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.

I. Introduction

Consider a homogeneous multi-agent system (MAS) with a leader agent and follower agents, in which each follower agent can access the relative output of all its neighboring agents (which may also include the leader agent). This information exchange is modeled using a communication graph. The output-feedback state consensus (OFSC) problem for this MAS is to find a local controller (for each agent) which, using the relative outputs, ensures that the state of all the follower agents converge to the state of the leader agent. Several works have addressed this problem when the agent dynamics is infinite-dimensional. In [1] and [2] the OFSC problem is solved for agent dynamics governed by a marginally stable wave equation having a simple eigenvalue at zero. Related consensus problems have been solved for agent dynamics described by abstract infinite-dimensional systems with strongly stable state operator and collocated input and output operators in [4] and by abstract second-order infinite-dimensional systems in [5]. In the absence of the leader agent, the OFSC problem (which is to ensure that all the follower states are eventually the same) is addressed in [3] and [11] for agent dynamics governed by parabolic PDEs with distributed input and boundary input, respectively. In [19] the solvability of the leaderless consensus problem in a network of parabolic PDEs using a static controller has been studied. While the above works consider identical agents, output consensus problem in a network of non-identical agents modeled by parabolic PDEs has been addressed in [6]. Some of the above works go beyond the consensus objective and also consider other objectives such as disturbance compensation.