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LMI stability conditions for fractional order systems. (English) Zbl 1189.34020
Summary: After an overview of the results dedicated to stability analysis of systems described by differential equations involving fractional derivatives, also denoted fractional order systems, this paper deals with Linear Matrix Inequality (LMI) stability conditions for fractional order systems. Under commensurate order hypothesis, it is shown that a direct extension of the second Lyapunov’s method is a tedious task. If the fractional order ν is such that 0<ν<1, the stability domain is not a convex region of the complex plane. However, through a direct stability domain characterization, three LMI stability analysis conditions are proposed. The first one is based on the stability domain deformation and the second one on a characterization of the instability domain (which is convex). The third one is based on generalized LMI framework. These conditions are applied to the gain margin computation of a CRONE suspension.
34A33Lattice differential equations
34D20Stability of ODE
93D20Asymptotic stability of control systems
26A33Fractional derivatives and integrals (real functions)
[1]Matignon, D.: Stability results on fractional differential equations with applications to control processing, Proceedings of computational engineering in systems and application multiconference 2, 963-968 (1996)
[2]Boyd, S.; Vandenberghe, L.: Convex optimization, (2004)
[3]Balakrishnan, V.; Kashyap, R. L.: Robust stability and performance analysis of uncertain systems using linear matrix inequalities, Journal of optimization theory and applications 100, No. 3, 457-478 (1999) · Zbl 0951.93060 · doi:10.1023/A:1022626120665
[4]V. Balakrishnan, Linear matrix inequalities in robust control: A brief survey, in: Proc. Math. Thy of Networks and Systems, Notre Dame, Indiana, August 2002
[5]Oustaloup, A.; Mathieu, B.: La commande CRONE du scalaire au multivariable, (1999)
[6]Hadid, S. B.; Alshamani, J. G.: Lyapunov stability of differential equations of non-integer order, Arab journal of mathematics 7, 1-17 (1986) · Zbl 0659.34056
[7]Momani, S.; Hadid: Lyapunov stability solution of fractional integrodifferential equations, International journal of mathematics and mathematical sciences 2004, No. 7, 2503-2507 (2004) · Zbl 1074.45006 · doi:10.1155/S0161171204312366
[8]Y. Li, Y. Chen, I. Podlubny, Yongcan Cao, Mittag-leffler stability of fractional order nonlinear dynamic systems, in: Paper Accepted to the Next IFAC Fractional Derivative and its Applications Workshop, FDA’08, 5–7 November, Ankara, Turkey, 2008
[9]S. Ladaci, E. Moulay, Lp-stability analysis of a class of nonlinear fractional differential equations, Journal of Automation and Systems Engineering (2008) (in press)
[10]J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, On a representation of fractional order systems: Interests for the initial condition problem, in: IFAC Fractional Derivative and its Applications Workshop, FDA’08, 5–7 November, Ankara, Turkey, 2008
[11]V.E. Tarasov, Fractional stability, 2007. arXiv.org >physics >arXiv:0711.2117. http://arxiv.org/abs/0711.2117
[12]Skaar, S. B.; Michel, A. N.; Miller, R. K.: Stability of viscoelastic control systems, IEEE transactions on automatic control 33, No. 4, 348-357 (1998) · Zbl 0641.93051 · doi:10.1109/9.192189
[13]C. Bonnet, J.R. Partington, Stabilization and nuclearity of fractional differential systems, in: Proceedings of the MTNS conference, Perpignan, France, 2000
[14]Bonnet, C.; Partington, J. R.: Analysis of fractional delay systems of retarded and neutral type, Automatica 38, 1133-1138 (2002) · Zbl 1007.93065 · doi:10.1016/S0005-1098(01)00306-5
[15]Matignon, D.: Stability properties for generalized fractional differential systems, ESAIM: proc. 5 (1998)
[16]Hwang, C.; Cheng, Y.: A numerical algorithm for stability testing of fractional delay systems, Automatica 42, 825-831 (2006) · Zbl 1137.93375 · doi:10.1016/j.automatica.2006.01.008
[17]Chen, Y.; Moore, K. L.: Analytical stablity bound for a class of delayed fractional-order dynamic systems, Nonlinear dynamics 29, 191-200 (2002) · Zbl 1020.34064 · doi:10.1023/A:1016591006562
[18]A. Benchellal, Modélisation des interfaces de diffusion à l’aide d’opérateurs d’intégration, Ph.D. Thesis, Poitiers University-France, 2008
[19]Deng, W.; Li, C.; Lu, J.: Stability analysis of linear fractional differential system with multiple time delay, Nonlinear dynamics 48, 409-416 (2007) · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0
[20]Lazarevic, M. P.: Finite time stability analysis of pda fractional control of robotic time-delay systems, Mechanics research communications 33, 269-279 (2006) · Zbl 1192.70008 · doi:10.1016/j.mechrescom.2005.08.010
[21]Chen, Y.; Ahn, H. S.; Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal processing 86, 2611-2618 (2006) · Zbl 1172.94385 · doi:10.1016/j.sigpro.2006.02.011
[22]Deif, A. S.: The interval eigenvalue problem, Zeitschrift fur angewandte mathematix und mechanik 71, No. 1, 61-64 (1991) · Zbl 0756.65056 · doi:10.1002/zamm.19910710117
[23]M. Moze, J. Sabatier, LMI tools for stability analysis of fractional systems, in: Proceedings of ASME 2005 IDET / CIE conferences, Long-Beach, September 24–28, 2005, pp. 1–9
[24]Ahn, H. S.; Chen, Y.; Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Applied mathematics and computation 187, 27-34 (2007) · Zbl 1123.93074 · doi:10.1016/j.amc.2006.08.099
[25]I. Petras, Y. Chen, B.M. Vinagre, I. Podlubny, Stability of linear time invariant systems with interval fractional orders and interval coefficients, in: Proceedings of the International Conference on Computation Cybernetics, ICCC04, Vienna, Austria, 30th August–1st September, 2004, pp. 1–4
[26]Matignon, D.; Prieur, C.: Asymptotic stability of linear conservative systems when coupled with diffusive systems, ESAIM: control, optimisation and calculus of variations 11, 487-507 (2005) · Zbl 1125.93030 · doi:10.1051/cocv:2005016 · doi:numdam:COCV_2005__11_3_487_0
[27]R. Malti, O. Cois, M. Aoun, F. Levron, A. Oustaloup, Computing impulse response energy of fractional transfer function, in: Proceedings of the 15th IFAC World Congress, Barcelona, Spain, July 21–26, 2002
[28]S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory, in: Studies in Applied Mathematics, vol. 15, Philadelphia, 1994 · Zbl 0816.93004
[29]S. Momani, R. El-Khazali, Stability analysis of composite fractional systems, in: Proceedings of the Intelligent Systems and Control Conference, November 19–22, Tampa, Florida, 2001
[30]Samko, A. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives, (1987)
[31]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equation, (1993)
[32]Podlubny, I.: Fractional-order systems and PIλDμ-controllers, IEEE transaction on automatic control 44, No. 1, 208-214 (1999) · Zbl 1056.93542 · doi:10.1109/9.739144
[33]Ben-Tal, A.; El Ghaoui, L.; Nemirovski, A.: Robustness, , 68-92 (2000)
[34]M. Chilali, Méthode LMI pour la synthèse multi-critère. Ph.D. Thesis, Université Paris IX-Dauphine, U.F.R. Mathématiques de la Décision, Paris, France, 1996
[35]O. Bachelier, Commande des systèmes linéaires incertains: Placement de pôles robuste en D-stabilité. Ph.D. Thesis, INSA, Toulouse, France, 1998
[36]X. Moreau, Intérêt de la Dérivation Non Entière en Isolation Vibratoire et son Application dans le Domaine de l’Automobile: la Suspension CRONE: Du Concept à la Réalisation, Ph.D. Thesis, Université de Bordeaux I, 1995
[37]Moreau, X.; Ramus-Serment, C.; Oustaloup, A.: Fractional differentiation in passive vibration control, International journal of nonlinear dynamics and chaos in engineering systems 29, No. 1–4, 343-362 (2002) · Zbl 1019.74028 · doi:10.1023/A:1016518118007
[38]C. Ramus-Serment, Synthèse d’un isolateur vibratoire d’ordre non entier fondée sur une architecture arborescente d’éléments viscoélastiques quasi-identiques, Ph.D. Thesis, Université de Bordeaux I, 2001