Citation
C. Enyioha, M.A. Rahimian, G. Pappas, A. Jadbabaie. "Controllability and Fraction of Leaders in Infinite Networks". CDC 2014, IEEE, 15, December, 2014.

Abstract
In this paper, we study controllability of a network of linear single-integrator agents when the network size goes to infinity. We first investigate the effect of increasing size by injecting an input at every node and requiring that network controllability Gramian remain well-conditioned with the increasing dimension. We provide theoretical justification to the intuition that high degree nodes pose a challenge to network controllability. In particular, the controllability Gramian for the networks with bounded maximum degrees is shown to remain well-conditioned even as the network size goes to infinity. In the canonical cases of star, chain and ring networks, we also provide closed-form expressions which bound the condition number of the controllability Gramian in terms of the network size. We next consider the effect of the choice and number of leader nodes by actuating only a subset of nodes and considering the least eigenvalue of the Gramian as the network size increases. Accordingly, while a directed star topology can never be made controllable for all sizes by injecting an input just at a fraction f < 1 of nodes; for path or cycle networks, the designer can actuate a non-zero fraction of nodes and spread them throughout the network in such way that the least eigenvalue of the Gramians remain bounded away from zero with the increasing size. The results offer interesting insights on the challenges of control in large networks and with high-degree nodes.

Electronic downloads

• https://www.src.org/library/publication/p071871/. (Link to paper on SRC website)
• http://dx.doi.org/10.1109/CDC.2014.7039571

• HTML

  C. Enyioha, M.A. Rahimian, G. Pappas, A. Jadbabaie. <a
  href="http://www.terraswarm.org/pubs/357.html"
  >Controllability and Fraction of Leaders in Infinite
  Networks</a>, CDC 2014, IEEE, 15, December, 2014.

• Plain text

  C. Enyioha, M.A. Rahimian, G. Pappas, A. Jadbabaie.
  "Controllability and Fraction of Leaders in Infinite
  Networks". CDC 2014, IEEE, 15, December, 2014.

• BibTeX

  @inproceedings{EnyiohaRahimianPappasJadbabaie14_ControllabilityFractionOfLeadersInInfiniteNetworks,
      author = {C. Enyioha and M.A. Rahimian and G. Pappas and A.
                Jadbabaie},
      title = {Controllability and Fraction of Leaders in
                Infinite Networks},
      booktitle = {CDC 2014},
      organization = {IEEE},
      day = {15},
      month = {December},
      year = {2014},
      abstract = {In this paper, we study controllability of a
                network of linear single-integrator agents when
                the network size goes to infinity. We first
                investigate the effect of increasing size by
                injecting an input at every node and requiring
                that network controllability Gramian remain
                well-conditioned with the increasing dimension. We
                provide theoretical justification to the intuition
                that high degree nodes pose a challenge to network
                controllability. In particular, the
                controllability Gramian for the networks with
                bounded maximum degrees is shown to remain
                well-conditioned even as the network size goes to
                infinity. In the canonical cases of star, chain
                and ring networks, we also provide closed-form
                expressions which bound the condition number of
                the controllability Gramian in terms of the
                network size. We next consider the effect of the
                choice and number of leader nodes by actuating
                only a subset of nodes and considering the least
                eigenvalue of the Gramian as the network size
                increases. Accordingly, while a directed star
                topology can never be made controllable for all
                sizes by injecting an input just at a fraction f <
                1 of nodes; for path or cycle networks, the
                designer can actuate a non-zero fraction of nodes
                and spread them throughout the network in such way
                that the least eigenvalue of the Gramians remain
                bounded away from zero with the increasing size.
                The results offer interesting insights on the
                challenges of control in large networks and with
                high-degree nodes. },
      URL = {http://terraswarm.org/pubs/357.html}
  }
  

Posted by Barb Hoversten on 10 Sep 2014.