×

Stability analysis for nonlinear fractional-order systems based on comparison principle. (English) Zbl 1281.34012

Summary: This work constructs a theoretical framework for the stability analysis of nonlinear fractional-order systems. A new definition, the generalized Caputo fractional derivative, is proposed for the first time. Based on that, the comparison principles for scalar and vector fractional-order systems are constructed, respectively. Furthermore, a sufficient theorem for stability analysis is proved, and how to use this theorem in stabilization is also discussed. Three examples have been presented to illustrate how to use the developed theory to analyze the stability and to design stabilization controllers. With the proposed method, the problems of stabilization and synchronization of the fractional-order chaotic fractional-order systems can be easily solved with linear feedback control.

MSC:

34A08 Fractional ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bagley, R.L., Calico, R.A., Guid, J.: Fractional order state equations for the control of viscoelastically damped structures. Control Dyn. 14, 304-311 (1991) · doi:10.2514/3.20641
[2] Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441-444 (1984) · Zbl 0532.93025 · doi:10.1109/TAC.1984.1103551
[3] Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253-265 (1971) · doi:10.1016/S0022-0728(71)80115-8
[4] Le Méhauté, A., Crepy, G.: Introduction to transfer and motion in fractal media: the geometry of kinetics. Solid State Ion. 9 & 10, 17-30 (1983)
[5] de Levie, R.: Fractals and rough electrodes. J. Electroanal. Chem. 281, 1-21 (1990) · doi:10.1016/0022-0728(90)87025-F
[6] Chen, G., Friedman, G.: An RLC interconnect model based on Fourier analysis. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 24, 170-183 (2005) · doi:10.1109/TCAD.2004.841065
[7] Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971) · JFM 30.0801.03
[8] Jenson, V.G., Jeffreys, G.V.: Mathematical Methods in Chemical Engineering, 2nd edn. Academic, New York (1971) · Zbl 0123.17501
[9] Cole, K. S., Electric conductance of biological systems, Cold Spring Harbor, New York
[10] Anastasio, T.J.: The fractional-order dynamics of Brainstem vestibule-oculumotor neurons. Biol. Cybern. 72, 69-79 (1994) · doi:10.1007/BF00206239
[11] Laskin, N.: Fractional market dynamics. Physica A 287, 482-492 (2000) · doi:10.1016/S0378-4371(00)00387-3
[12] Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136-1139 (1999) · doi:10.1103/PhysRevLett.82.1136
[13] Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I 42, 485-490 (1995) · doi:10.1109/81.404062
[14] Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339-351 (2003) · Zbl 1033.37019 · doi:10.1016/S0960-0779(02)00438-1
[15] Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55-61 (2004) · doi:10.1016/j.physa.2004.04.113
[16] Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685-688 (2006) · Zbl 1101.37307 · doi:10.1016/j.chaos.2005.04.037
[17] Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354, 305-311 (2006) · doi:10.1016/j.physleta.2006.01.068
[18] Ge, Z.-M., Ou, C.-Y.: Chaos in a fractional order modified Duffing system. Chaos Solitons Fractals 34, 262-291 (2007) · Zbl 1132.37324 · doi:10.1016/j.chaos.2005.11.059
[19] Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M., Chen, W.C., Lin, K.T., Kang, Y.: Chaos in the Newton-Leipnik system with fractional order. Chaos Solitons Fractals 36, 98-103 (2008) · Zbl 1152.37319 · doi:10.1016/j.chaos.2006.06.013
[20] Zhang, H.G., Huang, W., Wang, Z.L., Chai, T.Y.: Adaptive synchronization between two different chaotic systems with unknown parameters. Phys. Lett. A 350, 363-366 (2006) · Zbl 1195.93121 · doi:10.1016/j.physleta.2005.10.033
[21] Zhang, H.G., Liu, D.R., Wang, Z.L.: Controlling Chaos: Suppression, Synchronization and Chaotification. Springer, London (2009) · Zbl 1179.93004 · doi:10.1007/978-1-84882-523-9
[22] Zhang, H.G., Ma, T.D., Huang, G.B., Wang, Z.L.: Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans. Syst. Man Cybern., Part B 40, 831-844 (2010) · doi:10.1109/TSMCB.2009.2030506
[23] Fu, J., Zhang, H.G., Ma, T.D., Zhang, Q.L.: On passivity analysis for stochastic neural networks with interval time-varying delay. Neurocomputing 73, 795-801 (2010) · doi:10.1016/j.neucom.2009.10.010
[24] Matignon, D., Stability results on fractional differential equations with applications to control processing, Lille, France · Zbl 0863.93029
[25] Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple delays. Nonlinear Dyn. 48, 409-416 (2007) · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0
[26] Chen, Y.Q., Ahna, H.-S., Podlubny, I.: Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Process. 86, 2611-2618 (2006) · Zbl 1172.94385 · doi:10.1016/j.sigpro.2006.02.011
[27] Ahna, H.-S., Chen, Y.Q., Podlubny, I.: Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Appl. Math. Comput. 187, 27-34 (2007) · Zbl 1123.93074 · doi:10.1016/j.amc.2006.08.099
[28] Xing, S.Y., Lu, J.G.: Robust stability and stabilization of fractional-order linear systems with nonlinear uncertain parameters: an LMI approach. Chaos Solitons Fractals 42, 1163-1169 (2009) · Zbl 1198.93169 · doi:10.1016/j.chaos.2009.03.017
[29] Lu, J.G., Chen, G.: Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Trans. Autom. Control 54, 1294-1299 (2009) · Zbl 1367.93472 · doi:10.1109/TAC.2009.2013056
[30] Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193, 27-47 (2011) · doi:10.1140/epjst/e2011-01379-1
[31] Li, C.P., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71, 621-633 (2013) · Zbl 1268.34019 · doi:10.1007/s11071-012-0601-1
[32] Wang, X., Wang, M.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos 17, 033106 (2007) · Zbl 1163.37382 · doi:10.1063/1.2755420
[33] Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 26, 1125-1133 (2005) · Zbl 1074.65146 · doi:10.1016/j.chaos.2005.02.023
[34] Deng, W., Li, C.P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61-72 (2005) · doi:10.1016/j.physa.2005.01.021
[35] Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171-185 (2006) · doi:10.1016/j.physa.2005.06.078
[36] Yan, J., Li, C.: On chaos synchronization of fractional differential equation. Chaos Solitons Fractals 32, 725-735 (2007) · Zbl 1132.37308 · doi:10.1016/j.chaos.2005.11.062
[37] Ge, Z.-M., Ou, C.-Y.: Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35, 705-717 (2008) · doi:10.1016/j.chaos.2006.05.101
[38] Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57-70 (2008) · doi:10.1016/j.physa.2007.08.039
[39] Erjaee, G.H., Momani, S.: Phase synchronization in fractional differential chaotic systems. Phys. Lett. A 372, 2350-2354 (2008) · Zbl 1220.34004 · doi:10.1016/j.physleta.2007.11.065
[40] Wu, X., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25-35 (2009) · Zbl 1176.70029 · doi:10.1007/s11071-008-9416-5
[41] Wen, X.-J., Wu, Z.-M., Lu, J.-G.: Stability analysis of a class of nonlinear fractional-order systems. IEEE Trans. Circuits Syst. II 55, 1178-1182 (2008) · Zbl 1192.35094 · doi:10.1109/TCSII.2008.2002571
[42] Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965-1969 (2009) · Zbl 1185.93062 · doi:10.1016/j.automatica.2009.04.003
[43] Lakshmikantham, V., Matrosov, V.M., Sivasundaram, S.: Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems. Kluwer Academic, Dordrecht (1991) · Zbl 0721.34054 · doi:10.1007/978-94-015-7939-1
[44] Martynyuk, A.A.: Stability by Liapunov’s Matrix Function Method with Applications. Dekker, New York (1998) · Zbl 0907.34034
[45] Nersesov, S.G., Haddad, W.M.: On the stability and control of nonlinear dynamical systems via vector Lyapunov functions. IEEE Trans. Autom. Control 51, 203-215 (2006) · Zbl 1366.93553 · doi:10.1109/TAC.2005.863496
[46] Nersesov, S.G., Haddad, W.M.: Control vector Lyapunov functions for large-scale impulsive dynamical systems. Nonlinear Anal. 1, 223-243 (2007) · Zbl 1118.93348
[47] Stamova, I.M.: Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations. J. Math. Anal. Appl. 325, 612-623 (2007) · Zbl 1113.34058 · doi:10.1016/j.jmaa.2006.02.019
[48] Podlubny, I.: Fractional Differential Equations. Academic, New York (1999) · Zbl 0924.34008
[49] Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075-1081 (2007) · Zbl 1120.26003 · doi:10.1016/j.jmaa.2006.05.061
[50] Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag-Leffler function Eα,β and its derivative. Fract. Calc. Appl. Anal. 5, 491-518 (2002) · Zbl 1027.33016
[51] Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998) · Zbl 0902.35002
[52] Royden, H.L.: Real Analysis, 2nd edn. MacMillan, New York (1968) · Zbl 0704.26006
[53] Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) · Zbl 1215.34001 · doi:10.1007/978-3-642-14574-2
[54] Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control. Princeton University Press, New Jersey (2008) · Zbl 1142.34001
[55] Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.