Brief paper: Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems

Authors:
Hiroshi Ito;Sergey Dashkovskiy;Fabian Wirth
Affiliations:
Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, 820-8502, Japan;Department of Civil Engineering, University of Applied Sciences Erfurt, Germany;Institute of Mathematics, University of Würzburg, Germany
Venue:
Automatica (Journal of IFAC)
Year:
2012

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Abstract

This paper addresses the problem of verifying stability of networks whose subsystems admit dissipation inequalities of integral input-to-state stability (iISS). We focus on two ways of constructing a Lyapunov function satisfying a dissipation inequality of a given network. Their difference from one another is elucidated from the viewpoint of formulation, relation, fundamental limitation and capability. One is referred to as the max-type construction resulting in a Lipschitz continuous Lyapunov function. The other is the sum-type construction resulting in a continuously differentiable Lyapunov function. This paper presents geometrical conditions under which the Lyapunov construction is possible for a network comprising n=2 subsystems. Although the sum-type construction for general n2 has not yet been reduced to a readily computable condition, we obtain a simple condition of iISS small gain in the case of n=2. It is demonstrated that the max-type construction fails to offer a Lyapunov function if the network contains subsystems which are not input-to-state stable (ISS).