Graph Laplacian-Based Modeling of Distributed Consensus and Cooperative Control in Multi-Agent Engineering Systems
DOI:
https://doi.org/10.71426/jasm.v2.i2.pp140-146Keywords:
graph Laplacian, multi-agent systems, distributed consensus, cooperative control, algebraic connectivityAbstract
Graph Laplacians provide a compact mathematical bridge between communication topology and closed-loop behavior in distributed engineering systems. This paper develops a reproducible Laplacian based framework for first-order consensus, discrete time stability, switching communication graphs, link-loss robustness, and leader–follower cooperative formation control. For a 12-agent study, four canonical weighted topologies show a strong spectral ordering: algebraic connectivity λ2 increases from 0.068 for a path to 0.254 for a ring, 0.861 for a small-world mesh, and 12 for a complete graph, while measured 2% consensus settling time decreases from 19.76 s to 5.50 s, 2.53 s, and 0.33 s, respectively. Across 72 additional connected random graphs, log settling time versus log λ2 has slope −0.887 and R2 = 0.931, empirically supporting inverse spectral scaling. For the small-world mesh, the discrete consensus stability boundary is α < 2/λmax = 0.2720. Edge-removal tests reveal up to 29.26% loss of algebraic connectivity, and a Fiedler-vector edge sensitivity proxy correlates 0.956 with the measured λ2 loss. A switching-topology protocol settles in 2.56 s, while an eight-follower second order leader–follower system reaches 0.1 position-RMSE in 4.34 s. The study emphasizes spectral interpretation, limitations of topology-only modeling, and engineering tradeoffs among connectivity, communication burden, robustness, and control speed.
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DOI prefix: 10.71426
