  This paper proposes a data-driven framework to solve time-varying
optimization problems associated with unknown linear dynamical systems. Making
online control decisions to steer a system to the solution trajectory of a
time-varying optimization problem is a central goal in many modern engineering
applications. Yet, the available methods critically rely on a precise knowledge
of the system dynamics, thus requiring ad-hoc system identification and model
refinement phases. In this work, we leverage tools from behavioral theory to
show that the steady-state transfer function of a system can be computed from
control experiments without knowledge or estimation of the system model. Such
direct computation allows us to avoid the explicit model identification phase,
and is significantly more tractable than the direct model-based computation. We
leverage the data-driven representation to design a controller inspired from a
gradient-descent method that drives the system to the solution of an
unconstrained optimization problem, without any knowledge of time-varying
disturbances affecting the model equation. Results are tailored to cost
functions that are smooth and satisfy the Polyak-Lojasiewicz inequality.
Simulation results illustrate the technical findings.
