Vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication are emerging components of intelligent transport systems (ITS) based on which vehicles can drive in a cooperative way and, hence, significantly improve traffic flow efficiency. However, due to the high vehicle mobility, the unreliable vehicular communications such as packet loss and transmission delay can impair the performance of the cooperative driving system (CDS). In addition, the downstream traffic information collected by roadside sensors in the V2I communication may introduce measurement errors, which also affect the performance of the CDS.The goal of this paper is to bridge the gap between traffic flow modelling and communication approaches in order to build up better cooperative traffic systems. To this end, we aim to develop an enhanced cooperative microscopic (car-following) traffic model considering V2V and V2I communication (or V2X for short), and investigate how vehicular communications affect the vehicle cooperative driving, especially in traffic disturbance scenarios. For these purposes, we design a novel consensus-based vehicle control algorithm for the CDS, in which not only the local traffic flow stability is guaranteed, but also the shock waves are supposed to be smoothed. The IEEE 802.11p, the defacto vehicular networking standard, is selected as the communication protocols, and the roadside sensors are deployed to collect the average speed in the targeted area as the downstream traffic reference. Specifically, the imperfections of vehicular communication as well as the measured information noise are taken into account. Numerical results show the efficiency of the proposed scheme. This paper attempts to theoretically investigate the relationship between vehicular communications and cooperative driving, which is needed for the future deployment of both connected vehicles and infrastructure (i.e. V2X).