Fundamental Limits on Disturbance Propagation in Virtual Viscoelastic-based Multi-Agent Systems

In this paper, we investigate the performance deterioration of commensurate fractional-order consensus networks under exogenous stochastic disturbances. We formulate fractional-order differential equations for the network dynamics using Caputo derivatives and the Laplace transform, and employ the H
2
norm of the dynamical system as a performance measure. By developing a graph-theoretic methodology, we relate the structural specifications of the underlying graphs to the performance measure and explicitly quantify fundamental limits on the best achievable levels of performance in fractional-order consensus networks. We also establish new connections between the sparsity of the network and the performance measure, characterizing fundamental tradeoffs that reveal the interplay between the two. Finally, we provide numerical illustrations to verify our theoretical results, which could help in the design of robust fractional-order control systems in the presence of disturbances.