A. Annexe, Inégalités matricielles linéaires et compléments mathématiques A.2.2 Pseudo-inverses réflexives

A. Soit and A. +. N×m, A ? ? C m×n est une pseudo-inverse réflexive de A si elle vérifie (A.2.1)

Y. Boukal, M. Darouach, M. Zasadzinski, and N. Radhy, Robust $H_{\infty }$ Observer-Based Control of Fractional-Order Systems With Gain Parametrization, IEEE Transactions on Automatic Control, vol.62, issue.11
DOI : 10.1109/TAC.2017.2690140

URL : https://hal.archives-ouvertes.fr/hal-01643711

Y. Boukal, M. Darouach, M. Zasadzinski, and N. Radhy, Correction to the Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems & a new Enhanced LMI condition, Journal of Applied Nonlinear Dynamics

O. Lahoucine, Y. Boukal, M. Ganaoui, M. Darouach, M. Zasadzinski et al., A general fractional order heat transfer model for photovoltaic/thermal (PVT) hybrid systems and its observer design
URL : https://hal.archives-ouvertes.fr/hal-01242706

Y. Boukal, M. Zasadzinski, M. Darouach, and N. Radhy, Robust H ? Observer-based stabilization of disturbed uncertain fractional order systems using a two-step procedure . Lecture Notes in Electrical Engineering 357 : Theoretical Developments and Applications of Non-Integer Order Systems, Part : III -Chapter, pp.14-167, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01254955

Y. Boukal, M. Darouach, M. Zasadzinski, and N. E. Radhy, Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems, Journal of Applied Nonlinear Dynamics, vol.4, issue.2, pp.117-130, 2015.
DOI : 10.5890/JAND.2015.06.002

URL : https://hal.archives-ouvertes.fr/hal-01194310

M. Boukal, M. Zasadzinski, N. Darouach, and . Radhy, H ??? Dynamic Output Feedback Controller Design For Disturbed Fractional-Order Systems, B.2 Conférences internationales avec actes et comité de lecture 1, 2017.
DOI : 10.1016/j.ifacol.2017.08.2059

B. Annexe, M. Boukal, M. Darouach, N. Zasadzinski, and . Radhy, H ? observer-based-controller for fractional-order time-varying-delay systems, Liste des publications 2, 2017.

Y. Boukal, M. Zasadzinski, M. Darouach, and N. Radhy, Robust functional observer design for uncertain fractional-order time-varying delay systems, 2016 American Control Conference (ACC), pp.2741-2746, 2016.
DOI : 10.1109/ACC.2016.7525333

URL : https://hal.archives-ouvertes.fr/hal-01293998

Y. Boukal, M. Zasadzinski, M. Darouach, and N. Radhy, Stability and stabilizability analysis of fractional-order time-varying delay systems via diffusive representation, 2016 5th International Conference on Systems and Control (ICSC), pp.262-266, 2016.
DOI : 10.1109/ICoSC.2016.7507077

URL : https://hal.archives-ouvertes.fr/hal-01317099

Y. Boukal, M. Zasadzinski, M. Darouach, and N. Radhy, H ? filters design for fractional order time-varying delay systems, dans European Control Conference (ECC), pp.1243-1248, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01141213

O. Lahoucine, Y. Boukal, M. Ganaoui, M. Darouach, M. Zasadzinski et al., A general fractional order heat transfer model for photovoltaic/thermal (PVT) hybrid systems and its observer design, dans International Conference on Materials & Energy, p.2015
URL : https://hal.archives-ouvertes.fr/hal-01242706

Y. Boukal, M. Darouach, M. Zasadzinski, and N. E. Radhy, H ? observer design for linear fractional-order systems in time and frequency domains, European Control Conference (ECC), pp.2975-2980, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01017629

Y. Boukal, M. Darouach, M. Zasadzinski, and N. Radhy, Design of functional fractional-order observers for linear time-delay fractional-order systems in the time domain, ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014, p.2014
DOI : 10.1109/ICFDA.2014.6967356

URL : https://hal.archives-ouvertes.fr/hal-01017625

Y. Boukal, N. E. Radhy, M. Darouach, and M. Zasadzinski, Design of full and reduced orders observers for linear fractional-order systems in the time and frequency domains, 3rd International Conference on Systems and Control, pp.171-176, 2013.
DOI : 10.1109/ICoSC.2013.6750854

. S. Bibliographie-[-ac08-]-h, Y. Ahn, and . Chen, Necessary and sufficient stability condition of fractionalorder interval linear systems, Automatica, vol.44, pp.2985-2988, 2008.

Y. [. Ahn, I. Chen, and . Podlubny, Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Applied Mathematics and Computation, vol.187, issue.1, pp.27-34, 2007.
DOI : 10.1016/j.amc.2006.08.099

T. [. Achar, C. F. Hartley, and . Lorenzo, Theoretical Developments and Applications in Physics and Engineering, chapter The Caputo fractional derivative : initialization issues relative to fractional differential equations, pp.27-42, 2007.

[. Anh and R. Mcvinish, Fractional differential equations driven by L??vy noise, Journal of Applied Mathematics and Stochastic Analysis, vol.16, issue.2, pp.97-119, 1900.
DOI : 10.1155/S1048953303000078

]. R. Arg53, A propos d'une note de M. Pierre Humbert, C.R. Académie des Sciences, vol.236, pp.2031-2032, 1953.

J. [. Ahmad and . Sprott, Chaos in fractional-order autonomous nonlinear systems, Chaos, Solitons & Fractals, vol.16, issue.2, pp.339-351, 2003.
DOI : 10.1016/S0960-0779(02)00438-1

]. O. Bac98 and . Bachelier, Commande des Systèmes Linéaires Incertains : Placement de Pôles Robuste en D-Stabilité, 1998.

]. R. Bag79 and . Bagley, Application of Generalized Derivatives to Viscoelasticity, 1979.

]. R. Bag89 and . Bagley, The initial value problem for fractional order differential equations with constant coefficients, 1989.

R. [. Bagley and . Calico, The fractional order state equations for the control of viscoelastically damped structures, 30th Structures, Structural Dynamics and Materials Conference, pp.304-311, 1991.
DOI : 10.1115/1.3167615

M. [. Boutayeb and . Darouach, Observers for discrete-time systems with multiple delays, IEEE Transactions on Automatic Control, vol.46, issue.5, pp.746-750, 2001.
DOI : 10.1109/9.920794

M. Zasadzinski and N. E. Radhy, Design of functional fractional-order observers for linear time-delay fractional-order systems in the time domain, Fractional Differentiation and Its Applications (ICFDA), International Conference on, pp.1-6, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01017625

M. Zasadzinski and N. E. Radhy, H ? observer design for linear fractional-order systems in time and frequency domain, Proc. European Contr. Conf, pp.2975-2980, 2014.

M. [. Boukal, M. Darouach, N. E. Zasadzinski, and . Radhy, Unknown Input Observer Design for Linear Fractional-Order Time-Delay Systems, Journal of Applied Nonlinear Dynamics, vol.4, issue.2, pp.117-130, 2015.
DOI : 10.5890/JAND.2015.06.002

URL : https://hal.archives-ouvertes.fr/hal-01194310

Y. Boukal, M. Darouach, M. Zasadzinski, and N. E. Radhy, Robust H ? observerbased control of fractional-order systems with gain parametrization, IEEE Transactions on Automatic Control, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01643711

H. [. Besançon and . Hammouri, Reduced order observer for a class of non-uniformly observable systems, Proceedings of 1995 34th IEEE Conference on Decision and Control, 1995.
DOI : 10.1109/CDC.1995.478658

H. [. Boroujeni and . Momeni, Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems, Signal Processing, vol.92, issue.10, pp.922365-2370, 2012.
DOI : 10.1016/j.sigpro.2012.02.009

O. [. Bouafoura, N. B. Moussi, and . Braiek, A fractional state space realization method with block pulse basis, Signal Processing, vol.91, issue.3, pp.492-497, 2011.
DOI : 10.1016/j.sigpro.2010.04.010

]. M. Bou01 and . Boutayeb, Observer design for linear time-delay systems, Syst. & Contr. Letters, vol.44, pp.103-109, 2001.

J. [. Bonnet and . Partington, Stabilization of fractional exponential systems including delays, IFAC Proceedings Volumes, vol.33, issue.23, pp.345-353, 2001.
DOI : 10.1016/S1474-6670(17)36916-1

J. [. Bonnet and . Partington, Analysis of fractional delay systems of retarded and neutral type, Automatica, vol.38, issue.7, pp.1133-1138, 2002.
DOI : 10.1016/S0005-1098(01)00306-5

N. [. Boukal, M. Radhy, M. Darouach, . L. Zasadzinskibt83-]-r, P. J. Bagley et al., Design of full and reduced observers for linear fractional-order systems in the time and frequency domains A theoretical basis for the application of fractional calculus to viscoelasticity, Proc. International Conference on Systems and Control, pp.201-210, 1983.

P. [. Bagley and . Torvik, On the appearance of the fractional derivatives in the behaviour of real materials, J. Applied Mechanics, vol.41, pp.294-298, 1984.

P. [. Bagley and . Torvik, On the Fractional Calculus Model of Viscoelastic Behavior, Journal of Rheology, vol.30, issue.1, pp.918-925, 1986.
DOI : 10.1122/1.549887

]. M. Bus08 and . Bus?owicz, Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin Of The Polish Academy Of Sciences Technical Sciences, pp.237-240, 2008.

M. [. Boukal, M. Zasadzinski, N. E. Darouach, and . Radhy, H ? filters design for fractional-order time-varying delay systems, Proc. European Contr. Conf, pp.1243-1248, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01141213

]. Y. Bzdr16a, M. Boukal, M. Zasadzinski, N. E. Darouach, and . Radhy, Robust functional observer design for uncertain fractional-order time-varying delay systems, 2016.

]. Y. Bzdr16b, M. Boukal, M. Zasadzinski, N. Darouach, and . Radhy, Stability and stabilizability analysis of fractional-order time-varying delay systems via diffusive representation, Systems and Control (ICSC), 2016 5th International Conference on, pp.262-266, 2016.

]. M. Cap67 and . Caputo, Linear model of dissipation whose Q is almost frequency independent, Geophysical Journal of the Royal Astronomical Society, vol.13, pp.529-539, 1967.

H. [. Chen, I. Ahn, and . Podlubny, Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Processing, vol.86, issue.10, pp.2611-2618, 2006.
DOI : 10.1016/j.sigpro.2006.02.011

G. [. Caponetto and L. Dongola, Fortuna, and I. Petrá? s. Fractional Order Systems : Modeling and Control Applications, World scientific series on nonlinear science, Series A. World scientific, 2010.

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

P. [. Chilali, P. Gahinet, and . 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

[. Carlson and C. Halijak, Approximation of Fractional Capacitors(1/s)^(1/n)by a Regular Newton Process, Méthodes LMI pour l'Analyse et la Synthèse Multi-critère, pp.210-213, 1964.
DOI : 10.1109/TCT.1964.1082270

F. [. Caputo and . Mainardi, A new dissipation model based on memory mechanism, Pure and Applied Geophysics PAGEOPH, vol.19, issue.1, pp.134-137, 1971.
DOI : 10.1007/BF00879562

A. Charef, . Sun, and . Tsao, Fractal system as represented by singularity function, IEEE Transactions on Automatic Control, vol.37, issue.9, pp.1465-1470, 1992.
DOI : 10.1109/9.159595

[. Chen, B. M. Vinagre, and I. Podlubny, On Fractional Order Disturbance Observer, Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, pp.617-624, 2003.
DOI : 10.1115/DETC2003/VIB-48371

]. M. Dar00 and . Darouach, Existence and design of functional observers for linear systems, IEEE Trans. Aut. Control, vol.45, pp.940-943, 2000.

]. M. Dar01 and . Darouach, Linear functional observers for systems with delays in state variable, IEEE Trans. Aut. Control, vol.46, pp.491-496, 2001.

]. M. Dar07 and . Darouach, Unknown inputs observers design for delay systems, Asian Journal of Control, vol.9, pp.426-434, 2007.

]. M. Dar09 and . Darouach, Complements to full order observer design for linear systems with unknown inputs, Applied Mathematics Letters, vol.22, pp.1107-1111, 2009.

]. S. Das08 and . Das, Functional Fractional Calculus for System Identification and Controls, 2008.

H. [. Dadras and . Momeni, Fractional sliding mode observer design for a class of uncertain fractional order nonlinear systems, IEEE Conference on Decision and Control and European Control Conference, pp.6925-6930, 2011.
DOI : 10.1109/CDC.2011.6161100

]. L. Dor94 and . Dorckák, Numerical models for simulation the fractional-order control systems, 1994.

D. [. Dzielinski and . Sierociuk, OBSERVER FOR DISCRETE FRACTIONAL ORDER STATE-SPACE SYSTEMS, 2nd IFAC Workshop on Fractional Differentiation and its Applications, pp.511-516, 2006.
DOI : 10.3182/20060719-3-PT-4902.00085

M. [. Darouach and . Zasadzinski, State estimation for a class of singular systems, International Journal of Systems Science, vol.23, issue.4, pp.517-530, 1992.
DOI : 10.1109/TAC.1981.1102763

URL : https://hal.archives-ouvertes.fr/hal-00143555

]. M. Dzh66 and . Dzhrbashyan, Integral transforms and representations of functions in the complex domain, 1966.

M. [. Darouach, S. J. Zasadzinski, and . Xu, Full-order observers for linear systems with unknown inputs, IEEE Transactions on Automatic Control, vol.39, issue.3, pp.606-609, 1994.
DOI : 10.1109/9.280770

URL : https://hal.archives-ouvertes.fr/hal-00098125

]. N. Eng96 and . Engheta, On fractional calculus and fractional multipoles in electromagnetism, IEEE Trans. Antennas and Propagation, vol.44, pp.554-566, 1996.

]. A. Erd55a and . Erdélyi, Higher Transcendental Functions, 1955.

]. A. Erd55b and . Erdélyi, Higher Transcendental Functions, 1955.

]. A. Erd55c and . Erdélyi, Higher Transcendental Functions, 1955.

C. Farges, L. Fadiga, and J. Sabatier, analysis and control of commensurate fractional order systems, Mechatronics, vol.23, issue.7, pp.772-780, 2013.
DOI : 10.1016/j.mechatronics.2013.06.005

URL : https://hal.archives-ouvertes.fr/hal-00838800

C. [. Fadiga, J. Farges, K. Sabatier, and . Santugini, H ? output feedback control of commensurate fractional order systems, European Control Conference (ECC), pp.4538-4543, 2013.

J. [. Francis, G. Helton, and . Zames, H??? - optimal feedback controllers for linear multivariable systems, IEEE Trans. Aut. Control, vol.29, pp.888-900, 1984.
DOI : 10.1007/BFb0031065

K. [. Farshad and . Masoud, An efficient numerical algorithm for stability testing of fractional-delay systems, Moze, and J. Sabatier. Pseudo-state feedback stabilization of commensurate fractional order systems, pp.32-371730, 2009.

[. Fadiga, C. Sabatier, and . Farges, H??? state feedback control of commensurate fractional order systems, Proceedings of the 6th IFAC SSSC-FDA Joint Conference, pp.54-59, 2013.
DOI : 10.3182/20130204-3-FR-4032.00200

H. [. Fernando and . Trinh, A procedure for designing linear functional observers, Applied Mathematics Letters, vol.26, issue.2, pp.240-243, 2013.
DOI : 10.1016/j.aml.2012.09.002

H. [. Fernando, L. Trinh, and . Jennings, Functional Observability and the Design of Minimum Order Linear Functional Observers, IEEE Transactions on Automatic Control, vol.55, issue.5, pp.1268-1273, 2010.
DOI : 10.1109/TAC.2010.2042761

D. [. Fortmann and . Williamson, Design of low-order observers for linear feedback control laws, IEEE Transactions on Automatic Control, vol.17, issue.3, pp.301-308, 1972.
DOI : 10.1109/TAC.1972.1100006

S. [. Guermah, M. Djennoune, and . Bettayeb, Controllability and Observability of Linear Discrete-Time Fractional-Order Systems, International Journal of Applied Mathematics and Computer Science, vol.1, issue.2, pp.213-222, 2008.
DOI : 10.2478/v10006-008-0019-6

A. [. Gorenflo, F. Kilbas, S. V. Mainardi, and . Rogosin, Mittag-Leffler functions, related topics and applications, 2014.
DOI : 10.1007/978-3-662-43930-2

[. Humbert and R. Agarwal, Sur la fonction de mittag-leffler et quelques-unes de ses généralisations, Bull. Sci. Math, vol.77, issue.2, pp.180-185, 1953.

]. R. Hil00 and . Hilfer, Applications of Fractional Calculus in Physics, 2000.

C. [. Horn and . Johnson, Matrix Analysis, 1985.

T. Tom, C. F. Hartley, and . Lorenzo, Dynamics and control of initialized fractionalorder systems, Nonlinear Dynamics, vol.29, issue.1-4, pp.201-233, 2002.

C. [. Hartley and . Lorenzo, The Initialization Response of Linear Fractional-Order Systems With Constant History Function, Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C, pp.1321-1325, 2009.
DOI : 10.1115/DETC2009-87631

. T. Bibliographie-[-hlq95-]-t, C. F. Hartley, H. K. Lorenzo, and . Qammer, Chaos in a fractional order Chua's system, IEEE Trans. Circ. Syst. I : Fund. Theory & Appli, vol.42, pp.485-490, 1995.

T. Hartley, C. Lorenzo, J. C. Trigeassou, and N. Maamri, Equivalence of History-Function Based and Infinite-Dimensional-State Initializations for Fractional-Order Operators, IFAC Proceedings Volumes, pp.41014-31227, 1998.
DOI : 10.1115/1.4023865

]. I. Hor63, . Horowitzhot98-]-r, and . Hotzel, Synthesis of Feedback Systems Contribution à la Théorie Structurelle et à la Commande des Systèmes Linéaires Fractionnaires Superiority of transfer function over state-variable methods in linear time-invariant feedback system design, IEEE Trans. Aut. Control, vol.20, pp.84-97, 1963.

[. Humbert, Quelques résultats relatifs à la fonction de mittag-leffler

C. Rendus, H. Des, . De-l-academie-des-scienceshw79-]-g, E. Hardy, and . Wright, An introduction to the theory of numbers, pp.1467-1468, 1953.

J. [. Jesus and . Machado, Development of fractional order capacitors based on??electrolyte processes, Nonlinear Dynamics, vol.351, issue.8, pp.45-55, 2009.
DOI : 10.1007/s11071-008-9377-8

[. Jin and M. J. Tahk, Time-delayed state estimator for linear systems with unknown inputs, Int. J. Cont., Autom. Syst, vol.3, pp.117-121, 2005.

Y. [. Jiao and . Zhong, Robust stability for fractional-order systems with structured and unstructured uncertainties, Computers & Mathematics with Applications, vol.64, issue.10
DOI : 10.1016/j.camwa.2012.03.011

]. R. Kal60 and . Kalman, A new approach to linear filtering and prediction problems, ASME Trans. -Part D, J. Basic Engineering, vol.82, pp.34-45, 1960.

]. H. Kha92 and . Khalil, Nonlinear Systems, 1992.

]. S. Khls81a, N. A. Karunathilaka, R. Hampson, T. J. Leek, and . Sinclair, The impedance of the alkaline zinc-manganese dioxide cell. i. variation with state of charge, Journal of Applied Electrochemistry, vol.11, pp.365-372, 1981.

]. S. Khls81b, N. A. Karunathilaka, R. Hampson, T. J. Leek, and . Sinclair, The impedance of the alkaline zinc-manganese dioxide cell. ii. an interpretation of the data, Journal of Applied Electrochemistry, vol.11, pp.715-721, 1981.

A. [. Kreindler and . Jameson, Conditions for nonnegativeness of partitioned matrices, IEEE Transactions on Automatic Control, vol.17, issue.1, pp.147-148, 1972.
DOI : 10.1109/TAC.1972.1099894

S. Kawaji and H. S. Kim, Full order observer for linear descriptor systems with unknown-inputs, Proc. IEEE Conf. Decision & Control, 1995.

I. [. Khargonakar, K. Petersen, and . Zhou, Robust stabilization of uncertain linear systems: quadratic stabilizability and H/sup infinity / control theory, IEEE Transactions on Automatic Control, vol.35, issue.3, pp.356-361, 1990.
DOI : 10.1109/9.50357

]. A. Las99 and . Lasia, Modern Aspects of Electrochemistry, Kluwer Academic, 1999.

M. Lazarevi?, Further results on fractional order control of a mechatronic system, Scientific Technical Review, vol.206, p.2013, 1820.

G. [. Lu and . Chen, Robust stability and stabilization of fractional-order interval systems : an LMI approach, IEEE Trans. Aut. Control, vol.54, pp.1294-1299, 2009.

Y. [. Lu and . Chen, Robust stability and stabilization of fractional-order interval systems with the fractional-order ? : The 0 < ? < 1 case, IEEE Trans. Aut. Control, vol.55, pp.152-158, 2010.

Y. [. Li, I. Chen, and . Podlubny, Mittag???Leffler stability of fractional order nonlinear dynamic systems, Automatica, vol.45, issue.8, pp.1965-1969, 2009.
DOI : 10.1016/j.automatica.2009.04.003

Y. [. Li, I. Chen, and . Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag???Leffler stability, Computers & Mathematics with Applications, vol.59, issue.5, pp.1810-1821, 2010.
DOI : 10.1016/j.camwa.2009.08.019

A. Letnikov, [. Li, and M. Fu, Theory of differentiation of fractional order A linear matrix inequality approach to robust H ? filtering, Mat. Sb IEEE Trans. Sign. Proc, vol.3, issue.45, pp.18682338-2350, 1997.

. H. Lgc-+-14-]-y, H. B. Lan, C. X. Gu, Y. Chen, Y. P. Zhou et al., An indirect lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks, Neurocomputing, vol.136, pp.235-242, 2014.

T. [. Lorenzo and . Hartley, Initialization in fractional order systems, Proc. European Contr. Conf, 2001.

W. Li and Y. Hori, Vibration Suppression Using Single Neuron-Based PI Fuzzy Controller and Fractional-Order Disturbance Observer, IEEE Transactions on Industrial Electronics, vol.54, issue.1, pp.117-126, 2007.
DOI : 10.1109/TIE.2006.888771

R. [. Lanusse, P. Malti, and . Melchior, CRONE control system design toolbox for the control engineering community: tutorial and case study, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.15, issue.1990, p.37120120149, 1990.
DOI : 10.1016/j.conengprac.2006.11.018

J. Lofberg, YALMIP : a toolbox for modeling and optimization in MATLAB, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508), pp.284-289, 2004.
DOI : 10.1109/CACSD.2004.1393890

A. [. Lanusse, D. Oustaloup, and . Sutter, Multi-scalar CRONE control of multivariable plants, Wsc'96-Isiac, 1996.

Z. [. Liqiong and . Shouming, Finite-time stability analysis of fractional-order with multi-state time delay, International Journal of Information and Mathematical Sciences, vol.6, issue.4, pp.237-240, 2010.

M. [. Lancaster and . Tismenetsky, The Theory of Matrices, 1985.

J. Guo and L. , Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal. Physica A : Statistical Mechanics and its Applications, pp.107-118, 2006.

G. David and . Luenberger, Determining the state of a linear system with observers of low dynamic order, 1963.

G. David and . Luenberger, Observing the state of a linear system, IEEE transactions on military electronics, pp.74-80, 1964.

]. D. Lue66 and . Luenberger, Observers for multivariable systems, IEEE Trans. Aut. Control, vol.11, pp.190-197, 1966.

]. D. Lue71 and . Luenberger, An introduction to observers, IEEE Trans. Aut. Control, vol.16, pp.596-603, 1971.

S. Liang, Y. Wei, J. Pan, Q. Gao, and Y. Wang, Bounded real lemmas for fractional order systems, International Journal of Automation and Computing, vol.50, issue.1, pp.192-198, 2015.
DOI : 10.1109/TAC.2004.840475

]. A. Lya92a and . Lyapunov, The General Problem of Stability of Motion, p.1892

]. A. Lya92b and . Lyapunov, The General Problem of Stability of Motion, IJC, Lyapinov Centenary Issue, vol.55, issue.3, 1992.

]. A. Lya92 and . Lyapunov, The general problem of stability of motion, Int. J. Contr, vol.55, pp.531-773, 1992.

X. Lianglin, Z. Yun, and J. Tao, Stability analysis of linear fractional order neutral system with multiple delays by algebraic approach, World Academy of Science, Engineering and Technology, vol.5, issue.4, pp.758-761, 2011.

M. [. Mihailo and . Aleksandar, Finite-time stability analysis of fractional order time-delay systems : Gronwall's approach, Mathematical and Computer Modelling, vol.49, pp.3-4475, 2009.

M. [. Malti, F. Aoun, A. Levron, and . Oustaloup, Analytical computation of the -norm of fractional commensurate transfer functions, Automatica, issue.11, pp.472425-2432, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00668265

B. [. Matignon and . Andréa, Some results on controllability and observability of finite-dimensional fractional differential systems, Proc. Mathematical Theory of Networks and Systems Symposium, 1996.

D. Matignon and B. Andréa, Observer-based controllers for fractional differential systems, Proceedings of the 36th IEEE Conference on Decision and Control, 1997.
DOI : 10.1109/CDC.1997.649835

]. D. Mat94 and . Matignon, Représentation en Variables d'État de Modèles de Guides d'Ondes avec Dérivation Fractionnaire, 1994.

]. D. Mat96 and . Matignon, Stability results for fractional differential equations with applications to control processing, Proc. IEEE-IMACS Syst. Man Cyber. Conf, 1996.

. [. Mittag-leffler, Sur la nouvelle fonction E ? (x), C.R. Académie des Sciences, vol.137, pp.554-558, 1903.

. [. Mittag-leffler, Sur la repr??sentation analytique d???une branche uniforme d???une fonction monog??ne: cinqui??me note, Acta Mathematica, vol.29, issue.0, pp.101-182, 1905.
DOI : 10.1007/BF02403200

G. [. Mbodje and . Montseny, Boundary fractional derivative control of the wave equation, IEEE Transactions on Automatic Control, vol.40, issue.2, pp.378-382, 1995.
DOI : 10.1109/9.341815

P. [. Malti, P. Melchior, A. Lanusse, and . Oustaloup, Towards an object oriented CRONE Toolbox for fractional differential systems*, Proceedings of 18th IFAC World Congress, pp.10830-10835, 2011.
DOI : 10.3182/20110828-6-IT-1002.02443

URL : https://hal.archives-ouvertes.fr/hal-00668323

P. [. Malti, P. Melchior, A. Lanusse, and . Oustaloup, Object-oriented crone toolbox for fractional differential signal processing. Signal, Image and Video Processing, pp.393-400, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00804776

G. Montseny, Diffusive representation of pseudo-differential time-operators, ESAIM: Proceedings, vol.5
DOI : 10.1051/proc:1998005

]. N. Mra04 and . Mrani, Contribution à l'Étude des Systèmes Fractionnaires : Théorie et Applications, 2004.

M. Moze, J. Sabatier, and A. Oustaloup, LMI Tools for Stability Analysis of Fractional Systems, Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C, 2005.
DOI : 10.1115/DETC2005-85182

M. Moze, J. Sabatier, and A. Oustaloup, On fractional systems H ? -norm computation, Proc. IEEE Conf. Decision & Control, 2005.

J. [. Moze, A. Sabatier, and . Oustaloup, On bounded real lemma for fractional systems, Proc. Triennal IFAC World Congress, 2008.
DOI : 10.3182/20080706-5-KR-1001.02582

T. Nakamizo, Reduced-order functional estimator for linear stochastic systems . Statistical methods in control and signal processing, 1997.

. [. N-'doye, Généralisation du Lemme de Gronwall-Bellman pour la Stabilisation des Systèmes Fractionnaires, 2011.

I. N-'doye, M. Darouach, H. Voos, and M. Zasadzinski, Design of unknown input fractional-order observers for fractional-order systems, Int. J. Appli. Math. Comput. Sci, vol.23, pp.491-500, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00878442

]. I. Ndz12a, M. N-'doye, M. Darouach, and . Zasadzinski, Functional observers for fractionalorder systems with unknown inputs, Proc. International Conference on Systems and Control, 2012.

]. I. Ndz12b, M. N-'doye, M. Darouach, and . Zasadzinski, Observers design for linear and nonlinear fractional-order systems, Proc. IEEE Conf. Decision & Control

I. N-'doye, M. Darouach, M. Zasadzinski, and N. E. Radhy, Observers design for singular fractional-order systems, Proc. IEEE Conf. Decision & Control, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00651140

M. [. N-'doye, M. Darouach, N. E. Zasadzinski, and . Radhy, Robust stabilization of uncertain descriptor fractional-order systems, Automatica, vol.49, pp.1907-1913, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00878431

H. Nvd-+-13-]-i.-n-'doye, M. Voos, J. G. Darouach, N. Schneider, and . Knauf, H ? static output feedback control for a fractional-order glucose-insulin system, Fractional Differentiation and Its Applications, pp.266-271, 2013.

M. [. N-'doye, M. Zasadzinski, N. E. Darouach, and . Radhy, Observer-based control for fractional-order continuous-time systems, Proc. IEEE Conf. Decision & Control, 2009.

A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, Frequency-band complex noninteger differentiator : characterization and synthesis. Circuits and Systems I : Fundamental Theory and Applications, IEEE Transactions on, vol.47, issue.1, pp.25-39, 2000.

B. [. Olsson and . Newell, Wastewater Treatment Systems: Modelling, Diagnosis and Control, Water Intelligence Online, vol.4, issue.0, 1999.
DOI : 10.2166/9781780402864

]. A. Osl-+-08, J. Oustaloup, P. Sabatier, R. Lanusse, P. Malti et al., An overview of the crone approach in system analysis, modeling and identification , observation and control, Proc. of the 17th World Congress IFAC, pp.6-11, 2008.

]. A. Ous95 and . Oustaloup, La Dérivation Non Entière : Synthése et Applications, Hermes, 1995.

G. [. Owen and . Zamess, Duality theory of robust disturbance attenuation, Automatica, vol.29, issue.3, pp.695-705, 1993.
DOI : 10.1016/0005-1098(93)90064-Z

Y. [. Petrá?-s, B. M. Chen, and . Vinagre, Unsolved problems in the mathematics of systems and control, chapter Robust stability test for interval fractionalorder linear systems, pp.208-210, 2004.

Y. [. Petrá?-s, B. M. Chen, I. Vinagre, and . Podlubny, Stability of linear time invariant systems with interval fractional orders and interval coefficients, Second IEEE International Conference on Computational Cybernetics, 2004. ICCC 2004., 2005.
DOI : 10.1109/ICCCYB.2004.1437745

W. Press, B. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes in c : The art of scientific computing, 1992.

E. [. Pisano, Z. Usai, and M. Rapai?, Second-order sliding mode approaches to disturbance estimation and fault detection in fractional-order systems, IFAC Proceedings Volumes, vol.44, issue.1, pp.2436-2441, 2011.
DOI : 10.3182/20110828-6-IT-1002.00984

[. Roy, On the Realization of a Constant-Argument Immittance or Fractional Operator, IEEE Transactions on Circuit Theory, vol.14, issue.3, pp.264-274, 1967.
DOI : 10.1109/TCT.1967.1082706

J. [. Rosenblum and . Rovnyak, Hardy Classes and Operator Theory, 1985.

A. [. Radwan, A. S. Soliman, A. Elwakil, and . Sedeek, On the stability of linear systems with fractional-order elements, Chaos, Solitons & Fractals, vol.40, issue.5, pp.2317-2328, 2009.
DOI : 10.1016/j.chaos.2007.10.033

A. Si-ammour, S. Djennoune, and M. Bettayeb, A sliding mode control for linear fractional systems with input and state delays, Communications in Nonlinear Science and Numerical Simulation, vol.14, issue.5, pp.2310-2318, 2009.
DOI : 10.1016/j.cnsns.2008.05.011

J. [. Schmidt and . Drumheller, Dielectric Properties of Lithium Hydrazinium Sulfate, Physical Review B, vol.4, issue.12, pp.4582-4597, 1971.
DOI : 10.1002/pol.1966.150040514

J. Sabatier, C. Farges, M. Merveillaut, and L. Feneteau, On observability and pseudo state estimation of fractional order systems Multiobjective output-feedback control via LMI optimization, European J. Contr. IEEE Trans. Aut. Control, vol.18, issue.42, pp.260-271896, 1997.

C. [. Sundaram and . Hadjicotis, Comments on " Time-delayed state estimator for linear systems with unknown inputs On mittag-leffler type function and applications, Int. J. Cont., Autom. Syst. Integral Transforms and Special Functions, vol.3, issue.712, pp.646-64797, 1998.

J. [. Shen and . Lam, control of commensurate fractional-order systems, International Journal of Systems Science, vol.7, issue.3, pp.363-372, 2014.
DOI : 10.1049/iet-cta.2010.0746

J. Sabatier, M. Moze, and C. Farges, LMI stability conditions for fractional order systems, Computers & Mathematics with Applications, vol.59, issue.5, pp.1594-1609, 2010.
DOI : 10.1016/j.camwa.2009.08.003

URL : https://hal.archives-ouvertes.fr/hal-00368173

J. Sabatier, M. Moze, and C. Farges, LMI stability conditions for fractional order systems, Computers & Mathematics with Applications, vol.59, issue.5, pp.1594-1609, 2010.
DOI : 10.1016/j.camwa.2009.08.003

URL : https://hal.archives-ouvertes.fr/hal-00368173

J. Sabatier, M. Merveillaut, L. Fenetau, and A. Oustaloup, On Observability of Fractional Order Systems, Volume 4: 7th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B and C, pp.253-260, 2009.
DOI : 10.1115/DETC2009-87262

URL : https://hal.archives-ouvertes.fr/hal-00585437

M. [. Sabatier, A. Moze, and . Oustaloup, On fractional systems H ? -norm computation, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on, pp.5758-5763, 2005.

]. M. Sou96 and . Soula, Étude du Comportement Mécanique des Matériaux Viscoélastiques par les Dérivées Fractionnaires, Conservatoire National des Arts et Métiers de, 1996.

[. Trigeassou and N. Maamri, State space modeling of fractional differential equations and the initial condition problem, 2009 6th International Multi-Conference on Systems, Signals and Devices, pp.1-7, 2009.
DOI : 10.1109/SSD.2009.4956721

N. [. Trigeassou and . Maamri, Initial conditions and initialization of linear fractional differential equations, Signal Processing, vol.91, issue.3, pp.427-436, 2011.
DOI : 10.1016/j.sigpro.2010.03.010

[. Trigeassou, N. Maamri, and A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Symposium on Fractional Signal and Systems, 2009.
DOI : 10.1016/j.sigpro.2010.04.024

URL : https://hal.archives-ouvertes.fr/hal-00585448

J. Trigeassou, N. Maamri, and A. Oustaloup, The infinite state approach: Origin and necessity, Computers & Mathematics with Applications, vol.66, issue.5, pp.892-907, 2013.
DOI : 10.1016/j.camwa.2012.11.020

J. C. Trigeassou, N. Maamri, J. Sabatier, and A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Processing, vol.91, issue.3, pp.437-445, 2011.
DOI : 10.1016/j.sigpro.2010.04.024

URL : https://hal.archives-ouvertes.fr/hal-00585448

]. J. Tmso12a, N. Trigeassou, J. Maamri, A. Sabatier, and . Oustaloup, State variables and transients of fractional order differential systems, Computers & Mathematics with Applications, vol.64, issue.10, pp.3117-3140, 2012.

S. [. Trinh, T. Nahavandi, and . Tran, Algorithms for designing reduced-order functional observers of linear systems, International journal of innovative computing, information and control, vol.4, issue.2, pp.321-333, 2008.

]. C. Tsu04 and . Tsui, An overview of the applications and solutions of a fundamental matrix equation pair, Journal of the Franklin Institute, vol.341, issue.6, pp.465-475, 2004.

]. M. Vid69 and . Vidyasagar, On matrix measures and convex Liapunov functions, J. of Mathematical Analysis and Applications, vol.62, pp.90-103, 1969.

]. M. Vid93 and . Vidyasagar, Nonlinear Systems Analysis, 1993.

B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, Some Approximations of Fractional Order Operators used in Control Theory and Applications, Fractional Calculus & Applied Analysis, vol.3, pp.231-248, 2000.

A. [. Vidyasagar and . Vanelli, New relationships between input-output and Lyapunov stability, IEEE Transactions on Automatic Control, vol.27, issue.2, pp.481-483, 1982.
DOI : 10.1109/TAC.1982.1102937

]. L. Xd90, C. E. Xie, and . De-souza, Robust H ? control for linear time-invariant systems with norm-bounded uncertainty in the input matrix, Syst. & Contr. Letters, vol.14, pp.389-396, 1990.

]. L. Xd92, C. E. Xie, and . De-souza, Robust H ? control for linear systems with norm-bounded time-varying uncertainties, IEEE Trans. Aut. Control, vol.37, pp.1188-1191, 1992.

]. L. Xdf91, C. E. Xie, M. De-souza, and . Fu, H ? estimation for discrete-time linear uncertain systems, Int. J. Robust & Nonlinear Contr, vol.1, pp.111-123, 1991.

J. [. Xing and . Lu, Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: An LMI approach, Chaos, Solitons & Fractals, vol.42, issue.2, pp.1163-1169, 2009.
DOI : 10.1016/j.chaos.2009.03.017

J. [. Xu, S. Lu, C. Zhou, and . Yang, Design of observers for a class of discrete-time uncertain nonlinear systems with time delay, Journal of the Franklin Institute, vol.341, issue.3, pp.295-308, 2004.
DOI : 10.1016/j.jfranklin.2003.12.012

R. [. Yang and . Wilde, Observers for linear systems with unknown inputs, IEEE Transactions on Automatic Control, vol.33, issue.7, pp.677-681, 1988.
DOI : 10.1109/9.1278

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

M. [. Zemouche, G. I. Boutayeb, K. Bara, J. C. Zhou, K. Doyle et al., On observers design for nonlinear timedelay systems Robust and Optimal Control, Proc. IEEE American Control Conf, 1996.

E. [. Zasadzinski, M. Magarotto, and . Darouach, Unknown input reduced order observer for singular bilinear systems with bilinear measurements, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000.
DOI : 10.1109/CDC.2000.912866