Technical Brief

Stabilization Conditions for a Class of Fractional-Order Nonlinear Systems

[+] Author and Article Information
Sunhua Huang

Department of Electrical Engineering,
Northwest A&F University,
Yangling, Shaanxi 712100, China;
Key Laboratory of Agricultural Soil and
Water Engineering in Arid and Semiarid Areas,
Ministry of Education,
Northwest A&F University,
Yangling, Shaanxi 712100, China

Bin Wang

Department of Electrical Engineering,
Northwest A&F University,
Yangling, Shaanxi 712100, China;
Key Laboratory of Agricultural Soil and
Water Engineering in Arid and Semiarid Areas,
Ministry of Education,
Northwest A&F University,
Yangling, Shaanxi 712100, China
e-mail: binwang@nwsuaf.edu.cn

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 18, 2018; final manuscript received February 26, 2019; published online March 14, 2019. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 14(5), 054501 (Mar 14, 2019) (6 pages) Paper No: CND-18-1369; doi: 10.1115/1.4042999 History: Received August 18, 2018; Revised February 26, 2019

The stabilization problem of fractional-order nonlinear systems for 0<α<1 is studied in this paper. Based on Mittag-Leffler function and the Lyapunov stability theorem, two practical stability conditions that ensure the stabilization of a class of fractional-order nonlinear systems are proposed. These stability conditions are given in terms of linear matrix inequalities and are easy to implement. Moreover, based on these conditions, the method for the design of state feedback controllers is given, and the conditions that enable the fractional-order nonlinear closed-loop systems to assure stability are provided. Finally, a representative case is employed to confirm the validity of the designed scheme.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, New York.
Wen, X. J. , Wu, Z. M. , and Lu, J. G. , 2008, “ Stability Analysis of a Class of Nonlinear Fractional-Order Systems,” IEEE Trans. Circuits Syst. II Express Briefs, 55(11), pp. 1178–1182. [CrossRef]
Huang, S. H. , Zhang, R. F. , and Chen, D. Y. , 2015, “ Stability of Nonlinear Fractional-Order Time Varying Systems,” ASME J. Comput. Nonlinear Dyn., 11(3), p. 031007. [CrossRef]
Li, C. , Wang, J. C. , Lu, J. G. , and Ge, Y. , 2014, “ Observer-Based Stabilisation of a Class of Fractional Order Non-Linear Systems for 0 < α < 2 Case,” IET Control Theory Appl., 8(13), pp. 1238–1246. [CrossRef]
Yang, X. J. , Li, C. D. , Huang, T. W. , and Song, Q. K. , 2017, “ Mittag-Leffler Stability Analysis of Nonlinear Fractional-Order Systems With Impulses,” Appl. Math. Comput., 293, pp. 416–422.
Ma, Y. J. , Wu, B. W. , and Wang, Y. E. , 2016, “ Finite-Time Stability and Finite-Time Boundedness of Fractional Order Linear Systems,” Neurocomputing, 173, pp. 2076–2082. [CrossRef]
Huang, S. H. , and Wang, B. , 2017, “ Stability and Stabilization of a Class of Fractional-Order Nonlinear Systems for 0 < α < 2,” Nonlinear Dyn., 88(2), pp. 973–984. [CrossRef]
Yang, Y. , He, Y. , Wang, Y. , and Wu, M. , 2017, “ Stability Analysis for Impulsive Fractional Hybrid Systems Via Variational Lyapunov Method,” Commun. Nonlinear Sci. Numer. Simul., 45, pp. 140–157. [CrossRef]
He, B. B. , Zhou, H. C. , Chen, Y. Q. , and Kou, C. H. , 2018, “ Asymptotical Stability of Fractional Order Systems With Time Delay Via an Integral Inequality,” IET Control Theory Appl., 12(12), pp. 1748–1754. [CrossRef]
Liu, Q. , Xu, Y. , and Kurths, J. , 2018, “ Active Vibration Suppression of a Novel Airfoil Model With Fractional Order Viscoelastic Constitutive Relationship,” J. Sound Vib., 432, pp. 50–64. [CrossRef]
Xin, B. G. , and Zhang, J. Y. , 2015, “ Finite-Time Stabilizing a Fractional-Order Chaotic Financial System With Market Confidence,” Nonlinear Dyn., 79(2), pp. 1399–1409. [CrossRef]
Baleanu, D. , Golmankhaneh, A. K. , Golmankhaneh, A. K. , and Baleanu, M. C. , 2009, “ Fractional Electromagnetic Equations Using Fractional Forms,” Int. J. Theor. Phys., 48(11), pp. 3114–3123. [CrossRef]
Sun, H. , Abdelwahab, A. , and Onaral, B. , 1984, “ Linear Approximation of Transfer Function With a Pole of Fractional Order,” IEEE Trans. Autom. Control, 29(5), pp. 441–444. [CrossRef]
Asghar, M. , and Nasimullah , 2018, “ Performance Comparison of Wind Turbine Based Doubly Fed Induction Generator System Using Fault Tolerant Fractional and Integer Order Controllers,” Renewable Energy, 116(Pt. B), pp. 244–264. [CrossRef]
Kusnezov, D. , Bulgac, A. , and Dang, G. D. , 1999, “ Quantum Levy Processes and Fractional Kinetics,” Phys. Rev. Lett., 82(6), pp. 1136–1139. [CrossRef]
Zhou, P. , Cai, H. , and Yang, C. D. , 2016, “ Stabilization of the Unstable Equilibrium Points of the Fractional-Order BLDCM Chaotic System in the Sense of Lyapunov by a Single-State Variable,” Nonlinear Dyn., 84(4), pp. 2357–2361. [CrossRef]
Chen, L. P. , Pan, W. , Wu, R. C. , Machado, J. A. T. , and Lopes, A. M. , 2016, “ Design and Implementation of Grid Multi-Scroll Fractional-Order Chaotic Attractors,” Chaos, 26(8), p. 084303. [CrossRef] [PubMed]
Wang, H. H. , Sun, K. H. , and He, S. B. , 2015, “ Characteristic Analysis and DSP Realization of Fractional-Order Simplified Lorenz System Based on Adomian Decomposition Method,” Int. J. Bifurcation Chaos, 25(6), p. 1550085. [CrossRef]
Lei, Y. M. , Fu, R. , Yang, Y. , and Wang, Y. Y. , 2016, “ Dichotomous-Noise-Induced Chaos in a Generalized Duffing-Type Oscillator With Fractional-Order Deflection,” J. Sound Vib., 363, pp. 68–76. [CrossRef]
Chen, L. P. , He, Y. G. , Chai, Y. , and Wu, R. C. , 2014, “ New Results on Stability and Stabilization of a Class of Nonlinear Fractional-Order Systems,” Nonlinear Dyn., 75(4), pp. 633–641. [CrossRef]
Li, Y. , Chen, Y. Q. , and Podlubny, I. , 2009, “ Mittag-Leffler Stability of Fractional Order Nonlinear Dynamic Systems,” Automatica, 45(8), pp. 1965–1969. [CrossRef]
Matignon, D. , 1996, “ Stability Results for Fractional Differential Equations With Applications to Control Processing,” Comput. Eng. Syst. Appl., 2, pp. 963–968. https://www.researchgate.net/publication/2581881_Stability_Results_For_Fractional_Differential_Equations_With_Applications_To_Control_Processing
Chen, G. P. , and Yang, Y. , 2016, “ New Stability Conditions for a Class of Linear Time-Varying Systems,” Automatica, 71, pp. 342–347. [CrossRef]
Liu, S. , Zhou, X. F. , Li, X. Y. , and Jiang, W. , 2017, “ Asymptotical Stability of Riemann-Liouville Fractional Singular Systems With Multiple Time-Varying Delays,” Appl. Math. Lett., 65, pp. 32–39. [CrossRef]
Lu, J. G. , and Chen, G. R. , 2009, “ Robust Stability and Stabilization of Fractional Order Interval Systems: An LMI Approach,” IEEE Trans. Autom. Control, 54(6), pp. 1294–1299. [CrossRef]
Sabatier, J. , Moze, M. , and Farges, C. , 2010, “ LMI Stability Conditions for Fractional Order Systems,” Comput. Math. Appl., 59(5), pp. 1594–1609. [CrossRef]
Shahri, E. S. A. , Alfi, A. , and Machado, J. A. T. , 2015, “ An Extension of Estimation of Domain of Attraction for Fractional Order Linear System Subject to Saturation Control,” Appl. Math. Lett., 47, pp. 26–34. [CrossRef]
Wang, Z. L. , Yang, D. S. , and Zhang, H. G. , 2016, “ Stability Analysis on a Class of Nonlinear Fractional-Order Systems,” Nonlinear Dyn., 86(2), pp. 1023–1033. [CrossRef]
Trigeassou, J. , Maamri, N. , Sabatier, J. , and Oustaloup, A. , 2011, “ A Lyapunov Approach to the Stability of Fractional Differential Equations,” Signal Process., 91(3), pp. 437–445. [CrossRef]
Liu, S. , Jiang, W. , Li, X. Y. , and Zhou, X. F. , 2016, “ Lyapunov Stability Analysis of Fractional Nonlinear Systems,” Appl. Math. Lett., 51, pp. 13–19. [CrossRef]
Aldo, J. M. , Manuel, B. O. , Anand, S. , and Vicente, P. , 2019, “ Adaptive Robust Control of Fractional-Order Systems With Matched and Mismatched Disturbances,” Math. Comput. Simul., (in press).
Aghababa, M. P. , 2013, “ A Novel Terminal Sliding Mode Controller for a Class of Non-Autonomous Fractional-Order Systems,” Nonlinear Dyn., 73(1–2), pp. 679–688. [CrossRef]
Jakovljevic, B. , Pisano, A. , Rapaic, M. R. , and Usai, E. , 2016, “ On the Sliding-Mode Control of Fractional-Order Nonlinear Uncertain Dynamics,” Int. J. Robust Nonlinear Control, 26(4), pp. 782–798. [CrossRef]
Podlubny, I. , 1999, “ Fractional-Order Systems and PIλDμ Controllers,” IEEE Trans. Autom. Control, 44(1), pp. 208–214. [CrossRef]
Ding, D. S. , Qi, D. L. , and Wang, Q. , 2015, “ Non-Linear Mittag-Leffler Stabilisation of Commensurate Fractional-Order Non-Linear Systems,” IET Control Theory Appl., 9(5), pp. 681–690. [CrossRef]
Wang, F. J. , Liu, Z. , Zhang, Y. , and Chen, C. L. P. , 2017, “ Adaptive Fuzzy Visual Tracking Control for Manipulator With Quantized Saturation Input,” Nonlinear Dyn., 89(2), pp. 1241–1258. [CrossRef]
Wang, B. , Xue, J. Y. , Wu, F. J. , and Zhu, D. L. , 2016, “ Stabilization Conditions for Fuzzy Control of Uncertain Fractional Order Non-Linear Systems With Random Disturbances,” IET Control Theory Appl., 10(6), pp. 637–647. [CrossRef]
Chen, D. Y. , Zhao, W. L. , Sprott, J. C. , and Ma, X. Y. , 2013, “ Application of Takagi-Sugeno Fuzzy Model to a Class of Chaotic Synchronization and Anti-Synchronization,” Nonlinear Dyn., 73(3), pp. 1495–1505. [CrossRef]


Grahic Jump Location
Fig. 1

State trajectories of the fractional-order Chen system

Grahic Jump Location
Fig. 2

Stabilization of the nonlinear fractional-order Chen system with α = 0.5

Grahic Jump Location
Fig. 3

State trajectories of the controlled nonlinear fractional-order Chen system with α = 0.5



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In