A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation
Volume 6 Issue 3
May  2019

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 6.171, Top 11% (SCI Q1)
    CiteScore: 11.2, Top 5% (Q1)
    Google Scholar h5-index: 51, TOP 8
Turn off MathJax
Article Contents
Saliha Marir and Mohammed Chadli, "Robust Admissibility and Stabilization of Uncertain Singular Fractional-Order Linear Time-Invariant Systems," IEEE/CAA J. Autom. Sinica, vol. 6, no. 3, pp. 685-692, May 2019. doi: 10.1109/JAS.2019.1911480
Citation: Saliha Marir and Mohammed Chadli, "Robust Admissibility and Stabilization of Uncertain Singular Fractional-Order Linear Time-Invariant Systems," IEEE/CAA J. Autom. Sinica, vol. 6, no. 3, pp. 685-692, May 2019. doi: 10.1109/JAS.2019.1911480

Robust Admissibility and Stabilization of Uncertain Singular Fractional-Order Linear Time-Invariant Systems

doi: 10.1109/JAS.2019.1911480
More Information
  • This paper addresses the robust admissibility problem in singular fractional-order continuous time systems. It is based on new admissibility conditions of singular fractional-order systems expressed in a set of strict linear matrix inequalities (LMIs). Then, a static output feedback controller is designed for the uncertain closed-loop system to be admissible. Numerical examples are given to illustrate the proposed methods.

     

  • loading
  • [1]
    I. S. Jesus and J. A. T. Machado, "Fractional control of heat diffusion systems, "Nonlinear Dynamics, DOI: 10.1007/s11071-007-9322-2, 2008.
    [2]
    B. M. Vinagre, I. Petr$acute{a}check{s}$, P. Merchan, and L. Dorcak, "Two digital realizations of fractional controllers: Application to temperature control of a solid, " in Proc. ECC'01, Porto, Portugal, pp. 1764-1767.
    [3]
    P. Arena, R. Caponetto, L. Fortuna, and D. Porto, Nonlinear Non integer Order Circuits and Systems-An Introduction, Singapore: World Scientific, 2000.
    [4]
    A. Charef, "Modeling and analog realization of the fundamental linear fractional order differential equation, "Nonlinear Dynamics, vol. 46, pp. 195-210, 2006. doi: 10.1007/s11071-006-9023-2
    [5]
    M. Nakagava and K. Sorimachi, "Basic characteristics of a fractance device, "IEICE Trans. Fundamentals, no. 12, pp. 1814-1818, 1992. http://search.ieice.org/bin/summary.php?id=e75-a_12_1814&category=&year=1992&lang=E&abst=
    [6]
    M. F. Silva, J. A. T. Machado, A. M. Lopes, "Fractional order control of a hexapod robot, "Nonlinear Dynamics, vol. 38, pp. 417-433, 2004. doi: 10.1007/s11071-004-3770-8
    [7]
    B. M. Vinagre, Y. Q. Chen, I. Petráš, "Two direct Tustin discretization methods for fractional-order differentiator/integrator, "J. Franklin Institute, vol. 340, pp. 349-362, 2003. doi: 10.1016/j.jfranklin.2003.08.001
    [8]
    Ch. Tseng, "Design of FIR and ⅡR fractional order Simpson digital integrators, "Signal Processing, vol. 87, pp. 1045-1057, 2007. doi: 10.1016/j.sigpro.2006.09.006
    [9]
    K. B. Oldham, J. Spanier, The Fractional Calculus, New York: Academic Press, 1974.
    [10]
    M. S. Tavazoei and M. Haeri, "Chaos control via a simple fractional-order controller, "Physics Letters A, vol. 372, pp. 798-807, 2008. doi: 10.1016/j.physleta.2007.08.040
    [11]
    Q. L. Han, "A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, "Automatica, vol. 40, pp. 1791-1796, 2004. doi: 10.1016/j.automatica.2004.05.002
    [12]
    K. Tanaka, H. Ohtake, H. O. Wang, "Hamilton-Jacobi equation for descriptor systems, "IEEE Trans. Fuzzy Syst., vol. 15, pp. 333-341, 2007. doi: 10.1109/TFUZZ.2006.880005
    [13]
    L. Dai. Lecture Notes in Control and Information Sciences, vol. 118. Singular Control Systems. New York: Springer-Verlag, 1989.
    [14]
    A. Oustaloup, B. Mathieu, and P. Lanusse, "The CRONE control of resonant plants: application to a flexible transmission, "Eur. J. Control, vol. 1, no. 2, 1995.
    [15]
    J. A. Tenreiro Machado, "Special issue on fractional calculus and applications, "Nonlinear Dynamics, vol. 29, pp. 1-385, Mar. 2002.
    [16]
    S. B. Skaar, A. N. Michel, R. K. Miller, "Stability of viscoelastic control systems, "IEEE Trans. Automat. Control, vol. AC-33, no. 4, pp. 348-357, Apr. 1988.
    [17]
    D. Matignon, "Stability result on fractional differential equations with applications to control processing, " in Proc. IMACS-SMC, Lille, France, July 1996, pp. 963-968.
    [18]
    J. Sabatier, M. Moze, C. Farges, "LMI stability conditions for fractional order systems, "Computers and Mathematics with Applications, vol. 59, pp. 1594-1609, 2010. doi: 10.1016/j.camwa.2009.08.003
    [19]
    Y. Q. Chen, H. S. Ahn, I. Podlubny, "Robust stability check of fractional order linear time invariant systems with interval uncertainties, "Signal Processing, vol. 86, pp. 2611-2618, 2006. doi: 10.1016/j.sigpro.2006.02.011
    [20]
    I. Petr$acute{a}check{s}$, Y. Q. Chen, B. M. Vinagre, "Robust Stability Test for Interval Fractional Order Linear Systems, " V. D. Blondel and A. Megretski, Eds. Princeton, NJ: Princeton Univ. Press, ch. 6. 5, pp. 208-210, Jul. 2004.
    [21]
    N. Tan, O. F. Ozguven, M. M. Ozyetkin, "Robust stability analysis of fractional order interval polynomials, "ISA Transactions, vol. 48, no. 2, pp. 166-172, 2009. doi: 10.1016/j.isatra.2009.01.002
    [22]
    H. S. Ahn, Y. Q. Chen, I, "Podlubny. Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality, "Appl. Math. Comput., vol. 187, no. 1, pp. 27-34, 2007.
    [23]
    H. S. Ahn and Y. Q. Chen, "Necessary and sufficient stability condition of fractional-order interval linear systems, "Automatica, vol. 44, no. 11, pp. 2985-2988, 2008. doi: 10.1016/j.automatica.2008.07.003
    [24]
    J. G. Lu, G. R. Chen, "Robust stability and stabilization of fractional- order interval systems: An LMI approach, "IEEE Trans. Autom. Control, vol. 54, no. 6, pp. 1294-1299, Jun. 2009.
    [25]
    Y.-H. Lan, H.-X. Huang, Y. Zhou, "Observer-based robust control of a $(1 < alpha < 2)$ fractional-order uncertain systems: a linear matrix inequality approach, "IET Control Theory and Applications, vol. 6, no. 2, pp. 229-234, 2012. doi: 10.1049/iet-cta.2010.0484
    [26]
    J. G. Lu and Y. Q. Chen, "Robust stability and stabilization of fractional-order interval systems with the fractional-order $alpha$: The < alpha < 1$ case, "IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 152-158, 2010. doi: 10.1109/TAC.2009.2033738
    [27]
    I. N'Doye, M. Darouach, H. Voos, M. Zasadzinski, "On the robustness of linear and non-linear fractional-order systems with non-linear uncertain parameters, "IMA Journal of Mathematical Control and Information, pp. 1-18, 2015. doi: 10.1093/imamci/dnv022.
    [28]
    Y. H. Lan, Y. Zhou, "Non-fragile observer-based robust control for a class of fractional-order nonlinear systems, "Systems and Control Letters, vol. 62, no. 12, pp. 1143-1150, 2013. doi: 10.1016/j.sysconle.2013.09.007
    [29]
    S. Ibrir, "Sufficient conditions for domain stabilisability of uncertain fractional-order systems under static-output feedbacks, "IET Control Theory and Applications, doi: 10.1049/iet-cta.2016.0476.
    [30]
    Y. Ji, J. Qiu, "Stabilization of fractional-order singular uncertain systems, "ISA Transactions, vol. 56, pp. 53-64, 2015. doi: 10.1016/j.isatra.2014.11.016
    [31]
    S. Marir, M. Chadli, D. Bouagada, "A Novel Approach of Admissibility for Singular Linear Continuous-time Fractional-order Systems, "International Journal of Control, Automation and Systems, vol. 15, pp. 1-6, 2017. http://dx.doi.org/10.1007/s12555-016-0003-0
    [32]
    Y. Yao, J. Zhuang, S. Chang-Yin, "Sufficient and necessary condition of admissibility for fractional-order singular system, "Acta Automatica Sinica, vol. 39, no. 12, pp. 2160-2164, 2013. doi: 10.1016/S1874-1029(14)60003-3
    [33]
    I. N'Doye, M. Darouach, M. Zasadzinski, N. Radhy, "Robust stabilisation of uncertain descriptor fractional-order Systems, " Automatica, vol. 49, no. 6, pp. 1907-1913, June 2013.
    [34]
    M. Boukens, A. Boukabou, M. Chadli, "A real time self-tuning motion controller for mobile robot systems, "Acta Automatica Sinica, vol. 6, no. 1, pp. 84-96, 2019. doi: 10.1109/JAS.2018.7511216
    [35]
    S. Marir, M. Chadli, D. Bouagada, "New admissibility conditions for singular linear continuous-time fractional-order systems, "Journal of the Franklin Institute, vol. 354, pp. 752-766, 2017. doi: 10.1016/j.jfranklin.2016.10.022
    [36]
    I. Podlubny.Fractional Differential Equations. New-York: Academic, 1999.
    [37]
    M. Chadli, M. Darouach, "Novel bounded real lemma for discrete-time descriptor systems: appliccation to $H_{infty}$ control design, "Automatica, vol. 48, no. 2, pp. 449-453, 2012. doi: 10.1016/j.automatica.2011.10.003
    [38]
    S. Xu, J. Lam, Control and Filtering of Singular Systems. Springer, Berlin, 2006.
    [39]
    P. Khargonakar, I. Petersen, K. Zhou, "Robust stabilization of uncertain linear systems: quadratic stability and $H_{infty}$ control theory, "IEEE Trans. Automat. Contr., vol. 35, no. 3, pp. 3561, 1990.
    [40]
    S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994.

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(4)

    Article Metrics

    Article views (1027) PDF downloads(32) Cited by()

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return