Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-21T16:44:41.508Z Has data issue: false hasContentIssue false
  • English
  • Français

A SEPARATION PRINCIPLE FOR THE STABILISATION OF A CLASS OF FRACTIONAL ORDER TIME DELAY NONLINEAR SYSTEMS

Part of: Control systems Stability

Published online by Cambridge University Press:  30 August 2018

NADHEM ECHI Open the ORCID record for NADHEM ECHI [Opens in a new window] ,
IMED BASDOURI  and
HANEN BENALI
NADHEM ECHI*
Affiliation:
Department of Mathematics, Faculty of Sciences, Gafsa University, Zarroug Gafsa 2112, Tunisia email nadhemechi_fsg@yahoo.fr
IMED BASDOURI
Affiliation:
Department of Mathematics, Faculty of Sciences, Gafsa University, Zarroug Gafsa 2112, Tunisia email basdourimed@yahoo.fr
HANEN BENALI
Affiliation:
Department of Mathematics, Faculty of Sciences, Sfax University, BP 1171, Sfax 3000, Tunisia email hanenbenali2017@yahoo.com
*
✉nadhemechi_fsg@yahoo.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish a separation principle for a class of fractional order time-delay nonlinear differential systems. We show that a nonlinear time-delay observer is globally convergent and give sufficient conditions under which the observer-based controller stabilises the system.

Keywords

Riemann–Liouville fractional systemnonlinear time delay systemobserver designasymptotical stabilityLyapunov functional

MSC classification

Primary: 93C10: Nonlinear systems
Secondary: 93D15: Stabilization of systems by feedback 93D20: Asymptotic stability
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 161 - 173
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Baleanu, D., Ranjbar, A., Sadati, S. J., Delavari, H., Abdeljawad (Maraaba), T., Gejji, V. and Mustafa, O. G., ‘Lyapunov–Krasovskii stability theorem for fractional systems with delay’, Rom. J. Phys. 56(5–6) (2011), 636–643.Google Scholar
Benabdallah, A., ‘A separation principle for the stabilization of a class of time delay nonlinear systems’, Kybernetika 51(1) (2015), 99–111.Google Scholar
Benabdallah, A. and Echi, N., ‘Global exponential stabilisation of a class of nonlinear time-delay systems’, Internat. J. Systems Sci. 47(16) (2016), 3857–3863.Google Scholar
Boyd, S., Ghaoui, L. E., Feron, E. and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics, 15 (SIAM, Philadelphia, PA, 1994).Google Scholar
Chen, L., He, Y., Wu, R., Chai, Y. and Yin, L., ‘Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations’, Int. J. Control Autom. Syst. 12(3) (2014), 697–702.Google Scholar
De la Sen, M., ‘About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory’, Fixed Point Theory Appl. 2011 (2011), Article ID 867932.Google Scholar
Duarte-Mermoud, M. A., Aguila-Camacho, N., Gallegos, J. A. and Castro-Linares, R., ‘Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 650–659.Google Scholar
Echi, N., ‘Observer design and practical stability of nonlinear systems under unknown time-delay’, Preprint 2018. https://arxiv.org/abs/1806.03166.Google Scholar
Echi, N. and Benabdallah, A., ‘Delay-dependent stabilization of a class of time-delay nonlinear systems: LMI approach’, Adv. Difference Equ. 2017 (2017), Article ID 271.Google Scholar
Gahinet, P., Nemirovski, A., Laub, A. and Chilali, M., LMI Control Toolbox User’s Guide (The Mathworks, Natick, MA, 1995).Google Scholar
Gauthier, J. P., Hammouri, H. and Othman, S., ‘A simple observer for nonlinear systems—Application to bioreactors’, IEEE Trans. Automat. Control 37 (1992), 875–880.Google Scholar
Ibrir, S., ‘Observer-based control of a class of time-delay nonlinear systems having triangular structure’, Automatica J. IFAC 47 (2011), 388–394.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Application of Fractional Differential Equations (Elsevier, New York, NY, 2006).Google Scholar
Li, Y., Chen, Y. Q. and Podlubny, I., ‘Mittag–Leffler stability of fractional order nonlinear dynamic systems’, Automatica J. IFAC 45 (2009), 1965–1969.Google Scholar
Liu, S., Wu, X., Zhou, X. and Jiang, W., ‘Asymptotical stability of Riemann–Liouville fractional nonlinear systems’, Nonlinear Dynam. 86(1) (2016), 65–71.Google Scholar
Liu, S., Zhou, X. F., Li, X. and Jiang, W., ‘Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays’, Appl. Math. Lett. 65 (2017), 32–39.Google Scholar
Momani, S. and Hadid, S., ‘Lyapunov stability solutions of fractional integro-differential equations’, Int. J. Math. Math. Sci. 47 (2004), 2503–2507.Google Scholar
Podlubny, I., Fractional Diferential Equations (Academic Press, San Diego, CA, 1999).Google Scholar
Sadati, S. J. D., Baleanu, D., Ranjbar, A., Ghaderi, R. and Abdeljawad, T., ‘Mittag–Leffler stability theorem for fractional nonlinear systems with delay’, Abstr. Appl. Anal. 2010 (2010), Article ID 108651, 7 pages.Google Scholar
Wang, F. F., Chen, D. Y., Zhang, X. G. and Wu, Y., ‘The existence and uniqueness theorem of the solution to a class of nonlinear fractional order system with time delay’, Appl. Math. Lett. 53 (2016), 45–51.Google Scholar
Zhang, X., ‘Some results of linear fractional order time-delay system’, Appl. Math. Comput. 197 (2008), 407–411.Google Scholar