# The Car-Following Model and Its Applications in the V2X Environment: A Historical Review
## 3 Research on Car-Following Behavior in the V2X Environment
### Modelling the Car-Following Behavior
#### 3.2.7 Feedback Control Scheme  
Among the research on modeling car-following behavior affected by various kinds of information in the V2X environment, there is a special kind of model, i.e., the Coupled Map (CM) model. The CM model is essentially a discrete form of the OV model. Compared with the normal OV models, there are some unique advantages in the CM model and its extended models in combination with control items and the simplicity of their simulation algorithm. The CM model was proposed, and the validity of the feedback control item for alleviating the traffic jams was confirmed by [PERSON] et al. [103]. The CM model and its extended models can be subdivided into the chaotic CM models and the non-chaotic CM models. In the chaotic ones, the optimal velocity function is regarded as a chaotic map, and each vehicle in the system has its own chaotic state. In the non-chaotic CM models, the optimal velocity function remains the same form as that in the OV model. The first CM model proposed by [PERSON] et al. is one of the non-chaotic CM models, and it adopts the same time-delay feedback as the control scheme. Based on [PERSON]'s work, [PERSON] and [PERSON] adopted the relative velocity between the object vehicle and its preceding vehicles as the feedback control item [104], which is widely used in the following studies. Later, [PERSON] et al. incorporated the velocity of multiple preceding vehicles to improve the control scheme proposed by [PERSON] and [PERSON] and formed an extended CM model suitable for the V2X environment [105]. Specifically, from the perspective of collaborative control for a vehicle platoon, [PERSON] et al. designed a one-way feedback scheme, in which the vehicles in turn are controlled based on the information of their preceding vehicle. Later, [PERSON] et al. designed a control scheme based on the relative velocity between the two preceding vehicles and the object vehicle and, based on this, proposed an extended CM model [106]. [PERSON] et al. incorporated the Back Looking Effect and established an extended CM model with consideration of the following vehicle [107]. Based on the optimal velocity of the two preceding vehicles, [PERSON] et al. extended the optimal velocity function in the CM model and formed an improved CM model [107]. Then, [PERSON] et al. further introduced the headway and difference in optimal velocity between the object vehicle and its preceding vehicle into the control scheme to extend the CM model [108]. Considering that the penetration and reliability of V2X technology cannot be maintained at 100% all the time, [PERSON] et al. constructed a matrix to represent the communication connection state between vehicles in the system and, based on this, proposed an extended model with consideration of the headway and velocity information of multiple preceding vehicles [109]. The impacts of V2X technology penetration on car-following behavior were also explored by [PERSON] and [PERSON], utilizing numerical simulation based on the extended CM model considering the velocity fluctuation information of multiple preceding vehicles proposed by themselves [110]. As discussed in the previous sections, the delay exists in the system of which the vehicles are automatically or manually controlled. Considering this, [PERSON] et al. proposed an improved CM model with consideration of various safe headways and the corresponding delays [111]. Later, [PERSON] et al. considered the velocity of two preceding vehicles as well as the corresponding delay and further extended the CM model [112]. The results of the above research reveal that the control scheme designed based on various sources of information in the V2X environment can effectively alleviate traffic congestion by keeping the vehicle(s) operating at a relative steady state. From the perspective of traffic flow theory, the essence of traffic congestion is the generation and further evolution of disturbance. Under the ideal conditions, all vehicles in the system will maintain the initial steady state, and, thus, congestion never occurs.