Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. IV (2009), No. 4, pp. 348-348

An LMI Technique for the Global Stabilization of Nonlinear Polynomial
Systems

M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

Mohamed Moez Belhaouane, Riadh Mtar, Hela Belkhiria Ayadi, Naceur Benhadj Braiek
Laboratoire d’Etude et Commande Automatique de Processus - LECAP
Ecole Polytechnique de Tunisie (EPT), BP.743, 2078 La Marsa, Tunis, Tunisie.
E-mail: {moez.belhaouane, riadh.mtar, hela.ayadi, naceur.benhadj}@ept.rnu.tn

Abstract: This paper deals with the global asymptotic stabilization of nonlinear
polynomial systems within the framework of Linear Matrix Inequalities (LMIs). By
employing the well-known Lyapunov stability direct method and the Kronecker prod-
uct properties, we develop a technique of designing a state feedback control law
which stabilizes quadratically the studied systems. Our main goal is to derive suffi-
cient LMI stabilization conditions which resolution yields a stabilizing control law
of polynomial systems.

Keywords: Nonlinear Polynomial systems, Lyapunov method, Global stabilization,
Kronecker product, LMI approach.

1 Introduction

The control theoretician role may be viewed as one of developing methods that allows the control
engineer to make which seems relatively natural and physically motivated [1, 2]. Generally the main
and first object in the control theory is to ensure the stability and the convergence of the considered
system. In this context, the problem of stabilization of nonlinear systems has received a great deal of
attention and several methods have been proposed in the literature [3, 4, 5, 6]. However, the proposed
approaches remain restrictive to particular classes of nonlinear models, and there is no general method
for the analysis or synthesis of general nonlinear systems. That is the reason of continuing research on
study and control of nonlinear systems. The polynomial systems constitute an important class of non-
linear systems which has the advantage to describe the dynamical behavior of a large set of processes as
electrical machines and robot manipulators and has also the ability to approach any analytical nonlinear
system, since any analytical nonlinear function can be approximated by a polynomial expansion. Let’us
note that a lot of works have considered the modeling, analysis and control of the polynomial systems
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The main key of these developments is the description of the polyno-
mial system by using the Kronecker power of the state vector [17].

In the other hand, Linear Matrix Inequalities (LMIs) have emerged as a powerful formulation and
design technique for a variety of control problems [18, 19, 20]. Since solving LMI’s is a convex opti-
mization problem, such formulation offer a numerically tractable means of attacked problems that lack
an analytical solution. Besides, efficient interior-point algorithms are now available to solve the generic
LMI problems. They are applied to several important process control applications including control
structure selection, robust controller analysis and design, and optimal design of experiments [21, 22, 23].
Consequently, reducing a control design problem to an LMI can be considered as a practical solution [24].

The contribution of the present paper consists on the use of the Lyapunov method with a quadratic
candidate function, to derive a practice sufficient condition ensuring the global asymptotic stabilization
of the original equilibrium of a polynomial system. This condition is then reformulated in the form of an

Copyright c© 2006-2009 by CCC Publications



336 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

LMI feasibility problem which can be solved using the numerical software as MATLAB.

This paper is organized as follows: In section 2 the description of the studied systems and necessary
mathematical notations are introduced. Then, in the next section, the problem of stabilizing control
law synthesis of polynomial systems is investigated. The section 4 proposes an LMI formulation of
the obtained stabilization condition. An illustrative example is reported in section 5 to implement the
developed approach.

2 Studied polynomial systems and Mathematical Notations

2.1 Studied polynomial systems

The studied nonlinear polynomial systems are described by the following state equation:

Ẋ = f (X ) + GU, (1)

where f (X ) is a vectorial polynomial function of X .

f (X ) =
r∑

i=
FiX [i] =

r∑
i=

F̃iX̃ [i], (2)

with
X = [x, . . . , xn]T ∈Rn,

{
X [] = 
X [i] = X [i−] ⊗X = X ⊗X [i−] f or i ≥ , (3)

⊗
is the symbol of the Kronecker product [17].

X̃ [i]i=,...,r ∈ Rni where ni =
(

n + i − 
i

)
, is the nun-redundant Kronecker power of the state vector X

defined as

X̃ [] = X [] = X ,

∀ i ≥ , X̃ [i] = [xi, xi− x, ..., xi− xn, ..., xi− xn, ..., xi− x, ..., xin]T ,
(4)

i.e., the components of X̃ [i] are the same that those of X [i] with omission of the repeated terms.
Fi,i=,...,r ∈Rn×n

i
(resp. F̃i ∈Rn×ni ) are constant matrices.

The polynomial order r is considered odd: r = s − , with s ∈N∗.
U ∈Rm is the input vector and G is a constant (n×m) matrix.

2.2 Notations

In this section, we introduce some useful notations and needed rules and functions. Let the matrices
and vectors of the following dimensions

A(p×q), B(r ×s), C(q× f ), X (n×) ∈Rn, Y (m×) ∈Rm.
(i) We consider the following notations: In : (n×n) identity matrix; n×m : (n×m) zero matrix;

: zero matrix of convenient dimension; AT : transpose of matrix A; A > (A ≥ ) : symmetric
positive definite (semi-definite) matrix; eqk : q dimensional unit vector which has 1 in the k

th

element and zero elsewhere.



An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems 337

(ii) The relation between the redundant and the nun-redundant Kronecker power of the state vector X
can be stated as follows {

∀i ∈N ∃ ! Ti ∈ Rn
i×ni ,

X [i] = TiX̃ [i],
(5)

A procedure of the determination of the matrix Ti is given in [25].

(iii) The permutation matrix denoted Un×m is defined as

Un×m =
n∑

i=

m∑

k=

(
eni ·(emk )T

)
⊗

(
emk ·(eni )T

)
. (6)

This matrix is square (nm × nm) and has precisely a single 1 in each row and in each column.
Among the main properties of this matrix presented in [17], [11], we recall the following useful
ones

(B⊗A) = Ur×p(A⊗B)Uq×s, (7)
(X ⊗Y ) = Un×m(Y ⊗X ), (8)
∀i ≤ k X [k] = Uni×nk−i X [k]. (9)

(iv) An important vector valued function of matrix denoted vec(.) was defined in [17] as follows

A =
[

c c ... cq
]
∈Rp×q,

where
∀i ∈ {, ..., q} , ci ∈Rp are the columns of A

vec(A) =
[

cT c
T
 ... c

T
q

]T ∈Rpq.
We recall the following useful rules [17] of this function

vec(BAC) = (CT ⊗B)vec(A), (10)

vec(AT ) = Up×qvec(A). (11)

(v) A special function mat(n,m) (.) can be defined as follows
if V is a vector of dimension p = n.m then M = mat(n,m)(V ) is the (n × m) matrix verifying
V = vec(M).

(vi) For a polynomial vectorial function

a (X ) =
r∑

i=

AiX
[i], (12)

where X ∈Rn and Ai are (n×ni) constant matrices, we define the (υ ×υ) matrix M(a) as

M(a) =




M(A) M(A)  . . . 

 M(A)
. . .

. . .
...

...
. . .

. . .
. . . 

...
. . . Ms−,s−(As−) Ms−,s(As−)

 . . . . . .  Ms,s(As−)




, (13)



338 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

with υ = n + n + ... + ns.

• For j = , ..., s

M j, j(A j−) =




mat(n j−,n j )
(

AT j−
)

mat(n j−,n j )
(

AT j−
)

...

mat(n j−,n j )
(

AnT j−
)




, (14)

• For j = , ..., s − 

M j, j+(A j) =




mat(n j−,n j )
(

AT j
)

mat(n j−,n j )
(

AT j
)

...

mat(n j−,n j )
(

AnT j
)




, (15)

where Aik is the i
th row of the matrix Ak

Ak =
[

ATk A
T
k ... A

nT
k

]T
. (16)

(vii) We introduce the matrix R defined by

R = τ +[] ·U·H ·τ, (17)

where

τ = Diag
(
Ti,i=,...,s

)
, (18)

with τ + is the Moore-Penrose pseudo-inverse of τ and τ
+[]
 = τ

+
 ⊗τ + .

τ = Diag
(
Tj, j=,...,s

)
, (19)

U = Diag
(
Uni,i=,...,s×η

)
, (20)

H =




Iη 
η×η Iη
η×(η+η) Iη
...

. . .
ηs×(η+η+...+ηs−) Iηs




, (21)

for j = , ....., s : η j = n j ·
(

s∑
i=

ni
)

.

We note Γ the matrix defined by

Γ = (Iη  + Uη×η )
(
R+T RT − Iη 

)
, (22)



An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems 339

with η =
s∑

j=
n j =

s∑
j=

(
n + j − 

j

)
and R+ is the Moore-Penrose pseudo-inverse of R.

β = rank(Γ ) (23)

and Ci, i=,...,β are β linearly independent columns of Γ .

(iix) For a (n×l) matrix φ , we define Ds(φ ) the (υ ×υ ) matrix defined as

Ds(φ ) =




φ 
φ ⊗In

. . .
 φ ⊗Ins−


 . (24)

In the case where the matrix φ is square (l = n), the matrix Ds(φ ) is also square (υ × υ), with υ
is defined in (vi). As well, if φ is square and is symmetric positive definite, then so is Ds(φ ).

3 The Proposed Global Stabilization Condition of Controlled Polynomial
Systems

We consider the polynomial nonlinear systems defined by the equation (1). Our purpose is to deter-
mine a polynomial feedback control law

U = k(X ) =
r∑

i=
KiX [i], (25)

with Ki,i=,...,r are constant gains matrices which stabilizes asymptotically and globally the equilibrium
(X = ) of the considered system.
Applying this control law to the open-loop system (1), one obtains the closed-loop system

Ẋ = a(X ) = ( f + Gk)(X ),

=
r∑

i=
AiX [i],

(26)

where
Ai = Fi + GKi. (27)

Using a quadratic Lyapunov function V (X ) and computing the derivative V̇ (X ), lead to the sufficient
condition of the global asymptotic stabilization of the polynomial system, given by the following theorem
1.

Theorem 1. The nonlinear polynomial system defined by the equation (1) is globally stabilized by the
control law (25), if there exist

• an (n×n)-symmetric positive definite matrix P;

• arbitrary parameters µi,i=,...,β ∈R;

• gain matrices Ki,i=,...,r;



340 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

such that the (η ×η) symmetric matrix Q defined by
Q = τ T [DS(P)M( f ) + M( f )

T DS(P)]τ + τ T [DS(P)GM(k) + (DS(P)GM(k))
T ]τ

+
β∑

i=
µimat(η,η)(Ci),

(28)

be negative definite.
Where β and Ci,i=,...,β are defined in (23).

Proof. Consider the quadratic Lyapunov function

V (X ) = X T PX , (29)

Differentiating V (X ) along the trajectory of the system (26), we obtains

V̇ (X ) =
r∑

k=
(X T PAkX [k] + X [k]

T
ATk PX ),

= 
r∑

k=
X T PAkX [k].

(30)

Using the rule of the vec-function (10), the relation (30) can be written as

V̇ (X ) = 
r∑

k=
V Tk X

[k+], (31)

where

Vk = vec(PAk). (32)

We can write

r=s−∑
k=

V Tk X
[k+] =

s−∑
j=

V T jX
[ j+] +

s∑
j=

V T j−X
[ j], (33)

using the mat-function defined in section 2, one has

V̇ (X ) = [
s−∑
j=

X [ j]
T

mat(n j ,n j+)(V
T
 j)X

[ j+] +
s∑

j=
X [ j]

T
mat(n j ,n j )(V

T
 j−)X

[ j]]. (34)

Applying the following lemma [11]

Lemma 2. Consider a (n×nk) matrix A (k ∈N) and a (n×n) matrix P.
Let i and j two integers verifying i + j = k +  and i ≥ . Then

mat(ni, n j )(vec(PA)) = Uni−×n(P⊗Ini− ).M,
with

M =




mat(ni−,n j )
(
AT

)
mat(ni−,n j )

(
AT

)
...

mat(ni−,n j )
(
AnT

)


 ,

where Ai denotes the ith row of the matrix A

A =
[

AT AT ... AnT
]T



An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems 341

leads to the following relations

mat(n j ,n j+)(V
T
 j) = Un j−×n(P⊗In j− )M j, j+(A j), (35)

mat(n j ,n j )(V
T
 j−) = Un j−×n(P⊗In j− )M j, j(A j−), (36)

where M j, j+(A j) and M j, j(A j−) are defined respectively in (15) and (14) and Un j−×n is mentioned in
(20).
Using the results (35) and (36), the equality (34) can be expressed as

V̇ (X ) = [
s−∑
j=

X [ j]
T
Un j−×n(P⊗In j− )M j, j+(A j)X [ j+]

+
s∑

j=
X [ j]

T
Un j−×n(P⊗In j− )M j, j(A j−)X [ j]],

(37)

by means of the relation (9), one obtains

V̇ (X ) = [
s−∑
j=

X [ j]
T
(P⊗In j− )M j, j+(A j)X [ j+] +

s∑
j=

X [ j]
T
(P⊗In j− )M j, j(A j−)X [ j]]. (38)

Consequently, we obtain

V̇ (X ) = XT DS(P)M(a)X,
= XT (DS(P)M(a) + M(a)T DS(P))X,

(39)

with

X =
[

X T X []
T ··· X [s]T

]T
(40)

DS(P) and M(a) are defined respectively in (24) and (13).
Using the nun-redundant Kronecker product power form, the vector X can be written as

X = τX̃, (41)

where X̃ =
[

X̃ T X̃ []
T ··· X̃ [s]T

]T
∈Rη , η =

s∑
j=

n j and τ is defined in (18).

Then V̇ (X ) can be written in the following form

V̇ (X ) = X̃T τ T (DS(P)M(a) + M(a)
T DS(P))τX̃, (42)

A sufficient condition of the global asymptotic stability of the equilibrium (X = ) is that the quadratic
form V̇ (X ) is negative definite. This condition can be ensured if there exists a symmetric negative definite
Q ∈Rη×η such that

X̃T τ T (DS(P)M(a) + M(a)
T DS(P))τX̃ = X̃T QX̃, (43)

using the vec-function, the equality (43) can be expressed as

vecT
(
Q − τ T (DS(P)M(a) + M(a)

T DS(P))τ
)
X̃[] = . (44)

But, it can be easily checked that X̃[] can be written as

X̃[] = RX̃, (45)



342 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

where

X̃ =
[

X̃[]T . . . X̃[s+]T X̃[s+]T . . . X̃[s]T
]T

, (46)

and R is the matrix defined in (17). The proof of the relation (45) is given in [11].
Therefore the equality (44) yields the following equation

RT vec(S) = , (47)

with: S = Q − τ T
(
DS(P)M(a) + M(a)T DS(P)

)
τ.

The η -vector vec(S) solution of (47) can be expressed as

vec(S) =
(
R+T RT − Iη 

)
Y, (48)

where Y is an arbitrary vector of Rη

.

The matrix S is symmetric since Q is symmetric, then we can write

S = 

(S + ST ), (49)

and using the property (11) yields

vec(S) = 

(Iη  + Uη×η )vec(S) =

β∑
i=

µiCi, (50)

where
• β = rank

[
(Iη  + Uη×η )

(
R+T RT − Iη 

)]
,

• Ci,i=,...,β are β linearly independent columns of
(Iη  + Uη×η )

(
R+T RT − Iη 

)
, (51)

• µi,i=,...,β are arbitrary values.
Consequently, the symmetric matrix Q verifying (47) is of the following form

Q = τ T (DS(P)M(a) + M(a)
T DS(P))τ +

β∑
i=

µimat(η,η)(Ci). (52)

According to (26) and the following lemma [10]

Lemma 3. Let G ∈ Rn×m, k(.) a polynomial vectorial function defined in (25) and G.k(.) the resultant
product of G by k(.), then one has

M(G.k) = GM(k),

where G = Ds(G) and M(.) the matrix function defined in (13).

Thus, we can write
M(a) = M( f + Gk) = M( f ) + GM(k), (53)

finally, we obtain the following quadratic form of the symmetric matrix Q

Q = τ T [DS(P)M( f ) + M( f )
T DS(P)]τ + τ T [DS(P)GM(k) + M(k)

T GT DS(P)]τ

+
β∑

i=
µimat(η,η)(Ci).

(54)

If Q is negative definite, then the derivative V̇ (X ) is negative definite.
Which ends the proof.



An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems 343

4 Stabilizing Control Synthesis using the LMI approach

In this section we show how the stabilization problem stated by the theorem 1 can be formulated as
an LMI feasibility problem.
Let recall that our main problem is to find

• gain matrices Ki,i=,...,r;
• a (n×n) matrix P;
• real parameters µi,i=,...,β ;

such that
P > , (55)

τ T [DS(P)M( f ) + M( f )
T DS(P)]τ + τ T [DS(P)GM(k) + M(k)

T GT DS(P)]τ

+
β∑

i=
µimat(η,η)(Ci) < .

(56)

Note that this problem is nonlinear with respect of the unknown parameters P,Ki and µi, since the inequal-
ity (56) is bilinear on (P, Ki). To overcome this problem we make use of the known Schur’s complement
[18] and we exploit the separation lemma [26]. In this sequel we transform the BMI problem into LMI
problem as it is shown in the following development.
Making use of the following separation lemma [26]

Lemma 4. For any matrices A and B with appropriate dimensions and for any positive scalar ε > , one
has:

AT B + BT A ≤ ε AT A + ε −BT B,
one obtains

Q ≤ τ T [DS(P)M( f ) + M( f )T DS(P)]τ +
β∑

i=
µimat(η,η)(Ci)

+γ τ T DS(P)
T DS(P)τ + γ −τ T M(k)

T GT GM(k)τ,
(57)

with γ > .
Then, to ensure that the matrix Q is negative definite, it is sufficient to have

τ T [DS(P)M( f ) + M( f )
T DS(P)]τ +

β∑
i=

µimat(η,η)(Ci)

−τ T DS(P)
T (−γ I)DS(P)τ − τ T M(k)

T GT (−γ −I)GM(k)τ < .
(58)

Using the Generalized Schur’s complement, the inequality (58) is equivalent to



τ T (DS(P)M( f ) + M( f )
T DS(P))τ +

β∑
i=

µimat(η,η)(Ci) (DS(P)τ)T (GM(k)τ)T

DS(P)τ −γ −I 
GM(k)τ  −γ I


 < , (59)

when pre-and post-multiplying the inequality (59) by Ξ = diag(I, I, γ −I), we get



τ T (DS(P)M( f ) + M( f )
T DS(P))τ +

β∑
i=

µimat(η,η)(Ci) (DS(P)τ)T (GW(k)τ)T

DS(P)τ −γ −I 
GW(k)τ  −γ −I


 < , (60)

with W(k) = γ −M(k).
This new inequality is linear on the decision variables, and then we can state the following result.



344 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

Theorem 5. The equilibrium (X = ) of the system (1) is globally asymptotically stabilizable if there
exist

• a (n×n)-symmetric positive definite matrix P ;
• arbitrary parameters µi,i=,...,β ∈R ;
• gain matrices Ki,i=,...,r;
• a real γ >  ;

such that
P > , (61)

and



τ T (DS(P)M( f ) + M( f )
T DS(P))τ +

β∑
i=

µimat(η,η)(Ci) (DS(P)τ)T (GW(k)τ)T

DS(P)τ −γ −I 
GW(k)τ  −γ −I


 < . (62)

Thus, a stabilizing control law (25) for the considered polynomial system (1) can be characterized
by applying the following procedure

1. Solve the LMI feasibility problem i.e., find the matrices DS(P), W(k) and the parameters µi and γ
such that the inequalities (61), (62) are verified.

2. Extract the gain matrices Ki from the relation M(k) = γW(k).

This optimization problem can be carried out using MATLAB software. To provide the effectiveness of
the proposed approach, we consider the following numerical example.

5 Illustrative Example

Our aim in this section is to apply the proposed approach for the global stabilization of the following
polynomial system

{
ẋ = −x + x + x + xx − x


 + xx − xx


 + x


,

ẋ = −x + .x − x − .xx − x

 − xx + .xx


 − x


 + u.

(63)

Using the Kronecker product, this system can be described by the following compact state equation

Ẋ = FX + FX [] + FX [] + GU, (64)

with

F =
[

− 
− .

]
F =

[
   

− −.  

]
F =

[
−   −    
− −  .    −

]

and G =
[




]
.

We are interested with the stabilization of the origin equilibrium (X = ) of the system (64).
Let us note that the uncontrolled (U = ) non linear system is unstable since the matrix F has an unstable
eigenvalue.



An LMI Technique for the Global Stabilization of Nonlinear Polynomial Systems 345

Solving the optimization problem formulated by theorem 2, we obtain





µ = −.
µ = .
µ = .

; P =
[

. .
. .

]
; γ = ..

The searched gain matrices, extracted from M(k), are given by

K =
[

−. −.
]

K =
[

. .  
]

K =
[

−. −. −. −. . −. . .
]

Then a global stabilizing control law can be characterized for the studied system using the previous
developed method. This control law can be expressed as

U = KX + KX [] + KX []. (65)

The Figure 1 shows the behavior of the state variables x(t) and x(t) of the controlled system from ini-
tial conditions which were taken sufficiently far from the initial conditions (x() = −, x() = ). It
appears that the state variables converge into the origin point which confirm the asymptotic stability of
the controlled system.

0 0.2 0.4 0.6 0.8 1
−10

−5

0

5

10

15

t(s)

V
ar

ia
bl

es
 x

1 
an

d 
x2

 

 
Dynamic behaviour of the state variable x1
Dynamic behaviour of the state variable x2

Figure 1: Closed-loop responses of the system (64) with the control law (65).

6 Conclusion

In this paper, an original technique has been proposed for the global and asymptotic stabilization of
the nonlinear polynomial systems. This new stabilizing approach is based on the Lyapunov direct method
and elaborated algebraic developments using the Kronecker product properties. This development has
allowed the formulation of the system stabilization condition as an LMI feasibility problem, which res-
olution leads to a polynomial control law ensuring the quadratic stability in the whole state space of the



346 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

considered system. Further works, will consider extension of these results to the robust stabilization of
polynomial uncertain systems.

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Mohamed Moez Belhaouane was born in Tunis in 1980. He received the Electrical Engineer-
ing bachelor and the Master degree in Automatic Control from École Supérieure des Sciences et
Techniques de Tunis (ESSTT), in 2003 and 2005, respectively.
Currently, he is research member of the Processes Study and Automatic Control Laboratory
(LECAP) in the Polytechnic School of Tunisia and he is working toward Ph.D. degree in Electrical
Engineering. His research interests include analysis and control of nonlinear systems, polynomial
systems and robust control.
Riadh Mtar was born in Tunis in 1974. He received his Master degree of Systems Analysis
and Numerical Treatment from École Nationale d’ingénieurs de Tunis (ENIT), in 2005. He is
currently an contractual Assistant in École Supérieure de Commerce (ESC) and research member
of the Processes Study and Automatic Control Laboratory (LECAP) in the Polytechnic School
of Tunisia. Actually, he is preparing his Ph.D. dissertation in Electrical Engineering from École
Nationale d’ingénieurs de Tunis (ENIT). His research interests include analysis and control of
nonlinear polynomial discret systems.
Hela Belkhiria Ayadi received her engineer diplomas from the École Nationale d’ingénieurs de
Monastir (ENIM) in 1997 and Master degree of Automatic Control from École supérieure des
Sciences et Techniques de Tunis (ESSTT), in 1998. She completed her Ph.D. in Electrical En-
gineering in 2004 from École Nationale d’ingénieurs de Tunis (ENIT). She is now an assistant
professor in École supérieure des Sciences et Techniques de Tunis (ESSTT) and research member
of the Processes Study and Automatic Control Laboratory (LECAP) in the Polytechnic School of
Tunisia. Her research interests include robust analysis and control of nonlinear uncertain systems.



348 M. Moez Belhaouane, R. Mtar, H. Belkhiria Ayadi, N. Benhadj Braiek

Naceur Benhadj Braiek was born in Mahdia, Tunisia, in 1963. He obtained the Master of Elec-
trical Engineers and the Master of Systems Analysis and Numerical Processing, both from École
Nationale d’ingénieurs de Tunis (ENIT) in 1987, the Master of Automatic Control from Institut
Industriel de Nord (École Centrale de Lille) in 1988, the Ph.D. degree in Automatic Control from
Université des Sciences et Techniques de Lille, France, 1990, and the Doctorat d’état in Electrical
Engineering in 1995 from École Nationale d’ingénieurs de Tunis (ENIT). Now, he is Professor of
Electrical Engineering at the University of Tunis - École supérieure des Sciences et Techniques
de Tunis (ESSTT). He is also Director of the Processes Study and Automatic Control Laboratory
(LECAP) in the Polytechnic School of Tunisia. His domain of interest is related to the modeling,
analysis and control of nonlinear systems with applications on Electrical Processes.