A longitudinal autopilot system for flying vehicles, such as missiles and drones, is crucial for enhancing stability, reducing the risk of errors, improving efficiency, and providing better control in unpredictable situations. The advancement of such a system is imperative for advancing the aviation industry and ensuring successful missions carried out by these flying vehicles. The necessity for a reliable flight controller in agile missile applications motivates the design of this system. A state-space formulation was utilized to integrate a linear-quadratic regulator (LQR) method, a Luenberger observer, and proportional navigation algorithms. The design methodology aimed at minimizing a quadratic cost function while meeting performance objectives. The system was simulated using MATLAB, and the results demonstrate the autopilot's effectiveness in compensating for in-flight disturbances and noise caused by parametric uncertainties, environmental disturbances, and system non-linearities. The versatility of LQR optimization methods and the importance of robust control for stable and reliable missile systems in changing environments are emphasized in this observational study. Simulations and numerical analysis revealed a reduction in the miss distance, indicating the autopilot system's proficiency and robustness.