Analyzing Containment Control Performance for Fractional-Order Multi-Agent Systems via A Delay Margin Perspective

This article investigates the containment control performance analysis problem for double-integrator fractional-order multi-agent systems (MASs) with nonuniform time delays (NTDs). The primary focus is on evaluating the containment control performance by calculating the explicit delay margin. Firstly, the transfer function of the closed-loop error system is established by defining the containment control error. Then, the stability of the nonuniform delayed fractional-order closed-loop system is analyzed by using the frequency domain method, considering both undirected and directed communication topologies. Furthermore, the critical time delay (TD) is determined by generating the characteristic equation as a polynomial involving the sub-Laplacian matrix among the follower agents. Additionally, the containment convergence conditions for fractional-order MASs are derived. These conditions can be formulated based on delay margins and a set of inequalities involving the fractional order, eigenvalues of the Laplacian matrix, and control parameters. In summary, if all time delays (TDs) do not surpass the explicit delay margin, the containment control of the MAS is said to be realized; Otherwise, if all TDs surpass this margin, the MAS suffers from state divergence. Finally, two simulation examples are provided to verify the correctness of the theoretical results.