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A Left Eigenvector Producing a Smooth Lyapunov Function of ISS Networks

  • Hiroshi Ito
  • Björn S. Rüffer
Chapter
First Online:
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 480)

Abstract

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.

Keywords

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

Acknowledgements

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.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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