This paper is concerned with designing and numerically implementing a V2X (Vehicle-to-Vehicle and Vehicle-to-Infrastructure) control system architecture for a platoon of autonomous vehicles. The V2X control architecture integrates the well-known Intelligent Driver Model (IDM) for a platoon of Autonomous Driving Vehicles (ADVs) with Vehicle-to-Infrastructure (V2I) Communication. The main aim is to address practical implementation issues of such a system as well as the safety and security concerns for traffic environments. To this end, we first investigated a channel estimation model for V2I communication. We employed the IEEE 802.11p vehicular standard and calculated path loss, Packet Error Rate (PER), Signal-to-Noise Ratio (SNR), and throughput between transmitter and receiver end. Next, we carried out several case studies to evaluate the performance of the proposed control system with respect to its response to: (i) the communication infrastructure; (ii) its sensitivity to an emergency, inter-vehicular gap, and significant perturbation; and (iii) its performance under the loss of communication and changing driving environment. Simulation results show the effectiveness of the proposed control model. The model is collision-free for an infinite length of platoon string on a single lane road-driving environment. It also shows that it can work during a lack of communication, where the platoon vehicles can make their decision with the help of their own sensors. V2X Enabled Intelligent Driver Model (VX-IDM) performance is assessed and compared with the state-of-the-art models considering standard parameter settings and metrics.