### Jorge Cortés

#### Professor

Reachability metrics for bilinear complex networks

Y. Zhao, J. Cortés

Proceedings of the IEEE Conference on
Decision and Control, Osaka, Japan, 2015, pp. 4788-4793

### Abstract

Controllability metrics based on the controllability Gramian have
been widely used in linear control theory, and have recently seen
renewed interests in the study of complex networks of dynamical
systems. For example, the minimum eigenvalue and the trace of the
Gramian are related to the worst-case and average minimum input
energy, respectively, to steer the state from the origin to a target
state. This paper explores similar questions that remain unanswered
for bilinear control systems. In the context of complex networks,
bilinear systems characterize scenarios where an actuator not only
can affect the state of a node, but also can affect the strength of
the interconnections among some neighboring nodes. Under the
assumption that the infinity norm of the input is bounded by some
function of the network dynamic matrices, we derive a lower bound on
the minimum input energy to steer the state of a bilinear network
from the origin to any reachable target state based on the
generalized reachability Gramian of bilinear systems. We also
provide a lower bound on the average minimum input energy over all
target states on the unit hypersphere in the state space. Based on
the reachability metrics proposed, we propose an actuator selection
method that provides guaranteed minimum average input
energy. Finally, we show that the optimal actuator selection can be
found by maximizing a supermodular function under cardinality
constraints.

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Mechanical and Aerospace Engineering,
University of California, San Diego

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Ph: 1-858-822-7930

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