Necessary and Sufficient Conditions on Consensus of General Fractional-Order Multi-Agent Systems Over Directed Networks

This article tackles the consensus problem for a group of multi-agent systems communicating over a directed network. Our networked system consists of general fractional-order dynamic systems with order
$\alpha \in [1,2)$
in continuous time. This presents a more challenging issue than the previous research on simple single/double fractional-order integrators. Specifically, we investigate the consensus issue for agents described by fractional-order systems with general linear dynamics, which presents a challenge due to the limited applicability of existing tools for integer-order models. To address this, we utilize spectral graph theory and fractional-order systems theory to derive several equivalent conditions without conservatism for such systems where agents communicate through directed graphs. We develop a tractable convex programming algorithm for controller design based on the obtained results. We then demonstrate the effectiveness of our proposed approach through simulations on higher-order dynamic systems and fractional-order circuits.