Distributed robust stabilization of networked multi-agent systems with strict negative imaginary uncertainties

This paper deals with the distributed robust stabilization problem for networked
multi-agent systems with strict negative imaginary uncertainties (SNI). Communication among agents in the network is modelled by an undirected graph with at least one self-loop. A protocol based on relative state measurements of neighbouring agents and absolute state measurements of a subset of agents is considered. This paper shows how to design the protocol parameters such that the uncertain closed-loop networked multi-agent system is robustly stable against any SNI uncertainty within a certain set for various different network topologies. Tools from negative imaginary (NI) theory are used as an aid to simplify the problem and synthesise the protocol parameters. We show that a state, input and output transformation preserves the NI property of the network. Consequently, a necessary and sufficient condition for the transfer function matrix of the nominal closed-loop networked system to be NI and satisfy a DC gain condition is that multiple reduced-order equivalent systems be NI and satisfy a DC gain condition simultaneously. Based on the reduced-order systems, we derive sufficient conditions in an LMI framework which ensure the existence of a protocol satisfying the desired objectives. A numerical example is given to confirm the effectivenesses of the proposed results.