Safety is a critical concern in autonomous vehicle (AV) systems, especially when AI-based sensing and perception modules are involved. However, due to the black box nature of AI algorithms, it makes closed-loop analysis and synthesis particularly challenging, for example, establishing closed-loop stability and ensuring performance, while they are fundamental to AV safety. To approach this difficulty, this paper aims to develop new modeling, analysis, and synthesis tools for AI-based AVs. Inspired by recent developments in perception error models (PEMs), the focus is shifted from directly modeling AI-based perception processes to characterizing the perception errors they produce. Two key classes of AI-induced perception errors are considered: misdetection and measurement noise. These error patterns are modeled using continuous-time Markov chains and Wiener processes, respectively. By means of that, a PEM-augmented driving model is proposed, with which we are able to establish the closed-loop stability for a class of AI-driven AV systems via stochastic calculus. Furthermore, a performance-guaranteed output feedback control synthesis method is presented, which ensures both stability and satisfactory performance. The method is formulated as a convex optimization problem, allowing for efficient numerical solutions. The results are then applied to an adaptive cruise control (ACC) scenario, demonstrating their effectiveness and robustness despite the corrupted and misleading perception.