A note on the stability of fractional order systems. (English) Zbl 1168.34036

Summary: A new approach is suggested to investigate stability in a family of fractional order linear time invariant systems with order between 1 and 2. The proposed method relies on finding a linear ordinary system that possesses the same stability property as the fractional order system. In this way, instead of performing the stability analysis on the fractional order systems, the analysis is converted into the domain of ordinary systems which is well established and well understood. As a useful consequence, we have extended two general tests for robust stability check of ordinary systems to fractional order systems.


34D20 Stability of solutions to ordinary differential equations
26A33 Fractional derivatives and integrals
Full Text: DOI


[1] Anderson, B. D.O.; Bose, N. K.; Jury, E. I., A simple test for zeros of a complex polynomial in a sector, IEEE Trans. Automat. Control, 19, 437-438 (A1974)
[2] Chen, Y. Q.; Ahn, H.; Podlubny, I., Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Process., 86, 2611-2618 (2006) · Zbl 1172.94385
[3] Davison, E. J.; Ramesh, N., A note on the eigenvalues of a real matrix, IEEE Trans. Automat. Control, 15, April, 252-253 (1970)
[4] Delgado-Romero, J. J.D.; Gonzalez-Garza, R. S.; Hernandez-Morales, E.; Delgado-Romero, G., Robust stability of linear time invariant systems represented by an interval matrix, (1st International Conference in Control of Oscillations and Chaos, Proceedings, 3 August 27-29 (1997)), 545-548
[5] Hostetter, G. H., An improved test for the zeros of a polynomial in a sector, IEEE Trans. Automat. Control, 20, 433-434 (1975)
[6] Hwang, C.; Cheng, Y. C., A numerical algorithm for stability testing of fractional delay systems, Automatica, 42, 825-831 (2006) · Zbl 1137.93375
[8] Matignon, D., Stability results for fractional differential equations with applications to control processing, (Computational Engineering in Systems Applications. Computational Engineering in Systems Applications, Lille, France, IMACS, IEEE-SMC, vol. 2, July (1996)), 963-968
[9] Matignon, D., Stability properties for generalized fractional differential systems, ESAIM: Proc., 5, 145-158 (1998) · Zbl 0920.34010
[10] Ostalczyk, P., Nyquist characteristics of a fractional order integrator, J. Fract. Calculus, 19, 67-78 (2001) · Zbl 0993.93023
[11] Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F. M., Frequency-band complex non-integer differentiator: Characterization and synthesis, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 47, 1, 25-39 (2000)
[12] Oustaloup, A.; Moreau, X.; Nouillant, M., The CRONE suspension, Control Eng. Pract., 4, 8, 1101-1108 (1996)
[13] Petras, I.; Chen, Y. Q.; Vinagre, B. M., Robust stability test for interval fractional order linear systems, (Blondel, V. D.; Megretski, A., Unsolved Problems in the Mathematics of Systems and Control (2004), Princeton University Press: Princeton University Press Princeton, NJ), 208-210, (Chapter 6.5)
[14] Petras, I.; Chen, Y. Q.; Vinagre, B. M.; Podlubny, I., Stability of linear time invariant systems with interval fractional orders and interval coefficients, (Proceedings of the International Conference on Computation Cybernetics (ICCC04). Proceedings of the International Conference on Computation Cybernetics (ICCC04), Vienna Technical University, Vienna, Austria (2005)), 1-4
[15] Podlubny, I., Fractional-order systems and \(PI^λ D^μ\)-controllers, IEEE Trans. Automat. Control, 44, 1, 208-214 (1999) · Zbl 1056.93542
[16] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[17] Qiu, Z.; Muller, P. C.; Frommer, A., Ellipsoidal set-theoretic approach for stability of linear state-space models with interval uncertainty, Math. Comput. Simul., 57, 45-59 (2001) · Zbl 0985.65077
[18] Raynaud, H. F.; ZergaInoh, A., State-space representation for fractional order controllers, Automatica, 36, 1017-1021 (2000) · Zbl 0964.93024
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