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 8 Issue 11
Nov.  2021

IEEE/CAA Journal of Automatica Sinica

  • JCR Impact Factor: 7.847, Top 10% (SCI Q1)
    CiteScore: 13.0, Top 5% (Q1)
    Google Scholar h5-index: 64, TOP 7
Turn off MathJax
Article Contents
Mohammad Hejri, "Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid," IEEE/CAA J. Autom. Sinica, vol. 8, no. 11, pp. 1837-1851, Nov. 2021. doi: 10.1109/JAS.2021.1004183
Citation: Mohammad Hejri, "Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid," IEEE/CAA J. Autom. Sinica, vol. 8, no. 11, pp. 1837-1851, Nov. 2021. doi: 10.1109/JAS.2021.1004183

Global Practical Stabilization of Discrete-Time Switched Affine Systems via a General Quadratic Lyapunov Function and a Decentralized Ellipsoid

doi: 10.1109/JAS.2021.1004183
More Information
  • This paper addresses the problem of global practical stabilization of discrete-time switched affine systems via state-dependent switching rules. Several attempts have been made to solve this problem via different types of a common quadratic Lyapunov function and an ellipsoid. These classical results require either the quadratic Lyapunov function or the employed ellipsoid to be of the centralized type. In some cases, the ellipsoids are defined dependently as the level sets of a decentralized Lyapunov function. In this paper, we extend the existing results by the simultaneous use of a general decentralized Lyapunov function and a decentralized ellipsoid parameterized independently. The proposed conditions provide less conservative results than existing works in the sense of the ultimate invariant set of attraction size. Two different approaches are proposed to extract the ultimate invariant set of attraction with a minimum size, i.e., a purely numerical method and a numerical-analytical one. In the former, both invariant and attractiveness conditions are imposed to extract the final set of matrix inequalities. The latter is established on a principle that the attractiveness of a set implies its invariance. Thus, the stability conditions are derived based on only the attractiveness property as a set of matrix inequalities with a smaller dimension. Illustrative examples are presented to prove the satisfactory operation of the proposed stabilization methods.

     

  • loading
  • [1]
    D. Liberzon, Switching in Systems and Control, T. Basar, Ed. Bikhauser Boston, 2003.
    [2]
    G. S. Deaecto, J. C. Geromel, F. S. Garcia, and J. A. Pomilio, “Switched affine systems control design with application to DC-DC converters,” IET Control Theory and Applications, vol. 4, no. 7, pp. 1201–1210, 2009.
    [3]
    S. Baldi, A. Papachristodoulou, and E. B. Kosmatopoulos, “Adaptive pulse width modulation design for power converters based on affine switched systems,” Nonlinear Analysis:Hybrid Systems, vol. 30, pp. 306–322, 2018. doi: 10.1016/j.nahs.2018.07.002
    [4]
    V. L. Yoshimora, E. Assuncao, E. R. P. da Silva, and M. C. M. Teixeira, “Observer-based control design for switched affine systems and applications to DC-DC converters,” Journal of Control,Automation and Electrical Systems, vol. 24, no. 4, pp. 535–543, 2013. doi: 10.1007/s40313-013-0044-z
    [5]
    T. Wang, Y. Liu, X. Wang, and J. Li, “Robust sampling-based switching design for piecewise affine systems with application to DC-DC converters,” IET Control Theory &Applications, vol. 13, no. 9, pp. 1404–1412, 2019.
    [6]
    C. Albea, G. Garcia, and L. Zaccarian, “Hybrid dynamic modeling and control of switched affine systems: Application to DC-DC converters,” in Proc. IEEE 54th Annual Conf. Decision and Control, Osaka, Japan, Dec. 2015, pp. 2264–2269.
    [7]
    G. Beneux, P. Riedinger, J. Daafouz, and L. Grimaud, “Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters,” Automatica, vol. 99, pp. 82–91, 2019. doi: 10.1016/j.automatica.2018.10.015
    [8]
    M. Hejri, A. Giua, and H. Mokhtari, “On the complexity and dynamical properties of mixed logical dynamical systems via an automaton-based realization of discrete-time hybrid automaton,” Int. Journal of Robust and Nonlinear Control, vol. 28, no. 16, pp. 4713–4746, 2018. doi: 10.1002/rnc.4278
    [9]
    G. S. Deaecto and J. C. Geromel, “Stability analysis and control design of discrete-time switched affine systems,” IEEE Trans. Automatic Control, vol. 62, no. 8, pp. 4058–4065, Aug. 2017. doi: 10.1109/TAC.2016.2616722
    [10]
    L. N. Egidio and G. S. Deaecto, “Novel practical stability conditions for discrete-time switched affine systems,” IEEE Trans. Automatic Control, vol. 64, no. 11, pp. 4705–4710, 2019. doi: 10.1109/TAC.2019.2904136
    [11]
    C. Albea Sanchez, G. Garcia, H. Sabrina, W. P. M. H. Heemels, and L. Zaccarian, “Practical stabilisation of switched affine systems with dwell-time guarantees,” IEEE Trans. Automatic Control, vol. 64, no. 11, pp. 4811–4817, 2019. doi: 10.1109/TAC.2019.2907381
    [12]
    Z. Li, D. Ma, and J. Zhao, “Dynamic event-triggered L control for switched affine systems with sampled-data switching,” Nonlinear Analysis:Hybrid Systems, vol. 39, pp. 1–12, 2021.
    [13]
    S. Ding, X. Xie, and Y. Liu, “Event-triggered static/dynamic feedback control for discrete-time linear systems,” Information Sciences, vol. 524, 2020.
    [14]
    S. Ding and Z. Wang, “Event-triggered synchronization of discrete-time neural networks: A switching approach,” Neural Networks, vol. 125, 2020.
    [15]
    X. Xu and G. Zhai, “Practical stability and stabilization of hybrid and switched systems,” IEEE Trans. Automatic Control, vol. 50, no. 11, pp. 1897–1903, Nov. 2005. doi: 10.1109/TAC.2005.858680
    [16]
    X. Xu, G. Zhai, and S. He, “On practical asymptotic stabilizability of switched affine systems,” Nonlinear Analysis:Hybrid Systems, vol. 2, no. 1, pp. 196–208, 2008. doi: 10.1016/j.nahs.2007.07.003
    [17]
    X. Xu, G. Zhai, and S. He, “Some results on practical stabilizability of discrete-time switched affine systems,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 113–121, 2010.
    [18]
    V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Practical Stability of Nonlinear Systems. World Scientific, 1990.
    [19]
    A. Loría and E. Panteley, Stability, Told by Its Developers. London: Springer London, 2006, pp. 199–258.
    [20]
    M. Hejri, “Global practical stabilization of discrete-time switched affine systems via switched lyapunov functions and state-dependent switching functions,” Scientia Iranica, Transaction D, Computer Science & Electrical Engineering, DOI: 10.24200/SCI.2020.54524.3793, 2020.
    [21]
    M. Hejri, “On the global practical stabilization of discrete-time switched affine systems: Application to switching power converters,” Scientia Iranica, Transaction D, Computer Science & Electrical Engineering, DOI: 10.24200/SCI.2020.55427.4217, 2020.
    [22]
    L. Hetel and E. Fridman, “Robust sampled-data control of switched affine systems,” IEEE Trans. Automatic Control, vol. 58, no. 11, pp. 2922–2928, Nov. 2013. doi: 10.1109/TAC.2013.2258786
    [23]
    G. S. Deaecto and L. N. Egidio, “Practical stability of discrete-time switched affine systems,” in Proc. European Control Conf., Aalborg, Denmark, 2016, pp. 2048–2053.
    [24]
    C. A. Sanchez, A. Ventosa-Cutillas, A. S. A, and F. Gordillo, “Robust switching control design for uncertain discrete-time switched affine systems,” Int. Journal of Robust and Nonlinear Control, vol. 30, no. 17, pp. 7089–7102, 2020. doi: 10.1002/rnc.5158
    [25]
    P. Bolzern and W. Spinelli, “Quadratic stabilization of a switched affine system about a nonequilibrium point,” in Proc. American Control Conf. Boston, Massachusetts: IEEE, June 30–July 2 2004, pp. 3890–3895.
    [26]
    G. S. Deaecto and G. C. Santos, “State feedback H control design of continuous-time switched-affine systems,” IET Control Theory and Applications, vol. 9, no. 10, pp. 1511–1516, 2014.
    [27]
    G. S. Deaecto, “Dynamic output feedback H control of continuoustime switched affine systems,” Automatica, vol. 71, pp. 44–49, 2016. doi: 10.1016/j.automatica.2016.04.022
    [28]
    A. Poznyak, A. Polyakov, and V. Azhmyakov, Attractive Ellipsoids in Robust Control, T. Basar, Ed. Birkhauser, 2014.
    [29]
    C. Perez, V. Azhmyakov, and A. Poznyak, “Practical stabilization of a class of switched systems: Dwell-time approach,” IMA Journal of Mathematical Control and Information, vol. 32, no. 4, pp. 689–702, May 2014.
    [30]
    H. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 2003.
    [31]
    S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory. Society for Industrial and Applied Mathematics, SIAM, 1994.
    [32]
    J. Lofberg, “YALMIP: A toolbox for modeling and optimization in MATLAB,” in Proc. IEEE Int. Symp. Computer Aided Control Systems Design, Taipei, China, Sept. 2004, pp. 284–289.
    [33]
    G.-R. Duan and H.-H. Yu, LMIs in Control Systems: Analysis, Design and Applications. CRC Press, Taylor & Francis Group, 2013.
    [34]
    M. Kocvara and M. Stingl, PENBMI Users Guide (Version 2.1), www.penopt.com, March 5 2006.
    [35]
    M. J. Lacerda and T. da Silveira Gomide, “Stability and stabilizability of switched discrete-time systems based on structured Lyapunov functions,” IET Control Theory &Applications, vol. 14, no. 5, pp. 781–789, 2020.
    [36]
    A. Hassibi, J. How, and S. Boyd, “A path-following method for solving bmi problems in control,” in Proc. American Control Conf., San Diego, California, 1999, pp. 1385–1389.
    [37]
    W.-Y. Chiu, “Method of reduction of variables for bilinear matrix inequality problems in system and control designs,” IEEE Trans. Systems,Man,and Cybernetics:Systems, vol. 47, no. 7, pp. 1241–1256, Jul. 2017. doi: 10.1109/TSMC.2016.2571323
    [38]
    M. Hejri and A. Giua, “Hybrid modeling and control of switching DC-DC converters via MLD systems,” in Proc. IEEE 7th Int. Conf. Automation Science and Engineering, Trieste, Italy, Aug. 2011, pp. 714–719.

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(10)  / Tables(3)

    Article Metrics

    Article views (600) PDF downloads(38) Cited by()

    Highlights

    • Practical stability and Lyapunov stability are distinct concepts, and neither implies the other
    • A system may be practically stable with an acceptable performance yet still is Lyapunov unstable
    • New sufficient conditions are proposed for practical stabilization of switched affine systems
    • The stabilization is made through a shifted quadratic Lyapunov function and a shifted ellipsoid
    • The proposed stability conditions provide less conservative results than those in existing works

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return