Capability and limitation of max- and sum-type construction of Lyapunov functions for networks of iISS systems |
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Authors: | Hiroshi Ito Sergey Dashkovskiy Fabian Wirth |
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Affiliation: | 1. Department of Systems Design and Informatics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka, 820-8502, Japan;2. Department of Civil Engineering, University of Applied Sciences Erfurt, Germany;3. Institute of Mathematics, University of Würzburg, Germany;1. School of Science, Beijing Technology and Business University, Beijing 100048, PR China;2. State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing, 102206, PR China;1. School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith NSW 2751, Australia;2. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam, Hong Kong |
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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 subsystems. Although the sum-type construction for general has not yet been reduced to a readily computable condition, we obtain a simple condition of iISS small gain in the case of . 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). |
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