Stabilizability and Control Co-Design for Discrete-Time Switched Linear Systems

  • M. Fiacchini
  • M. Jungers
  • A. Girard
  • S. Tarbouriech
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 475)


In this work we deal with the stabilizability property for discrete-time switched linear systems. First we provide a constructive necessary and sufficient condition for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. Such a geometric condition is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with convexity. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity. The results are finally extended to the problem of the co-design of a control policy, composed by both the state feedback and the switching control law, for discrete-time switched linear systems. Constructive conditions are given in form of LMI that are necessary and sufficient for the stabilizability of systems which are periodic stabilizable.


  1. 1.
    Antunes, D., Heemels, W.P.M.H.: Linear quadratic regulation of switched systems using informed policies. IEEE Trans. Autom. Control 62, 2675–2688 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bertsekas, D.P.: Infinite-time reachability of state-space regions by using feedback control. IEEE Trans. Autom. Control 17, 604–613 (1972)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Birkhäuser (2008)Google Scholar
  4. 4.
    Blanchini, F.: Ultimate boundedness control for discrete-time uncertain systems via set-induced Lyapunov functions. IEEE Trans. Autom. Control 39, 428–433 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Blanchini, F.: Nonquadratic Lyapunov functions for robust control. Automatica 31, 451–461 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blanchini, F., Savorgnan, C.: Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions. Automatica 44, 1166–1170 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)CrossRefGoogle Scholar
  8. 8.
    Daafouz, J., Riedinger, P., Iung, C.: Stability analysis and control synthesis for switched systems : a switched Lyapunov function approach. IEEE Trans. Autom. Control 47, 1883–1887 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Deaecto, G.S., Geromel, J.C., Daafouz, J.: Dynamic output feedback H\(_\infty \) control of switched linear systems. Automatica 47(8), 1713–1720 (2011)Google Scholar
  10. 10.
    Fiacchini, M., Jungers, M.: Necessary and sufficient condition for stabilizability of discrete-time linear switched systems: a set-theory approach. Automatica 50(1), 75–83 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fiacchini, M., Tarbouriech, M.: Control co-design for discrete-time switched linear systems. Automatica 82, 181–186 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fiacchini, M., Girard, A., Jungers, M.: On the stabilizability of discrete-time switched linear systems: novel conditions and comparisons. IEEE Trans. Autom. Control 61(5), 1181–1193 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Geromel, J.C., Colaneri, P.: Stability and stabilization of discrete-time switched systems. Int. J. Control 79(7), 719–728 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jungers, R.M.: The Joint Spectral Radius: Theory and Applications. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Kolmanovsky, I., Gilbert, E.G.: Theory and computation of disturbance invariant sets for discrete-time linear systems. Math. Probl. Eng. 4, 317–367 (1998)CrossRefGoogle Scholar
  17. 17.
    Kruszewski, A., Bourdais, R., Perruquetti, W.: Converging algorithm for a class of BMI applied on state-dependent stabilization of switched systems. Nonlinear Anal. Hybrid Syst. 5, 647–654 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lee, J.W., Dullerud, G.E.: Uniformly stabilizing sets of switching sequences for switched linear systems. IEEE Trans. Autom. Control 52, 868–874 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liberzon, D.: Switching in Systems and Control. Birkh\(\ddot{\rm a}\)user, Boston (2003)CrossRefGoogle Scholar
  20. 20.
    Lin, H., Antsaklis, P.J.: Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Trans. Autom. Control 54(2), 308–322 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Margaliot, M.: Stability analysis of switched systems using variational principles: an introduction. Automatica 42, 2059–2077 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Molchanov, A.P., Pyatnitskiy, Y.S.: Criteria of asymptotic stability of differential and difference inclusions encounterd in control theory. Syst. Control Lett. 13, 59–64 (1989)CrossRefGoogle Scholar
  23. 23.
    Rubinov, A.M., Yagubov, A.A.: The space of star-shaped sets and its applications in nonsmooth optimization. In: Demyanov, V.F., Dixon, L.C.W. (eds.) Quasidifferential Calculus, pp. 176–202. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  24. 24.
    Sun, Z., Ge, S.S.: Stability Theory of Switched Dynamical Systems. Springer, Berlin (2011)CrossRefGoogle Scholar
  25. 25.
    VanAntwerp, J.G., Braatz, R.D.: A tutorial on linear and bilinear matrix inequalities. J. Process Control 10(4), 363–385 (2000)CrossRefGoogle Scholar
  26. 26.
    Wicks, M.A., Peleties, P., De Carlo, R.A.: Construction of piecewise Lyapunov functions for stabilizing switched systems. Proceedings of the 33rd IEEE Conference on Decision and Control, pp. 3492–3497 (1994)Google Scholar
  27. 27.
    Zhang, W., Abate, A., Hu, J., Vitus, M.P.: Exponential stabilization of discrete-time switched linear systems. Automatica 45(11), 2526–2536 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhang, W., Hu, J., Abate, A.: Infinite-horizon switched lqr problems in discrete time: a suboptimal algorithm with performance analysis. IEEE Trans. Autom. Control 57(7), 1815–1821 (2012)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • M. Fiacchini
    • 1
  • M. Jungers
    • 2
    • 3
  • A. Girard
    • 4
  • S. Tarbouriech
    • 5
  1. 1.University Grenoble Alpes, CNRS, Gipsa-labGrenobleFrance
  2. 2.Université de Lorraine, CRAN, UMR 7039LorraineFrance
  3. 3.CNRS, CRAN, UMR 7039LorraineFrance
  4. 4.Laboratoire des signaux et systèmes (L2S), CNRS, CentraleSupélecUniversité Paris-SudcedexFrance
  5. 5.LAAS-CNRSUniversité de ToulouseToulouseFrance

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