×

Finite-time stability of linear fractional-order time-delay systems. (English) Zbl 1411.93135

Summary: In this paper, a finite-time stability results of linear delay fractional-order systems is investigated based on the generalized Gronwall inequality and the Caputo fractional derivative. Sufficient conditions are proposed to the finite-time stability of the system with the fractional order. Numerical results are given and compared with other published data in the literature to demonstrate the validity of the proposed theoretical results.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C15 Control/observation systems governed by ordinary differential equations
34A08 Fractional ordinary differential equations
93C05 Linear systems in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Malek‐Zavarei, M, Jamshidi, M. Time‐Delay Systems: Analysis Optimization and Applications. Amsterdam, Netherlands: North‐Holland; 1987. · Zbl 0658.93001
[2] Lee, TN, Dianat, S. Stability of time‐delay systems. {\it IEEE Trans Autom Control}. 1981; 26( 4): 951‐ 953. · Zbl 0544.93052
[3] Mori, T. Criteria for asymptotic stability of linear time‐delay systems. {\it IEEE Trans Autom Control}. 1985; 30: 158‐ 161. · Zbl 0557.93058
[4] Hmamed, A. On the stability of time delay systems: new results. {\it Int J Control}. 1986; 43( 1): 321‐ 324. · Zbl 0613.34063
[5] Chen, J, Xu, D, Shafai, B. On sufficient conditions for stability independent of delay. {\it IEEE Trans Autom Control}. 1995; 40( 9): 1675‐ 1680. · Zbl 0834.93045
[6] Weiss, L, Infante, EF. On the stability of systems defined over finite time interval. {\it Proc Natl Acad Sci}. 1965; 54( 1): 44‐ 48. · Zbl 0134.30702
[7] Grujić, LT. Non‐Lyapunov stability analysis of large‐scale systems on time‐varying sets. {\it Int J Control}. 1975; 21( 3): 401‐ 405. · Zbl 0303.93010
[8] Grujić, LT. Practical stability with settling time on composite systems. {\it Automatika}. 1975; 9( 1): 1‐ 11.
[9] Lashirer, AMB, Storey, C. Final‐stability with some applications. {\it J Inst Math Appl}. 1972; 31( 3): 379‐ 410.
[10] Debeljkovic, DL, Lazarevic, MP, Milinkovic, SA, Jovanovic, MB. On practical stability of time delay system under perturbing forces. Paper presented at: AMSE 97; 1997; Melbourne, Australia.
[11] Debeljkovic, DL, Lazarevic, MP, Koruga, D, Tomaevic, S. Finite time stability analysis of linear time delay system: Bellman‐Gronwall approach. Paper presented at: IFAC International Workshop Linear Time Delay Systems; 1998; Grenoble, France.
[12] Lazarevic, MP, Debeljkovic, DL, Nenadic, ZL, Milinkovic, SA. Finite time stability of time delay systems. {\it IMA J Math Control Inf}. 2000; 17: 101‐ 109. · Zbl 0979.93095
[13] Debeljković, DL, Lazarević, MP, Koruga, D, Milinković, SA, Jovanović, MB, Jacić, LA. Further results on non‐Lyapunov stability of the linear nonautonomous systems with delayed state. {\it Facta Univ Ser Mech Autom Control Robotics}. 2001; 3( 11): 231‐ 241.
[14] Chen, C, Li, L, Peng, H, Yang, Y, Li, T. Finite‐time synchronization of memristor‐based neural networks with mixed delays. {\it Neurocomputing}. 2017; 235: 83‐ 89.
[15] Zhao, H, Li, L, Peng, H, et al. Finite‐time synchronization for multi‐link complex networks via discontinuous control. {\it Optik‐Int J Light Electron Opt}. 2017; 138: 440‐ 454.
[16] Zheng, M, Li, L, Peng, H, et al. Finite‐time synchronization of complex dynamical networks with multi‐links via intermittent controls. {\it Eur Phys J B}. 2016; 89: 43.
[17] Zheng, M, Wang, Z, Li, L, et al. Finite‐time generalized projective lag synchronization criteria for neutral‐type neural networks with delay. {\it Chaos, Solitons Fractals}. 2018; 107: 195‐ 203. · Zbl 1380.92008
[18] Zheng, M, Li, L, Peng, H, Xiao, J, Yang, Y, Zhao, H. Finite‐time stability and synchronization for memristor‐based fractional‐order Cohen‐Grossberg neural network. {\it Eur Phys J B}. 2016; 89: 204.
[19] Engheta, N. On fractional calculus and fractional multipoles in electromagnetism. {\it IEEE Trans Antennas Propag}. 1996; 44( 4): 554‐ 566. · Zbl 0944.78506
[20] Sun, HH, Abdelwahad, A, Onaral, B. Linear approximation of transfer function with a pole of fractional power. {\it IEEE Trans Autom Control}. 1984; 29( 5): 441‐ 444. · Zbl 0532.93025
[21] Laskin, N. Fractional market dynamics. {\it Phys A Stat Mech Its Appl}. 2000; 287( 3‐4): 482‐ 492.
[22] Oustaloup, A. La Dérivation Non Entiére. Paris, France: Hermes; 1995. · Zbl 0864.93004
[23] Naifar, O, Makhlouf, AB, Hammami, MA. Comments on “Lyapunov stability theorem about fractional system without and with delay”. {\it Commun Nonlin Sci Numer Simul}. 2016; 30: 360‐ 361. · Zbl 1489.34081
[24] Naifar, O, Makhlouf, AB, Hammami, MA. Comments on “Mittag‐Leffler stability of fractional order nonlinear dynamic systems [Automatica 45(8) (2009) 1965‐1969]”. {\it Automatica}. 2017; 75: 329. · Zbl 1351.93076
[25] Chen, L, Pan, W, Wu, R, He, Y. New result on finite‐time stability of fractional‐order nonlinear delayed systems. {\it J Comput Nonlinear Dyn}. 2015. https://doi.org/10.1115/1.4029784
[26] Chen, L, He, Y, Wu, R, Chai, Y, Yin, L. Robust finite time stability of fractional‐order linear delayed systems with nonlinear perturbations. {\it Int J Control Autom Syst}. 2014; 12( 3): 697‐ 702.
[27] Wu, R, Lu, Y, Chen, L. Finite‐time stability of fractional delayed neural networks. {\it Neurocomputing}. 2015. https://doi.org/10.1016/j.neucom.2014.07.060
[28] Wu, R‐C, Hei, X‐D, Chen, L‐P. Finite‐time stability of fractionalorder neural networks with delay. {\it Commun Theor Phys}. 2013; 60( 2): 189‐ 193.
[29] Chen, L, Chai, Y, Wu, R, Ma, T, Zhai, H. Dynamic analysis of a class of fractional‐order neural networks with delay. {\it Neurocomputing}. 2013; 111: 190‐ 194.
[30] Zhang, Z, Wei, J. Some results of the degenerate fractional differential system with delay. {\it Comput Math Appl}. 2011; 62: 1284‐ 1291. · Zbl 1228.34023
[31] Lazarević, MP, Spasić, AM. Finite‐time stability analysis of fractional order time‐delay systems: Gronwall’s approach. {\it Math Comput Model}. 2009; 49: 475‐ 481. · Zbl 1165.34408
[32] Wang, Q, Lu, D, Fang, Y. Stability analysis of impulsive fractional differential systems with delay. {\it Appl Math Lett}. 2015; 40: 1‐ 6. · Zbl 1319.34137
[33] Li, R, Cao, J, Wan, Y. Finite‐time stability analysis of fractional order delayed memristive neural networks. Paper presented at: 2016 International Joint Conference on Neural Networks (IJCNN); 2016; Vancouver, Canada.
[34] Pang, D, Jiang, W. Finite‐time stability analysis of fractional singular time‐delay systems. {\it Adv Diff Equ}. 2014; 2014: 259. · Zbl 1348.34135
[35] Lazarević, M. Finite time stability analysis of {\it P}{\it D}\^{}{{\it α}} fractional control of robotic time‐delay systems. {\it Mech Res Commun}. 2006; 33: 269‐ 279. · Zbl 1192.70008
[36] Podlubny, I. Fractional Differential Equations. San Diego, CA: Academic Press; 1999. · Zbl 0924.34008
[37] Ye, H, Gao, J, Ding, Y. A generalized Gronwall inequality and its application to a fractional differential equation. {\it J Math Anal Appl}. 2007; 328: 1075‐ 1081. · Zbl 1120.26003
[38] Li, M, Wang, JR. Finite time stability of fractional delay differential equations. {\it Appl Math Lett}. 2017; 64: 170‐ 176. · Zbl 1354.34130
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.