Another Note on Stability of Sliding Mode Dynamics in Suppression of Fractional Chaotic Systems

Abstract:

This paper revisits the stability analysis of sliding mode dynamics in suppression of a classof fractional chaotic systems by a different approach. Firstly, we convert fractional differential equationsinto infinite dimensional ordinary differential equations based on the continuous frequency distributedmodel of the fractional integrator. Then we choose a Lyapunov function candidate to proposethe stability analysis. The result applies to both the commensurate fractional systems and the incommensurateones.

• Periodical: Applied Mechanics and Materials (Volume 743)
• Pages: 303-306
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.743.303
• Online since: March 2015
• Authors: J. Yuan*, B. Shi, Yan Wang
• Keywords: Chaos Suppression, Fractional Calculus, Fractional Chaotic Systems, Frequency Distributedmodel

References

[1] Sabatier, J., C. Farges, and J.C. Trigeassou, Fractional systems state space description: some wrong ideas and proposed solutions. Journal of Vibration and Control, (2013).

DOI: 10.1177/1077546313481839

[2] Trigeassou, J. and N. Maamri. The initial conditions of Riemman-Liouville and Caputo derivatives: an integrator interpretation. in FDA'2010 Conference, Badajoz, Spain. (2010).

[3] Trigeassou, J.C., et al., State variables and transients of fractional order differential systems. Computers & Mathematics with Applications, 2012. 64(10): pp.3117-3140.

DOI: 10.1016/j.camwa.2012.03.099

[4] Trigeassou, J.C., et al., Transients of fractional-order integrator and derivatives. Signal, Image and Video Processing, 2012. 6(3): pp.359-372.

DOI: 10.1007/s11760-012-0332-2

[5] Trigeassou, J.C. and N. Maamri, Initial conditions and initialization of linear fractional differential equations. Signal Processing, 2011. 91(3): pp.427-436.

DOI: 10.1016/j.sigpro.2010.03.010

[6] Lorenzo, C.F. and T.T. Hartley, Initialization of Fractional-Order Operators and Fractional Differential Equations. Journal of Computational and Nonlinear Dynamics, 2008. 3(2): p.021101.

DOI: 10.1115/1.2833585

[7] Hartley TT, L.C., The error incurred using the Caputo-derivative Laplace transform. proceedings of the ASME international design engineering technical con- ference, DETC2007-87648, San Diego, CA, 30 August-2 September., (2009).

[8] C.F. Lorenzo, H.T.T., Time-varying initialization and Laplace transform of the Caputo derivative: with order between zero and one. Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA.

DOI: 10.1115/detc2011-47396

[9] Faieghi, M.R., H. Delavari, and D. Baleanu, A note on stability of sliding mode dynamics in suppression of fractional-order chaotic systems. Computers & Mathematics with Applications, 2013. 66(5): pp.832-837.

DOI: 10.1016/j.camwa.2012.11.015

[10] Trigeassou, J.C., et al., A Lyapunov approach to the stability of fractional differential equations. Signal Processing, 2011. 91(3): pp.437-445.

DOI: 10.1016/j.sigpro.2010.04.024

[11] Yuan J, Shi B, Ji W. Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems[J]. Advances in Mathematical Physics, (2013).

DOI: 10.1155/2013/576709

[12] Roopaei, M., B.R. Sahraei, and T. -C. Lin, Adaptive sliding mode control in a novel class of chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 2010. 15(12): pp.4158-4170.

DOI: 10.1016/j.cnsns.2010.02.017

[13] Dadras, S. and H.R. Momeni, Control of a fractional-order economical system via sliding mode. Physica A: Statistical Mechanics and its Applications, 2010. 389(12): pp.2434-2442.

DOI: 10.1016/j.physa.2010.02.025

[14] Chen, D. -y., et al., Control of a class of fractional-order chaotic systems via sliding mode. Nonlinear Dynamics, 2011. 67(1): pp.893-901.

DOI: 10.1007/s11071-011-0002-x

[15] Yin, C., S. -m. Zhong, and W. -f. Chen, Design of sliding mode controller for a class of fractionalorder chaotic systems. Communications in Nonlinear Science and Numerical Simulation, 2012. 17(1): pp.356-366.

DOI: 10.1016/j.cnsns.2011.04.024