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In this paper we study how high-gain anti-windup schemes can be used to implement projected dynamical systems in control loops that are subject to saturation on a (possibly unknown) set of admissible inputs. This insight is especially useful for the design of autonomous optimization schemes that realize a closed-loop behavior which approximates a particular optimization algorithm (e.g., projected gradient or Newton descent) while requiring only limited model information. In our analysis we show that a saturated integral controller, augmented with an anti-windup scheme, gives rise to a perturbed projected dynamical system. This insight allows us to show uniform convergence and robust practical stability as the anti-windup gain goes to infinity. Moreover, for a special case encountered in autonomous optimization we show robust convergence, i.e., convergence to an optimal steady-state for finite gains. Apart from being particularly suited for online optimization of large-scale systems, such as power grids, these results are potentially useful for other control and optimization applications as they shed a new light on both anti-windup control and projected gradient systems.