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1861-5252/ c© 2009 TSSD Transactions onSystems, Signals & Devices

Vol. 4, No. 1, pp.1-139

Stabilization Through Output Feedback

Control for Uncertain Switched Discrete

Time Systems

E. Maherzi,1 M. Besbes,1 J. Bernussou,2 and R. Mhiri1,3

1Reseaux et Machines Electriques (RME)Institut Superieur des Sciences Appliquees et de Technologie,centre urbain Tunis-nord, B.P. 676, 1080 Tunis, Tunisia.

2Laboratoire d’Analyse et d’Architecture des Systemes (LAAS)7 Avenue du Colonel Roch-31077 Toulouse Cedex 4, France.

3Faculte des Sciences de TunisCampus Universitaire B.P. 1060, Tunis, Tunisia.

Abstract This paper discusses the robust stabilization of discrete switchedsystems, focusing on the design of a robust static output feed-back control and dynamic output control based on a switchedobserver. The results are derived using the direct Lyapunovapproach and the polyquadratic function concept. The stabi-lization conditions are written through linear matrix inequalitiesrelations. The polyquadratic Lyapunov approach provides a con-structive way to tackle uncertainty in the switched framework.The feasibility is illustrated on an example of discrete uncertainswitched discrete time system.

Keywords: Robust control, stability and stabilization, switched discrete timesystems, output feedback control and observer control.

1. Introduction

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

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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 analysisfor systems with specific switching laws [19], stability analysis for systemswith 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 assumingthe 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-cient (but relatively non restrictive compared to the quadratic approach)stability condition for discrete switched systems is provided using thepolyquadratic approach recently proposed by Daafouz and Bernussou2001 [1] for stability analysis and stabilization control of Linear TimeVarying systems.

An extension of this works was presented in Millerioux and Daafouz2004 dealing with unknown input observers in the case of switched lineardiscrete time systems. A sufficient conditions of global convergence ofsuch kind of observers along with a systematic procedure to design thegains of the observers is proposed. A discussion about the existence ofsuch observers is provided [21].

However this paper proposes an other extension of this works in thecase when the switches are made among polytopic uncertain discretetime systems. The control investigated is of output feedback controltype: observer based which, of course, is a more realistic frameworkthan the state feedback control proposed which have been preliminaryworked in Maherzi et al., 2006 [4].

In part two of this paper we start by formulate problem of uncertaindiscrete 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 usingthe notions of polyquadratic stabilization and local polyquadratic stabi-lization. In fifth part interest is putting on the design and constructionof Luenberger’s observer. The synthesis of command law in the case ofclosed loop is developed in part six. Finally an illustrative example isdone in part seven.

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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 roughlyillustrated in (Fig.1) where the uncertainty domains are polytopic shapedwith different number of vertices to cope with maximal generality. Themodel can be stated as follows:

x(k + 1) =

M∑

l=1

ξl(k)

Nl∑

i(l)=1

αi(l)Ai(l)x(k) (1)

where l is the switching index. M the number of uncertain systemsdomain and Dl is the uncertainty domain for subsystem l defined by:

Dl =

Aα�Aα =

Nl∑

i(l)=1

αi(l)Ai(l), αi(l) ≥ 0,

Nl∑

i(l)=1

αi(l)(k) = 1

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

ξl(k) =

∣∣∣∣1 if the state matrix is defined into Dl domain0 else

SS1 SS2

SSM

A11 A12

A1N1

A21 A22

A2N2

AM1 AM2

AMNM

Fig. 1. Uncertain switched discrete time system.

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4 E. Maherzi et al.

3. Uncertain switched discrete time systemsstability

Starting from the results of [2, 5], the analysis of the polyquadraticstability of the uncertain switched discrete time systems is developed in[4] to propose the following criteria.

3.1 Polyquadratic stability

Theorem 1 The system described by (1) is polyquadratically stable ifthere exist H symmetric positive definite matrices S11 ...SMN and M ma-trices G1 ...GM of appropriates dimensions solutions of the LMIs:

[Gl + GT

l − Si(l)l GTl AT

i(l)l

Ai(l)lGl Si(j)j

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

(2)

H =

M∑

l=1

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

Proof: To prove the precedent condition we have to prove that if thiscondition (2) is verified then it is verified for any jumps realization inthe uncertainty domains. We suppose two dynamical matrices Al andAj defined respectively in the domain Dl and Dj .Let

Al =

Nl∑

i(l)=1

αi(l)Ai(l),

Nl∑

i(l)=1

αi(l) = 1

and

Aj =

Nj∑

i(j)=1

αi(j)Ai(j),

Nj∑

i(j)=1

αi(j) = 1

from(2) one gets

Nl∑

i(l)=1

αi(l)

[Gl + GT

l − Si(l)l GTl AT

i(l)l

Ai(l)lGl Si(j)j

]> 0

[Gl + GT

l −∑Nl

i(l)=1αi(l)Si(l)l GTl

∑Nl

i(l)=1αi(l)ATi(l)l∑Nl

i(l)=1αi(l)Ai(l)lGl

∑Nl

i(l)=1αi(l)Si(j)j

]> 0

[Gl + GT

l −∑Nl

i(l)=1αi(l)Si(l)l GTl AT

l

AlGl Si(j)j

]> 0

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Stabilization of uncertain switched discrete time systems 5

replacing∑Nl

i(l)=1αi(l)Si(l)l by Sl > 0 we obtain:

[Gl + GT

l − Si(l)l GTl AT

i(l)l

Ai(l)lGl Si(j)j

]> 0

again

Nj∑

i(j)=1

αi(j)

[Gl + GT

l − Sl GTl AT

l

AlGl Si(j)j

]> 0

[Gl + GT

l − Sl GTl AT

l

AlGl

∑Nj

i(j)=1αi(j)Si(j)j

]> 0

replacing∑Nj

i(j)=1αi(j)Si(j)j by Sj we obtain:

[Gl + GT

l − Sl GTl AT

l

AlGl Sj

]> 0

This concludes the proof.

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 maybe very heavy, computationally speaking, in the case of a large H num-ber. Applying the quadratic concept where a single Lyapunov functionis used for each of the uncertain sub-systems, a say local polyquadraticstability 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 andonly if it exist M symmetrical positive definite matrices S1 ...SM and Mmatrices G1 ...GM of appropriates dimensions solutions of the LMIs.

[Gl + GT

l − Sl GTl AT

i(l)l

Ai(l)lGl Sj

]> 0

∀(l, i(l), j),∈ (e × el × J)e = {1...M}, el = {1...Nl}J function of l.

(3)

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6 E. Maherzi et al.

4. Static output feedback control

Consider the uncertain switched discrete time system described by:

x(k + 1) =∑M

l=1ξl(k)(Alx(k) + Blu(k))

y(k) =∑M

l=1ξl(k)(Clx(k)).(4)

where [Al, Bl, Cl] ∈ Dl, with:

Dl =

Nl∑

i(l)=1

αi(l)[Ai(l), Bi(l), Ci(l)], αi(l) ≥ 0,

Nl∑

i(l)=1

αi(l)(k) = 1

The stabilization problem of the switched system through static out-put feedback consists in determining a control law of the form:

u(k) = Kly(k)x(k + 1) = (Alα + BlαKlClα)x(k)

x(k + 1) = Alαx(k) such Alα = (Alα + BlαKlClα)

4.1 Polyquadratic stabilization through staticoutput feedback

Introducing the dynamic matrix Alα in the condition (2) and after alinearizing change of variables the theorem 1 gives rise to the followingresult 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 , Mmatrices G1 ...GM , M matrices U1 ...UM and M matrices V1 ...VM ofappropriates dimensions solutions of the LMIs, LMEs :

[Gl + GT

l − Si(l)l (Ai(l)lGl + Bi(l)lUlCi(l)l)T

Ai(l)lGl + Bi(l)lUlCi(l)l Si(j)j

]> 0,

VlCi(l)l = Ci(l)lGl

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

H =∑M

l=1Nle = {1...M}, el = {1...Nl}, ej = {1...Nj}J function of l.

(5)

The output feedback gain is then defined by:

Kl = UlV−1l

An application of this theorem was proposed in [4] using the exampleproposed in [6] dealing with actuator break down problem.

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Stabilization of uncertain switched discrete time systems 7

4.2 Local polyquadratic Stabilisation through staticoutput feedback

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

Theorem 4 The system (4) can be locally polyquadratically stabilizedthrough 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 + GT

l − Sl (Ai(l)lGl + Bi(l)lUlCi(l)l)T

Ai(l)lGl + Bi(l)lUlCi(l)l Sj

]> 0,

VlCi(l)l = Ci(l)lGl

∀(l, i(l), j) ∈ (e × el × J)

H =∑M

l=1Nle = {1...M}, el = {1...Nl}J function of l.

(6)

The output feedback gain is then defined by:

Kl = UlV−1l

Remark 1 Introducing the equality constraints VlCi(l)l=Ci(l)lGl 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 conditionsdefined by (5) and (6) reduces to the classical state feedback stabilizingdetermination.The following parts presents the main results of this paper with an illus-trative example.

5. Main result: Observer design for Switchedsystem

This section is directed towards the design of a dynamic output controlbased on a switched observer for system (4).

Let first write the state space representation of the observer assumingthat the subsystems matrices are not uncertains (i.e at every time thedynamic system matrix is known), then:

x(k + 1) =∑M

l=1ξl(k)(∑Nl

i=1αi(l)(Ai(l)x(k)) + Blu(k) + Ll(y(k) − y(k)))

y(k) =∑M

l=1ξl(k)Clx(k).(7)

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8 E. Maherzi et al.

The observation matrices L must be computed to achieve stability ofthe estimation error; ε is the error between the switched system state(4) and the observer state (7).

ε(k) = x(k) − x(k) (8)

Using (7), the dynamic estimation error is:

ε(k + 1) =M∑

l=1

ξl(k)(Ai(l) − LlClε(k)) (9)

The estimation error is dynamically represented by an uncertain switcheddiscrete time system:

ε(k + 1) = Alαε(k), Alα = Alα − LlClα (10)

Applying condition (2), system (10) is polyquadratically stable if thereexist H symmetric positive definite matrices, S11 ...SMN , M matricesF1 ...FM and M matrices G1 ...GM of appropriates dimensions solutionof the LMIs :

[Gl + GT

l − Si(l)l (ATi(l)lGl − CT

i(l)lFl)T

ATi(l)lGl − CT

i(l)lFl Si(j)j

]> 0,

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

H =∑M

l=1Nle = {1...M}, el = {1...Nl}, ej = {1...Nj}J function of l.

(11)

And the observation matrix is given by:

Ll = (GTl )−1FT

l (12)

The observer (7) is obviously not feasible since for uncertain subsystemsthe dynamic A matrix is not known. As an attempt to overcome thisdifficulty one may think in choosing for the observer a dynamic matrixAnom which is inside the uncertainty domains at each time (for instance,the mean value with respect to the matrices associated with the verticesof the different uncertainty polytopic domains). To close the loop andrealize a dynamic output control a natural way is then to compute withthe polyquadratic approach a state feedback as in [4] and apply thisfeedback gain with the observer state, mimicking what is done with theseparation principle. Of course the closed loop stability is not impliedby such an approach and has to be checked using, for instance, thepolyquadratic analysis approach.

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Stabilization of uncertain switched discrete time systems 9

To support such an approach it is easy to prove that such controldetermined for non uncertain switched discrete time systems (nominalsystems) would provide a stabilizing gain for sufficiently small uncertain-ties around the nominal and it is also possible to give bounds for theseuncertainties.

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 dimensionsolution of the LMIs:

[Gl + GT

l − Sl (ATi(l)lGl − CT

i(l)lFl)T

ATi(l)lGl − CT

i(l)lFl Sj

]> 0,

∀(l, i(l), j) ∈ (e × el × J)

(13)

6. The closed loop stability

While choosing for the observer a dynamic A matrix which is insidethe uncertainty domains at each time, the observer state space represen-tation becomes:

x(k + 1) =∑M

l=1ξl(k)(Alx(k) + Blu(k) + Ll(y(k) − y(k)))

y(k) =∑M

l=1ξl(k)Clx(k)ε(k) = x(k) − x(k)

(14)

Using (14), the dynamic estimation error is:

ε(k + 1) =

M∑

l=1

ξl(k)(

Nl∑

i=1

αi(l)((Ai(l) − Al)x(k) + (Al − LlClε(k))) (15)

The control law is of the form:

u(k) = Klx(k) (16)

then:

x(k + 1) =M∑

l=1

ξl(k)(Nl∑

i=1

αi(l)(Ai(l) + BlKl)x(k) − BlKlε(k)) (17)

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10 E. Maherzi et al.

Now we consider the augmented system representing the dynamic of thestate and the estimation error:

(x(k + 1)ε(k + 1)

)=

∑M

l=1ξl(k)∑Nl

i=1αi(l)Φi(l)l

(x(k)ε(k)

)

with Φi(l)l =

[Ai(l)l + BlKl −BlKl

∆i(l)l Al − LlCl

]

and ∆i(l)l = Ai(l)l − Al

(18)

Using theorem 1 and replacing Gl by

[Gl1 Gl2

Gl3 Gl4

]The system described

by (18), is polyquadratically stabilizable with the state feedback gains Kl

and the observation gains Ll if there exist H symmetric positive definitematrices S11 ...SMN and M matrices G1 ...GM of appropriates dimensionssolutions of:

[Gl + GT

l − Si(l)l (.)(.)T Si(j)j

]> 0

with

(.) =

[GT

l1(Ai(l)l + BlKl) + GTl3∆i(l)l −GT

l1BlKl + GTl3Al − GT

l3LlCl

GTl2(Ai(l)l + BlKl) + GT

l4∆i(l)l −GTl2BlKl + GT

l4Al − GTl4LlCl

]

and ∆i(l)l = Ai(l)l − Al

(19)

7. Study cases

We propose to design a switched observer for the example proposedby Kothare, Balakrishnan and Morari 1996 [3], taken from a benchmarkwhere the physical system is constituted by two masses m1, m2 vehiclesconnected through a spring with stiffness k.

m m

m1 m2

x1 x2

u

k

Fig. 2. two masses vehicles

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Stabilization of uncertain switched discrete time systems 11

The discrete state space representation of the system is:

X(k + 1) = A

x1(k)x2(k)x3(k)x4(k)

+ Bu(k)

y(k) = x2(k)

A =

1 0 0.1 00 1 0 0.1−0.1k/M1 0.1k/M1 1 00.1k/M2 −0.1k/M2 0 1

, B =

00

0.1/M1

0

C =(0 1 0 0

)

An uncertainty is defined and related to the stiffness of the spring. Amass m can be added to one or to the two vehicles of masses m1 andm2. By adding and changing the masses, the overall system can berepresented as a switched system constituted by 4 sub-systems where:

M1 = m1, M2 = m2

M1 = m1 + m, M2 = m2

M1 = m1, M2 = m2 + m

M1 = m1 + m, M2 = m2 + m

01234

Fig. 3. Trajectory vehicles

Considering the uncertainly associated with k, the system becomesan uncertain switched discrete time system. The measurement x2 is theposition of m2. The design of a static output feedback control accordingto the polyquadratic approach given by (5) has failed. The followingpresents the results of the design of a dynamic output controller withobserver. For the observer design three different approaches have beenprocessed: the quadratic, polyquadratic and local polyquadratic one fordifferent values of m; remind that the quadratic approach means a sin-gle Lyapunov function for all the subsystems and, of course, for all the

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12 E. Maherzi et al.

vertices of the subsystems. The results are presented in table 1. Theused values for this experiment are kmin = 4, kmax = 10, m1 = m2 = 1.The results are coherent since polyquadratic implies local polyquadraticwhich, itself, implies quadratic. The result in (5) is used to design a statefeedback control from which the control is defined as u(k) = Klx(k). Thegain 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

)

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 thedesired path. The experiment is illustrated by Fig 3. The matrices Al forthe design of the switched observer are chosen for each sub-system as amean of its polyhedron vertices. The figure 4 represents the y(k) = x2(k)trajectory covering all uncertainly system rang starting from the initialcondition x2(0) = 4.

Fig. 4. Trajectory of x2

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Stabilization of uncertain switched discrete time systems 13

Table 1. Numerical results of each approach

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14 E. Maherzi et al.

8. Conclusion

The problem of designing a stabilizing output feedback is a complexone for switched systems with uncertain sub-systems. It is known thatdynamic output control design is difficult when the uncertainties arestructured ones, which is the case for polytopic uncertainty. In thispaper we propose sufficient conditions using polyquadratic stability tofind robust output feedback controls for this kind of systems. In the caseof dynamic output control with observer the design still incorporatesa level of heuristic due to the choice of the dynamic A matrix for theobserver ; This point deserves further research, for instance, on the designof a non observer based switched dynamic output controller with theprobable drawback of the definition of bilinear matrix inequalities BMIfor the control design and the loss of state estimation which is a goodfeature of the observer based output control since beside the control italso provides some possibility for detection and supervision.

References

[1] J. Daafouz and J. Bernussou. Parameter dependent Lyapunov functionsfor 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 basedapproach for linear switched systems. Int. Journal of Control, 75:1302-1310, 2002.

[3] M. V. Kothare, Balakrishnan and Morari. Robust Constrained ModelPredictive Control using LMI. Automatica, 32:1361-1379, 1996.

[4] E.Maherzi, J.Bernussou and R.Mhiri. Stability and Stabilization for un-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. IEEETrans. on Automatic Control, 47(11):1883-1887, 2002.

[6] P. Perez and J. Geromel. H2 control for discrete-time systems optimalityand robustness. Automatica, 22(4):397-411, 1993.

[7] E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen. Stability resultsfor switched controller systems. Automatica, 35:553-564, 1999.

[8] M. Mariton. Jump linear systems in automatic control. New York: MarcelDekker, 1990.

[9] Z.Sun, S.S. Ge and T.H. Lee. Controllability and reachability criteria forswitched 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.

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Stabilization of uncertain switched discrete time systems 15

[11] D. Cheng, L. Guo, Y. Lin and Y. Wang. Stabilization of switched linearsystems. IEEE Trans. on Automatic Control, 50(5):661-666, 2005.

[12] H. Ishii and B.A. Francis. Stabilizing a linear system by switching controlwith dwell time. IEEE Trans. on Automatic Control, 47(12):1962-1973,2002.

[13] D. Liberzon and A.S. Morse. Basic problem in stability and design ofswitched 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 switchedsystems 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 switchedautonomous linear Systems. Conf. on Decision and Control, 40th CDC,: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 ofcommon Lyapunov functions for stable linear switching systems. Conf.on Decision and Control, 37th CDC, :3723-3724, Tampa, 1998.

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[19] M.S. Branicky. Multiple Lyapunov Functions and Other Analysis Toolsfor 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 hybriddynamic systems. Dynamics of Continuous, Discrete and Impulsive Sys-tems, 8:359-373, 2001.

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BiographiesElyes Maherzi received the diploma of Master degree

from the “Universite de Montpellier II”, France, in 2003.

Currently, he is a PHD student at the “Institut National

des Sciences Appliques et de Technologie”, Tunisia. His

research interests include the robust control and switched

systems.

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16 E. Maherzi et al.

Mongi Besbes received the diploma degree in engi-

neering from the “Ecole Nationale d’Ingenieurs 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’Ingenieurs de Sfax”, Tunisia, in

2008. He is currently the head of Electrical Engineering

department at the “Ecole Superieure de Technologie et

d’Informatique, Tunis”, Tunisia. His research interests

include the uncertain systems and multimodels.

Jacques Bernussou received the diploma degree in

engineering from ENSEEIHT, Toulouse, France, in 1967.

He also received the “Docteur ingenieur” 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-

ulte des Sciences de Tunis”, Tunisia. He is the Head of the

Research Group on electrical engineering: “RME/AIA”

at the “Institut National des Sciences Appliquees et de

Technologie, Tunis” and a collaborative member of

CERES (University of Sherbrooke-Canada). He received

The Habilitation degree on Electrical Engineering from

the “Ecole Nationale d’Ingenieurs 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.