This paper proposes a data-driven framework to solve time-varying optimization problems associated with unknown linear dynamical systems. Making online control decisions to regulate a dynamical system to the solution of an 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 mandating a preliminary system identification phase before a controller can be designed. In this work, we leverage results from behavioral theory to show that the steady-state transfer function of a linear system can be computed from data samples without any knowledge or estimation of the system model. We then use this data-driven representation to design a controller, inspired by a gradient-descent optimization method, that regulates the system to the solution of a convex optimization problem, without requiring any knowledge of the time-varying disturbances affecting the model equation. Results are tailored to cost functions satisfy the Polyak-L ojasiewicz inequality.
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