Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations

Volume 10, Issue 10, pp 5182--5200

Publication Date: 2017-10-12

http://dx.doi.org/10.22436/jnsa.010.10.06

Authors

Ibtisam Kamil Hanan - Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia
Muhammad Zaini Ahmad - Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia
Fadhel Subhi Fadhel - Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, P. O. Box 47077, Baghdad, Iraq

Abstract

This paper focuses on the application of fractional backstepping control scheme for nonlinear fractional partial differential equation (FPDE). Two types of fractional derivatives are considered in this paper, Caputo and the Grünwald-Letnikov fractional derivatives. Therefore, obtaining highly accurate approximations for this derivative is of a great importance. Here, the discretized approach for the space variable is used to transform the FPDE into a system of fractional differential equations. The convergence of the closed loop system is guaranteed in the sense of Mittag-Leffler stability. An illustrative example is given to demonstrate the effectiveness of the proposed control scheme.

Keywords

Backstepping method, fractional Lyapunov function, fractional derivative, boundary control, fractional partial differential equation

References

[1] M. P. Aghababa, Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2670–2681.
[2] N. Aguila-Camacho, M. A. Duarte-Mermoud, J. A. Gallegos, Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2951–2957.
[3] A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W. Shatanawi, Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893–8903.
[4] D. Baleanu, J. A. T. Machado, A. C. Luo, Fractional dynamics and control, Springer, New York, (2011).
[5] T. A. Burton, Fractional differential equations and Lyapunov functionals, Nonlinear Anal., 74 (2011), 5648–5662.
[6] S. Dadras, H. R. Momeni, Passivity-based fractional-order integral sliding-mode control design for uncertain fractionalorder nonlinear systems, Mechatronics, 23 (2013), 880–887.
[7] D.-S. Ding, D.-L. Qi, Q. Wang, Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory Appl., 9 (2015), 681–690.
[8] C. Farges, L. Fadiga, J. Sabatier, \(H_\infty\) analysis and control of commensurate fractional order systems, Mechatronics, 23 (2013), 772–780.
[9] C. Farges, M. Moze, J. Sabatier, Pseudo-state feedback stabilization of commensurate fractional order systems, Automatica J. IFAC, 46 (2010), 1730–1734.
[10] V. Lakshmikantham, S. Leela, M. Sambandham, Lyapunov theory for fractional differential equations, Commun. Appl. Anal., 12 (2008), 365–376.
[11] Y. H. Lan, H. B. Gu, C. X. Chen, Y. Zhou, Y. P. Luo, An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks, Neurocomputing, 136 (2014), 235–242.
[12] Y.-H. Lan, Y. Zhou, LMI-based robust control of fractional-order uncertain linear systems, Comput. Math. Appl., 62 (2011), 1460–1471.
[13] Y. Li, Y. Chen, I. Podlubny, MittagLeffler stability of fractional order nonlinear dynamic systems, Automatica, 45 (2009), 1965–1969.
[14] Y. Li, Y.-Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821.
[15] J.-G. Lu, Y.-Q. Chen, W. Chen, Robust asymptotical stability of fractional-order linear systems with structured perturbations, Comput. Math. Appl., 66 (2013), 873–882.
[16] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, (1974).
[17] F. Padula, S. Alcántara, R. Vilanova, A. Visioli, \(H_\infty\) control of fractional linear systems, Automatica, 49 (2013), 2276– 2280.
[18] I. Pan, S. Das, Intelligent fractional order systems and control: an introduction, Springer, New York, (2012).
[19] I. Petráš, Fractional-order nonlinear systems: modeling, analysis and simulation, Springer, New York, (2011).
[20] J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus, Springer, Dordrecht, (2007).
[21] H. Sheng, Y. Chen, T. Qiu, Fractional processes and fractional-order signal processing: techniques and applications, Springer, New York, (2011).
[22] B. Shi, J. Yuan, C. Dong, Pseudo-state sliding mode control of fractional SISO nonlinear systems, Adv. Math. Phys., 2013 (2013), 7 pages.
[23] E. Sousa, How to approximate the fractional derivative of order \(1 < \alpha\leq 2\), Int. J. Bifurcation. Chaos, 2012 (2012), 7 pages.
[24] T. Takamatsu, H. Ohmori, Sliding Mode Controller Design Based on Backstepping Technique for Fractional Order System, SICE JCMSI, 9 (2016), 151–157.
[25] Y. Tang, X. Zhang, D. Zhang, G. Zhao, X. Guan, Fractional order sliding mode controller design for antilock braking systems, Neurocomputing, 111 (2013), 122–130.
[26] J. C. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov Stability of Linear Fractional Systems: Part 1-Definition of Fractional Energy, ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2013 (2013), 10 pages.
[27] J. C. Trigeassou, N. Maamri, A. Oustaloup, Lyapunov Stability of Linear Fractional Systems: Part 2-Derivation of stability condition, ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2013 (2013), 10 pages.
[28] J.-C. Trigeassou, N. Maamri, J. Sabatier, A. Oustaloup, A Lyapunov approach to the stability of fractional differential equations, Signal Processing, 91 (2011), 437–445.
[29] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2530–2538.
[30] X. Wen, Z. Wu, J. Lu, Stability analysis of a class of nonlinear fractional-order systems, IEEE Transactions on circuits and systems II: Express Briefs, 55 (2008), 1178–1182.
[31] J. Yu, H. Hu, S. Zhou, X. Lin, Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems, Automatica J. IFAC, 49 (2013), 1798–1803.
[32] Y. H. Yuan, Q. S. Sun, Fractional-order embedding multiset canonical correlations with applications to multi-feature fusion and recognition, Neurocomputing, 122 (2013), 229–238.
[33] X. F. Zhou, L. G. Hu, S. Liu, W. Jiang, Stability criterion for a class of nonlinear fractional differential systems, Appl. Math. Lett., 28 (2014), 25–29.

Downloads

XML export