\(\mathcal {S}\)-Procedure for Positive Switched Linear Systems and its Equivalence to Lyapunov–Metzler Inequalities

  • Junfeng ZhangEmail author
  • Tarek Raïssi
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)


This paper presents the \(\mathcal {S}\)-procedure characterization for the stabilization of positive switched linear systems and establishes the relationship between the \(\mathcal {S}\)-procedure and its equivalent Lyapunov–Metzler inequalities. First, a piecewise linear co-positive Lyapunov function is constructed for positive switched linear systems. Under the Lyapunov function, the \(\mathcal {S}\)-procedure stabilization for positive switched linear systems in the continuous-time context is explored under a min state switching law. The \(\mathcal {S}\)-procedure conditions are formulated in the form of linear programming. Finally, an equivalence relationship between \(\mathcal {S}\)-procedure and Lyapunov–Metzler inequalities is presented.


Positive switched linear systems \(\mathcal {S}\)-procedure Lyapunov–Metzler inequalities Linear programming 



The authors thank the anonymous reviewers and associate editor for their valuable suggestions and comments which have helped to improve the quality of the paper. This work was supported in part by the National Nature Science Foundation of China (61873314, 61503107, and 61703132), the Zhejiang Provincial Natural Science Foundation of China (S18F030001), and the Foundation of Key Laboratory of System Control and Information Processing, Ministry of Education, P.R. China.


  1. 1.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  2. 2.
    Ferrari-Trecate, G., Cuzzola, F.A., Mignone, D., Morari, M.: Analysis of discrete-time piecewise affine and hybrid systems. Automatica 38, 2139–2146 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fornasini, E., Valcher, M.E.: Stability and stabilizability criteria for discrete-time positive switched systems. IEEE Trans. Autom. Control. 57(5), 1208–1221 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley, New York (2000)CrossRefGoogle Scholar
  5. 5.
    Geromel, J.C., Colaneri, P.: Stability and stabilization of discrete time switched systems. Int. J. Control. 79(7), 719–728 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Geromel, J.C., Colaneri, P.: Stability and stabilization of continuous-time switched linear systems. SIAM J. Control. Optim. 45(5), 1915–1930 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heemels, W.P.M.H., Kundu, A., Daafouz, J.: On Lyapunov-Metzler inequalities and S-procedure characterizations for the stabilization of switched linear systems. IEEE Trans. Autom. Control. 62(9), 4593–4597 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hernandez-Vargas, E., Middleton, R., Colaneri, P., et al.: Discrete-time control for switched positive systems with application to mitigating viral escape. Int. J. Robust Nonlinear Control. 21(10), 1093–1111 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid system. IEEE Trans. Autom. Control. 43, 555–559 (1998)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jadbabaie, A., Lin, J., Morse, A.S.: Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2002)CrossRefGoogle Scholar
  12. 12.
    Shorten, R., Wirth, F., Leith, D.: A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Trans. Netw. 14(3), 616C629 (2006)CrossRefGoogle Scholar
  13. 13.
    Tong, Y., Wang, C., Zhang, L.: Stabilisation of discrete-time switched positive linear systems via time-and state-dependent switching laws. IET Control. Theory Appl. 6(11), 1603–1609 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Yakubovich, V.A.: Minimization of quadratic functionals under quadratic constraints and the necessity of a frequency condition in the quadratic criterion for absolute stability of nonlinear control systems. Sov. Math. Dokl. 14, 593–597 (1973)zbMATHGoogle Scholar
  15. 15.
    Zappavigna, A., Colaneri, P., Geromel, J.C., Middleton, R.: Stabilization of continuous-time switched linear positive systems. In: 2010 American Control Conference ACC, pp. 3275–3280 (2010)Google Scholar
  16. 16.
    Zhao, X., Zhang, L., Shi, P., Liu, M.: Stability of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.School of AutomationHangzhou Dianzi UniversityHangzhouChina
  2. 2.Key Laboratory of System Control and Information ProcessingMinistry of Education of ChinaShanghaiChina
  3. 3.Conservatoire National des Arts et Metiers (CNAM)Paris Cedex 03France

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