Modeling and stability analysis of live systems with time-varying dimension
Andrii Mironchenko
TL;DR
The paper develops a unified framework for live systems in which the state space dimension can vary over time, modeling arrivals and departures of components as impulses. It shows that input-to-state stability $ISS$ can be formulated for these systems and extends classical Lyapunov tools to handle configuration changes via dwell-time conditions. By deriving equivalences with standard stability properties and detailing special classes such as open multi-agent impulsive systems, unknown initial configurations, and switched systems, the work provides a practical foundation for analysis and design in networks with evolving topology. The illustrative example demonstrates ISS of a cascade interconnection under applicable dwell-time constraints, highlighting the approach's relevance to plug-and-play, reconfigurable, and hybrid control applications.
Abstract
A major limitation of the classical control theory is the assumption that the state space and its dimension do not change with time. This prevents analyzing and even formalizing the stability and control problems for open multi-agent systems whose agents may enter or leave the network, industrial processes where the sensors or actuators may be exchanged frequently, smart grids, etc. In this work, we propose a framework of live systems that covers a rather general class of systems with a time-varying state space. We argue that input-to-state stability is a proper stability notion for this class of systems, and many of the classic tools and results, such as Lyapunov methods and superposition theorems, can be extended to this setting.