, 1) is weakly detectable, then the system 2 : g(r,h(x),u) is an observer for (6.1), namely 0 e IP is asymptotically stable with respect to the error system è : f (x,u) -g(r -e,h(r, According to Definition 6.1, if uniformly on r and u

, We now start by stating the results given by Vidyasagar in [8a1. We consider the nonlinear system described by (6.1) and the following hypotheses

. G{l, 0) : 0, furthermore, there exist positive constants c1 and c2 such that for all r and u llV,/(", r)ll S ct

, 1) and the assumptions (?ll) and, (712) pertaini,ng to it. If we can fi,nd a teedback control u: l@) such that (i,) 1 is conti, Theorem 6.L Suppose system

K. Where-Ô-e, ii) n -0 fs a uni,fonnly asymptotically stable equilibri'um point of ù: T(x,z(u)

, 1) is weakly detectable, then n:0 and z:0 is a uni,formly asymptoti ,cally stable equili,brium point of the system ù: f@,te), p.2

, Proof : See Vidyasagar

, Tsinias ha^s relaxed condition (i) in Vidyasagar's theorem, and only considered 7(c) to be continuous

, 1) is wealcly detectable and i:: f (a,l@D is (locally) asurnptotically stable at 0 e W then the output feedbacle u:',tk) asymptotically stabilizes ù: f@,te)), 2 -eQ,h(x),tQD, Theorem 6.2 If system, p.64

, Chapitrc 7. Conclusions and future work

, Bibliographie

K. H. Adjallah, Contribution au di,agnosti,c de systèmes par obseruateur d'état, 1993.

A. Alleyne and J. K. Hedrick, Nonlinear adaptive control of active suspensions, IEEE Transactions on Control Systems Technology, vol.3, issue.1, pp.94-102, 1995.
DOI : 10.1109/87.370714

A. Alleyne, R. Liu, and H. Wright, On the limitations of force tracking control for hydraulic active suspensions, Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207), pp.43-47, 1998.
DOI : 10.1109/ACC.1998.694625

N. Ahmad, H. K. Atassi, and . Khalil, A separation principle for the stabilization of a class of nonlinear systems, IEEE Trans. on Automatic Control, vol.44, issue.9, pp.7672-1687, 1999.

S. Behtash, Robust output tracking for non-linear systems, International Journal of Control, vol.2, issue.87, pp.1381-1407, 1990.
DOI : 10.1007/978-1-4684-0068-7

O. Bernard, G. Sallet, and A. Sciandra, Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model, IEEE Transactions on Automatic Control, vol.43, issue.8, pp.431056-1065, 1998.
DOI : 10.1109/9.704977

T. Boukhobza, M. Djemai, and J. P. Barbot, Nonlinear Sliding Observer for Systems in Output and Output Derivative Injection Form, Proceedi,ngs of IFAC World Congress, pp.299-305, 1996.
DOI : 10.1016/S1474-6670(17)57995-1

C. I. Byrnes and A. Isidori, Local stabilization of minimum-phase nonlinear systems, Systems & Control Letters, vol.11, issue.1, pp.9-17, 1988.
DOI : 10.1016/0167-6911(88)90105-3

I. Christopher, A. Byrnes, and . Isidori, Asymptotic stabilization of minimum phase nonlinear systems, IEEE Trans. on Automatic Contror, vol.36, issue.10, pp.22-23, 1991.

F. Celle, J. P. Gauthier, D. , and G. Sallet, Synthesis of nonlinear observers: A harmonic-analysis approach, Mathematical Systems Theory, vol.13, issue.1, pp.29-322, 1989.
DOI : 10.1007/978-1-4612-1082-5

J. N. Chiasson and R. T. Novotnak, Nonlinear speed observer for the PM stepper motor, IEEE Transactions on Automatic Control, vol.38, issue.10, pp.1584-1588, 1993.
DOI : 10.1109/9.241582

G. Ciccarella, M. D. Mora, and A. Germani, A Luenberger-like observer for nonlinear systems, International Journal of Control, vol.6, issue.3, pp.537-556, 1993.
DOI : 10.1016/0167-6911(87)90021-1

S. V. Drakunov and I. V. Utkin, Sliding mode control in dynamic systems, International Journal of Control, vol.1, issue.4, pp.1029-1037, 1992.
DOI : 10.1016/0005-1098(76)90076-5

S. Drakunov and V. Utkin, Sliding mode observers. Tutorial, Proceedings of 1995 34th IEEE Conference on Decision and Control, pp.3376-3378, 1995.
DOI : 10.1109/CDC.1995.479009

C. Edwards and S. K. Spurgeon, On the development of discontinuous observers, International Journal of Control, vol.26, issue.5, pp.12-1229, 1994.
DOI : 10.1080/00207178708933870

C. Edwards, S. K. Spurgeon, and R. J. Patton, Sliding mode observers for fault detection and isolation, Automatica, vol.36, issue.4, pp.541-553, 2000.
DOI : 10.1016/S0005-1098(99)00177-6

A. Ellman, Leakage behaviour of four-way servovalve. In Proceed'ings of the ASME fl,uid power systems and technology diuision, pp.163-167, 1998.

A. Ellman and T. Virvalo, Formation of pressure gain in hydraulic servovalves and its significance in system behaviour, Proceedings of the ASME fl,uid power systems and technology di,uisi,on, pp.77-85, 1996.

B. Eryilmaz and B. H. Wilson, Combining leakage and orifice flows in a hydraulic servovalve model. Transactions of the ASME -Joumal of Dynamic Systems Measurement and Control, pp.576-579, 2000.
DOI : 10.1115/1.1286335

M. Feki and E. Richard, Including leakage flow in the servovalve steady state model. submitted to Tlansactions of the ASME -Journal of Dynamic Systems Measurement and Control, 2001.
DOI : 10.1080/02286203.2005.11442318

M. Feki, E. Richard, and F. Gomes-almeida, Commande en effort d'un vérin hydraulique par linéarization entrée/sortie, Journées Doctorales d'Automatiques, JDA'99, pp.181-184, 1999.

M. I22l, J. Feki, and . Vivalda, Sliding observer for a class of nonlinear systems with uncertainties. submitted to Systems & Control Letters, 2001.

M. Feki, Robust control of a hydraulic system force, Second Nonlinear Control Network (NCN) Workshop, 2000.

M. Feki and M. A. Hammami, Observer design for an electrohydraulic manipulator, IASTED Intl. Conf. on Control Appli, 2001.

M. Feki and J. Vivalda, Nonlinear observer design application to an electrohydraulic system, 7rd, Intt. Conf. on Appti Mathemati,cs and, 2000.

, L67

M. Feki and J. Vivalda, Sur la stabilisation des systèmes non linéaires par retour d'états estimés, Actes de Soci,été de Mathématiques Tunisienne, 2000.

M. Feki and J. Vivalda, A result on the stabilization of nonlinear systems by state detection. ln Fourth Nonlinear Control Networlc (NCN) Workshop, Shefield, 2001.

A. Rafael, C. E. Garcia, and . Attellis, Thajectory tracking in nonlinear systems via nonlinear reduced-order observers. Internati,onal Joumal of Contror, pp.685-715, 1995.

J. P. Gauthier and G. Bornard, Outils et modèles mathématiques pour I'automatique et Ia théorie des signals, chapter Stabilization des systèmes non linéaires, pp.307-324, 1981.

J. P. Gauthier and L. A. Kupka, Observability and Observers for Nonlinear Systems, SIAM Journal on Control and Optimization, vol.32, issue.4, pp.975-994, 1994.
DOI : 10.1137/S0363012991221791

J. P. Gauthier and G. Bornard, Observability for any u(t) of a class of nonlinear systems, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, pp.26922-926, 1981.
DOI : 10.1109/CDC.1980.271933

J. P. Gauthier, H. Hammouri, and S. Othman, A simple observer for nonlinear systems applications to bioreactors, IEEE Transactions on Automatic Control, vol.37, issue.6, pp.875-880
DOI : 10.1109/9.256352

V. Grag, Fault detection in nonlinear systems : An application to automated highway systems. Phd in mechanical engineering, 1995.

W. Hahn, Stability of Motion, 1967.
DOI : 10.1007/978-3-642-50085-5

M. A. Hammami, Stabilization of a class of nonlinear systems using an observer design, Proceedings of 32nd IEEE Conference on Decision and Control, pp.1954-1959, 1993.
DOI : 10.1109/CDC.1993.325537

H. Hammouri, K. Busawon, and M. Farza, Nonlinear observers for local uniform observable systems, SIAM Journal on Control and Optimization, 1999.
DOI : 10.1051/cocv:2003017

URL : http://www.esaim-cocv.org/10.1051/cocv:2003017/pdf

H. Hammouri, E. H. Kinnaert, and . Yaagoubi, Observer-based approach to fault detection and isolation for nonlinear systems, IEEE Transactions on Automatic Control, vol.44, issue.10, pp.441879-1884, 1999.
DOI : 10.1109/9.793728

A. Isidori, Nonlinear Control Systems, 1995.

V. Jurdjevic and J. P. Quinn, Controllability and stability, Journal of Differential Equations, vol.28, issue.3, pp.381-389, 1978.
DOI : 10.1016/0022-0396(78)90135-3

T. Kailath, Linear Systems. Printice-Hall, 1980.

K. Hassan and . Khalll, Nonlinear Systems, 1992.
URL : https://hal.archives-ouvertes.fr/hal-01698610

K. K. I42l, A. Lee, and . Arapostathis, Remarks on smooth feedback stabilization of nonlinear systems, Systems ?i Control Letters, vol.0, pp.41-44, 1988.

W. Lin, Feedback stabilization of general nonlinear control systems: A passive system approach, Systems & Control Letters, vol.25, issue.1, pp.4-52, 1995.
DOI : 10.1016/0167-6911(94)00056-2

W. Lin, Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback, Automatica, vol.32, issue.6, pp.2975-924, 1996.
DOI : 10.1016/0005-1098(96)00013-1

X. Lu and S. K. Spurgeon, Robust sliding mode control of uncertain nonlinear systems, Systems & Control Letters, vol.32, issue.2, pp.75-90
DOI : 10.1016/S0167-6911(97)00061-3

G. David and . Luenberger, Observer for multivariable systems, IEEE Trans. on Automatic Control, issue.2, pp.1190-197, 1966.

G. David and . Luenberger, An introduction to observers, IEEE Trans. on Automati,c Control, issue.6, pp.16596-602, 1977.

R. Marino, Feedback stabilization of single-input nonlinear systems, Systems & Control Letters, vol.10, issue.3, pp.201-206, 1988.
DOI : 10.1016/0167-6911(88)90053-9

R. Marino, Adaptive observers for single output nonlinear systems, IEEE Transactions on Automatic Control, vol.35, issue.9, pp.1054-1058, 1990.
DOI : 10.1109/9.58536

R. Marino and P. Tomei, Global adaptive observers for nonlinear systems via filtered transformations, IEEE Transactions on Automatic Control, vol.37, issue.8, pp.1239-1245, 1992.
DOI : 10.1109/9.151117

R. Marino and P. Tomei, Adaptive observers with arbitrary exponential rate of convergence for nonlinear systems, IEEE Transactions on Automatic Control, vol.40, issue.7, pp.1300-1304, 1995.
DOI : 10.1109/9.400471

E. Herbert and . Merritt, Hydraulic Control Systems, 1967.

S. Nicosia, P. Tomei, and A. Tornambè, Observer-based control law for a class of nonlinear systems, International Journal of Control, vol.51, issue.3, pp.355-566, 1990.

A. J. Nijmeijer, H. Vander, and . Schaft, Nonlinear Dynamical Control Systems, 1990.
DOI : 10.1007/978-1-4757-2101-0

N. Niksefat and N. Sepehri, Design and experimental evaluation of a robust force controller for an electro-hydraulic actuator via quantitative feedback theory, neeri,ng Pract'ice, pp.1335-1348, 2000.
DOI : 10.1016/S0967-0661(00)00075-7

J. O-'reilly, R. Outbib, and E. Richard, Obseruers for Linear Systems State feedback stabilization of an electropneumatic system . Transacti,ons of the ASME, Mathemati,cs in Sci,ence and EngineeringJournal of Dynami, vol.17057, pp.410-415, 1983.

R. Outbib and G. Sallet, Stabilizability of the angular velocity of a rigid body revisited, Systems & Control Letters, vol.18, issue.2, pp.93-98, 1992.
DOI : 10.1016/0167-6911(92)90013-I

J. William and I. Palm, Modeling Analysis and Control of Dynamical Systems, 1983.

R. V. Patel and M. Toda, Quantitative measures of robustness in multivariable systems, Proceedi,ngs of American Control Conference, 1980.

S. Raghavan and J. K. Hedrick, Observer design for a class of nonlinear systems, International Journal of Control, vol.109, issue.2, pp.515-528, 1994.
DOI : 10.1109/9.53535

R. Rajamani and J. Hedrick, Adaptive observers for active automotive suspensions: theory and experiment, IEEE Transactions on Control Systems Technology, vol.3, issue.1, pp.86-93, 1995.
DOI : 10.1109/87.370713

E. Richard and R. Outbib, State feedback stabilization of an electrohydraulic system, Proceedings of Thi,rd European Control Conference, pp.330-335, 1995.

E. Richard, S. Scavarda, D. Thomasset, and A. Oustaloup, Modélisation et commande des systèmes à fluide sous pression, 1999.

P. Seibert and R. Suarez, Global stabilization of nonlinear cascade systems, Systems & Control Letters, vol.14, issue.4, pp.347-352, 1990.
DOI : 10.1016/0167-6911(90)90056-Z

H. Sira-ramirez, On the dynamical sliding mode control of nonlinear systems, International Journal of Control, vol.42, issue.5, pp.1039-1261, 1993.
DOI : 10.1049/PBCE040E

J. J. Slotine, J. K. Hedrick, and E. A. Misawa, Nonlinear state estimation using sliding observers, 1986 25th IEEE Conference on Decision and Control, pp.332-339, 1986.
DOI : 10.1109/CDC.1986.267257

URL : http://dspace.mit.edu/bitstream/1721.1/14549/2/19117844-MIT.pdf

E. Jean-jacques and . Slotine, Sliding controller design for nonlinear systems, International Journal of Control, vol.40, issue.2, pp.42-434, 1984.

E. Jean-jacques, W. Slotine, . Li, and . Appli, , 1991.

G. A. Sohl and J. E. Bobrow, Experiments and simulations on the nonlinear control of a hydraulic servosystem, IEEE Trans. on Control Systems Technology, issue.3, pp.238-247, 1999.

E. D. Sontag, Mathematical Control theory, 1990.

E. D. Sontag and Y. Wang, On characterizations of the input-to-state stability property, Systems & Control Letters, vol.24, issue.5, pp.35-359, 1995.
DOI : 10.1016/0167-6911(94)00050-6

H. J. Sussman and P. V. Kokotovic, The peaking phenomenon and the global stabilization of nonlinear systems, IEEE Transactions on Automatic Control, vol.36, issue.4, pp.424-440, 1991.
DOI : 10.1109/9.75101

S. Tafazoli, C. W. De, P. D. Silva, and . Lawrence, Tracking control of an electrohydraulic manipulator in the presence of friction, IEEE Transactions on Control Systems Technology, vol.6, issue.3, pp.401-411, 1998.
DOI : 10.1109/87.668040

A. Teel and L. Praly, Global stabilizability and observability imply semi-global stabilizability by output feedback, Systems & Control Letters, vol.22, issue.5, pp.313-325, 1994.
DOI : 10.1016/0167-6911(94)90029-9

F. E. Thau, Observing the state of nonlinear dynamical systems, Intemational Journal of Control, vol.17, issue.3, pp.47-479

W. J. Thayer, Specification standards for electrohydraulic flow control servovalves, Technical Bulletin, vol.117

M. L. Thein and E. A. Misawa, Comparison of the sliding observer to several state estimators using a rotational inverted pendulum, Proceedings of 1995 34th IEEE Conference on Decision and Control, pp.3385-3390, 1995.
DOI : 10.1109/CDC.1995.479011

A. Tornambè, Use of asymptotic observers having-high-gains in the state and parameter estimation, Proceedings of the 28th IEEE Conference on Decision and Control, pp.79-794, 1989.
DOI : 10.1109/CDC.1989.70462

J. Tsinias, Further results on the observer design problem, Systems & Control Letters, vol.14, issue.5, pp.477-481, 1990.
DOI : 10.1016/0167-6911(90)90092-9

J. Tsinias, A generalization of Vidyasagar's theorem on stabilizability using state detection, Systems & Control Letters, vol.17, issue.1, pp.37-42
DOI : 10.1016/0167-6911(91)90096-W

I. Vadim and . Utkin, variable structure systems with sliding modes, IEEE Trans. on Automatic Control, AC, vol.22, issue.4, pp.212-222

I. Vadim and . Utkin, Sli,di,ng mode in control and opti,mization

M. S4l and . Vidyasagar, on the stabilization of nonlinear systems using state detection, IEEE Trans. on Automati,c Control, AC, vol.25, issue.3, pp.504-509, 1980.

M. Vidyasagar, Nonlinear Systems Analysis, 1993.

T. J. Viersma, Analysùs, Synthesis, and, Desi,gn of Hydraulic Seruosystems and Pi,pe- Iines, 1980.

M. [. Walcott, S. H. Corless, and . Zak, Comparative study of nonlinear state observation techniques, International Journal of Control, vol.45, issue.6, pp.2-09, 1987.

J. Warga, Optimal Control of Differential and Functi,onal Equations

J. Watton, Fluid Power Systems, Printice Hall, United Kingdom, 1989.

D. Williamson, Observation of bilinear systems with application to biological control, Automatica, vol.13, issue.3, pp.243-254
DOI : 10.1016/0005-1098(77)90051-6

Z. Bartosiewicz, Local observability of nonlinear systems, Systems & Control Letters, vol.25, issue.4, pp.295-298, 1995.
DOI : 10.1016/0167-6911(94)00074-6

M. Zeitz, The extended Luenberger observer for nonlinear systems, Systems & Control Letters, vol.9, issue.2, pp.149-156, 1987.
DOI : 10.1016/0167-6911(87)90021-1