robust pole placement

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43rd IEEE Conference on Decision and Control
December 14-17, 2004
Atlantis, Paradise Island, Bahamas
TuB12.3
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
-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.
EEMI
. Matrices
A
,
B
,
C
,
J
and
L
belong
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
.
E
is unknown.
the Hermitian expression
M + M
.The
Kronecker product is denoted by
H
(M )
||M ||
2
is the 2-norm
of matrix
M
induced by the Euclidean vector-norm,
i.e.
the
maximal singular value of
M
.I
n
⊗
.
is the identity matrix
is a null matrix of suitable dimension.
Matrix inequalities are considered in the sense of Löewner
i.e.
“
O
≺ 0
”(“
0
”) means negative (semi-)definite and
”) 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 (
0
”(“
0
•
)’ prevents us
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
aregion
from repeating the symmetric blocks. At last,
i
denotes the
imaginary unit.
D
of the complex plane. This property, known
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.
-stability, is an essential concept in this article. The
regions considered in this work are referred to as
D
EEMI
-
regions [2] (
E
xtended
E
llipsoidal
M
atrix
I
nequality)
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
A. The uncertain matrix
Consider a complex uncertain matrix
A
∈
C
n×n
defined
by :
A
= A +
E
with
E
= J EL .
(1)
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
Matrices
A
and
E
are both uncertain.
E
is an additve
uncertainty in which
E ∈
C
q×r
is unknown.
E
is assumed
to belong to
B(ρ)
, the ball of
(q × r)
complex matrices
D
U
-stabilizing the closed-loop system by a
static output feedback.
checking
||E||
2
≤ ρ
. Define matrix
M
by :
AJ
L O
A(θ) J(θ)
L(θ) O
.
LMI
(
L
inear
M
atrix
I
nequality)
M
=
=
(2)
D
U
-stabilization exist [4] and can
involve
P
arameter-
D
ependent
L
yapunov
M
atrix (
PDLM
)
[10], [17].
The paper is organized as follows : after this introduction,
Matrix
M
is assumed to belong to a polytope of matrices :
M =
˘
M = M(θ) ∈
C
(n+r)×(n+q)
|
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
,F e
Jerome.Bosche@esip.univ-poitiers.fr
X
N
)
M(θ)=
(θ
i
M
i
); θ ∈ Θ
(3)
i=1
where
Θ
is the set of all barycentric coordinates :
0-7803-8682-5/04/$20.00 ©2004 IEEE
869
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the second section recalls formulations of
to polytopes of matrices whereas
Notations :
We denote by
M
, the transpose conjugate
of
M
,by
of order
n
,
”
as
and are generically denoted by
conditions for robust
⎧
⎨
⎫
⎬
only if
there exists a set
P
of
k
HPD matrices
P
k
∈
C
n×n
,
⎡
⎣
⎤
⎦
∈{
k =1, ..., k
, such that :
U
(A, P
k
)=
II
θ
.
θ
N
N
II
d
⊗ A
U(P
k
)
Θ=
θ
=
IR
+
}
N
|
θ
i
=1
.
(4)
II
dn
≺
(8)
⎩
⎭
dn
II
⊗ A
i=1
d
Extreme matrices
M
i
,
i =1, ..., N
are the vertices of
M
:
with :
k
A
i
J
i
L
i
.
M
i
=
(5)
U(P
k
)=
(R
k
⊗ P
k
) .
(9)
O
k=1
Proof
: see [2].
.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
A
can read
A
= A(θ)+J(θ)EL(θ)
is only subject to
a classical norm-bounded uncertainty [18].
M
)then
A
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
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.
E
-regions
Definition 1:
[2] Let
EEMI
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
∈
C
2d×2d
(6)
⎩
D. The precise purpose
In this paper, we consider the closed-loop state matrix of
a multivariable system that can be written.
R
k
11
0,R
k
11
∈
C
d×d
∀k ∈{1, ..., k} .
The set of points
D
U
definedby:
<
2
4
ω
.
ω
k
3
5 ∈{
IR
+
}
k
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
D
U
=
:
z ∈
C
|∃ω =
| f
D
U
(ω, z)
X
k
(ω
k
ˆ
II
d
z
II
d
˜
R
k
»
II
d
z
II
d
–
=
;
=
) ≺ 0
(7)
k=1
∆
K
o
an uncertain one. The uncertain closed-loop state matrix is
then
is a nominal term and
is called an open
EEMI
-region of degree
d
.
When
k
=1
, this description reduces to
EMI
formu-
lation (
E
llipsoidal
M
atrix
I
nequality) proposed in [17].
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
-region is possibly nonconnected and can result
from the union of possibly complex and disjoint
EEMI
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
EMI
D
U
defined
as in (7).
K
o
is derived with calculating a bound on
||E||
2
, denoted by
ρ
, as large as possible, such that
A
o
in a
EEMI
-region
-subregions can be nonsymmetric (with respect to the
real axis).
Some necessary and sufficient conditions (NSC) for
D
U
-
stability of is guaranteed, for any radius
ρ ≤ ρ
and for
any
θ ∈ Θ
D
U
-
stability of real or complex matrices exist [2]. Such a NSC
is now recalled.
. 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.
D
U
-stability of a complex matrix
A reformulation of the conditions suggested in [2] is now
proposed :
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
Theorem 1:
[2] Let
D
U
be an
EEMI
-region as introduced
D
U
-stabilization condition
in the nominal case is presented.
870
in definition 1. A matrix
A ∈
C
n×n
is
D
U
-stable
if and
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Therefore matrix
and thus
B.
R
k
11
An
subregions. Such
C. Nominal
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.
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.
-region defined
as in (7). Let also
K
s
be a matrix associated with a static
state feedback such that
A
s
D
U
,bean
EEMI
=
A + BK
s
is
D
U
-stable.
The
LMI
system (13) is satisfied
if and only if
there
,madeupby
k
HPD matrices
P
k
∈
exist a set
P
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 :
Consider a multivariable realization, reachable and obser-
vable, with
n
states,
m
imputs and
p
outputs,
if
the condition
»
II
dn
II
d
⊗ A
s
O
–
»
–
II
dn
O
II
d
⊗ A
s
II
d
⊗ B
H
U
=
II
d
⊗ B
U(P
k
)
+
m + p>n
(12)
j»
O
–
(
II
d
⊗ G)
ˆ
−
II
dm
˜
ff
H
II
d
⊗ K
s
−
+
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
II
dm
j»
–
(
II
d
⊗ H)
ˆ
II
d
⊗ C O
˜
ff
O
II
dm
H
≺ 0
(15)
with
U(P
k
)
defined in (9).
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
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 :
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
Ψ
k
+
O
II
d
⊗ S
G
≺ 0
(17)
(•)
−
II
d
⊗ (G + G
)
with
2
4
3
5
EEMI
-region (which can
X
U
(A
s
,P
k
)
(R
k
10
⊗ P
k
B + R
k
11
⊗ A
s
P
k
B)
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
).
Ψ
k
=
k=1
X
(•)
(R
k
11
⊗ B
P
k
B)
k=1
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
and
S = K
o
C − K
s
.
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
N
v
=
II
dn
II
d
⊗ S
(18)
respectively the left and the right orthogonal comple-
ments of
O
and
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
U =
O
and
V
=
II
d
⊗ S
−
II
dm
.
(19)
II
dm
∀ k ∈{1, .., k}
LMI
system (13)
871
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Theorem 2:
Let a
LTI
system be modelled by the triple
of matrices (
A; B; C
)et
k
also be used for systems of class
k
B. Nominal
Define matrices
N
u
=
II
dn
LMI
First, notice that
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 :
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
)
N
v
Ψ
k
N
v
=
U
(A
o
,P
k
) .
(21)
It becomes clear that, in virtue of the elimination lemma
[21],
Theorem 3:
The
LMI
system (22) is satisfied
if
there
and
exist
N
sets
P
i
, each one made up by
k
HPD matrices
P
k,i
∈
C
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
LMI
system (13) is equivalent to condition (15).
∀ i ∈{1, ..., N}
for a given state feedback associated with the matrix
K
s
D
U
-stabilizing the pair
LMI
(24) is satisfied
.
It is the idea suggested in [1] and [16]. Although this
constraint is conservative, it allows, by checking the same
property of
P
(A(θ),B(θ))
.
2
4
U(P
i
)
O
II
d
⊗ L
i
O
3
5
D
U
-stability for
A
s
and
A
o
, to circumvent a
H
U
i
=
O
O O
O −
O
II
problem. Indeed, if the state feedback
K
s
which
D
U
-stabilizes the system is not initially calculated, the
condition (15) corresponds to a
dq
II
d
⊗ L
i
O
O
−γ
II
dr
<
:
2
4
3
5
ˆ
II
d
⊗ A
s
i
=
;
.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
BMI
+
H
F
O
O
−
II
dn
II
d
⊗ B
i
II
d
⊗ J
i
O
˜
<
:
2
4
O
O
3
5
d
⊗ G
ˆ
−
II
O O
˜
=
;
+
H
II
dm
O
O
II
d
⊗ K
s
O −
II
dm
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.
+
H
<
:
2
4
O
O
II
dm
O
O
3
5
II
d
⊗ H
ˆ
II
d
⊗ C
i
O O O O
˜
=
;
≺ 0 .
(24)
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
with
k
U(P
i
)=
(R
k
⊗ P
k
i
)
et
γ = ρ
−2
.
(25)
k=1
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.
An output feedback matrix is then given by :
K
o
= G
−1
H
(26)
Proof :
For any vector of barycentric coordinates
θ ∈ Θ
,
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
let us consider the convex combination
N
H
U
=
(θ
i
H
U
i
) .
B(ρ)
,
i.e.
which leads to the
i=1
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
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
.
Let us define the matrix
LMI
S = K
o
C−K
s
, the condition (27)
becomes with (26) :
system :
H
U
=
U
(
A
s
,P
k
(θ)) ≺ 0
U
(
A
o
,P
k
(θ)) ≺ 0
2
4
3
5
∀θ ∈ Θ .
(22)
U(P(θ))
II
d
⊗ S
G
O
O O
II
d
⊗ L
O
(•)
−
II
dq
⊗ (G + G
) O
O
O
with
−
II
dq
(•)
(•)
−γ
II
dr
U
(
A
•
,P
k
(θ)) =
<
:
2
4
3
5
ˆ
II
d
⊗ A
s
=
;
≺ 0 .
(28)
ˆ
II
dn
˜
k
X
»
II
dn
II
d
⊗
A
•
–
+
H
F
O
O
−
II
dn
II
d
⊗ B
II
d
⊗ J O
˜
II
d
⊗
A
•
(R
k
⊗ P
k
(θ))
≺ 0
(23)
k=1
872
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A
s
and
such that the
Here, it is important to emphasize that matrices
A
s
and
A
o
« share » the same set of Lyapunov matrices
BMI
A. Robust
L(θ)=L
Since
γ 0
, applying Schur’s lemma to (28) gives :
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
H
U
=
2
4
U(P(θ))
II
d
⊗ S
G
O
3
5
(•)
−
II
d
⊗ (G + G
)
+
H
˘
F
ˆ
II
d
⊗ A
s
−
II
dn
II
d
⊗ B
˜¯
).
Te s t :
Then we can define a convergence criterion which,
if it is satisfied,
apriori
picks up the « best »
K
o
(denoted
by
K
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.
(24), the robustness bound is imposed zero (
ρ
=0
+
»
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 :
2
4
U(P(θ))
3
5
H
U
=
II
d
⊗ S
G
O
(•)
−
II
d
⊗ (G + G
)
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 :
+
H
˘
F
ˆ
II
d
⊗ (A
s
+ Jρ EL) −
II
dn
II
d
⊗ B
˜¯
≺ 0 .
(30)
2
4
0 1 0 0
−10Γ −10δ 10Γ 10δ
0
3
5
; B =
2
4
0
0
0
1
3
5
;
Let consider
E = ρ E
,itcomes:
⎡
⎣
U(P(θ))
⎤
A(Γ,δ)=
0
0
1
II
d
⊗ S
G
O
Γ
δ −Γ −δ
H
U
=
⎦
»
–
−2 −5 −7 −2
−2, 5 , 3 , 5 −3, 3
(•)
−
II
d
⊗ (G + G
)
C =
.
+
H
F
II
d
⊗
A
s
II
dn
II
d
⊗ B
≺ 0 .
(31)
−
Then, the considered system belongs to the class
S
−
,i.e.
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)
.Itthen
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
.Inthesame
way, the viscous damping
δ
is also affected by such an
uncertainty and it is assumed to read
δ
|∆
Γ
|≤0, 03
leads to
LMI
system (22).
=
δ
0
+∆
δ
with
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 » .
δ
0
=0, 028
and
|∆
δ
|≤0, 001
.
The uncertain open-loop state matrix
A
can then be written :
2
4
3
5
=
0
1
0
0
−10∆
Γ
−10∆
δ
10∆
Γ
10∆
δ
A
= A(Γ
0
,δ
0
)+
0
0
0
1
∆
Γ
∆
δ
−∆
Γ
−∆
δ
A(Γ
0
,δ
0
)+
2
4
0 000
−10 0 10 0
0 000
10−10
3
5 ∆
Γ
+
2
4
0000
0 −10 0 10
0000
010−1
3
5 ∆
δ
.
and the describes polytope with four vertices.
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 ± 2
i
whereas
D
R
3
is a vertical half-plane defined by
The resolution method that we propose is as follows :
x<−4
. The three subregions are then disjoint and
D
U
is
symmetric. The optimization problem to be solved is :
max
Step 1 :
Several state feedbacks
K
s
, each one that
D
U
-
P
i
,F,G,H
ρ
), are computed by using
a beforehand specified eigenstructure assignment technique
and exploiting the
dof
on the eigenvalues and eigenvectors.
,
B(θ)
constraint (24).
The evolution process based on the genetic algorithms
873
under the
LMI
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the uncertain term is such that
The clustering region
stabilizes the pair (
A(θ)
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