3,755 research outputs found
Zero forcing number, constrained matchings and strong structural controllability
The zero forcing number is a graph invariant introduced to study the minimum
rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero
forcing number of a simple undirected graph. We complete this NP-hardness
result by showing that the non-equivalent problem of computing the zero forcing
number of a directed graph allowing loops is also NP-hard. The rest of the
paper is devoted to the strong controllability of a networked system. This kind
of controllability takes into account only the structure of the interconnection
graph, but not the interconnection strengths along the edges. We provide a
necessary and sufficient condition in terms of zero forcing sets for the strong
controllability of a system whose underlying graph is a directed graph allowing
loops. Moreover, we explain how our result differs from a recent related result
discovered by Monshizadeh et al. Finally, we show how to solve the problem of
finding efficiently a minimum-size input set for the strong controllability of
a self-damped system with a tree-structure.Comment: Submitted as a journal paper in May 201
The Observability Radius of Networks
This paper studies the observability radius of network systems, which
measures the robustness of a network to perturbations of the edges. We consider
linear networks, where the dynamics are described by a weighted adjacency
matrix, and dedicated sensors are positioned at a subset of nodes. We allow for
perturbations of certain edge weights, with the objective of preventing
observability of some modes of the network dynamics. To comply with the network
setting, our work considers perturbations with a desired sparsity structure,
thus extending the classic literature on the observability radius of linear
systems. The paper proposes two sets of results. First, we propose an
optimization framework to determine a perturbation with smallest Frobenius norm
that renders a desired mode unobservable from the existing sensor nodes.
Second, we study the expected observability radius of networks with given
structure and random edge weights. We provide fundamental robustness bounds
dependent on the connectivity properties of the network and we analytically
characterize optimal perturbations of line and star networks, showing that line
networks are inherently more robust than star networks.Comment: 8 pages, 3 figure
Minimum Input Selection for Structural Controllability
Given a linear system , where is an matrix
with nonzero entries, we consider the problem of finding the smallest set
of state variables to affect with an input so that the resulting system is
structurally controllable. We further assume we are given a set of "forbidden
state variables" which cannot be affected with an input and which we have
to avoid in our selection. Our main result is that this problem can be solved
deterministically in operations
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