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Volume 3 Issue 4
Oct.  2016

IEEE/CAA Journal of Automatica Sinica

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Yige Zhao, Yuzhen Wang and Haitao Li, "State Feedback Control for a Class of Fractional Order Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 3, no. 4, pp. 483-488, Oct. 2016.
Citation: Yige Zhao, Yuzhen Wang and Haitao Li, "State Feedback Control for a Class of Fractional Order Nonlinear Systems," IEEE/CAA J. Autom. Sinica, vol. 3, no. 4, pp. 483-488, Oct. 2016.

State Feedback Control for a Class of Fractional Order Nonlinear Systems


National Natural Science Foundation of China 61374065, 61374002, 61503225, 61573215

the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the Natural Science Foundation of Shandong Province ZR2015FQ003

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  • Using the Lyapunov function method, this paper investigates the design of state feedback stabilization controllers for fractional order nonlinear systems in triangular form, and presents a number of new results. First, some new properties of Caputo fractional derivative are presented, and a sufficient condition of asymptotical stability for fractional order nonlinear systems is obtained based on the new properties. Then, by introducing appropriate transformations of coordinates, the problem of controller design is converted into the problem of finding some parameters, which can be certainly obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, state feedback stabilization controllers making the closed-loop system asymptotically stable are explicitly constructed. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.


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  • [1]
    Podlubny I. Fractional Differential Equations. New York:Academic Press, 1998.
    Kilbas A A, Srivastava H H, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier Science, 2006.
    Manabe S. The non-integer integral and its application to control systems. Electrotechnical Journal of Japan, 1961, 6(3-4):83-87 http://cn.bing.com/academic/profile?id=594721390&encoded=0&v=paper_preview&mkt=zh-cn
    Ahn H S, Chen Y Q, Podlubny I. Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Applied Mathematics and Computation, 2007, 187(1):27-34 doi: 10.1016/j.amc.2006.08.099
    Tavazoei M S, Haeri M. A note on the stability of fractional order systems. Mathematics and Computers in Simulation, 2009, 79(5):1566-1576 doi: 10.1016/j.matcom.2008.07.003
    Chen Y Q, Ahn H S, Xue D Y. Robust controllability of interval fractional order linear time invariant systems. Signal Processing, 2006, 86(10):2794-2802 doi: 10.1016/j.sigpro.2006.02.021
    Daftardar-Gejji V, Babakhani A. Analysis of a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 2004, 293(2):511-522 doi: 10.1016/j.jmaa.2004.01.013
    Yu Y, Jiao Z, Sun C Y. Sufficient and necessary condition of admissibility for fractional-order singular system. Acta Automatica Sinica, 2013, 39(12):2160-2164 doi: 10.1016/S1874-1029(14)60003-3
    Lazarevic M P, Spasic A M. Finite-time stability analysis of fractional order time-delay systems:Gronwall's approach. Mathematical and Computer Modelling, 2009, 49(3-4):475-481 doi: 10.1016/j.mcm.2008.09.011
    Shen J, Lam J. Non-existence of finite-time stable equilibria in fractional-order nonlinear systems. Automatica, 2014, 50(2):547-551 doi: 10.1016/j.automatica.2013.11.018
    Lakshmikantham V, Leela S, Sambandham M. Lyapunov theory for fractional differential equations. Communications in Applied Analysis, 2008, 12(4):365-376 http://cn.bing.com/academic/profile?id=1763455823&encoded=0&v=paper_preview&mkt=zh-cn
    Burton T A. Fractional differential equations and Lyapunov functionals. Nonlinear Analysis:Theory, Methods & Applications, 2011, 74(16):5648-5662 http://cn.bing.com/academic/profile?id=1986795491&encoded=0&v=paper_preview&mkt=zh-cn
    Si-Ammour A, Djennoune S, Bettayeb M. A sliding mode control for linear fractional systems with input and state delays. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(5):2310-2318 doi: 10.1016/j.cnsns.2008.05.011
    Kamal S, Raman A, Bandyopadhyay B. Finite-time stabilization of fractional order uncertain chain of integrator:an integral sliding mode approach. IEEE Transactions on Automatic Control, 2013, 58(6):1597-1602 doi: 10.1109/TAC.2012.2228051
    Li Y, Chen Y Q, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8):1965-1969 doi: 10.1016/j.automatica.2009.04.003
    Li Y, Chen Y Q, Podlubny I. Stability of fractional-order nonlinear dynamic systems:Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications, 2010, 59(5):1810-1821 doi: 10.1016/j.camwa.2009.08.019
    Aguila-Camacho N, Duarte-Mermoud M A, Gallegos J A. Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 2014, 19(9):2951-2957 doi: 10.1016/j.cnsns.2014.01.022
    Qian C J, Lin W. Output feedback control of a class of nonlinear systems:a nonseparation principle paradigm. IEEE Transactions on Automatic Control, 2002, 47(10):1710-1715 doi: 10.1109/TAC.2002.803542
    Choi H L, Lim J T. Global exponential stabilization of a class of nonlinear systems by output feedback. IEEE Transactions on Automatic Control, 2005, 50(2):255-257 doi: 10.1109/TAC.2004.841886
    Zhang X F, Cheng Z L. Output feedback stabilization of nonlinear systems with delays in the input. Applied Mathematics and Computation, 2005, 167(2):1026-1040 doi: 10.1016/j.amc.2004.08.001
    Shen J, Lam J. State feedback H∞ control of commensurate fractionalorder systems. International Journal of Systems Science, 2014, 45(3):363-372 doi: 10.1080/00207721.2012.723055


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