##plugins.themes.bootstrap3.article.main##

Shadi Amiri Mohammad Keyanpour

Abstract

We investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional diffusion (FD) equation, where the interconnections are of Neumann type. We exploit the PDE back stepping method as a powerful tool for designing a controller to show the Mittag–Leffler stability of the FD-FODE cascade. Finally, numerical simulations are presented to verify the results.

Article Details

Keywords

Backstepping;, Stability;, Fractional-order cascaded systems.

References
1. Aamo, O.M., Smyshlyaev, A., and Krsti´c, M. Boundary control of the linearized Ginzburg-Landau model of vortex shedding, SIAM J. Control Optim. 43 (2005), 1953–1971.
2. Aamo, O.M., Smyshlyaev, A., Krstic, M. and Foss, B.A. Stabilization of a Ginzburg-Landau model of vortex shedding by output feedback bound ary control, Decision and Control, 2004. CDC. 43rd IEEE Conference, 3 (2004), 2409–2416.
3. Aguila-Camacho, N., Duarte-Mermoud, M. A. and Gallegos, J.A. Lyapunov functions for fractional order systems, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2951–2957.
4. Ahn, H.-S., Chen, Y.Q. and Podlubny, I. Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, Appl. Math. Comput. 187 (2007), 27–34.
5. Balogh, A. and Krstic, M. Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability, Eur. J. Control. 8(2) 2002, 165–175.
6. Balogh, A. and Krstic, M. Stability of partial difference equations governing control gains in infinite-dimensional backstepping, Systems Control Lett. 51 (2004), 151–164.
7. Chen, J., Zeng, Z. and Jiang, P. Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks Neural Networks, 51, (2014), 1–8.
8. Chen, J., Zhuang, B., Chen, Y.Q. and Cui, B. Backstepping-based boundary feedback control for a fractional reaction diffusion system with mixed or Robin boundary conditions, IET Control Theory Appl. 11(17) (2017), 2964–2976.
9. Ding, D., Qi, D., Meng, Y. and Xu, L. Adaptive Mittag-Leffler stabilization of commensurate fractional-order nonlinear systems Decision and Control (CDC), IEEE 53rd Annual Conference, (2014), 6920–6926.
10. Ding, D., Qi, D. and Wang, Q. Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory Appl. 9 (2015), 681–690.
11. Efe, M.O. ¨ Application of backstepping control technique to fractional or der dynamic systems, Fractional dynamics and control, 33–47, Springer, New York, 2012.
12. Ge, F., Chen, Y.Q. and Kou, C. Boundary feedback stabilisation for the time fractional-order anomalous diffusion system, IET Control Theory Appl. 10 (2016), 1250–1257.
13. Huang, S. and Wang, B. Stability and stabilization of a class of fractional order nonlinear systems for 0 < α < 2, Nonlinear Dynam. 88 (2017), 973–984.
14. Huang, S., Zhang, R. and Chen, D. Stability of nonlinear fractional-order time varying systems, J. Comput.Nonlin. Dyn. 11(3) (2016), 9 pp.
15. Krstic, M. Compensating actuator and sensor dynamics governed by diffusion PDEs, Systems Control Lett. 58 (2009), 372–377.
16. Krstic, M. Compensating a string PDE in the actuation or sensing path of an unstable ODE, IEEE Trans. Automat. Control 54 (2009), 1362–1368.
17. Krstic, M. and Smyshlyaev, A. Boundary control of PDEs: A course on backstepping designs SIAM, 2008.
18. Li, Y., Chen, Y.Q. and Podlubny, I. Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), 1965–1969.
19. Li, Y., and Chen, Y.Q. and Podlubny, I. Stability of fractional-order non linear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability, Comput. Math. Appl. 59 (2010), 1810–1821.
20. Lim, Y.-H., Oh, K.-K. and Ahn, H.-S. Stability and stabilization of fractional-order linear systems subject to input saturation, IEEE Trans. Automat. Control 58 (2013), 1062–1067.
21. Liu, W. Boundary feedback stabilization of an unstable heat equation, SIAM J. Control Optim. 42 (2003), 1033–1043.
22. Mandelbrot, B.B. The fractal geometry of nature, Schriftenreihe f¨ur den Referenten. [Series for the Referee] W. H. Freeman and Co., San Francisco, Calif., 1982.
23. Mbodje, B. and Montseny, G. Boundary fractional derivative control of the wave equation, IEEE Trans. Automat. Control 40 (1995), 378–382.
24. Murio, D.A. Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl. 56 (2008), 1138–1145.
25. Podlubny, I. Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
26. Smyshlyaev, A. and Krstic, M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. Automat. Control 49 (2004), 2185–2202.
27. Susto, G.A. and Krstic, M. Control of PDE–ODE cascades with Neumann interconnections, J. Franklin I. 347(1) 2010, 284–314.
28. Tang, S. and Xie, C. Stabilization for a coupled PDE-ODE control system, J. Franklin Inst. 348 (2011), 2142–2155.
29. Torvik, P.J. and Bagley, R.L. On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51(2) 1984, 294–298.
30. Zhou, H.-C. and Guo, B.-Z. Boundary feedback stabilization for an unstable time fractional reaction diffusion equation, SIAM J. Control Optim. 56 (2018), 75–101.
How to Cite
AmiriS., & KeyanpourM. (2020). On the stabilization of a coupled fractional ordinary and partial differential equations†. Iranian Journal of Numerical Analysis and Optimization, 10(1), 177-193. https://doi.org/10.22067/ijnao.v10i1.84902
Section
Research Article