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Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. (English) Zbl 1165.34408

Summary: A stability test procedure is proposed for linear nonhomogeneous fractional order systems with a pure time delay. Some basic results from the area of finite time and practical stability are extended to linear, continuous, fractional order time-delay systems given in state-space form. Sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems. A numerical example is given to illustrate the validity of the proposed procedure.

MSC:

34K20 Stability theory of functional-differential equations
26A33 Fractional derivatives and integrals
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[1] Zavarei, M.; Jamshidi, M., Time-Delay Systems: Analysis,Optimization and Applications (1987), North-Holland: North-Holland Amsterdam · Zbl 0658.93001
[2] Lee, T. N.; Diant, S., Stability of time delay systems, IEEE Trans. Automat. Control AC, 31, 3, 951-953 (1981) · Zbl 0544.93052
[3] Mori, T., Criteria for asymptotic stability of linear time delay systems, IEEE Trans. Automat. Control, AC, 30, 158-161 (1985) · Zbl 0557.93058
[4] Hmamed, A., On the stability of time delay systems: New results, Internat. J. Control, 43, 1, 321-324 (1986) · Zbl 0613.34063
[5] Chen, J.; Xu, D.; Shafai, B., On sufficient conditions for stability independent of delay, IEEE Trans. Automat Control AC, 40, 9, 1675-1680 (1995) · Zbl 0834.93045
[6] Weiss, L.; Infante, F., On the stability of systems defined over finite time interval, Proc. Natl. Acad. Sci., 54, 1, 44-48 (1965) · Zbl 0134.30702
[7] Grujić, Lj. T., Non-Lyapunov stability analysis of large-scale systems on time-varying sets, Internat. J. Control, 21, 3, 401-405 (1975) · Zbl 0303.93010
[8] Grujić, Lj. T., Practical stability with settling time on composite systems, Automatika (YU), T.P., 9, 1 (1975) · Zbl 0291.93038
[9] Lashirer, A. M.; Story, C., Final-stability with applications, J. Inst. Math. Appl., 9, 379-410 (1972)
[10] D.Lj Debeljković, M.P. Lazarević, Dj. Koruga, S. Tomašević, On practical stability of time delay system under perturbing forces, in: AMSE 97, Melbourne, Australia, October 29-31, 1997, pp. 447-450; D.Lj Debeljković, M.P. Lazarević, Dj. Koruga, S. Tomašević, On practical stability of time delay system under perturbing forces, in: AMSE 97, Melbourne, Australia, October 29-31, 1997, pp. 447-450
[11] Lj.D. Debeljković, M.P. Lazarević, S.A. Milinković, M.B. Jovanović, Finite time stability analysis of linear time delay system: Bellman-Gronwall approach, in: IFAC International Workshop Linear Time Delay Systems, Grenoble,France, 1998, pp. 171-176; Lj.D. Debeljković, M.P. Lazarević, S.A. Milinković, M.B. Jovanović, Finite time stability analysis of linear time delay system: Bellman-Gronwall approach, in: IFAC International Workshop Linear Time Delay Systems, Grenoble,France, 1998, pp. 171-176
[12] Lazarević, M. P.; Debeljković, Lj. D.; Nenadić, Z. Lj.; Milinković, S. A., Finite time stability of time delay systems, IMA J. Math. Control. Inform., 17, 101-109 (2000) · Zbl 0979.93095
[13] Lj.D. Debeljković, M.P. Lazarević, et al. Further results on non-lyapunov stability of the linear nonautonomous systems with delayed state, Journal Facta Uversitatis, Niš, Serbia,Yugoslavia, 2001, vol.3, No 11, pp. 231-241; Lj.D. Debeljković, M.P. Lazarević, et al. Further results on non-lyapunov stability of the linear nonautonomous systems with delayed state, Journal Facta Uversitatis, Niš, Serbia,Yugoslavia, 2001, vol.3, No 11, pp. 231-241
[14] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS - SMC Proceeding, July, Lille, France, 1996, pp. 963-968; D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS - SMC Proceeding, July, Lille, France, 1996, pp. 963-968
[15] D. Matignon, Stability properties for generalized fractional differential systems, ESAIM: Proceedings, 5, December,1998, pp. 145-158; D. Matignon, Stability properties for generalized fractional differential systems, ESAIM: Proceedings, 5, December,1998, pp. 145-158 · Zbl 0920.34010
[16] B.M. Vinagre, C.A. Monje, A.J. Calder’on, Fractional order systems and fractional order control actions, in: Lecture 3 of the IEEE CDC02 TW#2: Fractional Calculus Applications in Automatic Control and Robotics, 2002; B.M. Vinagre, C.A. Monje, A.J. Calder’on, Fractional order systems and fractional order control actions, in: Lecture 3 of the IEEE CDC02 TW#2: Fractional Calculus Applications in Automatic Control and Robotics, 2002
[17] Chen, Q.; Ahn, H.; Podlubny, I., Robust stability check of fractional order linear time invariant systems with interval uncertainties, Signal Processing, 86, 2611-2618 (2006) · Zbl 1172.94385
[18] Chen, Y. Q.; Moore, K. L., Analytical stability bound for delayed second order systems with repeating poles using Lambert function W, Automatica, 38, 5, 891-895 (2002) · Zbl 1020.93019
[19] Chen, Y. Q.; Moore, K. L., Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear Dynam., 29, 191-200 (2002) · Zbl 1020.34064
[20] Bonnet, C.; Partington, J. R., Stabilization of fractional exponential systems including delays, Kybernetika, 37, 345-353 (2001) · Zbl 1265.93211
[21] Bonnet, C.; Partington, J. R., Analysis of fractional delay systems of retarded and neutral type, Automatica, 38, 1133-1138 (2002) · Zbl 1007.93065
[22] Hotzel, R.; Fliess, M., On linear systems with a fractional derivation: Introductory theory and examples, Math. Comput. Simul., 45, 385-395 (1998) · Zbl 1017.93508
[23] Lazarević, M. P., Finite time stability analysis of PD \({}^\alpha\) fractional control of robotic time-delay systems, Mech. Res. Comm., 33, 269-279 (2006) · Zbl 1192.70008
[24] 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
[25] Lacroix, S. F., Traite Du Calcul Differential et du Calcul Integral, vol. 3 (1819), Paris Courcier, 409-410, 1819
[26] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[27] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 9, 1461-1477 (1996) · Zbl 1080.26505
[28] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[29] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[30] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent, Geophys. J. Royal Astronom. Soc., 13, 529-539 (1967)
[31] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and derivatives: Theory and Applications (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Switzerland · Zbl 0818.26003
[32] C. Lorenzo, T. Hartley, Initialization,conceptualization,and application, NASA_Tp, 1998-208415, December, 1998; C. Lorenzo, T. Hartley, Initialization,conceptualization,and application, NASA_Tp, 1998-208415, December, 1998
[33] Spasić, A.; Lazarević, M.; Krstic, D., Chapter: Theory of electroviscoelasticity, (Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering (2006), CRC Press-Taylor & Francis: CRC Press-Taylor & Francis Boca Raton-London-New York), 371-394
[34] A.M. Spasić, P.M. Lazarevi’c, Electroviscoelasticity of Liquid-Liquid Interfaces: Fractional-order model (new constitutive models of liquids), in: Lectures in rheology, Department of Mechanics, Faculty of Mathematics, University of Belgrade,Spring, 2004; A.M. Spasić, P.M. Lazarevi’c, Electroviscoelasticity of Liquid-Liquid Interfaces: Fractional-order model (new constitutive models of liquids), in: Lectures in rheology, Department of Mechanics, Faculty of Mathematics, University of Belgrade,Spring, 2004
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