Advertisement

Polyhedral Invariance for Convolution Systems over the Callier-Desoer Class

  • Jean Jacques LoiseauEmail author
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

BIBO stability is a central concept for convolution systems, introduced in control theory by Callier, Desoer and Vidyasagar, in the seventies. It means that a bounded input leads to a bounded output, and is characterized by the fact that the kernel of the system is integrable. We generalize this result in this chapter, giving conditions for the output of a convolution system to evolve in a given polyhedron, for any input evolving in another given convex polyhedron. The conditions are formulated in terms of integrals deduced from the kernel of the considered system and the given polyhedra. The condition is exact. It permits to construct exact inner and outer polyhedral approximations of the reachable set of a linear system. The result is compared to various known results, and illustrated on the example of a system with two delays.

Keywords

Convolution systems Callier-Desoer class Invariance Reachable set Polyhedra Approximations 

Notes

Acknowledgements

The author thanks very much Filippo Cacace and Joseph Winkin for their warm encouragements, which were crucial to produce this report.

References

  1. 1.
    Callier, F.M., Desoer, C.A.: An algebra of transfer functions for distributed linear time-invariant systems. IEEE Trans. Circuits Syst. 25, 651–662 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, H., Cheng, J., Zhong, S., Yang, J., Kang, W.: Improved results on reachable set bounding for linear systems with discrete and distributed delays. Adv. Differ. Equ. 145 (2015)Google Scholar
  3. 3.
    Chiasson, J., Loiseau, J.J. (eds.): Applications of Time Delay Systems. Springer, Berlin (2007)zbMATHGoogle Scholar
  4. 4.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: Conference Record of the Fifth Annual Symposium on Principles of Programming Languages. ACM Press, New York (1978)Google Scholar
  5. 5.
    Desoer, C.A., Callier, F.M.: Convolution feedback systems. SIAM J. Control 10, 737–746 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Academic Press, New York (1975)zbMATHGoogle Scholar
  7. 7.
    Falcone, P., Ali, M., Sjoberg, J.: Predictive Threat assessment via reachability analysis and set invariance theory. IEEE Trans. Intell. Transp. Syst. 12, 1352–1361 (2011)CrossRefGoogle Scholar
  8. 8.
    Fridman, E., Shaked, U.: On reachable sets for linear systems with delay and bounded peak inputs. Automatica 39, 2005–2010 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guéguen, H., Lefebvre, M.-A., Zaytoon, J., Nasri, O.: Safety verification and reachability analysis for hybrid systems. Annu. Rev. Control 33, 25–36 (2009)CrossRefGoogle Scholar
  10. 10.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. American Mathematical Society, Providence (1957)zbMATHGoogle Scholar
  11. 11.
    Hwang, I., Stipanović, D.M., Tomlin, C.J.: Polytopic approximations of reachable sets applied to linear dynamic games and to a class of nonlinear systems. In: Advances in Control, Communication Networks, and Transportation Systems, in Honor of Pravin Varaiya, pp. 3–19. Birkhäuser, Boston (2005)Google Scholar
  12. 12.
    Ignaciuk, P., Bartoszevicz, A.: Congestion Control in Data Transmission Networks. Sliding Modes and Other Designs. Springer, New York (2013)CrossRefGoogle Scholar
  13. 13.
    Lakkonen, P.: Robust regulation for infinite-dimensional systems and signals in the frequency domain. Ph.D. Thesis, Tampere University of Technology, Finland (2013)Google Scholar
  14. 14.
    Lygeros, J., Tomlin, C.J., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35, 349–370 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Meslem, N., Ramdani, N., Candau, Y.: Approximation garantie de l’espace d’état atteignable des systèmes dynamiques continus incertains. JESA J. Européen des Systèmes Automatisés 43, 241–266 (2009)Google Scholar
  16. 16.
    Moussaoui, C., Abbou, R., Loiseau, J.J.: On bounds of input-output systems. Reachability set determination and polyhedral constraints verification. In: Boje, S.O., Xia, X. (eds.) Proceedings of 19th IFAC World Congress, pp. 11012–11017. International Federation of Automatic Control, Cape Town (2014)Google Scholar
  17. 17.
    Moussaoui, C., Abbou, R., Loiseau, J.J.: Controller design for a class of delayed and constrained systems: application to supply chains. In: Seuret, A., Özbay, I., Bonnet, C., Mounier, H. (eds.) Low-Complexity Controllers for Time-Delay Systems, pp. 61–75. Springer, Berlin (2014)CrossRefGoogle Scholar
  18. 18.
    Olaru, S., Stanković, N., Bitsoris, G., Niculescu, S.-I.: Low complexity invariant sets for time-delay systems: a set factorization approach. In: Seuret, A., Özbay, H., Bonnet, C., Mounier, H. (eds.) Low-Complexity Controllers for Time-Delay Systems, pp. 127–139. Springer, New York (2014)CrossRefGoogle Scholar
  19. 19.
    Pecsvaradi, T., Narendra, K.S.: Reachable sets for linear dynamical systems. Inf. Control 19, 319–344 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Quadrat, A.: A lattice approach to analysis and synthesis problems. Math. Control Signals Syst. 18, 147–186 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Chichester (1970)CrossRefzbMATHGoogle Scholar
  22. 22.
    Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus. Springer, Berlin (2007)Google Scholar
  23. 23.
    Simon, H.A.: On the application of servomechanism theory in the study of production control. Econometrica 20, 247–268 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Varaya, P.: Reach set computation using optimal control. In: Inan, M.K., Kurshan, R.P. (eds.) Verification of Digital and Hybrid Systems, pp. 323–331. Springer, Berlin (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université Bretagne LoireÉcole Centrale de NantesNantes cedex 03France

Personalised recommendations