Stability criteria for a class of fractional order systems. (English) Zbl 1204.93106
Summary: This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems.
MSC:
| 93D25 | Input-output approaches in control theory |
| 34A08 | Fractional ordinary differential equations |
References:
| [1] | Sun, H., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A: Stat. Mech. Appl. 388, 4586–4592 (2009) · doi:10.1016/j.physa.2009.07.024 |
| [2] | Bertrand, N., Sabatier, J., Briat, O., Vinassa, J.M.: Fractional non-linear modelling of ultracapacitors. Commun. Nonlinear Sci. Numer. Simul. (2009). doi: 10.1016/j.cnsns.2009.05.066. In press · Zbl 1299.35307 |
| [3] | Tavazoei, M.S., Haeri, M., Jafari, S., Bolouki, S., Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations. IEEE Trans. Ind. Electron. 55(11), 4098–4101 (2008) · doi:10.1109/TIE.2008.925774 |
| [4] | Feliu-Batlle, V., Rivas Perez, R., Sanchez Rodrigue, L.: Fractional robust control of main irrigation canals with variable dynamic parameters. Control Eng. Pract. 15, 673–686 (2007) · doi:10.1016/j.conengprac.2006.11.018 |
| [5] | Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Application Multi-conference, vol. 2, IMACS, IEEE-SMC Proceedings, pp. 963–968, Lille, France (July 1996) |
| [6] | Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM: Proc. 5, 145–158 (1998) · Zbl 0920.34010 · doi:10.1051/proc:1998004 |
| [7] | Sabatier, J., Moze, M., Farges, C.: On stability of fractional order systems. In: Third IFAC Workshop on Fractional Differentiation and its Applications (FDA’08), Ankara, Turkey (November 2008) · Zbl 1189.34020 |
| [8] | Li, C., Zhao, Z.: Asymptotical stability analysis of linear fractional differential systems. J. Shanghai Univ. (Engl. Ed). 13(3), 197–206 (2009) · Zbl 1212.34009 · doi:10.1007/s11741-009-0302-1 |
| [9] | Deng, W., Li, C., Guo, Q.: Analysis of fractional differential equations with multi-orders. Fractals 15, 173–182 (2007) · Zbl 1176.34008 · doi:10.1142/S0218348X07003472 |
| [10] | Bonnet, C., Partington, J.R.: Coprime factorizations and stability of fractional differential systems. Syst. Control Lett. 41, 167–174 (2000) · Zbl 0985.93048 · doi:10.1016/S0167-6911(00)00050-5 |
| [11] | Chen, Y.Q., Moore, K.L.: Analytical stability bound for a class of delayed fractional order dynamic systems. Nonlinear Dyn. 29, 191–200 (2002) · Zbl 1020.34064 · doi:10.1023/A:1016591006562 |
| [12] | Deng, W., Li, C., Lü, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007) · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0 |
| [13] | 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 |
| [14] | Ahn, H.S., Chen, Y.Q.: Necessary and sufficient stability condition of fractional order interval linear systems. Automatica 44, 2985–2988 (2008) · Zbl 1152.93455 · doi:10.1016/j.automatica.2008.07.003 |
| [15] | Tavazoei, M.S., Haeri, M.: A note on the stability of fractional order systems. Math. Comput. Simul. 79, 1566–1576 (2009) · Zbl 1168.34036 · doi:10.1016/j.matcom.2008.07.003 |
| [16] | Chen, Y.Q., Ahn, 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 |
| [17] | Tavazoei, M.S., Haeri, M., Bolouki, S., Siami, M.: Stability preservation analysis for frequency based methods in numerical simulation of fractional order systems. SIAM J. Numer. Anal. 47, 321–338 (2008) · Zbl 1203.26012 · doi:10.1137/080715949 |
| [18] | Tavazoei, M.S., Haeri, M.: Rational approximations in the simulation and implementation of fractional order dynamics: A descriptor system approach. Automatica (2009). doi: 10.1016/j.automatica.2009.09.016. In press · Zbl 1213.93087 |
| [19] | Li, Y., Chen, Y.Q., 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 |
| [20] | 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) · doi:10.1109/TCSII.2008.2002571 |
| [21] | Tavazoei, M.S.: Comments on ”Stability analysis of a class of nonlinear fractional order systems”. IEEE Trans. Circuits Syst. II 56, 519–520 (2009) · doi:10.1109/TCSII.2009.2020944 |
| [22] | Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability. Comput. Math. Appl. (2009). doi: 10.1016/j.camwa.2009.08.019. In press · Zbl 1185.93062 |
| [23] | Petras, I.: Stability of fractional order systems with rational orders: A survey. Fract. Calc. Appl. Anal. 12, 269–298 (2009) · Zbl 1182.26017 |
| [24] | Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996) |
| [25] | Huang, T.Z.: Stability criteria for matrices. Automatica 34(5), 637–639 (1998) · Zbl 0937.93018 · doi:10.1016/S0005-1098(97)00234-3 |
| [26] | Hamamci, S.E.: An algorithm for stabilization of fractional order time delay systems using fractional order PID controllers. IEEE Trans. Automat. Contr. 52, 1964–1969 (2007) · Zbl 1366.93464 · doi:10.1109/TAC.2007.906243 |
| [27] | Hamamci, S.E.: Stabilization using fractional order PI and PID controllers. Nonlinear Dyn. 51(1–2), 329–343 (2008) · Zbl 1170.93023 · doi:10.1007/s11071-007-9214-5 |
| [28] | Bonnet, C., Partington, J.R.: Stabilization of some fractional delay systems of neutral type. Automatica 43, 2047–2053 (2007) · Zbl 1138.93049 · doi:10.1016/j.automatica.2007.03.017 |
| [29] | Li, C., Peng, G.: Chaos in Chen’s system with a fractional order. Chaos Solitons Fractals 22, 443–450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013 |
| [30] | Tavazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional order systems. Phys. Lett. A 367(1–2), 102–113 (2007) · Zbl 1209.37037 · doi:10.1016/j.physleta.2007.05.081 |
| [31] | Tavazoei, M.S., Haeri, M., Jafari, S.: Fractional controller to stabilize fixed points of uncertain chaotic systems: theoretical and experimental study. Proc. Inst. Mech. Eng., Part I, J. Syst. Control Eng. 222, 175–184 (2008) · doi:10.1243/09596518JSCE481 |
| [32] | Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087 |
| [33] | Tavazoei, M.S., Haeri, M.: Chaotic attractors in incommensurate fractional order systems. Phys. D: Nonlinear Phenom. 237, 2628–2637 (2008) · Zbl 1157.26310 · doi:10.1016/j.physd.2008.03.037 |
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.
