Almost Sure Convergence in Feedback Optimization via Stochastic Timescale Separation

In this paper, we study discrete-time feedback optimization in stochastic scenarios. Specifically, we consider a discrete-time stochastic nonlinear system where the goal is to optimize its steady-state performance while simultaneously controlling it. Moreover, in our framework, we deal with packet losses in the communication between the controller and the physical system. To address these challenges, we propose a gradient-based controller embedding sample-and-hold mechanisms to handle packet losses. We analyze the resulting stochastic closed-loop system through the lens of stochastic timescale separation. This perspective allows us to establish the almost sure global asymptotic convergence of the closed-loop system to a set in which first-order optimality conditions of the underlying nonconvex optimization problem are satisfied. Finally, we provide numerical simulations to support our theoretical findings.