This paper presents a robust model predictive control (MPC) framework that explicitly addresses the non-Gaussian noise inherent in deep learning-based perception modules used for state estimation. Recognizing that accurate uncertainty quantification of the perception module is essential for safe feedback control, our approach departs from the conventional assumption of zero-mean noise quantification of the perception error. Instead, it employs set-based state estimation with constrained zonotopes to capture biased, heavy-tailed uncertainties while maintaining bounded estimation errors. To improve computational efficiency, the robust MPC is reformulated as a linear program (LP), using a Minkowski-Lyapunov-based cost function with an added slack variable to prevent degenerate solutions. Closed-loop stability is ensured through Minkowski-Lyapunov inequalities and contractive zonotopic invariant sets. The largest stabilizing terminal set and its corresponding feedback gain are then derived via an ellipsoidal approximation of the zonotopes. The proposed framework is validated through both simulations and hardware experiments on an omnidirectional mobile robot along with a camera and a convolutional neural network-based perception module implemented within a ROS2 framework. The results demonstrate that the perception-aware MPC provides stable and accurate control performance under heavy-tailed noise conditions, significantly outperforming traditional Gaussian-noise-based designs in terms of both state estimation error bounding and overall control performance.