Controllability and Fraction of Leaders in Infinite Networks
C. Enyioha, M.A. Rahimian, G. Pappas, A. Jadbabaie

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.

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  • 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.

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