. Dans-cette-annexe, nous considérons leprobì eme dépendant d'un paramètre associé associéà la localisation des pôles Ceprobì eme consistè a déterminer, s'il existe, un correcteur garantissant que tous les pôles du système en boucle fermée sont dans une certaine région R(?) du plan complexe pour toutes les valeurs de ? ? [? ; ?], La famille R(?) de régions LMI est définie par (voir [CG96, Chi96] pour les différentes régions que peut représenter R(?))

A. Probì-eme, 2 Probì eme de placement de pôles dépendant d'un paramètre) Soient [? ; ?] un intervalle borné de R, P (p, ?) le système augmenté défini par (3.1), p.104

. Trouver, un correcteur K(p, ?) défini par (3.2) tel que, pour tout ? ? [? ; ?], tous les pôles de P (p, ?) ? K(p, ?) sont localisés dans R(?)

A. Lemme, Analyse de la localisation des pôles dépendant d'un paramètre) Soient [? ; ?] un intervalle borné de R, G(p, ?) le système défini par (3.3), page 108, p.202

A. Lemme, Soient [? ; ?] un intervalle borné de R, G(p, ?) le système défini par (3.3), page 108, et les données de toutes les propriétés i, i = 1, . . . , n p , soit G i (p, ?) les systèmes définis en (A.50) et ? i des entiers strictement positifs. S'il y a lieu, p.202

. Pour-i-=-1, n p , G i (p, ?) ? < ? i si la propriété i est une norme H ? ou G i (p, ?) 2 < ? i si la propriété i est une norme H 2

A. Théorème, Synthèse d'un correcteur multi?critères dépendant d'un paramètre par changement de variables paramétrisé) Soient [? ; ?] un intervalle borné de R, P (p, ?) le système augmenté défini par (3.1), page 104, et les données de toutes les propriétés i, [AG95] P. Apkarian and P. Gahinet. A convex characterization of gain-scheduled H ? controllers, IEEE Transactions Automatic Control, vol.21, issue.5, pp.40853-864, 1995.

P. Apkarian, P. Gahinet, and G. Becker, Self-scheduled H??? control of linear parameter-varying systems: a design example, Automatica, vol.31, issue.9, pp.311251-1261, 1995.
DOI : 10.1016/0005-1098(95)00038-X

]. D. Ala01 and . Alazard, Extracting physical tuning potentiometers from a complex control law : application to flexible aircraft flight control, AIAA Journal of Guidance, Control and Dynamics, 2001.

H. [. Apkarian and . Tuan, Parametrized LMIs in control theory, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), pp.152-157, 1998.
DOI : 10.1109/CDC.1998.760607

C. [. Boyd and . Barratt, Linear Controller Design : Limits of Performance, 1991.

F. D. Barb, A. Ben-tal, and A. Nemirovski, Robust Dissipativity of Interval Uncertain Linear Systems, SIAM Journal on Control and Optimization, vol.41, issue.6, pp.1661-1695, 2003.
DOI : 10.1137/S0363012901398174

]. G. Bec95 and . Becker, Parameter-dependent control of an under-actuated mechanical system, Conference on Decision and Control, pp.543-548, 1995.

X. Bombois, M. Gevers, G. Scorletti, and B. D. Anderson, Robustness analysis tools for an uncertainty set obtained by prediction error identification, Automatica, vol.37, issue.10, pp.371629-1636, 2001.
DOI : 10.1016/S0005-1098(01)00104-2

]. Bli02 and . Bliman, Nonconservative LMI approach to robust stability for systems with uncertain scalar parameters, Conference on Decision and Control, pp.305-310, 2002.

]. Bli03 and . Bliman, Stabilization of LPV systems, Conference on Decision and Control, pp.6103-6108, 2003.

]. Bli04 and . Bliman, An existence result for polynomial solutions of parameterdependent LMIs, Systems and Control Letters, vol.51, issue.3-4, pp.165-169, 2004.

. [. Bibliographie, A. Becker, and . Packard, Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Systems and Control Letters, vol.23, issue.3, pp.205-215, 1994.

F. [. Bamieh, M. A. Paganini, and . Dahleh, Distributed control of spatially invariant systems, IEEE Transactions on Automatic Control, vol.47, issue.7, pp.1091-1107, 2002.
DOI : 10.1109/TAC.2002.800646

G. Becker, A. Packard, D. Philbrick, and G. Balas, Control of parametricallydependent linear systems : a single quadratic Lyapunov approach, American Control Conference, pp.2795-2799, 1993.

G. Bombois, M. Scorletti, R. Gevers, P. Hildebrand, and . Van-den-hof, Data-based controller design at the cheapest experimental cost, Conference on Decision and Control, 2004.

A. [. Ben-tal and . Nemirovski, Lectures on Modern Convex Optimization : Analysis, Algorithms, and Engineering Applications
DOI : 10.1137/1.9780898718829

P. [. Braatz, J. C. Young, M. Doyle, and . Morari, Computational complexity of ?? calculation, IEEE Transactions on Automatic Control, vol.39, issue.5, pp.1000-1002, 1994.
DOI : 10.1109/9.284879

P. [. Chilali and . Gahinet, H/sub ???/ design with pole placement constraints: an LMI approach, IEEE Transactions on Automatic Control, vol.41, issue.3, pp.358-367, 1996.
DOI : 10.1109/9.486637

M. Chilali, P. Gahinet, and P. Apkarian, Robust pole placement in LMI regions, IEEE Transactions on Automatic Control, vol.44, issue.12, pp.2257-2270, 1999.
DOI : 10.1109/9.811208

]. E. Che82 and . Cheney, Approximation theroy, 1982.

]. M. Chi96 and . Chilali, Méthodes LMI pour l'analyse et la synthèse multi-critère, 1996.

[. Chou, A. L. Tits, V. Balakrishnan-ayres-de-castro, and F. Paganini, Stability multipliers and ?? upper bounds: connections and implications for numerical verification of frequency domain conditions, IEEE Transactions on Automatic Control, vol.44, issue.5, pp.906-913, 1999.
DOI : 10.1109/9.763207

]. D. Del84 and . Delchamps, Analytic feedback control and the algebraic Riccati equation, IEEE Transactions Automatic Control, vol.29, issue.11, pp.1031-1033, 1984.

]. C. Des69 and . Desoer, Slowly varying system ? x = A(t)x, IEEE Transactions Automatic Control, issue.146, pp.780-781, 1969.

S. [. Duc and . Font, Commande H ? et µ-analyse : des outils pour la robustesse . Hermès, 1999.

J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standard H/sub 2/ and H/sub infinity / control problems, IEEE Transactions on Automatic Control, vol.34, issue.8, pp.831-847, 1989.
DOI : 10.1109/9.29425

M. C. De-oliveira, J. Bernussou, and J. C. , A new discrete-time robust stability condition, Systems & Control Letters, vol.37, issue.4, pp.261-265, 1999.
DOI : 10.1016/S0167-6911(99)00035-3

]. J. Doy82 and . Doyle, Analysis of feedback systems with structured uncertainties, Proceedings of IEE, pp.242-250, 1982.

A. [. Doyle, K. Packard, and . Zhou, Review of LFT's, LMI's and µ, Conference on Decision and Control, pp.1227-1232, 1991.

C. [. Dettori and . Scherer, Robust stability analysis for parameter dependent systems using full block S-procedure, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), pp.2798-2799, 1998.
DOI : 10.1109/CDC.1998.757879

C. [. Dettori and . Scherer, New robust stability and performance conditions based on parameter-depedent multipliers, Conference on Decision and Control, pp.4187-4192, 2000.

P. [. Feron, P. Apkarian, and . Gahinet, Analysis and synthesis of robust control systems via parameter-dependent Lyapunov functions, IEEE Transactions on Automatic Control, vol.41, issue.7, pp.1041-1046, 1996.
DOI : 10.1109/9.508913

G. [. Fialho and . Balas, Adaptive vehicle suspension design using LPV methods, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), pp.469-474, 1998.
DOI : 10.1109/CDC.1998.760721

]. S. Fon95 and . Font, Méthodologie pour Prendre en Compte la Robustesse des Systèmes Asservis : Optimisation H ? et Approche Symbolique de la Forme Standard, 1995.

G. [. Fromion and . Scorletti, A theoretical framework for gain scheduling, International Journal of Robust and Nonlinear Control, vol.4, issue.10, pp.951-982, 2003.
DOI : 10.1002/rnc.748

G. [. Fromion, G. Scorletti, and . Ferreres, Nonlinear performance of a PI controlled missile: an explanation, International Journal of Robust and Nonlinear Control, vol.32, issue.8, pp.485-518, 1999.
DOI : 10.1002/(SICI)1099-1239(19990715)9:8<485::AID-RNC417>3.0.CO;2-4

A. [. Fan, J. C. Tits, and . Doyle, Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Transactions on Automatic Control, vol.36, issue.1, pp.25-38, 1991.
DOI : 10.1109/9.62265

]. M. Fu97 and . Fu, The real structured singular value is hardly approximable, IEEE Transactions Automatic Control, vol.42, issue.9, pp.1286-1288, 1997.

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

P. Gahinet, P. Apkarian, and M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Transactions on Automatic Control, vol.41, issue.3, pp.436-442, 1996.
DOI : 10.1109/9.486646

]. P. Gah96 and . Gahinet, Explicit controller formulas for LMI?based H ? synthesis, Automatica, vol.32, issue.7, pp.1007-1014, 1996.

F. Garofalo, G. Celentano, and L. Glielmo, Stability robustness of interval matrices via Lyapunov quadratic forms, IEEE Transactions on Automatic Control, vol.38, issue.2, pp.281-284, 1993.
DOI : 10.1109/9.250472

. R. Bibliographie-[-gj79-]-m, D. S. Garey, and . Johnson, Computers and Intractability : a guide to the theory of NP?completeness. A series of books in the mathematical sciences

P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. The Mathworks Partner Series, 1995.
DOI : 10.1109/cdc.1994.411440

F. [. Ghaoui, H. Oustry, and . Lebret, Robust Solutions to Uncertain Semidefinite Programs, SIAM Journal on Optimization, vol.9, issue.1, pp.33-52, 1998.
DOI : 10.1137/S1052623496305717

]. A. Hel03, . A. Helmerssonhj85-]-r, C. R. Horn, and . Johnson, On synthesis of practically valid gainscheduled controllers Matrix analysis, 1985.

P. [. Hiroe and . Khargonekar, Stability and ???/sub ???/ performance analysis of gain scheduled nonlinear control systems with /spl Dscr/-implementation under slowly varying command inputs, Proceedings of 35th IEEE Conference on Decision and Control, pp.1656-1662, 1996.
DOI : 10.1109/CDC.1996.572783

S. [. Iwasaki and . Hara, Generalization of Kalman-Yakubovich-Popov lemma for restricted frequency inequalities, American Control Conference, pp.3828-3833, 2003.

S. [. Iwasaki and . Hara, Generalized KYP lemma: unified frequency domain inequalities with design applications, IEEE Transactions on Automatic Control, vol.50, issue.1, pp.41-59, 2005.
DOI : 10.1109/TAC.2004.840475

G. [. Iwasaki, M. Meinsma, and . Fu, -procedure and finite frequency KYP lemma, Mathematical Problems in Engineering, vol.6, issue.2-3, pp.305-320, 2000.
DOI : 10.1155/S1024123X00001368
URL : https://hal.archives-ouvertes.fr/hal-00807794

R. [. Iwasaki and . Skelton, All controllers for the general H??? control problem: LMI existence conditions and state space formulas, Automatica, vol.30, issue.8, pp.1307-1317, 1994.
DOI : 10.1016/0005-1098(94)90110-4

G. [. Iwasaki and . Shibata, LPV system analysis via quadratic separator for uncertain implicit systems, IEEE Transactions on Automatic Control, vol.46, issue.8, pp.1195-1208, 2001.
DOI : 10.1109/9.940924

]. T. Iwa93 and . Iwasaki, A Unified Matrix Inequality Approach to Linear Control Design, 1993.

]. T. Iwa97 and . Iwasaki, Robust stability analysis with quadratic separator : Parametric time-varying uncertainty cases, Conference on Decision and Control, pp.4880-4885, 1997.

]. V. Jaker, Jakubovi? c. The S-procedure in nonlinear control theory. Vestnik Leningrad Univ. (russian) Vestnik Leningrad Univ, Math, issue.41, p.1971

F. [. Köse and . Jabbari, Control of LPV systems with partly-measured parameters, Proceedings of the 36th IEEE Conference on Decision and Control, pp.658-663, 1999.
DOI : 10.1109/CDC.1997.657570

K. , I. E. Köse, F. Jabbari, and W. E. Schmitendorf, A direct characterization of L 2 -gain controllers for LPV systems, Bibliographie IEEE Transactions Automatic Control, vol.221, issue.439, pp.1302-1307, 1998.

E. W. Kamen, P. P. Khargonekar, and A. Tannebaum, Control of slowly-varying linear systems, IEEE Transactions on Automatic Control, vol.34, issue.12, pp.1283-1285, 1989.
DOI : 10.1109/9.40776

. R. Kym-+-01-]-m, I. Katebi, M. Yamamoto, M. J. Matsuura, H. Grimble et al., Robust dynamic ship positioning control system design and applications H ? control of nonlinear systems : a convex characterization [LF03] X. Litrico and V. Fromion. Advanced control politics and optimal performance for an irrigation canal [LF05] X. Litrico and V. Fromion. H ? control of an irrigation canal pool with a mixed control politics. may, accepted for publication Analysis of LPV systems using a piecewise affine parameter-dependent Lyapunov function Analysis of linear parameter-varying systems using a non-smooth dissipative systems framework Analysis and control of linear parameter-varying systems, European Control Conference Conference on Decision and ControlLim99] S. LimMeg96] A. Megretski. L 2 BIBO output feedback stabilization with saturated control IFAC World Congress, pp.1257-1284, 1995.

I. Masubuchi, A. Kume, E. [. Shimemura, A. Megretski, . [. Rantzer et al., System analysis via integral quadratic constraints A dual formulation of mixed µ and on the losslessness of (d, g) scaling The projective method for solving linear matrix inequalities, Conference on Decision and Control American Control ConferencePac94] A. Packard. Gain scheduling via linear fractional transformations. Systems and Control LettersRan96] A. Rantzer. On te kalman-yakubovich-popov lemma. Systems and Control LettersRoc70] R.T. Rockafellar. Convex analysis. Princeton Landemarks in Mathematics and Physics, pp.1753-1758, 1970.

. C. Bibliographie-[-rp02-]-d, P. L. Ramos, and . Peres, An LMI condition for the robust stability of uncertain continuous-time linear systems, IEEE Transactions Automatic Control, vol.47, issue.4, pp.675-678, 2002.

L. [. Ran and . Rodman, On parameter dependence of solutions of algebraic riccati equations, Mathematics of Control, Signals, and Systems, vol.58, issue.3, pp.269-284, 1988.
DOI : 10.1007/BF02551288

J. [. Rugh and . Shamma, Research on gain scheduling, Automatica, vol.36, issue.10, pp.1401-1425, 2000.
DOI : 10.1016/S0005-1098(00)00058-3
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.409.6064

G. [. Rossignol, V. Scorletti, and . Fromion, Filter design under magnitude constraints is a finite dimensional convex optimization problem, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), pp.3575-3580, 2001.
DOI : 10.1109/CDC.2001.980414

G. [. Rossignol, V. Scorletti, and . Fromion, Filter design: a finite dimensional convex optimization approach, International Journal of Robust and Nonlinear Control, vol.6, issue.14, pp.1317-1335, 2003.
DOI : 10.1002/rnc.854

]. W. Rug91 and . Rugh, Analytical framework for gain scheduling, IEEE Control Systems Magazine, vol.11, issue.1, pp.79-84, 1991.

M. [. Safonov and . Athans, Robustness and computational aspects of nonlinear stochastic estimators and regulators, IEEE Transactions on Automatic Control, vol.23, issue.4, pp.717-725, 1978.
DOI : 10.1109/TAC.1978.1101825

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

]. M. Saf80 and . Safonov, Stability and Robustness of Multivariable Feedback Systems, 1980.

]. M. Saf83 and . Safonov, Propagation of conic model uncertainty in hierarchical systems, IEEE Transactions on Circuits and Systems, pp.388-396, 1983.

S. [. Shahruz and . Behtash, Design of controllers for linear parameter-varying systems by the gain scheduling technique, Journal of Mathematical Analysis and Applications, vol.168, issue.1, pp.195-217, 1992.
DOI : 10.1016/0022-247X(92)90199-N

]. C. Sch98 and . Scherer, Robust performance analysis for parameter dependent systems using tensor product splines, Conference on Decision and Control, pp.2216-2221, 1998.

]. C. Sch01 and . Scherer, LPV control and full block multipliers, Automatica, vol.73, issue.3, pp.361-375, 2001.

]. C. Sch03 and . Scherer, When are multiplier relaxations exact ?, IFAC Symposium on Robust Control Design, 2003.

]. G. Sco97 and . Scorletti, Approche Unifiée de l'Analyse et de la Commande des Systèmes par Formulation LMI, 1997.

L. [. Scorletti and . Ghaoui, Improved LMI conditions for gain scheduling and related control problems, International Journal of Robust and Nonlinear Control, vol.8, issue.10, pp.845-877, 1998.
DOI : 10.1002/(SICI)1099-1239(199808)8:10<845::AID-RNC350>3.0.CO;2-I

V. [. Scorletti and . Fromion, IntroductionàIntroductionà la commande multivariable des systèmes : méthodes de synthèse fréquentielle H ?, 2005.

V. [. Scorletti, S. Fromion, . [. Font, P. Scherer, M. Gahinet et al., Automatique fréquentielle : des critères graphiquesàgraphiquesà l'optimisation LMI Multiobjective Output-Feedback control via LMI optimization Analysis and design of gain scheduled control systems, JESA IEEE Transactions Automatic Control, vol.37, issue.427, pp.161-185896, 1988.

M. Sampei, T. Mita, M. Nakamichisp96a-]-s, I. Skogestad, . Postlethwaitesp96b-]-r et al., Multivariable Feedback Control, Analysis and Design Optimal control of pertubed linear static systems Interpolation of observer state feedback controllers for gain scheduling Stability preserving interpolation methods for the synthesis of gain scheduled controllers Frequency dependent model phase reduction using convex optimization Monotonic relaxation for robust control : new characterization Monotonic relaxations for robust control : new characterizations Biquadratic stability of uncertain linear systems Guaranteed cost control for rational parameterdependent discrete-time systems, American Control Conference Conference on Decision and ControlTO95] O. Toker and H. Ozbay. On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback American Control Conference Sano. A robust stabilization problem of fuzzy control systems and its application to backing up control of a truck-trailer. IEEE Transactions on Fuzzy Systems, pp.13-24579, 1914.

L. Vandenberghe, V. Balakrishnan, R. Wallin, and A. Hansson, On the implementation of primal-dual interior-point methods for semidefinite programming problems derived from the KYP lemma, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), pp.4658-4663, 1993.
DOI : 10.1109/CDC.2003.1272303

. [. Bibliographie, V. Wang, and . Balakrishnan, Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems, IEEE Transactions Automatic Control, vol.47, issue.5, pp.720-734, 2002.

]. J. Wil71 and . Willems, Least squares stationnary optimal control and the algebraic Riccati equation, IEEE Transactions Automatic Control, issue.166, pp.621-634, 1971.

F. Wu, X. H. Yang, A. Packard, and G. Becker, Induced l 2 -norm control for LPV systems with bounded parameter variations rates, International Journal of Robust and Nonlinear Control, vol.6, pp.9-10983, 1996.

A. [. Yu and . Sideris, H??? control with parametric Lyapunov functions, Systems & Control Letters, vol.30, issue.2-3, pp.57-69, 1997.
DOI : 10.1016/S0167-6911(96)00075-8

]. G. Zam81 and . Zames, Feedback and optimal sensitivity : model reference transformations , multiplicative seminorms, and approximate inverses, IEEE Transactions Automatic Control, AC, vol.26, issue.2, pp.301-320, 1981.

J. [. Zhou, K. Doyle, and . Glover, Robust and optimal control, 1996.

]. X. Zha02 and . Zhang, Parameter?dependent Lyapunov functions stability analysis of linear parameter?dependent dynamical systems, 2002.