4 research outputs found
On the Trade-off Between Controllability and Robustness in Networks of Diffusively Coupled Agents
In this paper, we demonstrate a conflicting relationship between two crucial
properties---controllability and robustness---in linear dynamical networks of
diffusively coupled agents. In particular, for any given number of nodes
and diameter , we identify networks that are maximally robust using the
notion of Kirchhoff index and then analyze their strong structural
controllability. For this, we compute the minimum number of leaders, which are
the nodes directly receiving external control inputs, needed to make such
networks controllable under all feasible coupling weights between agents. Then,
for any and , we obtain a sharp upper bound on the minimum number of
leaders needed to design strong structurally controllable networks with
nodes and diameter . We also discuss that the bound is best possible for
arbitrary and . Moreover, we construct a family of graphs for any
and such that the graphs have maximal edge sets (maximal robustness) while
being strong structurally controllable with the number of leaders in the
proposed sharp bound. We then analyze the robustness of this graph family. The
results suggest that optimizing robustness increases the number of leaders
needed for strong structural controllability. Our analysis is based on
graph-theoretic methods and can be applied to exploit network structure to
co-optimize robustness and controllability in networks.Comment: IEEE Transactions on Control of Network System
Strong Structural Controllability of Diffusively Coupled Networks: Comparison of Bounds Based on Distances and Zero Forcing
We study the strong structural controllability (SSC) of diffusively coupled
networks, where the external control inputs are injected to only some nodes,
namely the leaders. For such systems, one measure of controllability is the
dimension of strong structurally controllable subspace, which is equal to the
smallest possible rank of controllability matrix under admissible (positive)
coupling weights. In this paper, we compare two tight lower bounds on the
dimension of strong structurally controllable subspace: one based on the
distances of followers to leaders, and the other based on the graph coloring
process known as zero forcing. We show that the distance-based lower bound is
usually better than the zero-forcing-based bound when the leaders do not
constitute a zero-forcing set. On the other hand, we also show that any set of
leaders that can be shown to achieve complete SSC via the distance-based bound
is necessarily a zero-forcing set. These results indicate that while the
zero-forcing based approach may be preferable when the focus is only on
verifying complete SSC, the distance-based approach is usually more informative
when partial SSC is also of interest. Furthermore, we also present a novel
bound based on the combination of these two approaches, which is always at
least as good as, and in some cases strictly greater than, the maximum of the
two bounds. We support our analysis with numerical results for various graphs
and leader sets.Comment: Accepted to the 59th IEEE Conference on Decision and Contro
Improving Network Robustness through Edge Augmentation While Preserving Strong Structural Controllability
In this paper, we consider a network of agents with Laplacian dynamics, and
study the problem of improving network robustness by adding a maximum number of
edges within the network while preserving a lower bound on its strong
structural controllability (SSC) at the same time. Edge augmentation increases
network's robustness to noise and structural changes, however, it could also
deteriorate network controllability. Thus, by exploiting relationship between
network controllability and distances between nodes in graphs, we formulate an
edge augmentation problem with a constraint to preserve distances between
certain node pairs, which in turn guarantees that a lower bound on SSC is
maintained even after adding edges. In this direction, first we choose a node
pair and maximally add edges while maintaining the distance between selected
nodes. We show that an optimal solution belongs to a certain class of graphs
called clique chains. Then, we present an algorithm to add edges while
preserving distances between a certain collection of nodes. Further, we present
a randomized algorithm that guarantees a desired approximation ratio with high
probability to solve the edge augmentation problem. Finally, we evaluate our
results on various networks.Comment: American Control Conference (ACC)2020, Denve
Structural Robustness to Noise in Consensus Networks: Impact of Degrees and Distances, Fundamental Limits, and Extremal Graphs
We investigate how the graph topology influences the robustness to noise in
undirected linear consensus networks. We measure the structural robustness by
using the smallest possible value of steady state population variance of states
under the noisy consensus dynamics with edge weights from the unit interval. We
derive tight upper and lower bounds on the structural robustness of networks
based on the average distance between nodes and the average node degree. Using
the proposed bounds, we characterize the networks with different types of
robustness scaling under increasing size. Furthermore, we present a fundamental
trade-off between the structural robustness and the average degree of networks.
While this trade-off implies that a desired level of structural robustness can
only be achieved by graphs with a sufficiently large average degree, we also
show that there exist dense graphs with poor structural robustness. We then
show that, random k-regular graphs (the degree of each node is k) with n nodes
typically have near-optimal structural robustness among all the graphs with
size n and average degree k for sufficiently large n and k. We also show that
when k increases properly with n, random k-regular graphs maintain a structural
robustness within a constant factor of the complete graph's while also having
the minimum average degree required for such robustness.Comment: Accepted for publication in IEEE Transactions on Automatic Contro