Advertisement

Positive Stabilization of a Class of Infinite-Dimensional Positive Systems

  • M. Elarbi Achhab
  • Joseph J. WinkinEmail author
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

For a class of positive unstable infinite-dimensional linear systems, a method is described for computing a positively stabilizing state feedback, such that the resulting input trajectory remains in an affine cone. This design results in a possibly negative lower bound on the input, which makes the resulting closed-loop system stable and which maintains the nonnegativity of the state trajectory for specific initial states.

Keywords

Infinite dimensional systems Positive linear systems Positive stabilization State feedback Affine cone 

Notes

Acknowledgements

The authors wish to thank the following persons with whom they have worked jointly on dynamical analysis and control of positive systems for many years: B. Abouzaid (Ecole Nationale des Sciences Appliquées, Université Chouaib Doukkali, El Jadida., Morocco), Ch. Beauthier (Cenaero, Gosselies, Belgium), D. Dochain (Université Catholique de Louvain, Belgium), M. Laabissi (Université Chouaib Doukkali, El Jadida, Morocco) and V. Wertz (Université Catholique de Louvain, Belgium).

This chapter presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

References

  1. 1.
    Abouzaid, B., Winkin, J., Wertz, V.: Positive stabilization of infinite-dimensional linear systems. In: Proceedings of the 49th Conference on Decision and Control (CDC), Atlanta, GA, USA, 15–17 Dec 2010, Cd-Rom paper 0859, pp. 845–850Google Scholar
  2. 2.
    Achhab, M.E., Laabissi, M.: Feedback stabilization of a class of distributed parameter systems with control constraints. Syst. Control Lett. 45, 163–171 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Achhab, M.E., Winkin, J.: Stabilization of infinite dimensional systems by state feedback with positivity constraints. In: Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS 2014), Groningen, NL, 2014, pp. 379–384Google Scholar
  4. 4.
    Achhab, M.E., Winkin, J.: Work in progressGoogle Scholar
  5. 5.
    Aksikas, I., Winkin, J., Dochain, D.: Optimal LQ-feedback regulation of a nonisothermal plug flow reactor model by spectral factorization. IEEE Trans. Autom. Control 52(7), 1179–1193 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beauthier, Ch., Winkin, J.: LQ-optimal control of positive linear systems. Optim. Control Appl. Methods 31, 547–566 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Binid, A., Achhab, M.E., Laabissi, M., Abouzaid, B.: Positive observers for infinite dimensional positive linear systems (2016)Google Scholar
  8. 8.
    Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer (1995)Google Scholar
  9. 9.
    Dehaye, J.N., Winkin, J.: Positive stabilization of a diffusion system by nonnegative boundary control. In: Proceedings of POSTA 2016Google Scholar
  10. 10.
    De Leenheer, P., Aeyels, D.: Stabilization of positive linear systems. Syst. Control Lett. 44, 259271 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part II. Wiley Intersciences, New York (1951)zbMATHGoogle Scholar
  12. 12.
    Engel, K.J., Nagel, R.: A Short Course on Operator Semigroups. Springer (2006) (especially chapter VI)Google Scholar
  13. 13.
    Haddad, W.M., Chellaboina, V., Hui, Q.: Nonnegative and compartmental dynamical systems. Princeton University Press, Princeton, NJ (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Laabissi, M., Achhab, M.E., Winkin, J., Dochain, D.: Trajectory analysis of nonisothermal tubular reactor nonlinear models. Syst. Control Lett. 42, 169–184 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Laabissi, M., Achhab, M.E., Winkin, J., Dochain, D.: Positivity and invariance properties of nonisothermal tubular reactor nonlinear models. In: Benvenuti, L., De Santis, A., Farina, L. (eds.) Positive Systems (Proceedings of the 1rst Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 03), Grenoble, France), pp. 159–166. Lecture Notes in Control and Information Sciences. Springer, Berlin (2003)Google Scholar
  16. 16.
    Laabissi, M., Winkin, J., Beauthier, Ch.: On the positive LQ-problem for linear continuous-time systems. In: Commault, Ch., Marchand, N. (eds.) Positive Systems (Proceedings of the 2nd Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 06), Grenoble, France), pp. 295–302. Lecture Notes in Control and Information Sciences. Springer, Berlin (2006)Google Scholar
  17. 17.
    Roszak, B., Davison, E.J.: Necessary and sufficient conditions for stabilizability of positive LTI systems. Syst. Control Lett. 58, 474481 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (2000)zbMATHGoogle Scholar
  19. 19.
    Smith, H.L.: Monotone Dynamical Systems : An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society, Providence (1995)zbMATHGoogle Scholar
  20. 20.
    Winkin, J., Dochain, D., Ligarius, Ph.: Dynamical analysis of distributed parameter tubular reactors. Automatica 36(3), 349–361 (2000)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculté des Sciences El Jadida, Département de MathématiquesUniversité Chouaib DoukkaliEl JadidaMorocco
  2. 2.Department of Mathematics and naXysUniversity of NamurNamurBelgium

Personalised recommendations