A Left Eigenvector Producing a Smooth Lyapunov Function of ISS Networks
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.
KeywordsMonotone 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.
- 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
- 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
- 19.Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities: Theory and Applications Volume I: Ordinary Differential Equations. Academic Press, New York (1969)Google Scholar
- 21.Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2001)Google Scholar