C. 'est-bien-le, 2.3) qui réalise la meilleure performance L 2 . La spécification (S3) établie dans le chapitre précédent est donc satisfaite

. Bouclé-avec-le-retour-de-sortie, 1.7) d'abord avec le contrôleur adaptatif (6.2.1) à robustesse équivalente et stabilité asymptotique, puis (6.2.1) à robustesse améliorée et stabilité pratique. La période d'échantillonnage est fixée à 0.1s. Les conditions initiales de simulation sont les suivantes : ? 0 = [10 10 10

C. Adaptatif, 2.1) à robustesse équivalente Dans cette sous-section on considère une incertitude de ±33% autour de la valeur nominale sur les coefficients diagonaux de l'inertie J

K. ?. , K. !. , and K. Mt-b-sont-tracées, de la vitesse angulaire ! et de la vitesse des roues à réaction ! r , ainsi que celles des gains Les courbes obtenues pour l'axe z sont présentées ici, celles pour les axes x et y sont en annexe 3. L'analyse suivante concerne l'axe z mais s

. Autrement, 2.1) semble plus robuste

D. Nouveaux-paramètres-de-la-loi-adaptative-etant-donné-que-les-valeurs-de-k-?-min-et-k-!-min-sont-déterminées-par-celles-de and D. , nous avons modifié les contraintes (6.2.10) de sorte que K ? min soit proche de 0, 3(K ? 0 + F ? ) et K ! min de 0, 3(K ! 0 + F ! ) Puisque les relations (7.1.4) sont satisfaites pour tout k = x, y, z, les valeurs des nouveaux paramètres, données en annexe 5, et en particulier celles de D, fournissent

L. Loi-de-commande-adaptative-discrétisée, ), avec les paramètres donnés en annexe 5, peut maintenant être intégrée dans un simulateur complet d'un satellite de la filière MYRIADE, construit sur un noyau Fortran. Cette étape (et la validation des résultats obtenus) est nécessaire si l'objectif final, 2003.

. Afin, inertie sur l'attitude du satellite, l'attitude maximale restante après n ? 100s de simulation est tracée en fonction du ratio J/J dep sur les figures 7, et ce pour les 120 scénarii de déploiement. Les croix rouges correspondent au contrôleur statique

. Sur-la-figure, 9, on détecte facilement les moments auxquels ont lieu les sauts de guidage, que ce soit lorsque le satellite est contrôlé par le contrôleur statique

. La-loi-de, 1.7) est clairement observable puisqu'après chaque saut de guidage, l'attitude (en valeur absolue) décroît linéairement. L'attitude reprend sa valeur basse d

. Avec-le-contrôleur-adaptatif-robuste, 2.1), l'attitude ne décroît certes pas linéairement et des dépassements peuvent être observés, notamment sur l'axe x. Mais l'attitude reprend sa valeur basse d'avant le saut de guidage au bout de 89s en moyenne sur les courbes, soit 1min29s

. Avec-le-contrôleur-adaptatif-robuste, 2.1), les sauts de guidage n'entraînent aucune augmentation du couple de commande des roues. Les roues sont donc moins sollicitées qu

H. Leduc and C. , Pittet et D. Peaucelle -Adaptive attitude control of a microsatellite during payload deployment, 11th AHS Conference, 2017.

H. Leduc, D. Peaucelle, M. Lovera, and C. , Pittet -Robust adaptive magnetic control of satellites with uncertain parameters, 20th IFAC World Congress, 2017.

H. Leduc, D. Peaucelle, and C. , Pittet -LMI-based design of a robust direct adaptive attitude control for a satellite with uncertain parameters, IFAC 20th Symposium on Automatic Control in Aerospace, 2016.

H. Leduc, D. Peaucelle, and C. , Pittet -Adaptive control LMI-based for descriptor systems rational in the uncertainties, IFAC 12th International Workshop on Adaptation and Learning in Control and Signal Processing, 2016.

H. Leduc, D. Peaucelle, and C. , Pittet -LMI-based design of a structured direct adaptive satellite attitude control with actuator rate feedback, IEEE 54th Annual Conference on Decision and Control, 2015.

P. [. Apkarian and . Gahinet, A convex characterization of gain-scheduled H/sub ???/ controllers, IEEE Transactions on Automatic Control, vol.40, issue.5, pp.853-864, 1995.
DOI : 10.1109/9.384219

D. [. Astolfi, R. Karagiannis, and . Ortega, Nonlinear and adaptive control with applications, Communications and Control Engineering, 2007.
DOI : 10.1007/978-1-84800-066-7

]. P. Apk12 and . Apkarian, Elements de la théorie de la commande robuste, 2012.

A. [. Adams, S. Sparks, and . Banda, A gain scheduled multivariable design for a manual flight control system, [Proceedings 1992] The First IEEE Conference on Control Applications, pp.584-589, 1992.
DOI : 10.1109/CCA.1992.269809

]. K. Ast84 and . Aström, Analysis or rohrs counterexamples to adaptive control, IEEE Conference on decision and control, pp.982-987, 1984.

P. [. Biannic and . Apkarian, Missile autopilot design via a modified LPV synthesis technique, Aerospace Science and Technology, vol.3, issue.3, pp.153-160, 1999.
DOI : 10.1016/S1270-9638(99)80039-X

]. I. Bar07 and . Barkana, Simple adaptive control -a stable direct model reference adaptive control methodology -brief survey, IFAC Workshop on Adaptation and Learning in Control and Signal Processing, 2007.

G. [. Barker and . Balas, Comparing Linear Parameter-Varying Gain-Scheduled Control Techniques for Active Flutter Suppression, Journal of Guidance, Control, and Dynamics, vol.1, issue.5, pp.948-955, 2000.
DOI : 10.1016/0005-1098(96)00071-4

L. [. Boyd, E. Ghaoui, V. Feron, and . Balakrishnan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, 1994.
DOI : 10.1137/1.9781611970777

]. J. Bia10 and . Biannic, Contributions theoriques a la commande des systemes aeronautiques et spatiaux, Habilitation à diriger des recherches, 2010.

M. [. Bullo and . Murray, Proportional derivative control on the euclidean group, 1995.

R. [. Bianchi, C. Mantz, and . Christiansen, Gain scheduling control of variable-speed wind energy conversion systems using quasi-LPV models, Control Engineering Practice, vol.13, issue.2, pp.247-255, 2005.
DOI : 10.1016/j.conengprac.2004.03.006

C. [. Biannic, A. Ross, and . Knauf, Design and Robustness Analysis of Fighter Aircraft Flight Control Laws, European Journal of Control, vol.12, issue.1, pp.71-85, 2006.
DOI : 10.3166/ejc.12.71-85

[. Biannic, C. Roos, and C. Pittet, LPV analysis of switched controllers in satellite attitude control systems, AIAA Guidance, Navigation, and Control Conference, pp.1561-1566, 2011.
DOI : 10.1002/rnc.4590050604

[. Ben-yamin, I. Yaesh, and U. Shaked, Simplified adaptive control with guaranteed H 1 performance, 2007.

B. Clement, G. Duc, and S. Mauffrey, Aerospace launch vehicle control : A gain scheduling approach, Aerospace IFAC, 2002.
DOI : 10.3182/20020721-6-es-1901.01249

N. [. Cao and . Hovakimyan, Design and analysis of a novel l1 adaptive control architeture with guaranteed transient performance, IEEE Trans. on Auto. Control, pp.586-591, 2008.

A. [. Dydek, E. Annaswamy, and . Lavretsky, Adaptive Control and the NASA X-15-3 Flight Revisited, IEEE Control Systems Magazine, vol.30, issue.3, pp.32-48, 2010.
DOI : 10.1109/MCS.2010.936292

]. C. Des09 and . Desoer, Feedback systems ; input-output properties, Classics in Applied Mathematics, 2009.

]. M. Dje07 and . Djeziri, Diagnostic des systemes incertains par l'approche bond graph, 2007.

A. [. Doyle, K. Packard, and . Zhou, Review of LFTs, LMIs and µ, IEEE Conference on Decision and Control, pp.1227-1232, 1991.

C. [. Dettori and . Scherer, Robust stability analysis for parameter dependant systems using full block S-procedure, IEEE Conference on Decision and Control, pp.2798-2799, 1998.
DOI : 10.1109/cdc.1998.757879

]. O. De-weck, Attitude determination and control, Slides of the lecture at Massachusetts Institute of Technology, 2001.

D. [. Ebihara, D. Peaucelle, and . Arzelier, S-variable approach to LMI-based robust control, Communications and Control Engineering, 2015.
DOI : 10.1007/978-1-4471-6606-1

A. [. Fomin, V. Fradkov, and . Yakubovich, Adaptive control of dynamic plants, 1981.

]. A. Fra74 and . Fradkov, Adaptive stabilization of a linear dynamic plant, Autom. Remote Contr, pp.1960-1966, 1974.

A. [. Fan and . Tits, Characterization and efficient computation of the structured singular value, IEEE Transactions on Automatic Control, vol.31, issue.8, pp.734-743, 1986.
DOI : 10.1109/TAC.1986.1104388

P. [. Gahinet and . Apkarian, A linear matrix inequality approach toH??? control, International Journal of Robust and Nonlinear Control, vol.29, issue.4, pp.421-448, 1994.
DOI : 10.1002/rnc.4590040403

D. [. Gilbert, J. Henrion, D. Bernussou, and . Boyer, Polynomial LPV synthesis applied to turbofan engines, Control Engineering Practice, vol.18, issue.9, pp.1077-1083, 2010.
DOI : 10.1016/j.conengprac.2008.10.019

URL : http://www.laas.fr/~henrion/Papers/turbolpv_aca07.pdf

A. [. Hussain and . Annaswamy, Robust Adaptive Control in the Presence of Unmodeled Dynamics: A Counter to Rohrs's Counterexample, AIAA Guidance, Navigation, and Control (GNC) Conference, p.2013
DOI : 10.1109/TAC.1985.1104070

R. [. Hsu and . Costa, Adaptive control with discontinuous forgetting factor and saturation for imporved robustness, pp.1075-1080, 1986.

]. S. Hil13 and . Hillerin, Commande robuste de systemes non lineaires incertains, applications dans l'aerospatiale, 2013.

P. [. Hou and . Müller, Causal observability of descriptor systems, IEEE Transactions on Automatic Control, vol.44, issue.1, pp.158-163, 1999.
DOI : 10.1109/9.739111

Q. Hu, Sliding mode attitude control with L2-gain performance and vibration reduction of flexible spacecraft with actuator dynamics, Acta Astronautica, vol.67, issue.5-6, pp.572-583, 2010.
DOI : 10.1016/j.actaastro.2010.04.018

S. Hecker and A. Varga, Generalized LFT-Based Representation of Parametric Uncertain Models, European Journal of Control, vol.10, issue.4, pp.326-337, 2004.
DOI : 10.3166/ejc.10.326-337

P. [. Ioannou and . Kokotovic, An asymptotic error analysis of indentifiers and adaptive observers in the presence of parasitics, IEEE Conf. Decision and Control, vol.27, pp.921-927, 1982.

P. Ioannou and P. Kokotovi´ckokotovi´c, Adaptive systems with reduced models, 1983.
DOI : 10.1007/BFb0006357

]. A. Ilk15 and . Ilka, Gain-scheduled controller design, 2015.

G. [. Iwasaki and . Shibata, LPV system analysis via quadratic separator for uncertain implicit systems, IEEE Transactions on Automatic Control, vol.46, issue.8, pp.1195-1207, 2001.
DOI : 10.1109/9.940924

M. [. Ishihara and . Terra, Impulse controllability and observability of rectangular descriptor systems, IEEE Trans. Aut. Control, vol.46, issue.6, pp.991-994, 2001.

]. R. Kal58 and . Kalman, Design of self-optimizing control systems, ASME Transactions, vol.80, pp.468-478, 1958.

]. N. Kar84 and . Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica, vol.4, pp.373-385, 1984.

I. [. Kaufman, K. Barkana, and . Sobel, Direct adaptive control algorithms, 1998.
DOI : 10.1007/978-1-4612-0657-6

N. [. Kharisov, K. J. Hovakimyan, and . Aström, Comparison of architectures and robustness of model reference adaptive controllers and L1???adaptive controllers, International Journal of Adaptive Control and Signal Processing, vol.55, issue.2, pp.633-663, 2013.
DOI : 10.2514/6.2010-8015

I. [. Krstic, P. Kanellakopoulos, and . Kokotovic, Nonlinear and adaptive control design, 1995.

K. [. Kreisselmeier and . Narendra, Stable model reference adaptive control in the presence of bounded disturbances, IEEE Transactions on Automatic Control, vol.27, issue.6, pp.1169-1175, 1982.
DOI : 10.1109/TAC.1982.1103093

]. A. Kna09 and . Knauf, Modélisation sous forme lft et synthèse de correcteurs lft auto-séquencés de taille réduite et leurs implémentations aux applications de commande en aéronautique, 2009.

J. Krause and G. Stein, Structural limitations of model reference adaptive controllers [Lan74] I. Landau, A survey of model reference adaptive techniques -theory and applications, Automatica, vol.10, issue.4, pp.353-379, 1974.

R. [. Lindorff and . Carroll, Survey of adaptive control using Liapunov design, International Journal of Control, vol.18, issue.5, pp.897-914, 1973.
DOI : 10.1080/00207177308932569

L. [. Landau and . Dugard, Commande adaptative : aspects pratiques et théoriques, 1986.

]. A. Lev93 and . Levant, Sliding order and sliding accuracy in sliding mode control, Int. J. Control, vol.58, issue.6, pp.1247-1263, 1993.

S. [. Lublin, M. Grocott, and . Athans, H 2 (LQG) and H 1 control, The Control Handbook, pp.651-661, 1996.

W. [. Leith and . Leithead, Survey of gain-scheduling analysis and design, International Journal of Control, vol.73, issue.11, pp.1001-1025, 2000.
DOI : 10.1080/002071700411304

]. J. Lof15 and . Lofberg, Yalmip : A toolbox for modeling and optimization in matlab, 2015.

H. Leduc, D. Peaucelle, M. Lovera, and C. Pittet, Robust adaptive magnetic control of satellites with uncertain parameters, IFAC World Congress, p.2017
URL : https://hal.archives-ouvertes.fr/hal-01388231

H. Leduc, D. Peaucelle, and C. Pittet, LMI-based design of a structured direct adaptive satellite attitude control with actuator rate feedback, 2015 54th IEEE Conference on Decision and Control (CDC), p.2015
DOI : 10.1109/CDC.2015.7402732

URL : https://hal.archives-ouvertes.fr/hal-01388191

]. D. Lue77 and . Luenberger, Dynamic equations in descriptor form, IEEE Transactions on Automatic Control, vol.22, pp.312-321, 1977.

]. A. Lur57, Lur'e, Some nonlinear problems in the theory of automatic control, H.M. Stationery Off, 1957.

]. R. Luz14 and . Luzi, Commande variante dans le temps pour le contrôle d'attitude de satellites, 2014.

R. [. Liu, X. Wang, D. Zhang, and . Xu, Gain scheduling PD controller for variable pitch wind turbines, International power electronics and motion control conference, pp.2162-2167, 2012.

]. A. Lya92 and . Lyapunov, The general problem of the stability of motion, International journal of control, vol.55, issue.3, pp.531-773, 1992.

]. J. Mag05 and . Magni, User manual of the linear fractional representation toolbox version 2.0, Tech. report, ONERA -Systems Control and Flight Dynamics Depertment, 2005.

T. [. Masubuchi, M. Akiyama, and . Saeki, Synthesis of output feedback gain-scheduling controllers based on descriptor LPV system representation, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), pp.6115-6120, 2003.
DOI : 10.1109/CDC.2003.1272243

A. [. Megretski and . Rantzer, System analysis via integral quadratic constraints, IEEE Transactions on Automatic Control, vol.42, issue.6, pp.819-830, 1997.
DOI : 10.1109/9.587335

A. [. Narendra and . Annaswamy, A new adaptive law for robust adaptation without persistent excitation, IEEE Transactions on Automatic Control, vol.32, issue.2, pp.134-145, 1987.
DOI : 10.1109/TAC.1987.1104543

M. [. Nelson and . Balas, Model reference adaptive control of spacecraft attitude for a pnp satellite with unknown time varying input/output delays, Numerical algebra, pp.445-462, 2012.

L. [. Narendra and . Valavani, Direct and indirect model reference adaptive control, Automatica, vol.15, issue.6, pp.653-664, 1979.
DOI : 10.1016/0005-1098(79)90033-5

J. [. De-oliveira, L. Geromel, and . Hsu, LMI characterization of structural and robust stability: the discrete-time case, Linear Algebra and its Applications, vol.296, issue.1-3, pp.27-38, 1999.
DOI : 10.1016/S0024-3795(99)00086-5

A. [. Osburn, A. Whitaker, and . Kezer, New developmets in the design of model reference adaptive control systems, Institute of the Aerospace Sciences, 1961.

D. [. Pittet and . Arzelier, DEMETER : a benchmark for robust analysis and control of the attitude of flexible microsatellites, IFAC Symposium on Robust Control Design, pp.661-666, 2006.

B. [. Peaucelle, V. Andrievsky, A. Mahout, and . Fradkov, Robust Simple Adaptive Control with Relaxed Passivity and PID control of a Helicopter Benchmark, IFAC Proceedings Volumes, vol.44, issue.1, 2011.
DOI : 10.3182/20110828-6-IT-1002.01128

]. P. Par66 and . Parks, Liapunov redesign of model reference adaptive control systems, IEEE Trans. on Automatic Control, vol.11, issue.3, pp.362-367, 1966.

]. D. Pea00 and . Peaucelle, Formulation générique de problèmes en analyse et commande robuste par les fonctions de Lyapunov dépendant des paramètres, 2000.

]. D. Peh17a, Y. Peaucelle, Y. Ebihara, and . Hosoe, Robust observed-state feedback design for discrete-time systems rational in the uncertainties, Automatica, vol.76, pp.96-102, 2017.

C. [. Pittet and . Fallet, Gyroless attitude control of a flexible microsatellite, Proc DCSSS conference, 2002.

A. [. Peaucelle and . Fradkov, Robust adaptive -gain control of polytopic MIMO LTI systems ??? LMI results, Systems & Control Letters, vol.57, issue.11, pp.881-887, 2008.
DOI : 10.1016/j.sysconle.2008.04.005

. Plp-+-14-]-c, A. R. Pittet, D. Luzi, J. Peaucelle, J. Biannic et al., In flight results of adaptive attitude control law for a microsatellite, ESA Conference on Guidance, Navigation and Control Systems (Porto), 2014.

J. [. Pittet, C. Mignot, and . Fallet, LMI based multi-objective H/sub ???/ control of flexible microsatellites, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 1999.
DOI : 10.1109/CDC.2000.912340

K. [. Peterson and . Narendra, Bounded error adaptive control, IEEE Transactions on Automatic Control, vol.27, issue.6, pp.1161-1168, 1982.
DOI : 10.1109/TAC.1982.1103112

]. V. Pop62 and . Popov, Absolute stability of nonlinear systems of automatic control, Automation and remote control, pp.857-875, 1962.

]. L. Pra92 and . Praly, Adaptive regulation : Lyapunov design with a growth condition, International journal of adaptive control and signal processing, vol.6, pp.329-351, 1992.

. Pvsd-+-06-]-c, O. Poussot-vassal, L. Sename, P. Dugard, Z. Gaspar et al., A new semiactive suspension control strategy through LPV technique, control engineering practice, vol.16, issue.12, pp.1519-1534, 2006.

C. Poussot-vassal, O. Sename, L. J. Dugardrs00-]-w, J. C. Rugh, and . Shamma, Controle robuste LPV : application aux và c hicules automobiles, 2emes journees doctorales Reasearch on gain-scheduling, Automatica, vol.36, pp.1401-1425, 2000.

L. [. Rohrs, M. Valavani, G. Athans, and . Stein, Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics, IEEE Transactions on Automatic Control, vol.30, issue.9, pp.881-889, 1985.
DOI : 10.1109/TAC.1985.1104070

M. [. Shamma and . Athans, Analysis of gain scheduled control for nonlinear plants, IEEE Transactions on Automatic Control, vol.35, issue.8, pp.898-907, 1990.
DOI : 10.1109/9.58498

C. [. Syrmos, P. Abdallah, K. Dorato, and . Grigoriadis, Static output feedback???A survey, Automatica, vol.33, issue.2, pp.125-137, 1997.
DOI : 10.1016/S0005-1098(96)00141-0

URL : http://www.eece.unm.edu/faculty/chaouki/CONTROL/Tech_Reports/TR_EECE95_008.ps

]. O. Sen16 and . Sename, Robust and LPV control of MIMO systems, Slides of the lecture at Technologico de Monterrey, 2016.

]. J. Sha12, . J. Shammasid97-]-m, and . Sidi, Control of linear parameter varying systems with applications Spacecraft dynamics and control : A practical engineering approach, Journal of spacecraft and rockets, vol.34, pp.851-852, 1997.

I. [. Scherer and . Köse, Robustness with dynamic IQCs: An exact state-space characterization of nominal stability with applications to robust estimation, Automatica, vol.44, issue.7, pp.1666-1675, 2008.
DOI : 10.1016/j.automatica.2007.10.023

D. [. Sadabadi and . Peaucelle, From static output feedback to structured robust static output feedback: A survey, Annual Reviews in Control, vol.42, pp.11-26, 2016.
DOI : 10.1016/j.arcontrol.2016.09.014

URL : https://hal.archives-ouvertes.fr/hal-01342560

[. Tregouët, D. Arzelier, D. Peaucelle, C. Pittet, and L. Zaccarian, Reaction Wheels Desaturation Using Magnetorquers and Static Input Allocation, IEEE Transactions on Control Systems Technology, vol.23, issue.2, pp.525-539, 2014.
DOI : 10.1109/TCST.2014.2326037

M. [. Toh, R. H. Todd, and . Tutuncu, SDPT3 ??? A Matlab software package for semidefinite programming, Version 1.3, Optimization Methods and Software, vol.79, issue.1-4, pp.545-581, 1999.
DOI : 10.1287/moor.19.1.53

]. R. Wer02 and . Werking, Spacecraft attitude determination and control, 2002.

]. J. Wil72 and . Willems, Dissipative dynamical systems. part I : General theory. part II : Linear systems with quadratic supply rates, Arch. Rational Mach, Analysis, vol.45, pp.321-393, 1972.

]. F. Wu95 and . Wu, Control of linear parameter varying systems, 1995.

J. [. Xie and . Li, A robustness analysis of discrete-time direct model reference adaptive control, International Journal of Control, vol.78, issue.10, pp.1196-1204, 2006.
DOI : 10.1080/00207170500157369

]. V. Yak62 and . Yakubovich, The solution of certain matrix inequalities in automatic control theory, pp.620-623, 1962.

K. [. Yang and . Lum, Gain-scheduled flight control via state feedback, pp.3484-3489, 2003.

M. [. Young, J. C. Newlin, and . Doyle, mu analysis with real parametric uncertainty, [1991] Proceedings of the 30th IEEE Conference on Decision and Control, 1991.
DOI : 10.1109/CDC.1991.261579

A. [. Zanchettin, M. Calloni, and . Lovera, Robust Magnetic Attitude Control of Satellites, IEEE/ASME Transactions on Mechatronics, vol.18, issue.4, pp.1259-1268, 2013.
DOI : 10.1109/TMECH.2013.2259843

]. E. Zeh86 and . Zeheb, A sufficient condition of output feedback stabilization on uncertain systems, IEEE Trans. Aut. Control, vol.31, issue.11, pp.1055-1057, 1986.

W. [. Zhang and . Liu, Impulsive Mode Elimination for Descriptor Systems by a Structured P-D Feedback, IEEE Transactions on Automatic Control, vol.56, issue.12, pp.2968-2973, 2011.
DOI : 10.1109/TAC.2011.2160597

Y. [. Zhu, M. Xia, and . Fu, Adaptive Sliding Mode Control for Attitude Stabilization With Actuator Saturation, IEEE Transactions on Industrial Electronics, vol.58, issue.10, pp.4898-4907, 2011.
DOI : 10.1109/TIE.2011.2107719