Motivated by vision-based control of autonomous vehicles, we consider the problem of controlling a known linear dynamical system for which partial state information, such as vehicle position, is extracted from complex and nonlinear data, such as a camera image. Our approach is to use a learned perception map that predicts some linear function of the state and to design a corresponding safe set and robust controller for the closed loop system with this sensing scheme. We show that under suitable smoothness assumptions on both the perception map and the generative model relating state to complex and nonlinear data, parameters of the safe set can be learned via appropriately dense sampling of the state space. We then prove that the resulting perception-control loop has favorable generalization properties. We illustrate the usefulness of our approach on a synthetic example and on the self-driving car simulation platform CARLA.
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