c 2009 TSSD

1861-5252/ Transactions on
Systems, Signals & Devices
Vol. 4, No. 1, pp.1-139

F
Stabilization Through Output Feedback
Control for Uncertain Switched Discrete
Time Systems

3
OO
E. Maherzi,1 M. Besbes,1 J. Bernussou,2 and R. Mhiri1,3
1
Réseaux et Machines Electriques (RME)
Institut Supérieur des Sciences Appliquées et de Technologie,
centre urbain Tunis-nord, B.P. 676, 1080 Tunis, Tunisia.

Laboratoire d’Analyse et d’Architecture des Systèmes (LAAS)

7 Avenue du Colonel Roch-31077 Toulouse Cedex 4, France.

Faculté des Sciences de Tunis

Campus Universitaire B.P. 1060, Tunis, Tunisia.
Abstract This paper discusses the robust stabilization of discrete switched
systems, focusing on the design of a robust static output feed-
back control and dynamic output control based on a switched
observer. The results are derived using the direct Lyapunov
approach and the polyquadratic function concept. The stabi-
lization conditions are written through linear matrix inequalities
relations. The polyquadratic Lyapunov approach provides a con-
structive way to tackle uncertainty in the switched framework.
PR
The feasibility is illustrated on an example of discrete uncertain
switched discrete time system.

Keywords: Robust control, stability and stabilization, switched discrete time

systems, output feedback control and observer control.

1. Introduction
The literature has shown a growing interest on switched systems since
switched control systems exist widely in engineering technology and so-
cial systems [7, 8, 20]. Switched systems are an important class of hybrid
systems defined by a set whose elements are dynamic continuous (or dis-
crete) time models and a commutation law which governs, in time, the
jumps between the elements, defining a non stationary dynamic system.
Many important progress and remarkable achievements have been made
2 E. Maherzi et al.

on issues such as controllability, reachability and stabilizability [9, 10],

control and switching law design [11–14], optimal control [15, 16]...
Among others, stability analysis and stabilization control are two im-
portant topics. The basic problems considered include stability analysis

F
for systems with specific switching laws [19], stability analysis for systems
with arbitrary switching laws [13] and design of stabilizing switching laws
[14]. Many contributions to analyze stability of arbitrary switched sys-
tems use conservative arguments, the most pessimistic ones assuming
the existence of a common lyapunov function [17, 18]. Even if these con-
ditions are easily tractables, they can be used in a very few applications.
Some recent results are given in Daafouz et al., 2002 [2] where a suffi-

Varying systems.
OO
cient (but relatively non restrictive compared to the quadratic approach)
stability condition for discrete switched systems is provided using the
polyquadratic approach recently proposed by Daafouz and Bernussou
2001 [1] for stability analysis and stabilization control of Linear Time

An extension of this works was presented in Millerioux and Daafouz

2004 dealing with unknown input observers in the case of switched linear
discrete time systems. A sufficient conditions of global convergence of
such kind of observers along with a systematic procedure to design the
gains of the observers is proposed. A discussion about the existence of
such observers is provided [21].
However this paper proposes an other extension of this works in the
case when the switches are made among polytopic uncertain discrete
time systems. The control investigated is of output feedback control
type: observer based which, of course, is a more realistic framework
than the state feedback control proposed which have been preliminary
PR
worked in Maherzi et al., 2006 [4].
In part two of this paper we start by formulate problem of uncertain
discrete time switched systems. This formulation will allows us to dis-
cuss stability and stabilization for this kind of systems in part three.
The fourth part puts emphasis on static output feedback control using
the notions of polyquadratic stabilization and local polyquadratic stabi-
lization. In fifth part interest is putting on the design and construction
of Luenberger’s observer. The synthesis of command law in the case of
closed loop is developed in part six. Finally an illustrative example is
done in part seven.
Stabilization of uncertain switched discrete time systems 3

2. Problem formulation
We consider an uncertain discrete time switched system where the so-
called subsystems are uncertain with a polytopic uncertainty, as roughly
illustrated in (Fig.1) where the uncertainty domains are polytopic shaped

F
with different number of vertices to cope with maximal generality. The
model can be stated as follows:

M
X Nl
X
x(k + 1) = ξl (k) αi(l) Ai(l) x(k) (1)
l=1 i(l)=1

Dl =


OO
where l is the switching index. M the number of uncertain systems
domain and Dl is the uncertainty domain for subsystem l defined by:


Aα Aα =
Nl
X

i(l)=1
αi(l) Ai(l) , αi(l) ≥ 0,

Nl is the number of the vertices of the polyhedron Dl .

Nl
X

i(l)=1
αi(l) (k) = 1





1 if the state matrix is defined into Dl domain
ξl (k) = 
0 else

A11 A12 A21 A22

PR
A1N1 SS1 A2N2 SS2

AM 1 AM 2

AM NM SSM

Fig. 1. Uncertain switched discrete time system.

4 E. Maherzi et al.

3. Uncertain switched discrete time systems

stability
Starting from the results of [2, 5], the analysis of the polyquadratic
stability of the uncertain switched discrete time systems is developed in

F
[4] to propose the following criteria.

3.1 Polyquadratic stability

Theorem 1 The system described by (1) is polyquadratically stable if
there exist H symmetric positive definite matrices S11 ...SMN and M ma-

H=
XM

l=1
OO
trices G1 ...GM of appropriates dimensions solutions of the LM Is:

Gl + GTl − Si(l)l GTl ATi(l)l
Ai(l)l Gl Si(j)j

> 0, ∀(l, i(l), i(j), j) ∈ (e × el × ej × J)

Nl , e = {1...M }, el = {1...Nl }, ej = {1...Nj }

Proof: To prove the precedent condition we have to prove that if this

(2)

condition (2) is verified then it is verified for any jumps realization in

the uncertainty domains. We suppose two dynamical matrices Al and
Aj defined respectively in the domain Dl and Dj .
Let
XNl Nl
X
Al = αi(l) Ai(l) , αi(l) = 1
i(l)=1 i(l)=1
PR
and
Nj
X Nj
X
Aj = αi(j) Ai(j) , αi(j) = 1
i(j)=1 i(j)=1

from(2) one gets

Nl
X  
Gl + GTl − Si(l)l GTl ATi(l)l
αi(l) >0
Ai(l)l Gl Si(j)j
i(l)=1
" PN l PN l #
Gl + GTl − i(l)=1 α i(l) S i(l)l GT
l i(l)=1 α i(l) AT
i(l)l
PN l P Nl >0
i(l)=1 αi(l) Ai(l)l Gl i(l)=1 αi(l) Si(j)j
 P l 
Gl + GTl − N T T
i(l)=1 αi(l) Si(l)l Gl Al >0
Al Gl Si(j)j
Stabilization of uncertain switched discrete time systems 5
PN l
replacing i(l)=1 αi(l) Si(l)l by Sl > 0 we obtain:

 
Gl + GTl − Si(l)l GTl ATi(l)l
>0
Ai(l)l Gl Si(j)j

F
again
Nj
X  
Gl + GTl − Sl GTl ATl
αi(j) >0
Al Gl Si(j)j
i(j)=1

replacing
PN j
"

OO
Gl + GTl − Sl
Al Gl

i(j)=1 αi(j) Si(j)j

This concludes the proof.


GT AT
PN j l l
i(j)=1 αi(j) Si(j)j

by Sj we obtain:

Gl + GTl − Sl
Al Gl
GTl ATl
Sj

>0
#
>0

3.2 Local polyquadratic stability

The previous condition (Polyquadratic stability) is global; it asso-
ciates a Lyapunov function to each vertices of all sub-systems and may
be very heavy, computationally speaking, in the case of a large H num-
PR
ber. Applying the quadratic concept where a single Lyapunov function
is used for each of the uncertain sub-systems, a say local polyquadratic
stability criterion (more restrictive than the previous one but easier com-
putationally speaking) can be described by:

Theorem 2 The system (1) is locally polyquadratically stable if and

only if it exist M symmetrical positive definite matrices S1 ...SM and M
matrices G1 ...GM of appropriates dimensions solutions of the LM Is.
 
Gl + GTl − Sl GTl ATi(l)l
>0
Ai(l)l Gl Sj
∀(l, i(l), j), ∈ (e × el × J) (3)
e = {1...M }, el = {1...Nl }
J function of l.
6 E. Maherzi et al.

4. Static output feedback control

Consider the uncertain switched discrete time system described by:
PM
x(k + 1) = l=1 ξl (k)(Al x(k) + Bl u(k))
PM (4)

F
y(k) = l=1 ξl (k)(Cl x(k)).
where [Al , Bl , Cl ] ∈ Dl , with:
 
X Nl Nl
X 
Dl = αi(l) [Ai(l) , Bi(l) , Ci(l) ], αi(l) ≥ 0, αi(l) (k) = 1
 
i(l)=1 i(l)=1

OO
The stabilization problem of the switched system through static out-
put feedback consists in determining a control law of the form:
u(k) = Kl y(k)
x(k + 1) = (Alα + Blα Kl Clα )x(k)
x(k + 1) = Aelα x(k) such A
elα = (Alα + Blα Kl Clα )

4.1 Polyquadratic stabilization through static

output feedback
Introducing the dynamic matrix A elα in the condition (2) and after a
linearizing change of variables the theorem 1 gives rise to the following
result in terms of LMI and LME (Linear Matrix Equalities):
Theorem 3 The system (4) can be stabilized by a static output feed-
back if there exist H symmetric positive definite matrices S11 ...SMN , M
matrices G1 ...GM , M matrices U1 ...UM and M matrices V1 ...VM of
PR
appropriates dimensions solutions of the LMIs, LMEs :
 
Gl + GTl − Si(l)l (Ai(l)l Gl + Bi(l)l Ul Ci(l)l )T
> 0,
Ai(l)l Gl + Bi(l)l Ul Ci(l)l Si(j)j
Vl Ci(l)l = Ci(l)l Gl
∀(l, i(l), i(j), j) ∈ (e × el × ej × J) (5)
PM
H = l=1 N l
e = {1...M }, el = {1...N l}, ej = {1...N j}
J function of l.
The output feedback gain is then defined by:
Kl = Ul Vl−1

An application of this theorem was proposed in [4] using the example

proposed in [6] dealing with actuator break down problem.
Stabilization of uncertain switched discrete time systems 7

4.2 Local polyquadratic Stabilisation through static

output feedback
Extension of the preceding theorem is straightforward and the analo-
gous synthesis result of theorem 2:

F
Theorem 4 The system (4) can be locally polyquadratically stabilized
through output feedback if there exist M symmetric positive definite ma-
trices S1 ...SM , M matrices G1 ...GM and M matrices U1 ...UM of appro-
priates dimensions solutions of the LMIs:
 
Gl + GTl − Sl (Ai(l)l Gl + Bi(l)l Ul Ci(l)l )T

PM
H = l=1 N l
OO
Ai(l)l Gl + Bi(l)l Ul Ci(l)l
Vl Ci(l)l = Ci(l)l Gl
∀(l, i(l), j) ∈ (e × el × J)

e = {1...M }, el = {1...N l}
J function of l.
The output feedback gain is then defined by:
Kl = Ul Vl−1
Sj
> 0,

(6)

Remark 1 Introducing the equality constraints Vl Ci(l)l =Ci(l)l Gl in-

deed increases the sufficiency in the relations for the control design. For-
tunately in the case when Ci(l)l = I ∀(l, i(l)) ∈ (e × el ) the conditions
defined by (5) and (6) reduces to the classical state feedback stabilizing
determination.
The following parts presents the main results of this paper with an illus-
PR
trative example.

5. Main result: Observer design for Switched

system
This section is directed towards the design of a dynamic output control
based on a switched observer for system (4).
Let first write the state space representation of the observer assuming
that the subsystems matrices are not uncertains (i.e at every time the
dynamic system matrix is known), then:
PM PN l
xb(k + 1) = l=1 ξl (k)( i=1 αi(l) (Ai(l) x
b(k)) + Bl u(k) + Ll (y(k) − yb(k)))
PM
yb(k) = l=1 ξl (k)Cl xb(k).
(7)
8 E. Maherzi et al.

The observation matrices L must be computed to achieve stability of

the estimation error; ε is the error between the switched system state
(4) and the observer state (7).

ε(k) = x(k) − x
b(k) (8)

F
Using (7), the dynamic estimation error is:
M
X
ε(k + 1) = ξl (k)(Ai(l) − Ll Cl ε(k)) (9)
l=1

The estimation error is dynamically represented by an uncertain switched

discrete time system:

OO elα ε(k), A
ε(k + 1) = A elα = Alα − Ll Clα

Applying condition (2), system (10) is polyquadratically stable if there

exist H symmetric positive definite matrices, S11 ...SMN , M matrices
F1 ...FM and M matrices G1 ...GM of appropriates dimensions solution
of the LMIs:
"
Gl + GTl − Si(l)l (ATi(l)l Gl − Ci(l)l
T
Fl )T
#
> 0,
(10)

T
ATi(l)l Gl − Ci(l)l Fl Si(j)j
∀(l, i(l), i(j), j) ∈ (e × el × ej × J) (11)
PM
H = l=1 N l
e = {1...M }, el = {1...N l}, ej = {1...N j}
J function of l.
And the observation matrix is given by:
PR
Ll = (GTl )−1 FlT (12)

The observer (7) is obviously not feasible since for uncertain subsystems
the dynamic A matrix is not known. As an attempt to overcome this
difficulty one may think in choosing for the observer a dynamic matrix
Anom which is inside the uncertainty domains at each time (for instance,
the mean value with respect to the matrices associated with the vertices
of the different uncertainty polytopic domains). To close the loop and
realize a dynamic output control a natural way is then to compute with
the polyquadratic approach a state feedback as in [4] and apply this
feedback gain with the observer state, mimicking what is done with the
separation principle. Of course the closed loop stability is not implied
by such an approach and has to be checked using, for instance, the
polyquadratic analysis approach.
Stabilization of uncertain switched discrete time systems 9

To support such an approach it is easy to prove that such control

determined for non uncertain switched discrete time systems (nominal
systems) would provide a stabilizing gain for sufficiently small uncertain-
ties around the nominal and it is also possible to give bounds for these
uncertainties.

F
Remark 2 We can note that for the local polyquadratic stability de-
scribed by (2), condition (11) becomes: System (10) is locally polyquadrat-
ically stable if there exist M symmetric positive definite matrices, S1 ...SM ,
M matrix F1 ...FM and M matrices G1 ...GM of appropriate dimension
solution of the LMIs:
"
T
OO
Gl + GTl − Sl (ATi(l)l Gl − Ci(l)l
T
Ai(l)l Gl − Ci(l)l Fl
∀(l, i(l), j) ∈ (e × el × J)

6. The closed loop stability

Sj

While choosing for the observer a dynamic A

T
Fl )T
#
> 0,

b matrix which is inside

the uncertainty domains at each time, the observer state space represen-
(13)

tation becomes:
PM
xb(k + 1) = l=1 ξl (k)(Abl x
b(k) + Bl u(k) + Ll (y(k) − yb(k)))
PM
yb(k) = l=1 ξl (k)Cl x
b(k) (14)
ε(k) = x(k) − xb(k)
PR
Using (14), the dynamic estimation error is:

M
X XNl
ε(k + 1) = ξl (k)( bl )x(k) + (A
αi(l) ((Ai(l) − A bl − Ll Cl ε(k))) (15)
l=1 i=1

The control law is of the form:

u(k) = Kl x
b(k) (16)

then:

M
X XNl
x(k + 1) = ξl (k)( αi(l) (Ai(l) + Bl Kl )x(k) − Bl Kl ε(k)) (17)
l=1 i=1
10 E. Maherzi et al.

Now we consider the augmented system representing the dynamic of the

state and the estimation error:
   
x(k + 1) PM PN l x(k)
= l=1 ξl (k) i=1 αi(l) Φi(l)l

F
ε(k + 1)   ε(k)
Ai(l)l + Bl Kl −Bl Kl (18)
with Φi(l)l = bl − Ll Cl
∆i(l)l A
b
and ∆i(l)l = Ai(l)l − Al

 
Gl1 Gl2
Using theorem 1 and replacing Gl by The system described

Gl + GTl − Si(l)l (.)

with "
(.)T Si(j)j
OO 
>0
Gl3 Gl4
by (18), is polyquadratically stabilizable with the state feedback gains Kl
and the observation gains Ll if there exist H symmetric positive definite
matrices S11 ...SMN and M matrices G1 ...GM of appropriates dimensions
solutions of:


#
bl − GT Ll Cl
GTl1 (Ai(l)l + Bl Kl ) + GTl3 ∆i(l)l −GTl1 Bl Kl + GTl3 A
(.) = l3
bl − GT Ll Cl
GTl2 (Ai(l)l + Bl Kl ) + GTl4 ∆i(l)l −GTl2 Bl Kl + GTl4 A l4
and ∆i(l)l = Ai(l)l − A bl
(19)

7. Study cases
PR
We propose to design a switched observer for the example proposed
by Kothare, Balakrishnan and Morari 1996 [3], taken from a benchmark
where the physical system is constituted by two masses m1 , m2 vehicles
connected through a spring with stiffness k.

x1 x2

m k m
u
m1 m2

Fig. 2. two masses vehicles

Stabilization of uncertain switched discrete time systems 11

The discrete state space representation of the system is:

 
x1 (k)
 x2 (k) 
X(k + 1) = A  
 x3 (k)  + Bu(k)

F
x4 (k)

y(k) = x2 (k)
   
1 0 0 0.1 0
0 1 0.1 
0  0 
A=
 −0.1k/M1
 , B=
 

0.1k/M1 0 1 0.1/M1 
0.1k/M2

OO −0.1k/M2 1
C= 0 1 00


0 0

An uncertainty is defined and related to the stiffness of the spring. A

mass m can be added to one or to the two vehicles of masses m1 and
m2 . By adding and changing the masses, the overall system can be
represented as a switched system constituted by 4 sub-systems where:

M 1 = m1 , M 2 = m2

M1 = m1 + m, M2 = m2
M 1 = m1 , M 2 = m2 + m
M1 = m1 + m, M2 = m2 + m
PR
4 3 2 1 0

Fig. 3. Trajectory vehicles

Considering the uncertainly associated with k, the system becomes

an uncertain switched discrete time system. The measurement x2 is the
position of m2 . The design of a static output feedback control according
to the polyquadratic approach given by (5) has failed. The following
presents the results of the design of a dynamic output controller with
observer. For the observer design three different approaches have been
processed: the quadratic, polyquadratic and local polyquadratic one for
different values of m; remind that the quadratic approach means a sin-
gle Lyapunov function for all the subsystems and, of course, for all the
12 E. Maherzi et al.

vertices of the subsystems. The results are presented in table 1. The

used values for this experiment are kmin = 4, kmax = 10, m1 = m2 = 1.
The results are coherent since polyquadratic implies local polyquadratic
which, itself, implies quadratic. The result in (5) is used to design a state
feedback control from which the control is defined as u(k) = Kl x b(k). The

F
gain K is computed for m = 1.1

K1 = −44.7395 36.0760 −14.5427 −5.3980


K2 = −105.9129 86.6016 −30.8068 −13.0689

OO
K3 = −40.2343 32.0069 −14.4219 −4.7267

K4 = −94.4552 76.1556 −30.5207 −11.5165

Numerical experiments are given in Fig. 4. Changing m1 and m2 in the

desired path. The experiment is illustrated by Fig 3. The matrices Abl for
the design of the switched observer are chosen for each sub-system as a
mean of its polyhedron vertices. The figure 4 represents the y(k) = x2 (k)
trajectory covering all uncertainly system rang starting from the initial
condition x2 (0) = 4.
PR

Fig. 4. Trajectory of x2
Stabilization of uncertain switched discrete time systems 13

Table 1. Numerical results of each approach

F
OO
PR
14 E. Maherzi et al.

8. Conclusion
The problem of designing a stabilizing output feedback is a complex
one for switched systems with uncertain sub-systems. It is known that
dynamic output control design is difficult when the uncertainties are

F
structured ones, which is the case for polytopic uncertainty. In this
paper we propose sufficient conditions using polyquadratic stability to
find robust output feedback controls for this kind of systems. In the case
of dynamic output control with observer the design still incorporates
a level of heuristic due to the choice of the dynamic A matrix for the
observer ; This point deserves further research, for instance, on the design

OO
of a non observer based switched dynamic output controller with the
probable drawback of the definition of bilinear matrix inequalities BMI
for the control design and the loss of state estimation which is a good
feature of the observer based output control since beside the control it
also provides some possibility for detection and supervision.

References
[1] J. Daafouz and J. Bernussou. Parameter dependent Lyapunov functions
for discrete time systems with time varying parametric uncertainties. Sys-
tems & Control Letters, 43:355-359, 2001.
[2] J. Daafouz, G. Millerioux and C. Iung. A poly-quadratic stability based
approach for linear switched systems. Int. Journal of Control, 75:1302-
1310, 2002.
[3] M. V. Kothare, Balakrishnan and Morari. Robust Constrained Model
Predictive Control using LMI. Automatica, 32:1361-1379, 1996.
[4] E.Maherzi, J.Bernussou and R.Mhiri. Stability and Stabilization for un-
PR
certain switched systems, a polyquadratic Lyapunov approach. Int. Jour-
nal on Sciences and Techniques of Automatic control , 1:61-74, 2007.
[5] J. Daafouz, Riedinger, and C. Iung. Stability analysis and control synthe-
sis for switched systems: A switched Lyapunov function approach. IEEE
Trans. on Automatic Control, 47(11):1883-1887, 2002.
[6] P. Perez and J. Geromel. H2 control for discrete-time systems optimality
and robustness. Automatica, 22(4):397-411, 1993.
[7] E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen. Stability results
for switched controller systems. Automatica, 35:553-564, 1999.
[8] M. Mariton. Jump linear systems in automatic control. New York: Marcel
Dekker, 1990.
[9] Z.Sun, S.S. Ge and T.H. Lee. Controllability and reachability criteria for
switched linear systems. Automatica, 38(5):775-786, 2002.
[10] G. Xie and L. Wang. Controllability and stabilizability of switched linear-
systems. Systems & Control Letters, 48(2):135-155, 2003.
Stabilization of uncertain switched discrete time systems 15

[11] D. Cheng, L. Guo, Y. Lin and Y. Wang. Stabilization of switched linear

systems. IEEE Trans. on Automatic Control, 50(5):661-666, 2005.
[12] H. Ishii and B.A. Francis. Stabilizing a linear system by switching control
with dwell time. IEEE Trans. on Automatic Control, 47(12):1962-1973,
2002.

F
[13] D. Liberzon and A.S. Morse. Basic problem in stability and design of
switched systems. IEEE Control Systems Magazinz, 19(5):59-70, 1999.
[14] Z. G. Li, C.Y. Wen and Y.C. Soh. Stabilization of a class of switched
systems via designing switching laws. IEEE Trans. on Automatic Control,
46(4):665-670,2001.
[15] A. Giua, C. Seatzu and C. Van Der Mee. Optimal control of switched
autonomous linear Systems. Conf. on Decision and Control, 40th CDC,

OO
:2472-2477, Orlando-USA, 2001.
[16] A. Raymond and R.A. DeCarlo. Optimal control of switching systems.
Automatica, 41:11-27, 2005.
[17] R.N. Shorten and K. S. Narendra. On the stability and existence of
common Lyapunov functions for stable linear switching systems. Conf.
on Decision and Control, 37th CDC, :3723-3724, Tampa, 1998.
[18] M.A. Wicks, P.A. Peleties and R.A. DeCarlo. Switched controller syn-
thesis for the quadratic stabilization of a pair of unstable linear systems.
European Journal of Control, 4:140-147, 1998.
[19] M.S. Branicky. Multiple Lyapunov Functions and Other Analysis Tools
for Switched and Hybrid Systems. IEEE Trans. on Automatic Control,
43(4), 1998.
[20] X. Liu, X. Shen and Y. Zhang. Stability analysis of a class of hybrid
dynamic systems. Dynamics of Continuous, Discrete and Impulsive Sys-
tems, 8:359-373, 2001.
[21] G. Millerioux and J. Daafouz. Unknown input observers for switched
linear discrete time systems. American Control Conference, proceedings
PR
6(30):5802-5805, july 2004.

Biographies
Elyes Maherzi received the diploma of Master degree
from the “Université de Montpellier II”, France, in 2003.
Currently, he is a PHD student at the “Institut National
des Sciences Appliqués et de Technologie”, Tunisia. His
research interests include the robust control and switched
systems.
16 E. Maherzi et al.

Mongi Besbes received the diploma degree in engi-

neering from the “Ecole Nationale d’Ingénieurs de Tu-
nis”, Tunisia, in 1993. He also received the Aggrega-
tion in Electrical Engineering in 1996 and the PHD from
the “Ecole Nationale d’Ingénieurs de Sfax”, Tunisia, in

F
2008. He is currently the head of Electrical Engineering
department at the “Ecole Supérieure de Technologie et
d’Informatique, Tunis”, Tunisia. His research interests
include the uncertain systems and multimodels.

Jacques Bernussou received the diploma degree in

OO
engineering from ENSEEIHT, Toulouse, France, in 1967.
He also received the “Docteur ingénieur” and the “Doc-
torat d’Etat” degrees both from the University Paul
Sabatier, Toulouse, in 1968 and 1970, respectively. He
is currently at the LAAS-CNRS as ”Directeur de recher-
che”. He is the author of more than one hundred research
papers and many books (Pergamon Press, North Holland,
Hermes). His research interests include large scale inter-
connected systems, optimal control, and robust control
theory applied to electrical energies network, spatial and aeronautics.

Radhi Mhiri is currently a full Professor at the “Fac-

ulté des Sciences de Tunis”, Tunisia. He is the Head of the
Research Group on electrical engineering: “RME/AIA”
at the “Institut National des Sciences Appliquées et de
Technologie, Tunis” and a collaborative member of
CERES (University of Sherbrooke-Canada). He received
PR
The Habilitation degree on Electrical Engineering from
the “Ecole Nationale d’Ingénieurs de Tunis”, in 2000. He
received also the Diploma degree “ICT for teaching” in
September 2001 (DUESS UTICEF Strasbourg University. His research in-
cludes automatic control, slinding mode control, computer controlled systems,
softcomputing, biosystems and e-learning.