Convex Stability Conditions for Interconnected/Networked Linear Multi-agent Systems

This paper considers the stability of large-scale interconnected/networked systems. Verifiable convex sufficient stability conditions for certain classes of interconnection structures and subsystem dynamics, which are derived by applying topological graph separation, involve positive realness of certain transfer functions. These positive realness conditions are transformed into linear matrix inequality conditions in terms of a state-space realization of the subsystem dynamics, a Lyapunov matrix, and multipliers. We show that the conditions can be extended to the design of stabilizing decentralized controllers that optimize local LQR objectives. The proposed stability tests are demonstrated in several numerical examples of interconnected systems.