Scalable robustness of interconnected systems subject to structural changes

Scalability in Nonlinear Network Systems Affected by Delays and Disturbances

“…Essentially, Corollary 1 states that, if there are disturbances on the network, and the matrix measure induced by |•| ∞ is considered to study stability, then not only the system is stable, as shown in [33], but it is also scalable. Also, we note that ( 14) includes, as a special case, the models studied in [22] where linear, delay-free, couplings are considered.…”

“…[20], that offer a generalization of string stability to (linear and distubance-free) networks with regular topologies. Finally, other results include [21], where sufficient conditions for the scalability of delay-free leaderless networks with homogeneous agents interacting over regular topologies are introduced, and the more recent [22] where, by considering linear systems, the assumption of homogeneous agents was removed.…”

Robustness of the terminal behaviour of resistive electrical networks

“…We note that the matrix on the right-hand side of (31) is of dimension h. As the number of updates is typically small, this yields a computationally efficient approach for computing the updated total effective resistance. Remark 3.5: Due to the link between the total effective resistance and performance of consensus networks (Lemma 3.5), the result of Theorem 3.6 can be used to study the robustness of network performance with respect to structural changes, i.e., changes in the network structure, similar to results in [19], [20]. Using the above, Theorem 3.1, and ( 19) the effective resistance between node 1 and 3 in G can be computed as Reff (1, 3) = 8 5 .…”

“…Remark 3.1: It is clear from (20) that the effective resistance increases when the conductance of one of the edges is decreased. This is well-known from Rayleigh's monotonicity theorem, e.g.…”