Finite-Dimensional Controllers for Consensus in a Leader-Follower Network of Marginally Unstable Infinite-Dimensional Agents

We consider the output-feedback state consensus problem for a homogeneous multi-agent system consisting of one leader agent and $N$ follower agents. The dynamics of each of these agents is governed by a single-input single-output regular linear system (RLS), with the input to the leader agent being zero at all times. The transfer function of this RLS has all its poles in the closed left-half of the complex-plane, with only a finite number of them lying on the imaginary axis. Each follower agent can access the relative output of all its neighboring agents and some of the follower agents can access the relative output of the leader. The communication graph associated with the exchange of relative outputs is directed. Under this setting, we first establish that a controller solves the above leader-follower consensus problem if and only if it can simultaneously stabilize $N$ regular linear systems. We then adapt a recently developed frequency-domain technique to construct a stable finite-dimensional output-feedback controller which solves the simultaneous stabilization problem, and hence the consensus problem. We demonstrate the efficacy of our controller design technique using a numerical example.