H∞ control design for synchronisation of identical linear multi-agent systems

In this paper, we study the state synchronisation problem of multi-agent systems subject to external additive perturbations. We consider high-order linear time-invariant multi-agent systems whose communication topology is encoded by an undirected and connected graph. We propose an H∞ control design technique based on a decentralised output feedback controller. We give sufficient conditions to ensure state synchronisation with bounded L2 gain using a Lyapunov-based approach. These conditions are characterised in terms of matrix inequalities. Since these matrix inequality conditions are non-convex and cannot be solved straightforwardly, we propose a relaxation technique and an effective numerical procedure to design a suitable controller with guaranteed performance on the multi-agent distributed closed loop.