On a class of multi-input laplacian controllable graphs

In this paper we consider the Laplacian controllability problem of combining two special threshold graphs. We form this combination by connecting two identical and connected threshold graphs which have exactly one repeated degree. With appropriate selection of connecting vertices for combination we show how to use the minimum number of inputs to render the combined graph Laplacian controllable. Numerical examples are provided to demonstrate our work.

[1]  Ming Cao,et al.  Interacting with Networks: How Does Structure Relate to Controllability in Single-Leader, Consensus Networks? , 2012, IEEE Control Systems.

[2]  H.G. Tanner,et al.  On the controllability of nearest neighbor interconnections , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[3]  N. Mahadev,et al.  Threshold graphs and related topics , 1995 .

[4]  Charles R. Johnson,et al.  Matrix Analysis, 2nd Ed , 2012 .

[5]  R. Merris.  Degree maximal graphs are Laplacian integral , 1994 .

[6]  Magnus Egerstedt,et al.  Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective , 2009, SIAM J. Control. Optim..

[7]  Jason J. Molitierno.  Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs , 2012 .

[8]  Bahman Gharesifard,et al.  Graph Controllability Classes for the Laplacian Leader-Follower Dynamics , 2015, IEEE Transactions on Automatic Control.

[9]  Cesar O. Aguilar,et al.  Laplacian controllability classes for threshold graphs , 2015 .

[10]  M. Kanat Camlibel,et al.  Upper and Lower Bounds for Controllable Subspaces of Networks of Diffusively Coupled Agents , 2014, IEEE Transactions on Automatic Control.

[11]  Anirban Banerjee,et al.  On the normalized spectrum of threshold graphs , 2016, 1610.08816.

[12]  R. Merris.  Laplacian graph eigenvectors , 1998 .

[13]  Shun-Pin Hsu.  Controllability of the multi-agent system modeled by the threshold graph with one repeated degree , 2016, Syst. Control. Lett..

[14]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[15]  Giuseppe Notarstefano,et al.  Controllability and Observability of Grid Graphs via Reduction and Symmetries , 2012, IEEE Transactions on Automatic Control.

[16]  Charles Delorme,et al.  Laplacian eigenvectors and eigenvalues and almost equitable partitions , 2007, Eur. J. Comb..

[17]  Shun-Pin Hsu.  A necessary and sufficient condition for the controllability of single-leader multi-chain systems , 2017 .

[18]  Mehran Mesbahi,et al.  On the Controllability Properties of Circulant Networks , 2013, IEEE Transactions on Automatic Control.

[19]  Shun-Pin Hsu,et al.  Constructing a controllable graph under edge constraints , 2017, Syst. Control. Lett..

[20]  Giuseppe Notarstefano,et al.  On the Reachability and Observability of Path and Cycle Graphs , 2011, IEEE Transactions on Automatic Control.