Abstract
This paper presents the stability analysis problem of fractional-order nonlinear systems with time-varying delay. After formulating the problem and selecting the nonlinear model as the system under study, stability analysis and expression of the sufficient conditions for fractional-order nonlinear systems with time-varying delay are obtained using two different methods. In these methods, sufficient conditions for stability of fractional-order nonlinear systems are found in the form of satisfying some inequalities based on norms of nonlinear functions in the system and in terms of linear matrix inequality according fractional-order and nonlinear functions. In each case, despite the presence of time-varying delay, the system stability is ensured by meeting the stability sufficient conditions in terms of an inequality of functions and system parameters. Finally, numerical examples are given to determine the effectiveness of the proposed theorem.
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26 September 2020
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References
Boukal Y, Zasadzinski M, Darouach M, Radhy NE (2016) Stability and stabilizability analysis of fractional order time variant delay systems via diffusive representation. In: Proceedings of the 5th international conference on systems and control, Cadi Ayyad University, Marrakesh, Morocco, pp 25–27
Boulares H, Ardjouni A, Laskri Y (2016) Stability in delay nonlinear differential equations. Rend Circ Mat Palermo. https://doi.org/10.1007/s12215-016-0230-5
Buslowicz M (2008) Stability of linear continuous-time fractional order systems with delays of the retarded type. Bull Pol Ac Tech 56:319–324
Chen Y, Ahn HS, Podlubny I (2005) Robust stability check of fractional order linear time invariant systems with interval uncertainties. In: IEEE international conference on mechatronics and automation, vol 1, pp 210–215
Gao Z (2014) A graphic stability criterion for non-commensurate fractional order time-delay systems. Nonlinear Dyn 78:2101–2111
Hartley TT, Lorenzo CF (2008) Fractional order system identification based on continuous order-distributions. Sig Process 83:2287–2300
Luo J, Li G, Liu H (2014) Linear control of fractional-order financial chaotic systems with input saturation. Discret Dyn Nat Soc 2014:1–8. https://doi.org/10.1155/2014/802429 (Special Issue)
Matignon D (1996) Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems and application multiconference. IMACS, IEEE-SMC Proceedings, Lille, France 2 pp 963–968
Mohammadzadeh A, Ghaemi S, Kaynak O, Khanmohammadi S (2018) Robust predictive synchronization of uncertain fractional-order time-delayed chaotic systems. Soft Comput 26:1–16
Pakzad MA, Pakzad S, Nekoui MA (2014) On stability of linear time-invariant fractional order systems with multiple delays. In: American Control Conference (ACC), Portland, Oregon, USA, pp 4–6
Petras I (2009) Stability of fractional order systems with rational orders. Fract Calc Appl Anal 12:269–298
Petras I (2010) Fractional order nonlinear systems, modeling, analysis and simulation. Springer, Nonlinear Physical Science
Petras I, Chen YQ, Vinagre BM (2004) Robust Stability test for interval fractional order linear systems. In: Blondel VD, Megretski A (eds) Princeton Univ Press, Princeton, pp 208–210, ch 6.5
Petras I, Chen YQ, Vinagre BM, Podlubny I (2005) Stability of linear time invariant systems with interval fractional orders and interval coefficients. In: Proceedings of International Conference Computational Cybern (ICCC’04), Vienna, Austria pp 1–4
Podlubny I (1999) Fractional order systems and PI-D controllers. IEEE Trans Automatic Control 44:208–214
Shu Y, Zhu Y (2018) Stability analysis of uncertain singular systems. Soft Comput 22:5671–5681
Tavazoei MS, Haeri M (2008a) Stabilization of unstable fixed points of chaotic fractional order systems by a state fractional PI controller. Euro J Control 14:247–257
Tavazoei MS, Haeri M (2008b) Synchronization of chaotic fractional order systems via active sliding mode controller. Phys A Statist Mech Appl 387:57–70
Tavazoei MS, Haeri M (2009) A note on the stability of fractional order systems. Math Comput Simul 79:1566–1576
Tavazoei MS, Haeri M, Nazari N (2008) Analysis of undamped oscillations generated by marginally stable fractional order systems. Signal Process 88:2971–2978
Veselinova M, Kiskinov H, Zahariev A (2016) About stability conditions for retarded fractional differential systems with distributed delays. Commun Appl Anal 20:325–334
Wang F, Yang H, Yang Y (2019) Swarming movement of dynamical multi-agent systems with sampling control and time delays. Soft Comput 23:707–714
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Rahmanipour, P., Ghadiri, H. Stability analysis for a class of fractional-order nonlinear systems with time-varying delays. Soft Comput 24, 17445–17453 (2020). https://doi.org/10.1007/s00500-020-05118-w
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DOI: https://doi.org/10.1007/s00500-020-05118-w