Jorge Cortés
Professor
Cymer Corporation Endowed Chair
Gramian-based reachability metrics for bilinear networks
Y. Zhao, J. Cortés
IEEE Transactions on Control of Network Systems 4 (3) (2017), 620-631
Abstract
This paper studies Gramian-based reachability metrics for bilinear
control systems. In the context of complex networks, bilinear
systems capture scenarios where an actuator not only can affect the
state of a node but also interconnections among nodes. Under the
assumption that the input's infinity norm is bounded by some
function of the network dynamic matrices, we derive a Gramian-based
lower bound on the minimum input energy required to steer the state
from the origin to any reachable target state. This result
motivates our study of various objects associated to the
reachability Gramian to quantify the ease of controllability of the
bilinear network: the minimum eigenvalue (worst-case minimum input
energy to reach a state), the trace (average minimum input energy to
reach a state), and its determinant (volume of the ellipsoid
containing the reachable states using control inputs with no more
than unit energy). We establish an increasing returns property of
the reachability Gramian as a function of the actuators, which in
turn allows us to derive a general lower bound on the reachability
metrics in terms of the aggregate contribution of the individual
actuators. We conclude by examining the effect on the worst-case
minimum input energy of the addition of bilinear inputs to
difficult-to-control linear symmetric networks. We show that the bilinear
networks resulting from the addition of either inputs at a finite
number of interconnections or at all self loops with weight
vanishing with the network scale remain
difficult-to-control. Various examples illustrate our results.
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Mechanical and Aerospace Engineering,
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