A Left Eigenvector Producing a Smooth Lyapunov Function of ISS Networks

  • Hiroshi ItoEmail author
  • Björn S. Rüffer
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)


For a class of monotone nonlinear systems, it is shown that a continuously differentiable Lyapunov function can be constructed implicitly from a left eigenvector of vector fields. The left eigenvector which is a continuous function of state variables is deduced from a right eigenvector which represents a small gain condition. It is demonstrated that rounding off the edges of n-orthotopes, which is the maximization of state variables, yields level sets of the Lyapunov function. Applying the development to comparison systems gives continuously differentiable input-to-state Lyapunov functions of networks consisting of input-to-state systems which are not necessarily monotone.


Monotone nonlinear systems Lyapunov functions Input-to-state stability Perron-Frobenius theory Small-gain condition 



The work of H. Ito was supported in part by JSPS KAKENHI Grant Number 17K06499. B. S. Rüffer has been supported by ARC grant DP160102138.


  1. 1.
    Dashkovskiy, S., Ito, H., Wirth, F.: On a small-gain theorem for ISS networks in dissipative Lyapunov form. European J. Control 17, 357–369 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.: An ISS small-gain theorem for general networks. Math. Control Signals Syst. 19, 93–122 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dashkovskiy, S., Rüffer, B.S., Wirth, F.: Small gain theorems for large scale systems and construction of ISS Lyapunov functions. SIAM J. Control Optim. 48, 4089–4118 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dirr, G., Ito, H., Rantzer, A., Rüffer, B.S.: Separable Lyapunov functions: constructions and limitations. Discret. Contin. Dyn. Syst. B 20(8), 2497–2526 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Freeman, R.A., Kokotović, P.V.: Robust Nonlinear Control Design: State-Space and Lyapunov Techniques. Birkhäuser, Boston (1996)CrossRefGoogle Scholar
  6. 6.
    Grüne, L.: Asymptotic Behavior of Dynamical and Control Systems Under Perturbation and Discretization. Springer, Berlin (2002)CrossRefGoogle Scholar
  7. 7.
    Isidori, A.: Nonlinear Control Systems II. Springer, London (1999)CrossRefGoogle Scholar
  8. 8.
    Ito, H.: State-dependent scaling problems and stability of interconnected iISS and ISS systems. IEEE Trans. Autom. Control 51(10), 1626–1643 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ito H.: Utility of iISS in composing Lyapunov functions. In: Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems, pp. 723–730. Toulouse, France (2013)Google Scholar
  10. 10.
    Ito H.: An implicit function approach to Lyapunov functions for interconnections containing non-ISS components. In: Proceedings of the 2nd IFAC Conference on Modelling, Identification and Control of Nonlinear Systems, pp. 254–259. Guadalajara, Mexico (2018)CrossRefGoogle Scholar
  11. 11.
    Ito, H., Dashkovskiy, S., Wirth, F.: Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems. Automatica 48(6), 1197–1204 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ito, H., Jiang, Z.P.: Necessary and sufficient small gain conditions for integral input-to-state stable systems: a Lyapunov perspective. IEEE Trans. Autom. Control 54, 2389–2404 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ito, H., Jiang, Z.P., Dashkovskiy, S., Rüffer, B.S.: Robust stability of networks of iISS systems: construction of sum-type Lyapunov functions. IEEE Trans. Autom. Control 58, 1192–1207 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jiang, Z.P., Mareels, I., Wang, Y.: A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems. Automatica 32, 1211–1215 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jiang, Z.P., Teel, A.R., Praly, L.: Small-gain theorem for ISS systems and applications. Math. Control Signals Syst. 7, 95–120 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jiang Z.P., Wang Y.: A generalization of the nonlinear small-gain theorem for large-scale complex systems. In: Proceedings of the 7th World Congress on Intelligent Control and Automation, pp. 1188–1193 (2008)Google Scholar
  17. 17.
    Kellett, C.M.: Classical converse theorems in Lyapunov’s second method. Discret. Contin. Dyn. Syst. B 20, 2333–2360 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Khalil, H.K.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2002)zbMATHGoogle Scholar
  19. 19.
    Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities: Theory and Applications Volume I: Ordinary Differential Equations. Academic Press, New York (1969)Google Scholar
  20. 20.
    Liu, T., Hill, D.J., Jiang, Z.P.: Lyapunov formulation of ISS small-gain in continuous-time dynamical networks. Automatica 47, 2088–2093 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2001)Google Scholar
  22. 22.
    Rüffer, B.S., Kellett, C.M., Weller, S.R.: Connection between cooperative positive systems and integral input-to-state stability of large-scale systems. Automatica 46, 1019–1027 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sontag, E.D.: Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34, 435–443 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sontag, E.D.: Comments on integral variants of ISS. Syst. Control Lett. 34, 93–100 (1998)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sontag, E.D., Wang, Y.: On characterizations of input-to-state stability property. Syst. Control Lett. 24, 351–359 (1995)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Kyushu Institute of TechnologyIizuka, FukuokaJapan
  2. 2.The University of NewcastleCallaghanAustralia

Personalised recommendations