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Continuous-Time Compartmental Switched Systems

  • Maria Elena ValcherEmail author
  • Irene Zorzan
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

In this chapter we investigate state-feedback and output-feedback stabilization of compartmental switched systems, under the additional requirement that the resulting switched system is in turn compartmental. Necessary and sufficient conditions for the solvability of the two problems are given. Subsequently, affine compartmental switched systems are considered, and a characterization of all the switched equilibria that can be “reached” under some stabilizing switching law \(\sigma \) is provided.

Keywords

Compartmental system Linear/affine switched system Stabilization Switched equilibrium point 

References

  1. 1.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Set Valued Maps and Viability Theory. Springer (1984)Google Scholar
  2. 2.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)zbMATHGoogle Scholar
  3. 3.
    Blanchini, F., Colaneri, P., Valcher, M.E.: Co-positive Lyapunov functions for the stabilization of positive switched systems. IEEE Trans. Autom. Control 57(12), 3038–3050 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blanchini, F., Colaneri, P., Valcher, M.E.: Switched linear positive systems. Found. Trends Syst. Control 2(2), 101–273 (2015)CrossRefGoogle Scholar
  5. 5.
    Bolzern, P., Colaneri, P., De Nicolao, G.: On almost sure stability of discrete-time Markov jump linear systems. In: Proceedings of 43rd IEEE Conference on Decision and Control, Atlantis, Paradise Island, The Bahamas, pp. 3204–3208 (2004)Google Scholar
  6. 6.
    Bru, R., Romero, S., Sánchez, E.: Canonical forms for positive discrete-time linear control systems. Linear Algebra Appl. 310, 49–71 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brualdi, R.A., Ryser, H.J.: Combinatorial Matrix Theory. Cambridge University Press (1991)Google Scholar
  8. 8.
    de Leenheer, P., Aeyels, D.: Stabilization of positive linear systems. Syst. Control Lett. 44, 259–271 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fornasini, E., Valcher, M.E.: Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Trans. Autom. Control 55(8), 1933–1937 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Frobenius, G.F.: Über Matrizen aus nicht Negativen Elementen. Sitzungsber. Kon. Preuss. Akad. Wiss. Berlin, 1912, pp. 456–477, Ges. Abh., Springer, vol. 3:546–567 (1968)Google Scholar
  11. 11.
    Gantmacher, F.R.: The Theory of Matrices. Chelsea Pub. Co. (1960)Google Scholar
  12. 12.
    Gurvits, L., Shorten, R., Mason, O.: On the stability of switched positive linear systems. IEEE Trans. Autom. Control 52(6), 1099–1103 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Haddad, W.M., Chellaboina, V., Hui, Q.: Nonnegative and Compartmental Dynamical Systems. Princeton University Press (2010)Google Scholar
  14. 14.
    Hou, S.P., Meskin, N., Haddad, W.M.: A general multicompartment lung mechanics model with nonlinear resistance and compliance respiratory parameters. In: Proceedings of 2014 American Control Conference, Portland, OR, pp. 566–571 (2014)Google Scholar
  15. 15.
    Knorn, F., Mason, O., Shorten, R.N.: On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica 45(8), 1943–1947 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li, H., Haddad, W.M.: Optimal determination of respiratory airflow patterns using a nonlinear multicompartment model for a lung mechanics system. Comput. Math. Meth. Med. 2012(165946) (2012)Google Scholar
  17. 17.
    Lin, L.: Stabilization analysis for economic compartmental switched systems based on quadratic Lyapunov function. Nonlinear Anal. Hybrid Syst. 2, 1187–1197 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mason, O., Shorten, R.N.: On linear copositive Lyapunov functions and the stability of switched positive linear systems. IEEE Trans. Autom. Control 52(7), 1346–1349 (2007)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Minc, H.: Nonnegative Matrices. Wiley, New York (1988)zbMATHGoogle Scholar
  20. 20.
    Schneider, H.: The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties. Linear Algebra Appl. 84, 161–189 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Smith, H.L.: Linear Compartmental Systems—The Basics (2006)Google Scholar
  22. 22.
    Son, N.K., Hinrichsen, D.: Robust stability of positive continuous time systems. Numer. Funct. Anal. Opt. 17(5 & 6), 649–659 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Taussky, O.: A recurring theorem on determinants. Am. Math. Mon. 56(10), 672–676 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Valcher, M.E.: Controllability and reachability criteria for discrete time positive systems. Int. J. Control 65, 511–536 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Valcher, M.E., Zorzan, I.: Stability and stabilizability of continuous-time compartmental switched systems. IEEE Trans. Autom. Control. 61(12), 3885–3897 (2016). doi: 10.1109/TAC.2016.2525016
  26. 26.
    Valcher, M.E., Zorzan, I.: On the stabilizability of continuous-time compartmental switched systems. In: Proceedings of the 54th IEEE Conf. on Decision and Control, pp. 4246–4251, Osaka, Japan (2015)Google Scholar
  27. 27.
    Valcher, M.E., Zorzan, I.: New results on the solution of the positive consensus problem. In: Proceedings of the 55th IEEE Conf. on Decision and Control, pp. 5251–5256, Las Vegas, Nevada (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversità di PadovaPadovaItaly

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