. Massachusetts ; Addison Wesley, 1986.

[10] J. C. Geromel, M. C. de Oliveira, and L. Hsu. LMI characterization

of structural and robust stability. Linear Algebra and its Applications,
285 :69–80, 1998.

[11] H. Kimura.

Pole assignment by gain output feedback.

IEEE

Transactions on Automatic Control, pages 509–516, 1975.

[12] G. P. Liu and R. J. Patton. Robust control design based on parametric

eigenvalue assignment. International Journal of Systems Science,
29(1) :65–74, 1998.

[13] D. Mehdi, E. Boukas, and O. Bachelier.

Static output feedback

design for uncertain linear discrete time systems. IMA Journal of
Mathematical Control and Information, 21(1) :1–13, 2004.

[14] B. C. Moore.

On the flexibility offered by state feedback in

multivariable systems beyond closed loop eigenvalue assignment.
IEEE Transactions on Automatic Control, 21 :689–692, 1976.

[15] N. K. Nichols.

Robustness in partial pole assignment.

IEEE

Transactions on Automatic Control, AC-32(8) :727–732, Aout 1987.

[16] D. Peaucelle and D. Arzelier. An effi cient numerical solution for

H

2

static output feedback synthesis. Europeen Control Conference,

Septembre 2001.

[17] D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou. A new

robust D-stability condition for real convex polytopic uncertainty.
Systems and Control Letters, 40(1) :21–30, mai 2000.

[18] I. R. Petersen. A stabilization algorithm for a class of uncertain linear

systems. Systems and Control Letters, 8 :351–357, 1987.

[19] G. Roppenecker and J. O’Reilly. Parametric ouptut feedback control-

ler design. Automatica, 25(2) :259–265, 1989.

[20] J. Rosenthal, J.M. Schumacher, and J.C. Willems. Generic eigenvalue

assignment by memoryless real output feedback. Systems and Control
Letters, 26 :253–260, 1995.

[21] R. E. Skelton, T. Iwasaki, and K. Grigoriadis. A unified approach

to linear control design. Taylor and Francis series in Systems and
Control, 1997.

[22] V.L. Syrmos and F.L. Lewis. Output feedback eigenstructure assign-

ment using two sylvester equations. IEEE Transactions on Automatic
Control, 38(3) :495–499, Mars 1993.

[23] E. J. Davison S. H. Wang. On pole assignment in linear multivariable

systems using output feedback. IEEE Transactions on Automatic
Control, pages 515–518, Aout 1975.

[24] X.A. Wang. Grassmannian, central projection, and output feedback

pole assignment of linear systems. IEEE Transactions on Automatic
Control, 41(6) :786–794, 1996.

[25] L. Xie. Output feedback

H

∞

control of systems with parameter

uncertainties.

International Journal of Control, 63(4) :741–750,

1996.

874

Authorized licensed use limited to: Biblioteka Glowna i OINT. Downloaded on May 20, 2009 at 05:48 from IEEE Xplore. Restrictions apply.