Finite time stability of linear fractional order dynam-ical system with variable delays
DOI:
https://doi.org/10.14419/ijamr.v7i2.9167Published:
2018-03-08Keywords:
Dynamical System, Fractional Calculus, Finite Time Stability, Time Varying Delays.Abstract
In this paper, some sufficient condition ensuring finite time stability are derived for a class of linear fractional order dynamical system with variable delay using generalized Gronwall inequality as well as classical Bellman-Gronwall inequality.
References
[1] F. Amato, M. Ariola, C. Cosentino, C. T. Abdallah, and P. Dorato, Necessary and sufficient conditions for finite-time stability of linear systems. Proc. American Control Conference, Denver, Colorado, USA (2003), 4452–4456.
[2] E. K. Boukas and N. F. Al-Muthairi, Delay-dependent stabilization of singular linear systems with delays. International Journal of Innovative Computing, Information and Control, 2(2), (2006), 283–291.
[3] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, II. Geophys. J. R. Astron. Soc. 13, (1967), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.
[4] Y. Q. Chen and K. L. Moore, Analytical Stability bound for a class of delayed fractional order dynamic system, Nonlin. Dyn. Vol. 29, (202), 191-200.
[5] S. Das, Functional fractional calculus, Academic press, San Diego (1999).
[6] D. L. Debeljkovi, M. P. Lazarevic, D. Koruga, S. A. Milinkovic, M. B. Jovanovic, and L. A. Jacic, Further Results on Non-Lyapunov stability of the linear nonautonomous systems with delayed state, Facta Universitatis, Mechanics, Automatic Control and Robotic series, 3(11), (2001), 231–241.
[7] D. L. Debeljkovi, S. B. Stojannovic, G. V. Simeunovic and N. J. Dimitrijevic, Further results on stability of singular time delay systems in the sense of non-Lyapunov: A new delay dependent conditions, Automatic Control and Information Sciences, 2(1), (2014) 13–19.
[8] D. L. Debeljkovic, S. B. Stojanovic, and A. M. Jovanovic, Further results on finite time and practical stability of linear continuous time delay systems. FME Transactions, 41(3), (2013), 241–249.
[9] W. Deng, Smoothness and stability of the solution for nonlinear fractional differential equation, Nonlinear Anal. 72(3-4), (2010), 1768- 1777. https://doi.org/10.1016/j.na.2009.09.018.
[10] W. Deng, and C. Li, Numerical schemes for fractional ordinary differential equations. Numerical Modelling, Dr. Peep Miidla (Ed), ISBN: 978-953-51-0219-9, InTech, Retrieved from http://cdn.intechopen.com/pdfs-wm/33091.pdf, (2012).
[11] W. Deng, C. Li, and J. Lü, Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 48(4), (2007), 409–416. https://doi.org/10.1007/s11071-006-9094-0.
[12] P. Denghao, and J. Wei, Finite time stability analysis of fractional singular time delay systems, Advances in difference equation, 2014/1/259.
[13] C. A. Desoer, and M. Vidyasagar, (1975). Feedback Systems: Input- Output properties. Academic Press, New York, (1975).
[14] L. Dugard, and E. I. Verriest, Stability and control of time delay systems Springer-Verlag, (1997).
[15] N. M. F. Ferreira, F. B. Duarte, F. Miguel, M. G. Marcos, and J. A. T. Machado, Application of fractional calculus in the dynamical analysis and control of mechanical manipulators. Fractional calculus and applied analysis, 11(1), (2008), 91-113.
[16] J. K. Hale, Functional differential equations, Springer, New York, (1971). https://doi.org/10.1007/978-1-4615-9968-5.
[17] K. Gu, V. I. Kharitonov, and J. Chen, Stability of time delay systems, Boston, Birkhauser, (2003). https://doi.org/10.1007/978-1-4612-0039-0.
[18] R. Hilfer, Application of fractional calculus in Physics, World Scientific Singapore, (2000) https://doi.org/10.1142/3779.
[19] Y. Hong, Y. Xu, and J. Huang, Finite time control for robot manipulation, Systems and Control Letters, 46, (2002), 243–253. https://doi.org/10.1016/S0167-6911(02)00130-5.
[20] Y. Hong, Finite time Stabilization and Stabilizability of a class of controllable systems. Systems and Control Letters, 46, (2002), 231– 236. https://doi.org/10.1016/S0167-6911(02)00119-6.
[21] G. D. Hu, and M. Z. Liu, The weighted logarithm matrix norm and bounds of the matrix exponential. Linear Algebra Appl., 390, (2004), 145–154. https://doi.org/10.1016/j.laa.2004.04.015.
[22] G. D. Hu and T. Mitsui, Bounds of the matrix eigenvalues and its exponential by lyapunov equation. Kybernetika, 48(5), (2012), 865– 878.
[23] P. C. Jackreece, Finite time stability of linear control system with multiple delays, Control Theory and Informatics, Vol. 6(3), (2016), 69-72.
[24] P. C. Jackreece, Finite time stability of linear control system with constant delay in the state, International Journal of Mathematics and Statistics Invention(IJMI), Vol.5(2), (2017), 49-52.
[25] H. Jia, X. Cao, X. Yu, and P. Zhang, A simple approach to determine power system delay margin, in Proceedings of the IEEE PES general meeting, Montreal, Quebec, (2007), 1-7 https://doi.org/10.1109/PES.2007.385467.
[26] N. A. Kablar, and D. L. Debeljkovic, Non-Lyapunov stability of linear singular systems: Matrix measure approach. Preprints 5th IFAC Symposium on Low Cost Automation, Shenyang, China, September 8- 10, TS13, (1998), 16–20.
[27] E. Kaslik, and S. Sivasundaram, An analytical and numerical methods for the stability analysis of linear fractional delay differential equations Journal of computational and Applied Mathematics, 236(16), (2012), 4027-4041. https://doi.org/10.1016/j.cam.2012.03.010.
[28] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, New York, (2006).
[29] V. V. Kulish, and J. L. Lage, Application of fractional calculus to Mechanics, Journal of Fluids Engineering, Vol. 124, No. 3, (2002), 803-806. https://doi.org/10.1115/1.1478062.
[30] M. P. Lazarevic, Finite time stability analysis of fractional control of robotic time delay systems, Mech. Res. Commun.33 (2), (2006), 269- 279. https://doi.org/10.1016/j.mechrescom.2005.08.010.
[31] M. P. Lazarevic, and D. L. Debeljkovi, Finite time stability analysis of linear autonomous fractional order systems with delayed state, Asian Journal of Control, 7(4), (2005), 440–447. https://doi.org/10.1111/j.1934-6093.2005.tb00407.x.
[32] M. P. Lazarevic, A. Obradovic and V. Vasic, Robust finite time stability analysis of fractional order time delay systems: New results, Advances in dynamical systems and control (2010), 101-106, retrieved from www.wsea.us/e-library/conference/2010/Tunisia/control.
[33] T. N. Lee, and S. Diant, Stability of time delay system, IEEE Trans. Automat. Control AC 31(3), (1981), 951-953. https://doi.org/10.1109/TAC.1981.1102755.
[34] C. Li, K. Chen, J. Lu, and R. Tang, Stability and Stabilization analysis of fractional order linear systems subject to actuator saturation and distribution, IFAC paper online 50-1, (2017), 9718-9723.
[35] Y. Li, Y. Chen, and I. Podlubny, Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag- Leffler stability, Comput. Math. Appl. 59(5), (2010), 1810-1821. https://doi.org/10.1016/j.camwa.2009.08.019.
[36] P. L. Liu, Exponential stability for linear time delay systems with delay dependence. Journal of the Franklin Institute, 340, (2003), 481– 488. https://doi.org/10.1016/j.jfranklin.2003.10.003.
[37] P. L. Liu, Robust exponential stabilization for uncertain systems with state and control Delay. International Journal of Systems Science, 34(12–13), (2003), 675–682. https://doi.org/10.1080/00207720310001640719.
[38] D. Matignon, Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications, (1996), 963–968.
[39] F. A. Mohd, S. Manoj, and J. Renu, An application of fractional calculus in electrical engineering, Advanced Engineering Technology and Application, 5(2), (2016), 41-45. https://doi.org/10.18576/aeta/050204.
[40] S. Momani, and S. Hadid, Lyapunov stability solution of fractional integrodifferential equations, Int. J. math. Math. Sc. 47, (2004), 2503- 2507.
[41] T. Mori, Criteria for Asymptotic stability of linear time delay systems, IEEE trans. Automat. Control, AC 30, (1985), 158-161.
[42] E. Moulay, M. Dambrine, N. Yeganefar, and W. Perruquetti, Finite Time Stability and Stabilization of time delay systems. Systems and control letters, 57, (2008), 561–566. https://doi.org/10.1016/j.sysconle.2007.12.002.
[43] M. Naber, Time fractional Schrodinger equation, J. Maths. Phys. 45, (2004), 3339-3352. https://doi.org/10.1063/1.1769611.
[44] K. B. Oldham, and J. Spanier, The fractional calculus, Academic Press, New York, (1974).
[45] I. Podlubny, Fractional differential equation, Academic Press, San Diego, (1999).
[46] Y. Ryabov, and A. Puzenko, Damped oscillation in view of the fractional oscillator equation, Phys. Rev., 66, (2002), 184-201. https://doi.org/10.1103/PhysRevB.66.184201.
[47] J. Sabatier, M. Moze, and C. Farges, Stability conditions for fractional order systems, Comput. Math. Appl. 59(5), (2010), 1594-1609. https://doi.org/10.1016/j.camwa.2009.08.003.
[48] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives theory and applications, Gordon and Breach, New York, (1993).
[49] Y. Shen, L. Zhu, and Q. Guo, Finite time boundedness analysis of uncertain neural networks with time delay: An LMI approach. Proc. 4th Int Symp. Neural Networks, Nanjing, China. (2007), 904–909. https://doi.org/10.1007/978-3-540-72383-7_105.
[50] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, Vol. 23, (2002), 397-404.
[51] C. Tunç, and E. Biçer, Stability to a Kind of Functional differential equations of second order with multiple delays by fixed points, Abstract and Applied Analysis, 2014(3).
[52] H. Ye, J. Gao, and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328(2007), 1075-1081. https://doi.org/10.1016/j.jmaa.2006.05.061.
[53] F. Yu, Integrable coupling system of fractional differential equations, Eur. Phys. J. Spec. Top., (2011), 193, 27-47.
[54] M. Zavarei, and M. Jamshidi, Time Delay systems: Analysis, Optimization and Applications, North-Holland, Amsterdam, (1987).
[55] X. Zhang, Some results of linear fractional order time delay system, Appl. Math. Comp., 197, (2008), 407-411. https://doi.org/10.1016/j.amc.2007.07.069.
[56] S. Zhou, and J. Lam, Robust stabilization of delay singular systems with linear fractional parametric uncertainties. Circuits Systems Signal Processing, 22(6), (2003), 578–588. https://doi.org/10.1007/s00034-003-1218-x.
License
Authors who publish with this journal agree to the following terms:
                          [1]           Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
                          [2]           Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
                          [3]           Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).