Controllability of the multi-agent system modeled by the threshold graph with one repeated degree

Yang

2019

It is well known that if a network topology is a path or line and the states of vertices or nodes evolve according to the consensus policy, then the network is Laplacian controllable by an input connected to its terminal vertex. In this work a path is regarded as the resulting graph after interconnecting a finite number of two-vertex antiregular graphs and then possibly connecting one more vertex. It is shown that the single-input Laplacian controllability of a path can be extended to the case of interconnecting a finite number of k-vertex antiregular graphs with or without one more vertex appended, for any positive integer k. The methods to interconnect these antiregular graphs and to select the vertex for connecting the single input that renders the network Laplacian controllable are presented as well.

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\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ Λ_{k} : = {0, 1, \dots, k} ∖ {\bar{κ}} . \end{matrix}

where

\begin{matrix} right left right left right left right left right left right left0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em3pt \\ \bar{κ} = ⌈\frac{k}{2}⌉, \end{matrix}

and a full set of orthogonal eigenvectors of

ℒ_{A}^{false(k false)}

can be obtained using the following lemma [17, 29]. Lemma 1 Let the ( i,j )th entry of matrices

T^{false(m false)}

t_{i j}^{false(m false)}

for each

m \in falsefalse{1, 2, 3, 4 falsefalse}

.…”

Section: Resultsmentioning

confidence: 99%

Generalising Laplacian controllability of paths

Yang

2019

show abstract

“…A full set of orthogonal eigenvectors of

{scriptL}_{A}^{false(k false)}

can be obtained using the following lemma [17, 18]. Lemma 1 Let the ( i,j )th entry of matrices

T^{false(m false)}

t_{i j}^{false(m false)}

for each

m \in falsefalse{1, 2, 3, 4 falsefalse}

.…”

Section: Resultsmentioning

confidence: 99%

“…A basic question that motivates the study is: given the numbers of vertices and controllers, how do we connect these vertices and controllers to create a Laplacian controllable graph? To the best of our knowledge, there are only several specific families with fully or partially explored controllability properties, including paths [12], grids [13], circulant graphs [14], complete graphs [15], multi‐chain graphs [16] and threshold graphs [17, 18]. The successful exploration is due to the special structures of their Laplacian matrices that facilitate the application of the Popov–Belevitch–Hautus (PBH) test to the Laplacian eigenspaces of graphs.…”

Section: Introductionmentioning

confidence: 99%

Laplacian controllable graphs based on connecting two antiregular graphs

Yang

2018

“…A necessary and sufficient condition for a binary control vector to ensure its controllability was provided in [26]. This result was later extended to the multi‐input case where more vertices with the same degree were considered [27]. The Laplacian controllability of hybrid graphs constructed by connecting one antiregular graph to another antiregular graph [28] or to a path graph [29] was studied.…”

Section: Introductionmentioning

confidence: 99%

Minimal Laplacian controllability problems of threshold graphs

2019