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Stability Analysis of Neutral Type Time-Delay Positive Systems

  • Yoshio EbiharaEmail author
  • Naoya Nishio
  • Tomomichi Hagiwara
Chapter
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 471)

Abstract

This chapter is concerned with asymptotic stability analysis of neutral type time-delay positive systems (TDPSs). We focus on a neutral type TDPS represented by a feedback system constructed from a finite-dimensional LTI positive system and the pure delay, and give a necessary and sufficient condition for the stability. In the case where we deal with a retarded type TDPS, i.e., if the direct-feedthrough term of the finite-dimensional LTI positive system is zero, it is well known that the retarded type TDPS is stable if and only if its delay-free finite-dimensional counterpart is stable. In the case of neutral type TDPS, i.e., if the direct-feedthrough term is nonzero, however, we clarify that the neutral type TDPS is stable if and only if its delay-free finite-dimensional counterpart is stable and the direct-feedthrough term is Schur stable. Namely, we need additional condition on the direct-feedthrough term.

Keywords

Asymptotic stability Time-delay positive systems Neutral type 

Notes

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 25420436.

References

  1. 1.
    Ait Rami, M., Jordan, A.J., Schonlein, M., Schonlein, M.: Estimation of linear positive systems with unknown time-varying delays. Eur. J. Control 19(3), 179–187 (2013)Google Scholar
  2. 2.
    Bellman, R., Cooke, K.L.: Differential Difference Equations. Academic Press, New York (1963)zbMATHGoogle Scholar
  3. 3.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)zbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Briat, C.: Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: \(L_1\)-gain and \(L_\infty \)-gain characterization. Int. J. Robust Nonlinear Control 23(17), 1932–1954 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ebihara, Y., Peaucelle, D., Arzelier, D.: \(L_1\) gain analysis of linear positive systems and its application. In: Proceedings Conference on Decision and Control, pp. 4029–4034 (2011)Google Scholar
  7. 7.
    Ebihara, Y., Peaucelle, D., Arzelier, D.: Analysis and synthesis of interconnected positive systems. IEEE Trans. Autom. Control 62(2), 652–667 (2017)Google Scholar
  8. 8.
    Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. Wiley (2000)Google Scholar
  9. 9.
    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
  10. 10.
    Haddad, W.M., Chellaboina, V.: Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst. Control Lett. 51(5), 355–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hagiwara, T., Kobayashi, M.: Concatenated solutions of delay-differential equations and their representation with time-delay feedback systems. Int. J. Control 84(6), 1126–1139 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  13. 13.
    Horn, R.A., Johnson, C.A.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kaczorek, T.: Positive 1D and 2D Systems. Springer, London (2001)zbMATHGoogle Scholar
  15. 15.
    Kim, J.H., Hagiwara, T., Hirata, K.: Spectrum of monodromy operator for a time-delay system with application to stability analysis. IEEE Trans. Autom. Control 60(12), 3385–3390 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mason, O., Shorten, R.: On linear copositive Lyapunov function and the stability of switched positive linear systems. IEEE Trans. Autom. Control 52(7), 1346–1349 (2007)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Najson, F.: On the Kalman-Yakubovich-Popov lemma for discrete-time positive linear systems: A novel simple proof and some related results. Int. J. Control 86(10), 1813–1823 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rantzer, A.: Scalable control of positive systems. Eur. J. Control 24(1), 72–80 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rantzer, A.: On the kalman-yakubovich-popov lemma for positive systems. IEEE Trans. Autom. Control 61(5), 1346–1349 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Shen, J., Lam, J.: \({L}_\infty \)-gain analysis for positive systems with distributed delays. Automatica 50(2), 547–551 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Shorten, R., Mason, O., King, C.: An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions. Linear Algebra Appl. 430, 34–40 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tanaka, T., Langbort, C.: The bounded real lemma for internally positive systems and \(H_\infty \) structured static state feedback. IEEE Trans. Autom. Control 56(9), 2218–2223 (2011)CrossRefGoogle Scholar
  23. 23.
    Valcher, M.E., Misra, P.: On the stabilizability and consensus of positive homogeneous multi-agent dynamical systems. IEEE Trans. Autom. Control 59(7), 1936–1941 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Yoshio Ebihara
    • 1
    Email author
  • Naoya Nishio
    • 1
  • Tomomichi Hagiwara
    • 1
  1. 1.Department of Electrical EngineeringKyoto UniversityNishikyo-ku, KyotoJapan

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