Laplacian Controllability of Interconnected Graphs

We consider the controllability of a graph constructed by interconnecting a finite number of single-input Laplacian controllable (SILC) graphs. We first study the interconnection realized by the composite graph of two SILC graphs, called the structure graph and the cell graph, respectively. Suppose the cell graph is Laplacian controllable by an input connected to some special vertex called the composite vertex. The composite graph is constructed by interconnecting several identical cell graphs through the composite vertices such that connecting these composite vertices alone forms the structure graph. We prove that the structure graph is SILC by an input connected to some vertex of the graph if and only if the composite graph is SILC by the same input connected to the same vertex. The second part of this article generalizes the path structure by viewing it as a serial interconnection of two-vertex antiregular graphs, with or without appending a terminal one-vertex path. We show that its SILC property is preserved if we increase the number of vertices of the antiregular graphs and/or that of the terminal path. Examples are provided to illustrate the novel class of SILC graphs we propose.

[1]  Shun-Pin Hsu,et al.  Laplacian controllable graphs based on connecting two antiregular graphs , 2018, IET Control Theory & Applications.

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

[3]  Magnus Egerstedt,et al.  Graph Theoretic Methods in Multiagent Networks , 2010, Princeton Series in Applied Mathematics.

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

[5]  EgerstedtMagnus,et al.  Controllability of Multi-Agent Systems from a Graph-Theoretic Perspective , 2009 .

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

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

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

[9]  Wen-Chyuan Yueh.  EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES , 2005 .

[10]  Manfredi Maggiore,et al.  Necessary and sufficient graphical conditions for formation control of unicycles , 2005, IEEE Transactions on Automatic Control.

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

[12]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

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

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

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

[16]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

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

[18]  Suk-Geun Hwang,et al.  Cauchy's Interlace Theorem for Eigenvalues of Hermitian Matrices , 2004, Am. Math. Mon..

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

[20]  Irene Sciriha,et al.  Controllability of undirected graphs , 2014 .

[21]  S. Hsu.  Minimal Laplacian controllability problems of threshold graphs , 2019, IET Control Theory & Applications.

[22]  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).

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

[24]  M. Kanat Camlibel,et al.  Controllability of diffusively-coupled multi-agent systems with general and distance regular coupling topologies , 2011, IEEE Conference on Decision and Control and European Control Conference.

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

[26]  HUA BAI.  THE GRONE-MERRIS CONJECTURE , 2011 .

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

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

[29]  Zhen Wang,et al.  Interconnection topologies for multi-agent coordination under leader-follower framework , 2009, Autom..

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

[31]  Shun-Pin Hsu,et al.  Generalising Laplacian controllability of paths , 2019, IET Control Theory & Applications.

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

[33]  R. Bapat.  On the adjacency matrix of a threshold graph , 2013 .

[34]  Russell Merris,et al.  ANTIREGULAR GRAPHS ARE UNIVERSAL FOR TREES , 2003 .