ROBUST FEEDBACK LINEARIZATION AND GH CONTROLLER FOR A QUADROTOR UNMANNED AERIAL VEHICLE

Transcription

1 Journal o ELECTRICAL ENGINEERING, VOL. 7, NO. 1, 6, 7 ROBUST FEEDBACK LINEARIZATION AND GH CONTROLLER FOR A QUADROTOR UNMANNED AERIAL VEHICLE Abdellah Mokhtari Abdelaziz Benallegue Boubaker Daachi In this paper, a mixed robust eedback linearization with linear GH controller is applied to a nonlinear quadrotor unmanned aerial vehicle. An actuator saturation and constrain on state space output are introduced to analyse the worst case o control law design.the results show that the overall system becomes robust when weighting unctions are chosen judiciously. Perormance issues o the controller are illustrated in a simulation study that takes into account parameter uncertainties and external disturbances as well as measurement noise. K e y w o r d s: UAV, GH, sensitivity, robust linearization 1 INTRODUCTION The H control problem with continuous-time measurement output or linear systems has been studied in the last decade by many researchers such as Kwakernaak [1] and Grimble [] in polynomial orm and by van der Schat [3] and Ball [] or nonlinear systems. For the hovering control o helicopters, many control methods have been proposed including linear approaches such as LQG [], H design [6],[7] and the nonlinear approaches such as sliding mode [8], backstepping technique [9], and input/output linearization [1]. Even tough the design o controllers to achieve a linear input-output response or nonlinear systems has been well researched [11], the conventional input-output linearization techniques will perorm very poorly when it comes to output tracking as it will render the unstable internal dynamics unobservable [1]. Hence the nonlinear system must have stable zero dynamics or the input-output linearized system to be internally stable [11]. Following this context, classical eedback linearization may have poor robustness properties and cannot be easily combined with a H control law. So a robust nonlinear eedback is proposed to robustly control an uncertain nonlinear system around an operating point on using an appropriate approach or stability and robustness, W stability [13]. Inertial Navigation System (INS) and GPS are used to calculate a position and orientation o the vehicle. They are especially suitable or guidance and navigation o an autonomous Unmanned Aerial Vehicle (UAV) [1]. This paper proposes an attempt to apply linear H outer control o helicopter quadrotor with plant uncertainty combined with a robust eedback linearization inner controller. The plant to be controlled is described by six-degree-o-reedom nonlinear dynamics with plant uncertainties due to the variation o moments o inertia and payload operation. Successul application o the autonomous quadrotor depends on its level o controllability and lying qualities. The overall inner outer controller should improve tracking perormance and disturbance rejection capability. The process disturbance represents not only the uncertainty in the operating conditions, but the lack o precision in the system model. It degrades the robustness and perormance o control systems and the estimation o unknown dynamics seems to be diicult. The approach that allows or worst-case disturbances is the H controller. Uncertainty bounds step can be used to deine simple sensitivity and complementary sensitivity weights. These weights are chosen to maximize the disturbance attenuation properties. The disturbances attempt to maximize perormance index, while the control attempts to minimize. The analysis o this problem has been primarily in the requency domain. This analysis can be carried out in time domain, and naturally extends H theory to inite-time and non-linear systems [1]. In this work we mention the polynomial solution based on Diophantine equations [16]. DYNAMIC QUADROTOR Using Newton law and reerring to V. Mistler et al [17] [18], the general MIMO nonlinear system (Fig.1) can be represented in orm: ẋ = F(x) + G 1 (x)w + G (x)ū y 1 = H 1 (x) + K 1 (x)ū (1) y = H (x) + K 1 (x)w University o Science and Technology Oran Algérie, Laboratoire de Mécanique Appliquée, Laboratoire de Robotique de Versailles 1-1 av de l Europe, 781 Velizy, France, LISSI Vitry Creteil; mokhtari@robot.uvsq.r ISSN c 6 FEI STU

2 Journal o ELECTRICAL ENGINEERING 7, NO. 1, 6 1 [ ] 1 G (x) = P H 1 (x) = [,,,,x,y,z,ψ] T H (x) = x K 1 (x) = [ I ] K 1 (x) = [I 1 1 ] Fig. 1. The quadrotor helicopter where x R n is the state vector, ū R m is the control input, w R n is the noise and unknown perturbation vector, y 1 R l is the controlled output, y R p is the available measure vector. The ollowing hypothesis hold: (A1) the unctions F(x), G 1 (x), G (x), H 1 (x), H (x), K 1 (x), K 1 (x) are piecewise continuous. (A) F() =, H 1 () = and H () = or almost every t. (A3) H1 T (x)k 1 (x) =, K1(x)K T 1 (x) = I, K 1 (x)g T 1 (x) =, K 1 (x)k1(x) T = I. where [ ] wb w =, w p and where w b is the noise vector o size 1. w p is composed o aerodynamic orces disturbances [A x,a y,a z ] T and aerodynamic moment disturbances [A p,a q,a r ] T. They act on the UAV and are computed rom the aerodynamic coeicients C i as A i = 1 ρ airc i W (ρ air is the air density, W is the velocity o the UAV with respect to the air), (C i depend on several parameters like the angle between airspeed and the body ixed reerence system, the aerodynamic and geometric orm o the wing). The rotor is the primary source o control and propulsion or the UAV. The Euler angle orientation to the low provides the orces and moments to control the altitude and position o the system. The absolute position is described by three coordinates (x,y,z ),and its attitude by Euler angles (ψ,θ,φ), under the conditions ( π ψ < π) or yaw, ( π < θ < π ) or pitch and ( π < φ < π ) or roll. The state vector and other parameters are deined as: x = (x,y,z,ψ,θ,φ,ẋ,ẏ,ż,ζ 1,ξ, ζ 1 and ξ are deined in (1) F(x) = [ 1 (x),.., 1 (x)] T G 1 (x) = 3 1 M P 1 ψ, θ, φ) T The real control signals (u 1,u,u 3,u ) have been replaced by (ū 1,ū,ū 3,ū ) to avoid singularity in lie transormation matrices when using eedback linearization [17]. In that case u 1 has been delayed by a double integrator. The others control signals will keep unchanged. u 1 = ζ 1 + mg ζ = ξ ξ = ū 1 u = ū u 3 = ū 3 u = ū Let the state vector be written into the orm: so one can have: with () x = (x 1,x,...,x 1 ) T (3) 1 (x) = x 7, (x) = x 8, 3 (x) = x 9 (x) = x 1, (x) = x 13, 6 (x) = x 1 7 (x) = g 7 1x 1, 8 (x) = g 8 1x 1, 9 (x) = g + g 9 1(x 1 + mg) 1 (x) = x 11, 11 (x) = 1(x) x 1 13 (x) = P x 13 + P 3 x 1x 13 x 1 x 1 1 (x) x 1 x 13 x 1 P 1 = 1 d M 1 = 1 m I 3 3 g 1 3 g 1 d d d g 13 3 g 13 g 1 g 1 3 g 1 P = p 11 p 1 p 31 p 3 P 3 = p 311 p 31 p 313 p 31 p 3 p 33 p 331 p 33 p 333

3 A. Mokhtari A. Benallegue B. Daachi: ROBUST FEEDBACK LINEARIZATION AND GH CONTROLLER FOR A... 1 C is the orce to moment scaling actor P = g3 1 g 1 I x, I y, I z represent the diagonal coeicient o inertia g3 13 g 13 matrix o the system. g 1 g3 1 g 1 S(.) = sin(.), C(.) = cos(.), T(.) = tan(.), S e (.) = sec(.). g 7 1 = 1 m (Cx 6Cx Sx + Sx 6 Sx ) g 8 1 = 1 m (Cx 6Sx Sx Cx Sx 6 ) 3 ROBUST FEEDBACK LINEARIZATION (INNER CONTROLLER) g 1 = d I x g 9 1 = 1 m (Cx Cx 6 ) g 1 3 = dsx 6 I y Cx ; g 13 3 = dcx 6 I y ; g 1 3 = dsx Sx 6 I y Cx g 1 = Cx 6 I z Cx ; g 13 = Sx 6 I z ;g 1 = Sx Cx 6 I z Cx p 11 = Sx Sx 6 Cx 6 (I 1 I 3 ) p 1 = Sx Cx (Cx 6 ) (I 1 I 3 ) Cx Sx 6 (I 1 ) p 31 = Sx 6 Cx 6 (Cx ) (I + I 3 I 1 ) + Sx 6 Cx 6 (I 1 + I 3 ) p 3 = I Sx 6 Cx 6 p 311 = T(x )(Cx 6 ) (I 1 I 3 ) + T(x )(1 + I 3 ) p 31 = Sx 6 Cx 6 (I 3 I 1 ) p 313 = (Cx 6 ) S e (x )(I 3 I 1 ) + S e (x )(1 I 3 ) p 31 = Sx 6 Cx 6 Sx (I 1 + I 3 ) p 3 = Cx (Cx 6 ) (I 3 I 1 ) Cx (1 I 1 ) p 33 = Sx 6 Cx 6 (I 1 I 3 ) p 331 = S e (x )(1 + I 3 ) + (Cx 6 ) S e (x )(I 1 I 3 )+ (Cx 6 ) (I + I 3 I 1 ) Cx (I + I 3 ) p 33 = Sx 6 Cx 6 Sx ( I 1 + I 3 ) p 333 = T(x )(Cx 6 ) (I 3 I 1 ) + T(x )(1 I 3 ) I 1 = I y I x I z ; I = I y I z ; I 3 = I z I x I x I y g is the gravity constant (g = 9.81ms ); d is the distance rom the center o mass to the rotors; u 1 is the resulting thrust o the our rotors deined as u 1 = (F 1 + F + F 3 + F ) u is the dierence o thrust between the let rotor and the right rotor deined as u = d (F F ) u 3 is the dierence o thrust between the ront rotor and the back rotor deined as u 3 = d(f 3 F 1 ) u is the dierence o torque between the two clockwise turning rotors and the two counter-clockwise turning rotors deined as u = C(F 1 F + F 3 F ) The robust eedback linearization method used in this context is based on Sobolev norm deined as [ h W = h T (t)h(t)dt + ḣ T (t)ḣ(t)dt ] 1. () It transorms a nonlinear system into its tangent linearized system around an operating point. Then, under state eedback and change o coordinates deined by ū(x,v) = α(x) + β(x)v z = φ(x) α(x) = α c (x) + β c (x)ltφ c (x) β(x) = β c (x)r 1 () φ(x) = T 1 φ c (x) where L =. αc x x=,t = φc x x=,r = 1, α c (x) = 1 (x)b(x), β c (x) = 1 (x) then the nonlinear system is transormed into a ollowing one with A = F(x) x () satisies α [ ] φ ż = Az + B v + x G 1(x) x=φ 1 (z) (6) x=,b = G (). Note that equation x φ x==, x x== I 1 1, β() = I. For the quadrotor helicopter the input-output decoupling problem is solvable or the nonlinear system by means o static eedback. The vector relative degree {r 1,r,r 3,r } is given by r 1 = r = r 3 = ; r = and we have b(x) = [ L r1 h 1(x) L r h (x) L r3 h 3(x) φ c (x) = [φ c1 (x),φ c (x),φ c3 (x),φ c (x) ] T h 1 (x) = x L h 1 (x) = x 7 = ẋ φ c1 (x) = L h 1(x) = Ax m + g7 1x 1 = ẍ L 3 h 1(x) =... x L r h (x) ] T

4 Journal o ELECTRICAL ENGINEERING 7, NO. 1, 6 3 φ c (x) = L h (x) = Ay with h (x) = y L h (x) = x 8 = ẏ m + g8 1(x,x,x 6 )x 1 = ÿ L 3 h (x) =... y h 3 (x) = z L φ c3 (x) = h 3 (x) = x 9 = ż L h 3(x) = Az m + g + g9 1x 1 = z L 3 h 3(x) =... z [ ] h (x) = x φ c (x) = L h (x) = ẋ (x) = = L g1 L r1 1 h 1 (x) = 1 m (Cx 6Cx Sx + Sx 6 Sx ) 1 = L g L r1 1 h 1 (x) = d (x 1 Sx 6 Cx Sx x 1 Cx 6 Sx ) mi x 13 = L g3 L r1 1 h 1 (x) = d mi y ( x 1 Cx Cx ) 1 = 1 = L g1 L r 1 h (x) = 1 m (Cx 6Sx Sx Cx Sx 6 ) = L g L r 1 h (x) = d (x 1 Sx 6 Sx Sx + x 1 Cx 6 Cx ) mi x 3 = L g3 L r 1 h (x) = d mi y ( x 1 Sx Cx ) = L g L r 1 h (x) = 31 = L g1 L r3 1 h 3 (x) = 1 m (Cx Cx 6 ) 3 = L g L r3 1 h 3 (x) = d mi x (x 1 Sx 6 Cx ) 33 = L g3 L r3 1 h 3 (x) = d mi y (x 1 Sx ) 3 = L g L r3 1 h 3 (x) = In act the system in equation () is still nonlinear because o w vector. One seeks a controller which ensures the compensated system to be internally asymptotically stable and its output to tend asymptotically toward a desired trajectory even in the presence o external disturbance. In this context the linear GH is proposed. H OPTIMAL CONTROL (OUTER CONTROLLER) H synthesis methods take into account in an explicit manner some speciication o robustness. The issue here is to take maximum guaranty or a synthesized control law on a chosen model to work eectively on the physical system. For that a transer unction amily is considered where the nominal model W nom = A 1 B constitutes the center. We assume that it is possible to choose these sets o transer unction contain the real system. Hence i the stability and perormance o the closed loop system are obtained and demonstrated or all W i elements then it will be also or the real system [19]. Let the transer unction o the uncertain system be W = (A + D p P p ) 1 (B + D p 1 F p ) (7) where D p P p and D p 1 F p are modelling errors on A and B. D p, P p and F p are characterized by low and pass ilter respectively and 1, are the non structured uncertainty. It is assumed that disturbances are bounded and there exists a unction V which veriies V V 1 < 1 (8) Hence the minimization criterion is written as: J = (Pp S + F p M) Φ (P p S + F p M) (9) The cost unction o GH lead to eigenvalues problem which lead to the minimization o (Pp S +F p M)A 1 D where Φ = A 1 D D A 1. The weighting unctions can be represented as P p (z 1 ) = P 1 d P n, F p (z 1 ) = P 1 d F n (1) where P d is strictly schur and P n (). The polynomials P n and F n are chosen to assure that the polynomial 1 = L g1 L r 1 h (x) = = L g L r 1 h (x) = 3 = L g3 L r 1 h (x) = d I y (Sx 6 S e x ) = L g L r 1 h (x) = 1 I z (Cx 6 S e x ) L c = P n B F n A (11) veriies L c L c > on z = 1. I we can write L c = L 1 L with L 1 strictly minimal phase, L a non-minimal phase and L s is Schur polynomial satisying L s = L z n where n = deg(l ), then the control law procedure is summarized as ollow:

5 A. Mokhtari A. Benallegue B. Daachi: ROBUST FEEDBACK LINEARIZATION AND GH CONTROLLER FOR A /(Pc*D/A) Magnitude (db) 1 1 T S M 1/(Fc*D/A) Magnitude (db) Frequency (rad/sec) Fig.. Sensitivity S, M and T and inverse o weightings or x, y, z Frequency (rad/sec) Fig. 3. Cost unction or x, y, z 6 1/(Pc*D/A) 1/(Fc*D/A) Magnitude (db) T M S Magnitude (db) Frequency (rad/sec) Fig.. Sensitivity S, M, T and inverse weightings or ψ Frequency (rad/sec) Fig.. Cost unction or ψ compute (G,H,F ) through Diophantine equations F AP d + L G = L s P n D (1) F BP d L H = L s F n D (13) compute the eigenvalue/eigenvector equation (N 1,F 1, F 1s,λ) with F 1s is schur polynomial satisying F 1s = F 1 z n1 where n 1 = deg(f 1 ). compute the control law L N 1 + F 1 λl s = F 1s F (1) C = (H + KB) 1 (G KA), K = F1s 1 N 1P d (1) The main dierence between H and GH is that the irst one uses iteration algorithm to compute control law whereas the second one uses eigenvalue/eigenvector problem to get solution which is easier to compute. The error sensibility S, the control sensibility M and the complementary sensibilityt are deined as ollow: S = A(H + KB),M = A(G KA) L 1 L s D L 1 L s D T = B (G KA) L 1 L s D Finally, the GH controller has been computed with the ollowing constrain: Forces must be greater than or equal to zero and less than 1 ( F i 1N) which systematically lead to (u 1 ). This is due to actuator output limits. The altitude z must be less than or equal zero (z ) since the reerence rame is upside down. APPLICATION TO QUADROTOR The nominal transer matrix is computed or I y = 1.16;I z = 1.16;I x = 1.16;m = ;d =.1 with

6 Journal o ELECTRICAL ENGINEERING 7, NO. 1, 6 x(m) z(m) measured desired y(m) ψ(rad) Fig. 6. Trajectories x, y, z and ψ θ(rad) φ(rad) Fig. 7. Trajectories θ and φ F1(N) 1 F(N) 1 (x d x )(m)..1.1 (y d y )(m)..1.1 F3(N) 1 1 F(N) 1 1 (z d z )(m) ψ d ψ)(rad). 1 1 x Fig. 8. Applied orces Fig. 9. Tracking errors F1 (N) 8 6 F (N) 8 6 x error.. y error F3 (N) 6 F (N) z error.. ψ error Fig. 1. Applied orces Fig. 11. Tracking errors inputs v 1, v, v 3, v and outputs x, y, z, ψ. W nom = s s. s s The reerence trajectory is chosen as: x d = y d = z d =.a(t ) Γ(t )+.7a(t T ) Γ(t T ).7a(t T ) Γ(t T ).a(t 3T ) Γ(t 3T ).a(t T ) Γ(t T ).7a(t T ) Γ(t T )+.7a(t 6T ) Γ(t 6T ) +.a(t 7T ) Γ(t 7T )

7 6 A. Mokhtari A. Benallegue B. Daachi: ROBUST FEEDBACK LINEARIZATION AND GH CONTROLLER FOR A... ψ d =.b(t ) Γ(t )+.7b(t T ) Γ(t T ).7b(t T ) Γ(t T ).b(t 3T ) Γ(t 3T ).b(t T ) Γ(t T ).7b(t T ) Γ(t T )+.7(t 6T ) Γ(t 6T )+.b(t 7T ) Γ(t 7T ) With Γ(t αt) = or t αt and Γ(t αt) = 1 or t > αt; T = s, a =..1 3 m/s and b =..1 3 rad/s so the steady state o the trajectories will be x d = 1 m y d = 1 m; z d = 1 m ψ d = 1 rad. With a chosen weighting unction F n, P n and P d, the controller C is given. The sensitivity S, the control sensitivity M, the complementary sensitivity T and the cost unction are represented: Output: x, y, z Output ψ Poles o C Zeros o C 7.18 ±.63i ±.76i ± i ±.199i 7.18 ±.63i.778 ±.796i ±.1779i.1676 ±.1779i.131 ±.13i.388 λ x = λ y = λ z = γ x = γ y = γ z = Poles o C ψ Zeros o C ψ ±.33i ±.38i λ ψ =.13; γ ψ = 1 λ ψ =.966 Case o Aerodynamic orce disturbances: An aerodynamic orce disturbances has been taken or A x =. N,A y =. N and A z = 1. N occurring at s, s and 6 s respectively, the results o (x,y,z,ψ) and (θ,φ) are shown in Fig. and Fig. 6. The behavior o the applied orces and aerodynamic orce disturbances is shown in Fig. 7. The tracking errors are shown in Fig. 8. Case o Aerodynamic moment disturbances: For the aerodynamic moments A p =.8 Nm, A q =.8 Nm and A r =. Nm occurring at s, s and 6 s respectively the results o the applied orces are represented in Fig. 9. Tracking errors are represented in Fig. 1. It is noted rom Fig. to Fig. 8 that the system when subjected to aerodynamic orce disturbance [A x,a y,a z ] and % uncertainties on mass and inertia,the mixed inner-outer controller satisying results even without block disturbances estimation. This can be shown rom tracking error trajectories which vanished ater a inite time with a perect convergency. This is due not only to the robustness o the H controller but also to robust eedback linearization which preserve the good robustness properties. This is shown at the time s on x, s on y, and at the time 6 s on z trajectory when the disturbances occurs. The robustness o the system can be conirmed by tracking errors trajectories (Fig. 8). The magnitude disturbances are limited by actuator saturation between and 1 N. However despite the overshoots on orces in igure(fig. 7) which lead to saturation the system remain stable. However, when subjected to aerodynamic moments disturbances [A P,A q,a r ] and % uncertainties on mass and inertia, the results are shown in Fig. 9, Fig. 1. It is seen that the inner-outer controller shows eiciency to overcome easily disturbances on z and ψ, better than on x and y. The orces in Fig. 9 relect perectly the relation between control input (ū 1,ū,ū 3,ū ) and (F 1,F,F 3,F ). The computed sensitivity S in Fig. 1 and Fig. 3 show a low gain at low requency and a gain oscillating near db at high requency, however the complementary sensitivity T shows a gain o db at low requency and a low gain at high requency. This conirms that the resulting design is appropriate. Reerences [1] KWAKERNAAK, H.: A polynomial approach to Hinini optimization o control systems, Modelling, Robustness and Sensitivity Reduction (Ruth F. Curtain, ed.), Springer, Berlin, 1987, pp [] GRIMBLE, M. J. JOHNSON, M. A.: H robust control design- A tutorial review, IEE Comput. Contr. Eng. J.,, No. 6, 7-81, (1991). [3] Van der SHAFT, A.J.: : L -gain analysis o nonlinear systems and nonlinear state eedback H control, IEEE Trans. Automat. Contr. 37 No. 6 (June 199), [] BALL, J. A. HELTON, J. W. WALKER, M. L.: Hinini control or nonlinear systems with output eedback, IEEE Trans. Automat. Contr. 38 No. (April 1993), 6-9. [] MORRIS, J. C. NIEUWSTADT, M. V. BENDOTTI, P.: Identiication and Control o a Model Helicopter in Hover, Proceedings o American Control Conerence (199), [6] LIN, F. ZHANG, W. BRANDT, R. D. : Robust Hovering Control o a PVTOL Aircrat, IEEE Trans. on Contr. Sys. Tech. 7 No. 3 (1999), [7] BENDOTTI, P. MORRIS, J. C.: Robust Hover Control or a Model Helicopter, Proceedings o American Control Conerence 1 (1997), [8] PIEPER, J. K.: Application o SLMC: TRC Control o a Helicopter in Hover, Proceedings o American Control Conerence (199),

8 Journal o ELECTRICAL ENGINEERING 7, NO. 1, 6 7 [9] MAHONY, R. HAMEL, T. DZUL, A.: Hover Control via Lyapunov Control or an Autonomous Model Helicopter, Proceedings o Conerence on Decision and Control (1999), [1] MAHONY, R. LOZANO, R. : Almost Exact Path Tracking Control or an Autonomous Helicopter in Hover Maneuvers, Proceedings o IEEE International Conerence on Robotics and Automation (), 1-1. [11] ISIDORI, A.: Nonlinear Control Systems - An Introduction, Springer-Verlag, New York, NY, second edition, [1] BENVENUTI, L. DIGIAMBERARDINO, P. FARINA, L.: Trajectory Tracking or a PVTOL Aircrat: A Comparative Analysis, Proceedings o the 3th Conerence on Decision and Control, Kobe, Japan December [13] BOURLÈS, H. COLLEDANI, F.: W-stability and local input-output stability results, IEEE Trans. Automat.Contr. AC- (199), [1] SASIADEK, J. Z. HARTANA, P.: Sensor Fusion or Navigation o an Autonomous Unmanned Aerial Vehicle, Proceedings o the IEEE International Conerence on Robotics & Automation New Orleans, LA, April. [1] CHICHKA, D. F.: A Disturbance Attenuation Approach to a Class o Uncertain Systems, PHD in Aerospace Engineering, University o Caliornia Los Angeles, 199. [16] KWAKERNAAK, H.: H Optimization Theory and Applications to Robust Control Design, Annual Reviews in Control 6, pp.-6,. [17] MISTLER, V. BENALLEGUE, A. M SIRDI, N. K.: Exact Linearization and Noninteracting Control o a Rotors Helicopter via Dynamic Feedback, IEEE 1th IEEE International Workshop on Robot-Human Interactive Communication Bordeaux and Paris (1). [18] MOKHTARI, A. BENALLEGUE, A.: Dynamic eedback controller o Euler angles and wind parameters estimation or a quadrotor unmanned aerial vehicle, IEEE Int. Con. on Robotics and Automation (ICRA ) (). [19] GROCOTT, S. C. MAcMARTIN, D. G. MILLER, D. W. : Experimental implementation o a multiple model technique or robust control o the MACE test article, In Proceedings o 3rd International Conerence on Adaptive Structures, San Diego, CA (199). Received 1 April Abdellah Mokhtari received his electronics Eengineer diploma rom Electronic Institute, University o Sciences and Technology Oran in 198. He received MEng in 1986 and MSc in 1988 rom Wales University, UK. He was head o Marine Institute in Oran until 3. As a doctor o automatic control and robotics at the university o Sciences and Technology Oran, he is currently a researcher in robust control area in Laboratory o Robotic o Versailles, France. Abdelaziz Benallegue born in 196, received the BS degree in electronics engineering rom Algiers National Polytechnic School, Algeria, in 1987 he received the MS degree in automatic control and robotics rom University o Pierre and Marie Curie in Paris. In 1991 he received the PhD degree in automatic control and robotics rom University o Pierre and Marie Curie. Between he worked as associate proessor o automatic control and robotics at the University o Pierre and Marie Curie. At present he is proessor o automatic control and robotics at the University o Versailles (France). He is a member o the National Center o Scientiic Research - France. Boubaker Daachi born in 1973 received the BS degree in computer science rom University o Seti, Algeria. In 1998 he received the MS degree in automatic control and robotics rom University o Pierre and Marie Curie, Paris, France. In he received the PhD degree in robotics rom University o Versailles, France. Now he is associate proessor o automatic control and robotics at the University o Paris 1 Val de Marne. He is a member o PRC Identiication - CNRS France. SLOVART G.T.G. s.r.o. GmbH E X P O R T - I M P O R T E X P O R T - I M P O R T o periodicals and o non-periodically printed matters, book s and CD - ROM s Krupinská PO BOX 1, 8 99 Bratislava,Slovakia tel.: , ax.: gtg@internet.sk, SLOVART G.T.G. s.r.o. GmbH E X P O R T - I M P O R T