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This paper introduces a learning-based optimal control strategy enhanced with nonmodel-based state estimation to manage the complexities of lane-changing maneuvers in autonomous vehicles. Traditional approaches often depend on comprehensive system state information, which may not always be accessible or accurate due to dynamic traffic environments and sensor limitations. Our methodology dynamically adapts to these uncertainties and sensor noise by iteratively refining its control policy based on real-time sensor data and reconstructed states. We implemented an experimental setup featuring a scaled vehicle equipped with GPS, IMUs, and cameras, all processed through an Nvidia Jetson AGX Xavier board. This approach is pivotal as it addresses the limitations of simulations, which often fail to capture the complexity of dynamic real-world conditions. The results from real-world experiments demonstrate that our learning-based control system achieves smoother and more consistent lane-changing behavior compared to traditional direct measurement approaches. This paper underscores the effectiveness of integrating Adaptive Dynamic Programming (ADP) with state estimation techniques, as demonstrated through small-scale experiments. These experiments are crucial as they provide a practical validation platform that simulates real-world complexities, representing a significant advancement in the control systems used for autonomous driving.