  In this paper we consider a recently developed distributed optimization
algorithm based on gradient tracking. We propose a system theory framework to
analyze its structural properties on a preliminary, quadratic optimization
set-up. Specifically, we focus on a scenario in which agents in a static
network want to cooperatively minimize the sum of quadratic cost functions. We
show that the gradient tracking distributed algorithm for the investigated
program can be viewed as a sparse closed-loop linear system in which the
dynamic state-feedback controller includes consensus matrices and optimization
(stepsize) parameters. The closed-loop system turns out to be not completely
reachable and asymptotic stability can be shown restricted to a proper
invariant set. Convergence to the global minimum, in turn, can be obtained only
by means of a proper initialization. The proposed system interpretation of the
distributed algorithm provides also additional insights on other structural
properties and possible design choices that are discussed in the last part of
the paper as a starting point for future developments.
