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Volume 8 Issue 11
Nov.  2021

IEEE/CAA Journal of Automatica Sinica

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Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2021  >  8(11): 1837-1851
M. 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: M. 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
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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

  • Department of Electrical Engineering, Sahand University of Technology, Sahand New Town, Tabriz 51335-1996, Iran

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    Abstract
  • 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.

     

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    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

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