Cyber-physical description and CPS-based pinning approach of mixed traffic

Abstract

In this paper, a cyber-physical description of mixed traffic that can reflect the interaction between vehicles is first proposed. By treating vehicles as nodes and the cyber-physical interaction between vehicles as directed edges between nodes, the network of vehicles in traffic is obtained. The small-world and scale-free characteristics of mixed traffic are found under this description. Then the CPS-based pinning approach for achieving indirect control of HVs is proposed. Besides the control algorithm of pinning nodes, the impact of the spatial distribution of automated vehicles on traffic is studied. The approach is compatible with the existing control of automated vehicle in mixed vehicle platoon. Numerical simulation results under three-lane scenario illustrated the effectiveness of the proposed pinning approach.

1 INTRODUCTION

Mixing of automated driving vehicles and manual driving vehicles will certainly happen. Studies have shown that the mixing of automated vehicles could relieve traffic congestion[1]. Thus, the mixing of automated driving vehicles has brought new opportunities for the development of transport system. Most control strategies, however, focus on the traffic consisting of connected automated vehicles (CAVs). Control of mixed traffic, consisting of human-driven vehicles(HVs) and CAVs, remains a major challenge[2]. The new challenge cannot be easily solved under the existing research, and the new opportunities have not been well utilized.

If all vehicles are CAVs, the traffic situation will be better than it is now. However, before this goal is achieved, mixed traffic consisting of CAVs and HVs will be the form of future traffic. The main motivation of this paper is to achieve the purpose of indirect control of other HVs through direct control of CAV in mixed traffic before CAVs are fully popularized. The idea of this scheme is that by changing the motion state of CAVs, the HVs that follow them have to change the motion state accordingly, so as to achieve the purpose of indirect control of HVs. Based on the current research on mixed traffic, the use of CAVs to indirectly control the use of HVs can be further explored.

However, heterogeneous vehicles in mixed traffic have large differences in the types, scope, and real-time nature of information acquisition[3, 4], the decision mechanisms between the human driver and the driving automation system are also fundamentally different[5]. Moreover, besides physical interaction of vehicles, the mixing of CAVs will lead to more frequent cyber interaction between vehicles in traffic. This makes the cyber-physical characteristics of mixed traffic more obvious than the homogeneous case. If these characteristics could be described, it might help to reveal the mechanism of vehicle driving in mixed traffic, and provide a basis for the control of mixed traffic. Therefore, a new description of mixed traffic might be needed.

A lot of current research focuses on the design of CAV control strategies and the impact of CAV penetration on traffic. There are few studies about the impact of spatial distribution of automated vehicles on traffic. [6] mentioned that the position of CAVs in the mixed traffic stream does not bring significant difference on damping stochastic behavior of HVs. But they did not take into account the sudden braking or acceleration of vehicles, which may have a negative influence on the platoon. From the perspective of the pinning approach, the role of CAV in mixed traffic can also be brought into full play.

When traffic system utilizes modern wireless communication technologies, which have greatly promoted the development of intelligent transportation system, it could be considered as a cyber-physical system, in which all vehicles communicate via vehicular networking[7]. The applications of CPS in transportation systems could alleviate traffic congestion, and achieve the coordination and intelligence of the traffic system[8]. Since ”Cyber” is synonymous with a mixed composition of computation, storage, software, and networking[9], the realization of T-CPS benefits from the development of Vehicular Ad Hoc Network (VANET) technology. It is predicted that by 2040, VANET-based autonomous vehicles will account for 75% of the total road traffic[10]. In the related research of VANET, the vehicle is regarded as the node in the communication network[11]. [12]-[13] proposed a hybrid DSRC-cellular vehicle-connected network structure to meet the requirements of heterogeneous vehicle applications. The ADHOC MAC protocol based on distributed time division multiple access (DTDMA) proposed by [14]-[15] effectively solves the broadcast storm problem. The VeMAC protocol proposed by [16] can minimize the reduction in throughput caused by the relative movement of vehicles, while providing reliable broadcast services. These researches laid the foundation for the work of this paper. Therefore, in this paper, a pinning control strategy to bring CAVs in mixed traffic into full play to optimize traffic is proposed. The main contribution of this paper is:

• a. A cyber-physical description of traffic proposed, then the regular network of homogeneous traffic and the small-world and scale-free characteristics of mixed traffic are found;
• b. Based on the description above, the Cyber-Physical-Based (CPS-based) pinning approach is proposed. This approach provides a novel method for mixed traffic that is compatible with the existing control of automated vehicle;
• c. Besides the control algorithm of pinning nodes, the impact of the spatial distribution of automated vehicles on traffic is studied, and better control effect is achieved by selecting the pinning nodes when the control strategy of CAVs is determined.

The remainder of the paper is organized as following. Section 3 introduces weighted directed graph to describe mixed traffic, and the characteristics of homogeneous traffic and the small-world/scale-free characteristics of mixed traffic are also provided. The pinning approach is given in Section 4, which includes pinning node control strategy and pinning node selection strategy. In Section 5, the examples of CAV input, HV input and pinning node selection are given. The illustration of effectiveness of the proposed method is presented in Section 6. Section 7 summarizes the main conclusions.

2 LITERATURE REVIEW

2.1 Pinning control

By controlling some selected entities in the system to indirectly control other entities, then to achieve the desired development of the system, the idea is first been proposed in complex network. This control method is named pinning control: selecting the pinning nodes and exerting control on them, to indirectly control other nodes that are not directly pinned by the coupling between the nodes in complex network[17]. It is of vital importance to study how dense the pinning nodes should be for controlling the system, and how the pinning nodes influence the dynamics of the system[18]. Moreover, different pinning schemes for different topological networks are analytically and numerically investigated[17].

Pinning control method has been applied in different scenarios. In mixed traffic, pinning control means controlling the whole traffic system by adding controllers to CAVs in traffic. Besides theoretical analysis in a complex network, the pinning control method has been applied in aircraft conflict network[19], road traffic network[20], vehicle platoon[21, 22] etc. Moreover, [23] shows that when the penetration rate and the control strategy of each vehicle are certain, the purpose of optimizing the traffic can also be achieved by selecting the distribution of the auto-driving vehicles in the platoon. This shows that in mixed traffic, the selection strategy of pinning nodes will also have an impact on the control results.

2.2 Mixed traffic

Research on mixed traffic has achieved some results. For example, Panichpapiboon designed a self-organizing information release system architecture to ensure that an effective information transmission mechanism is formed between connected vehicles (CVs) for mixed traffic optimization[24]. Yang use speed control to make the connected and automated vehicles (CAVs) in the traffic flow maintains stable driving and guides the HV's car-following behavior to alleviate the speed oscillations generated by it, thereby achieving the purpose of reducing fuel consumption[25]. This can also be regarded as adopting the pinning method. Knorr analyzed that CVs in mixed traffic flow can promote traffic congestion suppression. By periodically sending the dynamic change information of the traffic flow, it can warn drivers in real time to pay attention to the upcoming traffic congestion[26].

Different from the usual auto-driving vehicle control strategy, the CAV control strategy design in mixed traffic needs to take the influence of CAV to traffic into account additionally. Wang designed an adaptive driving strategy for CAV to suppress disturbances, thereby ensuring string stability of mixed traffic[27]. Jiang proposed an eco-driving system for the mixed traffic situation at signalized intersections, which has a smooth effect on traffic congestion and is robust[28]. Ghiasi used the data collected by the CVs(connected vehicles) in the mixed platoon to design a CAV control strategy to indirectly control the CVs and HVs in the platoon[29], Chen proposed a framework which separates the traffic flow into ”$$1+n$$” microstructures consisting of one leading CAV and $$n$$ following HVs[30].

Recent studies have demonstrated that the mixing of automated vehicles could relieve traffic congestion[1]. Hybrid of heterogeneous vehicles has brought new opportunities for the development of transportation system. Research on mixed traffic has gradually grown up to be a hot spot[31]-[33].

3 CYBER-PHYSICAL DESCRIPTION OF HOMOGENEOUS AND MIXED TRAFFIC

In transportation system, when a vehicle is driving, it will use the information of its surrounding vehicles to make decisions. The application of communication technology and advanced computer technology in transportation has resulted in communications between vehicles in the transportation system. It can be observed that with the increasing role of communication in transportation, the cyber-physical characteristics of transportation systems become more and more obvious (Figure 1).

By mapping the entities of the physical space into the cyber space, we can get abstract nodes. There are physical and cyber based information flow between these nodes. It should be noted that the process that the driver perceives through the senses or the autonomous driving system obtains the information of other vehicles through on-board sensors is abstracted into physical information flow. The process by which the autonomous driving system obtains information about other vehicles through communication is abstracted into cyber information flow. Thus, we could use a directed graph to describe vehicles and their interactions.

We regard the vehicles in traffic as nodes in the cyber space. Suppose there are $$d$$ lanes. There are $$N^s$$ vehicles in the $$s$$-th lane ($$1\le s\le d$$). Then the vehicles in the traffic could be described by a directed graph $$\bar{\mathcal {G}}=\lbrace \bar{\mathcal {V}},\bar{\mathcal {E}}\rbrace$$, where

$${\fontsize{9}{11}{\selectfont{ \begin{eqnarray} \bar{\mathcal {V}}=\lbrace (1,1),(1,2),\ldots ,(1,N^1),(2,1),(2,2),\ldots ,(2,N^2),\ldots ,(d,N^d)\rbrace , \nonumber\\ \end{eqnarray}}}}$$(1)

is the node set, $$(s,k)$$ represents the $$k$$-th vehicle in the $$s$$-th lane. $$\bar{\mathcal {E}}\subset \bar{\mathcal {V}}\times \bar{\mathcal {V}}$$ is the edge set, the pair $$((k_1,i),(k_2,j))\in \bar{\mathcal {E}}$$ means that there is a directed edge from node $$(k_1,i)$$ to node $$(k_2,j)$$. When $$((k_1,i),(k_2,j))\in \bar{\mathcal {E}}$$, node $$(k_1,i)$$ is called as the neighbour of node $$(k_2,j)$$. $$N=\sum _{h=1}^dN^h$$.

Reorder the nodes in $$\bar{\mathcal {V}}$$ to $$\mathcal {V}=\lbrace 1,2,\ldots ,N\rbrace$$ according to the specific rules (e.g. according to the longitudinal position of the vehicle etc.), then the corresponding edge set $$\mathcal {E}\subset \mathcal {V}\times \mathcal {V}$$ could be derived from $$\bar{\mathcal {E}}$$. It is easy to proof that the graph $$\mathcal {G}=\lbrace \mathcal {V},\mathcal {E}\rbrace$$ and $$\bar{\mathcal {G}}$$ are equivalent.

When the platoon is in a stable state, the topology of the graph will not change. At this time, as the CAV penetration rate increases from 0 to 1, the graph will gradually change.

3.1 Homogeneous traffic: regular network

When a vehicle is not connected to the IoV, the driver (whether a human or an autonomous driving system) makes decisions usually based on the state of the surrounding vehicles. When we model in the form of a graph, the nodes in it are only connected to its nearly neighboring nodes. This network is known as nearest-neighbor coupled network.

When $$N$$ is large enough, the adjacency matrix $$\mathcal {A}$$ of a nearest-neighbor coupled graph will be sparse. This is because each node in the graph is connected to only a few neighbors. For example, in Figure 2a, each row of $$\mathcal {A}$$ has only 4 non-zero elements. In Figure 2b, it should be noted that an HV does not interact with all the vehicles in its adjacent lanes. When an HV is far away from a vehicle in its adjacent lane, the HV will not use the information of that vehicle. A similar situation can be seen in Figure 4b. In nearest-neighbor coupled graph composed of HVs, the interactions are all in physical space.

FIGURE 1

Cyber-physical view of vehicles

FIGURE 2

Nearest-neighbor coupled network

FIGURE 3

Globally coupled network

FIGURE 4

Small-world network

When a vehicle is connected to the IoV, it can obtain information about other vehicles in the network. Its perception range is also wider than HV. Furthermore, if it is self-driving, theoretically it can use the information of all vehicles in the network to take decisions, and their cyber-physical interaction become more frequent and more complex. Thus, when each vehicle in traffic is CAV, then the network becomes a globally coupled network (Figure 3).

The adjacency matrix $$\mathcal {A}$$ of a globally coupled graph satisfies: each row has one and only one zero element, which is the element on the diagonal. In globally coupled network composed of CAVs, the physical and cyber interactions are both exist.

It is pointed out that these two diagrams are dynamic. This happens because there are still lane changes, merging, and diversion in traffic. However, when the vehicles in traffic consist of homogeneous vehicles, the type of the dynamic network is kept unchanged.

3.2 Mixed traffic: From small-world to scale-free

Small-world model is first proposed by Watts and Strogtz in 1998[34]. It is the intermediate result of the transition from a completely regular network to a completely random graph. Small-world networks could be discovered in reality. For example, people a person often contact are usually friends, relatives, colleagues etc., but there may also be a small number of netizens who are far away in foreign countries. In this way, this person's social network has the characteristics of small-world.

When the proportion of CAV in mixed traffic is not large, the network obtained also meets the characteristics of the small-world: each node is connected to its neighbor nodes first, but there are also a small number of nodes (CAV) connected to other nodes (other CAVs) far away. When these relationships in the physical space are mapped to the cyber space, Figure 4b can be obtained.

Note that the edge connecting HV and CAV is usually one-way, from HV to CAV. This is because the perception range of CAV is usually wider than that of HV, so CAV can adjust its own control strategy according to HV's behavior, while HV may ignore CAVs far away from it.

In mixed traffic, if each vehicle is HV or CAV at random, it is more appropriate to use the NW small-world model to describe. NW small-world is a small-world model obtained by randomly adding edges in the nearest-neighbor coupled graph (Figure 4a)[35]. The random mixing of CAV in mixed traffic can be regarded as the process of adding nodes randomly and adding edges in the graph. Through this process, when the proportion of CAV is not high, we get the NW small world model (Figure 4b).

The scale-free model is proposed by Barabási and Albert in 1999[36]. It has two essential characteristics: 1. Growth, that is, the scale of the network is constantly expanding; 2. Preferential attachment, that is, new nodes are more inclined to connect with those with more connections.

In Figure 5, one-way arrows pointing from HV to CAV could be found. This is because autopilot systems can usually use radars, cameras etc. to perceive the surrounding environment, and the perception range is generally wider than that of humans. Therefore, it may happen that one CAV uses the information of an HV in the adjacent lane, but this HV does not use the information of the CAV, thus generating such a one-way arrow.

FIGURE 5

Scale-free in mixed traffic

In fact, the transportation network is not a static network. The number of vehicles in traffic will increase or decrease: this leads to changes in the number of nodes in the network; and vehicles may take measures such as lane changes, merging, and diversion in traffic: this leads to changes in the network topology. When a vehicle enters traffic, besides connecting with the surrounding vehicles, it also preferentially connects to other CAVs in the traffic if its a CAV, no matter how far the actual distance is (Figure 5). CAV is also a node with higher connectivity in mixed traffic. In this way, when the scale of mixed traffic grows, both the growth characteristics and the preferential attachment characteristics are satisfied. This satisfies the scale-free feature.

To sum up, it can be summarized as follows: when all vehicles in traffic are HVs, the traffic network could be considered as nearest neighbor coupled network. When the number of CAVs in traffic is not large, the traffic network could be considered as NW small-world network. As the percentage of CAVs in traffic increases, the network gradually evolves into a scale-free network. Finally, when all vehicles in traffic are CAVs, the traffic network could be seen as globally coupled network.

4 CPS-BASED PINNING APPROACH

The basic idea of pinning control is to control the complex network by controlling some special nodes in the network[18]. Research of pinning control strategy is especially focused on small-world network and scale-free network. In Section 3, we have illustrated that the complex network formed by the projection of mixed traffic entities has these two characteristics. Pinning control is essentially synchronous control, and the network topology plays an important part in determining the dynamic characteristics of the network. The discovery of the small-world and scale-free characteristics of complex networks has led people to draw attention to the relationship between network topology and network synchronization behavior[37]-[39].

By controlling the CAVs in mixed traffic to indirectly influence HVs to achieve the purpose of optimizing traffic, is the basic idea of mixed traffic pinning approach. It has two aspects: 1. How to design the control strategy of the pinning nodes (CAV); 2. How to choose the pinning nodes (how to adjust the distribution of CAV in mixed traffic) to optimize the traffic network.

4.1 Pinning nodes control strategy

We use the second-order multi-agent system to describe the kinematic behavior of the vehicle. Vehicles in mixed traffic are considered in the directed graph $$\bar{\mathcal {G}}$$ first. To simplify the discussion, we assume that the leader in each lane follows one virtual leader. Besides, the CAVs in the same lane follow the same virtual leader. The introduction of the virtual leaders is to more easily express the current road restrictions on vehicle driving (such as the speed limit). Since CAVs and the leading vehicle in each lane can obtain information on roadside signs and road conditions ahead, we made this assumption. The virtual leader of lane $$k$$ ($$1\le k\le d$$) is given by:

$$\begin{equation} \dot{\bm {x}}_0^k(t)=\bm {f}(t,\bm {x}_0^k(t)), \end{equation}$$(2)

and the followers are described by:

$$\begin{eqnarray} \dot{\bm {x}}_i^k(t) &=&\bm {f}(t, \bm {x}_i^k(t))+\sum _{h=1}^{d}\sum _{j=1}^{N^h}a_{ij}^{kh} \Gamma (\bm {x}_j^h(t)-\bm {x}_i^k(t))+\bm {u}_i^k(t-\tau ),\nonumber\\ a_{\min } &\le& \Vert \bm {u}_i^k\Vert \le a_{\max }, v_{\min }\le \Vert \bm {v}_i^k\Vert \le v_{\max }. \end{eqnarray}$$(3)

in which $$\bm {s}_i^k=[x_i^k-x_i^{0,k},y_i^k-y_i^{0,k}]^T$$, $$\bm {v}_i^k=[v_{ix}^k,v_{iy}^k]^T$$ are the position and velocity states of the $$i$$-th vehicle in lane $$k$$, respectively. $$x^k$$ is the longitudinal position, $$y^k$$ is the lateral position, $$x_i^{0,k}$$ is the desired longitudinal gap between the $$i$$-th vehicle and the virtual leader in lane $$k$$, $$y_i^{0,k}$$ is the desired lateral gap between the $$i$$-th vehicle and the virtual leader in lane $$k$$. $$\bm {x}_i^k\triangleq [(\bm {s}_i^k)^T,(\bm {v}_i^k)^T]^T\in \mathbb {R}^4$$. $$\bm {F}_{a,i}^k(t)=[F_{ax,i}^{k},F_{ay,i}^{k}]^T$$, $$\bm {F}_{l,i}^k(t)=[F_{lx,i}^k,F_{ly,i}^k]^T$$. $$\bm {F}_{r,i}^k(t)\triangleq \sum _{h=1}^{d}\sum _{j=1}^{N^h}a_{ij}^{kh} \Gamma (\bm {x}_j^h(t)-\bm {x}_i^k(t))$$. $$\bm {f}(t, \bm {x}_i^k(t))\triangleq \bm {F}_{a,i}^k(t)+\bm {F}_{l,i}^k(t)$$. $$\bm {u}_i^k(t)$$ is the pinning control input. $$\Gamma$$ is the inner coupling matrix(usually a positive definite matrix), which represents the coupling relationship between the state variables of each node. $$\tau$$ is the information delay, $$a_{ij}^{kh}=1$$ means node $$(k,i)$$ could obtain information from $$(h,j)$$, $$a_{ij}^{kh}=0$$ means node $$(k,i)$$ could not obtain information from $$(h,j)$$. $$a_{ii}^{kk}=0$$.

The definitions of these three forces are:

• $$\bm {F}_a$$ reflects the tendency of drivers to accelerate to their desired speed;
• $$\bm {F}_l$$ aims to keep vehicles close to the center of the lane;
• $$\bm {F}_r$$ represents the influence of the vehicles in the adjacent lane.

Usually, the information used by $$F_a$$, $$F_r$$ and $$F_l$$ comes from physical space. While a vehicle is a CAV, then it could obtain information from other CAVs, then the information used by $$F_r$$ might come from cyber space. Moreover, the information used by the pinning control input comes from cyber space, too. Let $$\bar{\mathcal {V}}_{\text{pin}}$$ as the set of CAVs in $$\bar{\mathcal {G}}$$, denote $$g_i^k\ge 0$$ as the pinning gain, where

$$\begin{equation} {\left\lbrace \begin{aligned} &g_i^k(t)>0, (k,i)\in \bar{\mathcal {V}}_{\text{pin}}\text{ and }h_i^k\ge h_{\text{safe}}\\ &g_i^k(t)=0, \text{otherwise} \end{aligned} \right.}, \end{equation}$$(4)

$$i\in \bar{\mathcal {V}}$$, $$h_i^k$$ is the distance between node $$(i,k)$$ and its preceding vehicle in the same lane, $$h_{\text{safe}}$$ is the safe distance. Then the pinning control input is

$$\begin{equation} \bm {u}_i^k(t)=g_i^k\Gamma (\bm {x}_0^k(t)-\bm {x}_i^k(t)). \end{equation}$$(5)

The assumptions of the model are given as follows:

• a. Communication conditions are ideal, which means that there will be no packet loss or communication interruption in the communication between CAVs;
• b. To simplify the discussion, the information delay of all CAVs is the same.

4.2 Stability analysis of mixed traffic

Assuming that the leader in each lane follows the same virtual leader. Besides, the CAVs in mixed traffic also follow the same virtual leader. Vehicles in mixed traffic are considered in the directed graph $$\mathcal {G}$$. Then the virtual leader is given by:

$$\begin{equation} \dot{\bm {x}}_0(t)=\bm {f}(t,\bm {x}_0(t)), \end{equation}$$(6)

and the followers are described by:

$$\begin{equation} \begin{aligned} \dot{\bm {x}}_i&=\bm {f}(t, \bm {x}_i(t))+\sum _{j=1}^{N}a_{ij}\Gamma (\bm {x}_j(t)-\bm {x}_i(t))+\bm {u}_i(t-\tau ),\\ & a_{\min }\le \Vert \bm {u}_i\Vert \le a_{\max }, v_{\min }\le \Vert \bm {v}_i\Vert \le v_{\max }, \end{aligned} \end{equation}$$(7)

in which $$\bm {s}_i=[x_i-x_i^0,y_i-y_i^0]^T$$, $$\bm {v}_i=[v_i^x,v_i^y]^T$$ are the position and velocity states of the $$i$$-th vehicle, respectively. $$x$$ is the longitudinal position, $$y$$ is the lateral position, $$x_i^0$$ is the desired longitudinal gap between the $$i$$-th vehicle and the virtual leader, $$y_i^0$$ is the desired lateral gap between the $$i$$-th vehicle and the virtual leader. $$\bm {x}_i\triangleq [\bm {s}_i^T,\bm {v}_i^T]^T\in \mathbb {R}^4$$. $$\bm {F}_i^a(t)=[F_i^{ax},F_i^{ay}]^T$$ reflects the tendency of drivers to accelerate to their desired speed, $$\bm {F}_i^l(t)=[F_i^{lx},F_i^{ly}]^T$$ aims to keep vehicles close to the center of the lane(see in Figure 6). $$\bm {F}_i^r(t)\triangleq \sum _{j=1}^{N}a_{ij}\Gamma (\bm {x}_j(t)-\bm {x}_i(t))$$. $$\bm {f}(t, \bm {x}_i(t))\triangleq \bm {F}_i^a(t)+\bm {F}_i^l(t)$$. $$\bm {u}_i(t)$$ is the pinning control input. $$\Gamma$$ is the inner coupling matrix, which represents the coupling relationship between the state variables of each node. $$a_{ij}=1$$ means node $$i$$ could obtain information from $$j$$, $$a_{ij}=0$$ means node $$i$$ could not obtain information from $$j$$. $$a_{ii}=0$$.

FIGURE 6

Simplified schematics of different forces

Let $$\mathcal {V}_{\text{pin}}$$ as the set of CAVs in $$\mathcal {G}$$, denote $$g_i\ge 0$$ as the pinning gain, where

$$\begin{equation} {\left\lbrace \begin{aligned} &g_i(t)>0,\quad i\in \mathcal {V}_{\text{pin}}\text{ and }h_i\ge h_{\text{safe}}\\[-4pt] &g_i(t)=0,\quad \text{otherwise} \end{aligned} \right.}, \end{equation}$$(8)

$$i\in \mathcal {V}$$, $$h_i$$ is the distance between node $$i$$ and its preceding vehicle in the same lane. Then the pinning control input is

$$\begin{equation} \bm {u}_i(t)=g_i\Gamma (\bm {x}_0(t)-\bm {x}_i(t)). \end{equation}$$(9)

It is obvious that (7) and (3) are the same system, the difference is only in the serial numbers of the vehicles in the traffic. When trying to achieve mixed traffic consistency, it is better to use (7) to describe the vehicles.

Define $$\bm {e}_i(t)=\bm {x}_i(t)-\bm {x}_0(t)$$, then

$$\begin{equation} \bm {u}_i(t)=-g_i\Gamma \bm {e}_i(t), \end{equation}$$(10)

and (7) could be written as:

$$\begin{equation} \dot{\bm {e}}_i=\bm {F}_i(t, \bm {x}_i(t),\bm {x}_0(t))+\sum _{j=1}^{N}a_{ij}\Gamma (\bm {e}_j(t)-\bm {e}_i(t))+\bm {u}_i(t-\tau ), \end{equation}$$(11)

where $$\bm {F}_i(t, \bm {x}_i(t),\bm {x}_0(t))=\bm {f}(t, \bm {x}_i(t))-\bm {f}(t, \bm {x}_0(t))$$. Let

$$\begin{eqnarray} && \bm {e}(t) = \left[\bm {e}_1^T(t),\bm {e}_2^T(t),\ldots ,\bm {e}_N^T(t)\right]^T,\nonumber\\ &&\quad \bm {F}(t, \bm {x}(t),\bm {x}_0(t))=\left[\bm {F}_1^T(t, \bm {x}_1(t),\bm {x}_0(t)),\right.\nonumber\\ &&\left.\quad \bm {F}_2^T(t, \bm {x}_2(t),\bm {x}_0(t)),\ldots ,\bm {F}_N(t, \bm {x}_N^T(t),\bm {x}_0(t))\right]^T, \end{eqnarray}$$(12)

and combine (10), (11) and (12), then (11) could be rewritten as

$$\begin{equation} \dot{\bm {e}}(t)=\bm {F}(t,\bm {x}(t),\bm {x}_0(t))+(\mathcal {A}-\mathcal {L})\Gamma \bm {e}(t)-G\Gamma \bm {e}(t-\tau ), \end{equation}$$(13)

where

$$\begin{equation} \begin{aligned} \mathcal {L}&=\text{diag}{\left\lbrace \sum _{j=1}^Na_{1j},\sum _{j=1}^Na_{2j},\ldots ,\sum _{j=1}^Na_{Nj}\right\rbrace} ,\\ G&=\text{diag}\lbrace g_1,g_2,\ldots ,g_N\rbrace , \mathcal {A}=(a_{ij})_{N\times N}. \end{aligned} \end{equation}$$(14)

It is necessary and reasonable to make the following assumption:

Assumption For system (7) with pinning control input (9) and isolated node(6), the function $$\bm {f}(\cdot )$$ satisfies:

$${\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray} {\left(\bm {f}(x)-\bm {f}(y)-U_1(x-y)\right)}^T{\left(\bm {f}(x)-\bm {f}(y)-U_2(x-y)\right)}\le 0, \nonumber\\ \end{eqnarray}}}}$$(15)

for $$\forall x,y\in \mathbb {R}^4$$, where $$U_1$$ and $$U_2$$ are constant symmetric matrices with appropriate dimensions.

Remark 1.This assumption is weaker than Lipschitz's continuity, which can cover more cases. In practice, the position of the vehicle will not change suddenly. Moreover, unless there is an emergency, sudden changes in velocity and acceleration rarely occur. Therefore, we made this assumption.

The following lemma is helpful to the conclusion of this paper:

Lemma 1.[40] For any constant matrix $$Z>0$$, scalar $$\tau _2>\tau _1\ge 0$$, $$\alpha \ne 0$$ and vector function $$x:[t-\tau _2,t-\tau _1]\rightarrow \mathbb {R}^n$$, one has

$$\begin{eqnarray} && {e^{\alpha \tau _2}-e^{\alpha \tau _1}\over \alpha }\int _{t-\tau _2}^{t-\tau _1}e^{\alpha (s-t)}x^T(s)Zx(s)\text{d}s\nonumber\\ &&\quad \ge\, \int _{t-\tau _2}^{t-\tau _1}x^T(s)\text{d}sZ\int _{t-\tau _2}^{t-\tau _1}x(s)\text{d}s. \end{eqnarray}$$(16)

Then the main conclusion of this subsection is as follows:

Theorem 1.Under the assumptions mentioned above, and let $$\tau$$ be given. For system (13), the Lyapunov-Krasovskii function $$V(t)$$ is defined as follows:

$$\begin{equation} V(t)=V_1(t)+V_2(t)+V_3(t), \end{equation}$$(17)

where

$$\begin{equation} \begin{aligned} V_1(t)&=\bm {e}^T(t)P\bm {e}(t),\\ V_2(t)&=\int _{t-\tau }^te^{\alpha (s-t)}\bm {e}^T(s)Q\bm {e}(s)\text{d}s,\\ V_3(t)&=\int _{-\tau }^0\int _{t+\theta }^te^{\alpha (s-t)}\dot{\bm {e}}^T(s)R\dot{\bm {e}}(s)\text{d}s\text{d}\theta . \end{aligned} \end{equation}$$(18)

For given constant $$\alpha$$, if there exists positive definite matrices $$P,Q,R,X_1,X_2$$, such that

$$\begin{equation} \Phi ={\left[\begin{matrix} \Phi _{11} &\quad \Phi _{12} &\quad \Phi _{13} &\quad \Phi _{14}\\[5pt] * &\quad \Phi _{22} &\quad \Phi _{23} &\quad \Phi _{24}\\[5pt] * &\quad * &\quad \Phi _{33} &\quad \Phi _{34}\\[5pt] * &\quad * &\quad * &\quad \Phi _{44} \end{matrix}\right]}<0, \end{equation}$$(19)

where

$$\begin{eqnarray} \begin{aligned} \Phi _{11}=&\alpha P+Q-{\alpha \over e^{\alpha \tau }-1}R+X_1(\mathcal {A}\\ &-\,\mathcal {L})\Gamma +\Gamma ^T(\mathcal {A}-\mathcal {L})^TX_1-\bar{U}_1\\ \Phi _{12}=&{\alpha \over e^{\alpha \tau }-1}R+{1\over 2}(X_1G\Gamma +\Gamma ^TG^TX_1), \Phi _{13}=X_1-\bar{U}_2,\\ \Phi _{14}=&P-X_1+{1\over 2}(X_2(\mathcal {A}-\mathcal {L})\Gamma +\Gamma ^T(\mathcal {A}-\mathcal {L})^TX_2),\\ \Phi _{22}=&-e^{\alpha \tau }Q-{\alpha \over e^{\alpha \tau }-1}R, \Phi _{24}=-{1\over 2}(X_2G\Gamma +\Gamma ^TG^TX_2),\\ \Phi _{33}=&-I_{4N\times 4N}, \Phi _{34}=X_2,\Phi _{44}=\tau R-2X_2, \end{aligned} \end{eqnarray}$$(20)

the remaining block matrices are all zero matrices with appropriate dimensions. Note that

$$\begin{equation} \begin{aligned} \bar{U}_1=&{(I_{N\times N}\otimes U_1)^T(I_{N\times N}\otimes U_2)\over 2}\\ & +\,{(I_{N\times N}\otimes U_2)^T(I_{N\times N}\otimes U_1)\over 2},\\ \bar{U}_2=&-{(I_{N\times N}\otimes U_1)^T+(I_{N\times N}\otimes U_2)^T\over 2}, \end{aligned} \end{equation}$$(21)

then the following inequality holds:

$$\begin{equation} V(t)\le e^{-\alpha t}V(0). \end{equation}$$(22)

Proof.Taking the derivative of $$t$$ in $$V(t)$$, it could be obtained that

$$\begin{eqnarray} \begin{aligned} \dot{V}_1(t)&=2\bm {e}^T(t)P\dot{\bm {e}}(t),\\ \dot{V}_2(t)&=\bm {e}^T(t)Q\bm {e}(t)-e^{-\alpha \tau }\bm {e}^T(t-\tau )Q\bm {e}(t-\tau )-\alpha V_2(t),\\ \dot{V}_3(t)&=\tau \dot{\bm {e}}^T(t)R\dot{\bm {e}}(t)-\int _{t-\tau }^te^{\alpha (s-t)}\dot{\bm {e}}^T(s)R\dot{\bm {e}}(s)\text{d}s-\alpha V_3(t). \end{aligned} \nonumber\\ \end{eqnarray}$$(23)

According to Lemma 1, it holds that

$$\begin{eqnarray} && -\int _{t-\tau }^te^{\alpha (s-t)}\dot{\bm {e}}^T(s)R\dot{\bm {e}}(s)\text{d}s\nonumber\\ &&\quad\le\, {\alpha \over 1-e^{\alpha \tau }}{\left(\bm {e}^T(t)-\bm {e}^T(t-\tau )\right)} R{\left(\bm {e}(t)-\bm {e}(t-\tau )\right)},\qquad \end{eqnarray}$$(24)

then

$$\begin{eqnarray} \dot{V}_3(t) &\le& \tau \dot{\bm {e}}^T(t)R\dot{\bm {e}}(t)+ {\alpha \over 1-e^{\alpha \tau }}\nonumber\\ &&\times\, {\left(\bm {e}^T(t) -\bm {e}^T(t-\tau )\right)}R{\left(\bm {e}(t)-\bm {e}(t-\tau )\right)}-\alpha V_3(t). \nonumber\\ \end{eqnarray}$$(25)

According to assumption, it could be proofed that

$${\fontsize{9.5}{11.5}{\selectfont{ \begin{eqnarray} {\left[\begin{matrix} \bm {e}(t)\\[5pt] \bm {F}(t,\bm {x}(t),\bm {x}_0(t)) \end{matrix}\right]}^T {\left[\begin{matrix} \bar{U}_1 &\quad \bar{U}_2\\[5pt] * &\quad I_{4N\times 4N} \end{matrix}\right]} {\left[\begin{matrix} \bm {e}(t)\\[5pt] \bm {F}(t,\bm {x}(t),\bm {x}_0(t)) \end{matrix}\right]}\le 0.\nonumber\\ \end{eqnarray}}}}$$(26)

$$\Box$$

Moreover, according to (13), for given matrix $$X_1$$, $$X_2$$, we have

$${\fontsize{9}{11}{\selectfont{ \begin{eqnarray} && 2{\left(\bm {e}^T(t)X_1+\dot{\bm {e}}^T(t)X_2\right)} \left(-\dot{\bm {e}}(t)+\bm {F}(t,\bm {x}(t),\bm {x}_0(t))\right.\nonumber\\ &&\left.\quad +\,(\mathcal {A}-\mathcal {L})\Gamma \bm {e}(t)-G\Gamma \bm {e}(t-\tau )\right)=0. \end{eqnarray}}}}$$(27)

Let $$\bm {e}(t-\tau )=\bm {e}_{\tau }(t)$$, $$\zeta =[\bm {e}^T,\bm {e}_{\tau }^T,\bm {F}^T,\dot{\bm {e}}^T]^T$$, combining (23), (25), (26) and (27) yields

$$\begin{equation} \begin{aligned} \dot{V}(t)+\alpha V(t)\le &\dot{V}(t)+\alpha V(t)-\bm {e}^T\bar{U}_1\bm {e}-2\bm {e}^T\bar{U}_2\bm {F}-\bm {F}^T\bm {F}\\ =\,&\bm {e}^T\Phi _{11}\bm {e}+2\bm {e}^T\Phi _{12}\bm {e}_{\tau }+2\bm {e}^T\Phi _{13}\bm {F}+2\bm {e}^T\Phi _{14}\dot{\bm {e}} \\ &+\,\bm {e}_{\tau }^T\Phi _{22}\bm {e}_{\tau } +2\bm {e}_{\tau }^T\Phi _{23}\bm {F}+2\bm {e}_{\tau }^T\Phi _{24}\dot{\bm {e}}\\ &+\,2\bm {F}^T\Phi _{34}\dot{\bm {e}}+\dot{\bm {e}}^T\Phi _{44}\dot{\bm {e}}\\ =\,&\zeta ^T\Phi \zeta . \end{aligned} \end{equation}$$(28)

It is obvious that when $$\Phi <0$$, $$\zeta ^T\Phi \zeta \le 0$$. Therefore,

$$\begin{equation} V(t)\le e^{-\alpha t}V(0). \end{equation}$$(29)

Remark 2.Theorem 1 shows that in order to achieve the mixed traffic consistency, in addition to adjusting the coupling strength $$a_{ij}$$ between vehicles, it is necessary to design an appropriate pinning gain $$g_i$$. The former is common in the existing CAV control algorithm research, while the latter is a unique problem of mixed traffic. However, since the transportation network is not a static network, the adjacency matrix $$\mathcal {A}$$ is usually dynamic. We can dynamically adjust $$a_{ij}$$ and $$g_i$$ according to the actual situation to meet the needs, but the amount of calculation will become extremely large. The numerical complexity of Theorem 1 is $$O(N^3)$$, where $$N$$ is the number of vehicles in the traffic. There are usually hundreds of vehicles in traffic at the same time, and each vehicle needs to make decisions and implement control strategies in a short time. In practical applications, it is almost impossible to determine the pinning gain $$g_i$$ by solving linear matrix inequalities. The pinning nodes control strategy must be determined earlier.Therefore, in this paper, we assume that these parameters have been determined earlier.

String stability refers to non-amplifying propagation of perturbation on state (speed, acceleration etc.), error (spacing error, time gap error etc.), or control signal through a string of vehicles. This stability criterion guarantees that any oscillation will be attenuated when propagating upstream[41]. The relationship between exponentially stable and string stable is as follows.

Theorem 2.[42]. If the following conditions are satisfied:

• 1. $$h$$ is locally Lipscitz in $$\Omega$$ in its arguments, that is, when $$y_i\in \Omega$$, $$z_i\in \Omega$$, $$i=\lbrace 1,2,\ldots ,r\rbrace$$,
  $$\begin{equation} \begin{aligned} &\Vert h(y_1,\ldots ,y_r)-h(z_1,\ldots ,z_r)\Vert \\ & \le l_1\Vert y_1-z_1\Vert +\cdots +l_r\Vert y_r-z_r\Vert . \end{aligned} \end{equation}$$(30)
• 2. The origin of $$\dot{x}=h(x,0,\ldots ,0)$$ is locally exponentially stable in $$\Omega$$.

Then for sufficiently small $$l_i,i=2,\ldots ,r$$, the interconnected system is locally string stable in $$\Omega$$.

Remark 3.Theorem 2 gives the conclusion that exponential stable can derive string stable. Therefore, if Theorem 1 holds, then the system is string stable.

4.3 Influences of adding CAVs

In this subsection, some conventions about symbols must be proposed. First, the following lemma is proposed:

Lemma 2.[44]. If Assumption satisfies, then (9) globally and asymptotically synchronizes with the isolated node (6) if

$$\begin{equation} \mathcal {J}(L+G)\triangleq \min _{1\le i\le N}\text{Re}(\lambda _i(L+G))>{\alpha \over m}, \end{equation}$$(31)

where the corresponding Laplacian matrix $$L=(l_{ij})_{N\times N}$$ is defined as:

$$\begin{equation} \begin{aligned} &l_{ij}=-a_{ij}\le 0, i\ne j,\\ &l_{ii}=\sum _{j=1,j\ne i}^{N}a_{ij}, i=1,2,\ldots ,n. \end{aligned} \end{equation}$$(32)

$$\text{Re}(\lambda _i(L+G))$$ is the real part of the $$i$$-th eigenvalue of the matrix $$L+G$$, the constant $$m$$ is known as the coupling strength.

Remark 4.Lemma shows that, to control a complex network, a good pinning strategy is to select a pinning control matrix $$G$$ such that $$\text{Re}(\lambda _i(L+G))$$ is as large as possible and the coupling strength $$m$$ is as small as possible[45]. The determination of $$G$$ includes two parts: 1. Design the pinning control algorithm to make the value of $$g_i$$ meet the requirements; 2. Select pinning nodes to make the arrangement of $$g_i$$ meet the requirements.

Many conclusions in complex networks can be used in mixed traffic. For example, the pinning control for multi-weighted complex dynamical networks with fixed and switching topologies could be found in[46], and the pinning control for leader-following second-order time-delay systems could be found in [47] etc. These methods have a common feature: to ensure that $$g_i$$ is large enough. First we introduce Theorem 1 to illustrate the mixing of CAVs helps traffic consistency.

Definition 1.A matrix $$A=(a_{ij})_{N\times N}$$ is called $$M-matrix$$ if

$$\begin{equation} a_{ij}\le 0, i\ne j, i,j=1,2,\ldots ,N. \end{equation}$$(33)

Definition 2.A matrix $$A=(a_{ij})_{N\times N}$$ is called $$non-negative matrix$$ if

$$\begin{equation} a_{ij}\ge 0,\forall i,j, \end{equation}$$(34)

and we note that $$A\succcurlyeq 0$$. Moreover, if

$$\begin{equation} a_{ij}>0,\forall i,j, \end{equation}$$(35)

then $$A$$ is a $$positive matrix$$, and we note that $$A\succ 0$$. We also note that $$A\succcurlyeq (\succ )B$$ if $$A-B\succcurlyeq (\succ )0$$.

Theorem 3.Let $$G_i=\text{diag}(0,\dots ,g_i,\ldots ,0), g_i>0$$, then $$\mathcal {J}(L+G_i)\ge \mathcal {J}(L)$$.

Proof.After defining $$L$$, it is easy to prove according to Gerschgorin's theorem that the eigenvalues of $$L+G_i$$ all have non-negative real parts, and at least one eigenvalue has a positive real part. And $$L$$ is an M-matrix. According to [48], the matrix $$L+G_i$$ could be expressed as $$L+G_i=s_iI_N-\bar{B}_i$$, where $$\bar{B}_i\succcurlyeq 0$$, $$s_i>\rho (B_i)$$, $$\rho (B_i)$$ is the spectral radius of matrix $$B_i$$. And the matrix $$L$$ could be expressed as $$L=\bar{s}I_N-\bar{B}$$, where $$B\succcurlyeq 0$$, $$\bar{s}>\rho (B)$$.$$\Box$$

Let $$s=\max \lbrace \bar{s},s_i\rbrace$$, we have $$L=sI_N-B$$, $$L+G_i=sI_N-B_i$$, where

$$\begin{equation} B={\left[\begin{matrix} s-l_{11} &\quad -l_{12} &\quad \cdots &\quad -l_{1N}\\[7pt] -l_{21} &\quad s-l_{22} &\quad \cdots &\quad -l_{2N}\\[7pt] \vdots &\quad \vdots &\quad &\quad \vdots \\[7pt] -l_{N1} &\quad -l_{N2} &\quad \cdots &\quad s-l_{NN} \end{matrix}\right]}, \end{equation}$$(36)

$$\begin{equation} B_i={\left[\begin{matrix} s-l_{11} &\quad \cdots &\quad -l_{1i} &\quad \cdots &\quad -l_{1N}\\[7pt] \vdots &\quad &\quad \vdots &\quad &\quad \vdots \\[7pt] -l_{i1} &\quad \cdots &\quad s-l_{ii}-g_i &\quad \cdots &\quad -l_{iN}\\[7pt] \vdots &\quad &\quad \vdots &\quad &\quad \vdots \\[7pt] -l_{N1} &\quad \cdots &\quad -l_{Ni} &\quad \cdots &\quad s-l_{NN}\\[7pt] \end{matrix}\right]}. \end{equation}$$(37)

It is obvious that $$B\succcurlyeq B_i$$ and $$B_i\ne B$$, then we have $$\rho (B_i)\le \rho (B)$$, and $$s-\rho (B_i)\ge s-\rho (B)$$[45]. Then we have

$$\begin{equation} Lx=\lambda x\Longleftrightarrow (sI_N-B)x=\lambda x\Longleftrightarrow Bx=(s-\lambda )x. \end{equation}$$(38)

Since $$B$$ is a non-negative matrix, $$\rho (B)$$ is an eigenvalue of $$B$$[48]. It could be concluded from (38) that there exist an eigenvalue of $$A$$(noted as $$\lambda _k$$) that $$\lambda _k=s-\rho (B)$$.

From (38), we can see that the eigenvalues of $$L$$ and $$B$$ have a one-to-one correspondence. If the eigenvalue of $$L$$ is $$\lambda _i$$, the eigenvalue corresponding to $$B$$ is $$\gamma _i(i=1,2,\ldots ,N$$), we have:

$$\begin{equation} \text{Re}(\gamma _i)\le |\gamma _i|\le \rho (B)=\gamma _k, \end{equation}$$(39)

and

$$\begin{equation} \text{Re}(\gamma _i)=\text{Re}(s-\lambda _i)=s-\text{Re}(\lambda _i). \end{equation}$$(40)

Combining (39) and (40), we have

$$\begin{equation} \text{Re}(\lambda _i)\ge s-\rho (B)=\text{Re}(\lambda _k). \end{equation}$$(41)

Thus we have $$\mathcal {J}(L)=\lambda _i=s-\rho (B)$$. Similarly, we can get $$\mathcal {J}(L+G_i)=s-\rho (B_i)$$. Then $$\mathcal {J}(L+G_i)\ge \mathcal {J}(L)$$.

Remark 5.According to Theorem 1, not only $$\mathcal {J}(L+G_i)\ge \mathcal {J}(L)$$, the following conclusions can also be drawn through similar derivations:

• (1) $$\mathcal {J}(L+G_i+G_j)\ge \mathcal {J}(L+G_i), i\ne j$$;
• (2) $$\mathcal {J}(L+\bar{G}_i)\ge \mathcal {J}(L+G_i), \bar{g}_i\ge g_i$$.

Thereby, the controllability of the weakly-connected network can be increased by either adding a pinning node or increasing one of the pinning gains. In mixed traffic, these two methods correspond to: (1) increasing the amount of CAVs; (2) adjusting the control algorithm of CAVs to have a greater pinning gain. Increasing the pinning gain may cause the CAV itself to lose stability, therefore, global consistency and individual stability should be considered comprehensively when designing the CAV control algorithm.

In the last two paragraphs of this subsection, the issue of pinning node selection will be briefly discussed. There are two pinning nodes selection strategies: random pinning and specific pinning. [43] shows specific pinning have better control results than random pinning. Current existing pinning node selection strategy[49]-[50] is only applicable to the situation where the network topology remains unchanged, and therefore is not applicable in the mixed traffic scenario. Since the number of vehicles in actual traffic usually very large, the computational cost of giving the optimal distribution of CAVs might be too large. Moreover, it is difficult to control the spatial distribution in real traffic environment. Therefore, it is preferable to give a sub-optimal node selection method with much less computational overhead and realizable. In [45], the authors proposed that a more effective strategy is to distribute the pinning nodes evenly throughout the network.

Inspired by this conclusion, if the CAVs as evenly distribute as possible, the effect of the pinning control strategy might be improved. At present, specific implementation at least has two ways. The first is to send a signal to some selected vehicles in the platoon to switch them from manual driving mode to automated driving mode. This method requires sufficient number of vehicles with automated driving functions in the platoon. The second is to artificially mix automated vehicles in the platoon, for example, at the entrance of a certain road, an automated vehicle is issued every time a certain number of vehicles pass or a certain time passes. Through these methods that can be taken, the spatial distribution strategy has the possibility of realization, rather than just theoretical analysis. The numerical simulation in [45] supports this conjecture. In the following examples, we will establish the effectiveness of this strategy through numerical simulation.

5 A THREE-LANES MIXED TRAFFIC EXAMPLE

The two-dimensional car-following model was proposed to describe the influence of surrounding vehicles[51]-[53]. It is necessary to make some simplifying assumptions:

• there exists and only exists one virtual leader;
• the longitudinal and lateral movements of the vehicle are considered as independent with each other;
• the influence of $$\bm {F}^a(t)$$ is mostly longitudinal, thus we ignore its lateral influence, and let $$F^{ay}=0$$;
• the influence of $$\bm {F}^l(t)$$ is mostly lateral, thus we ignore its longitudinal influence, and let $$F^{lx}=0$$.

Under these assumptions, we have:

$$\begin{equation} \begin{aligned} {\left[\begin{matrix} a_x(t)\\[7pt] a_y(t) \end{matrix}\right]}={\left[\begin{matrix} F^{ax}+F^{rx}\\[7pt] F^{ry}+F^{ly} \end{matrix}\right]}, \end{aligned} \end{equation}$$(42)

where $$\dot{\bm {v}}=[a_x,a_y]^T$$.

5.1 CAV

Since there are not many studies on the two-dimensional car-following model, the specific forms of $$\bm {F}^a(t)$$, $$\bm {F}^r(t)$$ and $$\bm {F}^l(t)$$ in this paper mostly refer to [51]. When a vehicle is a CAV, then we have

$$\begin{equation} F^{ax}=(V-v^x(t))c_1. \end{equation}$$(43)

$$V$$ is the desired speed, $$v^x$$ is the longitudinal speed of the vehicle, and $$c_1$$ is the sensitivity to own speed. $$c_1$$ could also be regarded as the pinning gain.

$$\begin{equation} F^{ly}=-v^y(t)k_1-(y(t)-y^l)k_2. \end{equation}$$(44)

$$v^y$$ is the lateral speed of the vehicle, $$y$$ is the lateral position of the vehicle, $$y^l$$ is the lateral position of the center of lane $$l$$, $$k_1$$ is the sensitivity to lateral speed, and $$k_2$$ is the sensitivity to distance to lane's center.

$$\begin{equation} \bm {F}^r(t)=\sum _{k\in \mathcal {N}}\bm {F}_k^r(t). \end{equation}$$(45)

$$\mathcal {N}\subset \mathcal {V}$$ is the index set of vehicles that will influence the vehicle, and

$$\begin{equation} \bm {F}_k^r(t)=\bm {f}_k(\bm {v}_k(t),\bm {v}(t),\bm {s}_k(t),\bm {s}(t)). \end{equation}$$(46)

Since the $$k$$th vehicle will affect the vehicle only when it is close enough to the vehicle, we define $$\mathcal {N}$$ as $$\mathcal {N}=\lbrace k:\Vert \bm {s}_k-\bm {s}\Vert \le s^*\rbrace$$, where $$s^*$$ is the repulsive force's nominal width, $$\Vert \cdot \Vert$$ is the Euclidean distance. Specifically, $$\bm {f}_k$$ is defined as:

$$\begin{equation} \bm {f}_k=\bm {Q}\hat{\bm {r}}_k^*\min {\left\lbrace 0,\Delta \bm {v}_k^*(t)c_2+(\Vert \bm {s}_k-\bm {s}\Vert -s^*)c_3\right\rbrace} . \end{equation}$$(47)

$$c_2$$ is the sensitivity to velocity difference, $$c_3$$ is the sensitivity to spacing,

$$\begin{equation} \bm {Q}={\left[\begin{matrix} 1 &\quad 0\\[7pt] 0 &\quad q \end{matrix}\right]}\triangleq {\left[\begin{matrix} 1 &\quad 0\\[7pt] 0 &\quad {v^x\tau _r+s_r\over s^*} \end{matrix}\right]}. \end{equation}$$(48)

$$\tau _r$$ is the repulsive force's sensitivity to speed, and $$s_r$$ is the repulsive force's jam range,

$$\begin{equation} \hat{\bm {r}}_k^*={\bm {r}_k^*\over \Vert \bm {r}_k^*\Vert }, \bm {r}_k^*=\bm {Q}^{-1}(\bm {s}_k-\bm {s}), \end{equation}$$(49)

and

$$\begin{equation} \Delta \bm {v}_k^*=\bm {Q}^{-1}(\bm {v}_k-\bm {v})\cdot \hat{\bm {r}}_k^*. \end{equation}$$(50)

5.2 HV

When a vehicle is a HV, the form of $$\bm {F}^r$$ and $$\bm {F}^l$$ are the same with CAV. For HV, we still apply the intelligent driver model(IDM)[54] to describe $$\bm {F}^a$$:

$$\begin{equation} F^{ax}(t)=a{\left(1-{\left({v^x(t)\over v^x_f}\right)}^\theta -{\left({h^*(v^x(t),\Delta v^x(t))\over h(t)}\right)}^2\right)}, \end{equation}$$(51)

where

$$\begin{equation} h^*(v^x(t),\Delta v^x(t))=h_0+v^x(t)T+{v^x(t)\Delta v^x(t)\over 2\sqrt {ab}}, \end{equation}$$(52)

$$v^x_f$$ is the free-flow longitudinal velocity, $$\theta$$ is the acceleration exponent, $$h(t)$$ is the longitudinal spacing between the vehicle and its preceding vehicle in the same lane, $$h^*$$ is the desired headway of the driver in the current state, $$h_0$$ is the safe spacing between the vehicle and its preceding vehicle in the same lane, $$T$$ is the safe time headway, $$a$$ is the maximum acceleration, and $$b$$ is the desired deceleration.

5.3 Lane changing

When describing that the vehicle in the lane $$l$$ changes to the lane $$l^{^{\prime }}$$, replace the $$l$$ in (44) with $$l^{^{\prime }}$$. Then the position change of this vehicle will still be calculated and updated according to (7) after substituting (44).

5.4 The difference between HV and CAV

The difference between HV and CAV is not only reflected in the difference of the longitudinal model. In fact, this difference is not decisive, because we can use other models to characterize HV and CAV. The essential difference between HV and CAV is the information they used. CAV can directly know the limit of the road (e.g. the desired speed $$V$$) ahead and information from other CAVs through V2I/V2V communications, then make a better decision, while HV can only respond to the states(e.g. velocity) of the vehicle in front of it before seeing the traffic sign on the side of the road.

5.5 Pinning nodes selection

In Section 3, we have explained that in a directed graph, a more effective pinning strategy should be to make the pinning nodes evenly distributed in the entire network. When applying evenly distribution in the three-lane scenario in Figure 7, it is as following:

FIGURE 7

Evenly distribution in three-lanes, weighted directed graph

Select the nodes of the middle lane as the pinning nodes, the number of nodes that can be affected is significantly more than that of other lanes, for example (Figure 8): Therefore, the concentration of all CAVs in the middle lane can be considered as one of the alternative pinning strategies. By applying all CAVs at the middle of the platoon might reach better results. Moreover, applying all CAVs at the begin of the platoon also might be taken into account. However, not all pinning strategies could be realized in current mixed traffic. It is almost impossible to gather all CAVs together in mixed traffic. Since the concentration of all CAVs in the middle lane can be achieved by setting up an accommodation lane, and evenly distribution of CAVs in traffic can be achieved by the methods mentioned before. In the numerical simulation, we will only consider the evenly distribution, random distribution and the concentration of all CAVs in the middle lane.

6 SIMULATION RESULTS

We first give the definition of the penetrate rate $$\delta$$ as follows:

Definition 3.Assuming that there are $$K$$ CAVs in the platoon, and the remaining vehicles are HVs. There are $$N$$ vehicles in the platoon. Then the penetration rate is defined as

$$\begin{equation} \delta \triangleq {K\over N}. \end{equation}$$(53)

Similar to [55], we proposed index as follows:

$$\begin{equation} J^v={1\over (T_e-T_0)(N-[N\delta ])}\sum _{i=1}^Nd_i\int _{T_0}^{T_e} (v_i-v_{i-1})^2\text{d}t, \end{equation}$$(54)

where

$$\begin{equation} d_i={\left\lbrace \begin{aligned} &0,\quad \text{if} \text{$i$-th vehicle is CAV}\\ &1,\quad \text{if} \text{$i$-th vehicle is HV} \end{aligned} \right.}. \end{equation}$$(55)

The smaller of the value of $$J^v$$, the better the consistency of the HVs.

Besides consistency, the comfort of driver also should be considered. We then introduce the $$RMS$$ function to evaluate human driver's driving comfort[56]. $$RMS$$ (noted as $$J^a$$) is as follows:

$$\begin{equation} J^a={1\over (T_e-T_0)(N-[N\delta ])}\sum _{i=1}^Nd_i\int _{T_0}^{T_e}u_i^2(t)\text{d}t. \end{equation}$$(56)

The smaller of the value of $$J^a$$, the better the comfort of the HVs.

The purpose of pinning control is to keep all vehicles in the platoon in the same state. This includes: (1) In a given initial state, the vehicle state can achieve the same; (2) When a disturbance occurs in the traffic, the propagation of the disturbance can be suppressed, and then the vehicle state can be the same. When all vehicles are in the same state, their velocity and acceleration will not fluctuate unless there is a disturbance. On the other hand, when a disturbance occurs, the faster the vehicle state converges, the smaller the fluctuations in velocity and acceleration. Therefore, in this way, the values of $$J_a$$ and $$J_v$$ can also be used to measure the stability of the entire system: the smaller their value, the more stable the system; the smaller their value after disturbance occurs, the stronger the system's ability to suppress the disturbance, thus the more string stable the system.

Referring to [51] and [54], the parameter values of CAV and HV are as following (Tables 1–3):

TABLE 1. Parameters of CAV

Parameters	$$k_1$$	$$k_2$$	$$c_1$$	$$c_2$$	$$c_3$$	$$\tau _r$$	$$S_r$$	$$s^\star$$
Value	1.1	0.3	0.12	$${40\over 64}$$	$${12\over 64}$$	0.9	$${210\over 9}$$	2.5

TABLE 2. Parameters of HV(longitudinal)

Parameters	$$a$$	$$b$$	$$v^y_f$$	$$T$$	$$\theta$$	$$h_0$$
Value	1	2	120	1.5	4	2

TABLE 3. Parameters of HV(lateral)

Parameters	$$k_1$$	$$k_2$$	$$\tau _r$$	$$S_r$$	$$s^\star$$
Value	1	0.25	0.8	$${196\over 9}$$	2.8

$$v_{\min }=0$$, $$v_{\min }=v_f^y$$, $$a_{\min }=-5$$ m/s, $$a_{\min }=3$$ m/s. In Sections 6.1– 6.5, $$\tau =0.1$$ s.

6.1 Effect of mixing CAV into traffic

First of all, we need to verify through simulation experiments that under the leadership of CAV, the indicators of HVs in the mixed platoon are better than in the HV platoon. We set the virtual leader drive at a constant speed, at a constant speed after decelerating at a constant speed, and at a constant speed after accelerating at a constant speed, respectively. Then we conducted experiments in the above three scenarios. The initial position of the first vehicle in the platoon is placed at 0 m. In Figures 9 and 10, the initial velocity and expected velocity of all vehicles are 80 km/h. In Figure 10, the expected velocity of all vehicles at 800 m becomes 60 km/h. In Figure 11, the initial velocity and expected velocity of all vehicles are 60 km/h, and the expected velocity of all vehicles at 800 m becomes 80 km/h. Note that in the following three sets of experiments, only the first vehicle in the mixed platoon is CAV. $$T_0^i$$ is the time when the $$i$$th vehicle reaches 800 m, and $$T_e^i$$ is the time when it reaches 900m. The changes of related indicators with the increase of vehicles, velocity and acceleration are as following:

FIGURE 8

Select the nodes of the middle lane as the pinning nodes

FIGURE 9

Leading vehicle at constant speed

FIGURE 10

Leading vehicle decelerate

FIGURE 11

Leading vehicle accelerate

Figure 9 illustrates that in a stable platoon, the impact of the presence of CAV on the indicators is negligible. However, Figures 10 and 11 illustrate that when the leading vehicle is a CAV, the disturbance from the head of the platoon can be better suppressed. Figure 10 also illustrates that deceleration behavior has the greatest impact on comfort.

Then we set the experimental road as a three-lane road (Figure 12): The platoon on each lane is composed of 10 HVs. The initial speed and speed limit are in the tables, and the platoon is initially stable. Other conditions are the same as the previous set of experiments. The value of indicators are as following(The unit of speeds are all km/h) (Tables 4 and 5):

TABLE 4. Value of $$J_v$$, HV platoon

Speed limit
Initial speed	40	60	80	100
60	6.2259	0.0000	8.8985	39.4254
65	10.2562	0.4847	5.1431	30.6538
70	15.6168	2.0546	2.3406	22.9760
75	22.6011	4.8768	0.5954	16.3511
80	31.9033	9.1166	0.0001	10.9018

TABLE 5. Value of $$J_a$$, HV platoon

Speed limit
Initial speed	40	60	80	100
60	0.0609	0.0000	0.0176	0.0547
65	0.0863	0.0019	0.0090	0.0415
70	0.1148	0.0068	0.0035	0.0292
75	0.1389	0.0129	0.0008	0.0197
80	0.1562	0.0193	0.0000	0.0143

In subsequent experiments, the conditions not mentioned are the same as the previous set of experiments. Each of our random experiments will be conducted 50 times, and the average of the indicators will be compared.

6.2 Influence of speed limit

To illustrate the effectiveness of pinning nodes selecting methods, first we set $$\delta ={1\over 3}$$. The value of indicators when applying different pinning methods are as following(The unit of speeds are all km/h):

The known conclusions can be inferred from the data in the above tables, thus the two indicators proposed above are effective. These conclusions are:

• a. The presence of CAVs can improve the consistency and comfort of the HVs;
• b. When the vehicle speed changes greatly, the comfort and consistency will deteriorate;
• c. The greater the current speed of the vehicle, the greater the impact of vehicle acceleration and deceleration on comfort and consistency, which will affect traffic safety;
• d. When a stable platoon moves forward at a constant speed, the distribution of CAVs has negligible effect on the platoon.

By comparing Tables 6–8 and 9–11, it can be seen that of the three methods, centralizing all CAVs in the middle lane had the worst effect. In fact, when all CAVs concentrated in the middle lane, the graph of the vehicles will become a regular network. Since centralizing all CAVs in the middle lane had the worst effect, it could be concluded that the pinning control is not effective in regular network. Therefore, this method will not be considered in the rest of the paper. In addition, the values of related indicators can also explain that the evenly distribution of CAVs in the mixed platoon can further optimize traffic compared to random distribution.

TABLE 6. Value of $$J_v$$, random distribution

Speed limit
Initial speed	40	60	80	100
60	1.3758	0.0000	2.3853	12.6425
65	2.1964	0.1207	1.5404	10.3501
70	4.0411	0.5023	0.7192	8.2806
75	6.0785	1.2425	0.1690	5.5927
80	7.6540	2.6169	0.0000	3.9865

TABLE 7. Value of $$J_v$$, all CAVs in the middle lane

Speed limit
Initial speed	40	60	80	100
60	6.2515	0.0000	8.9185	39.4918
65	10.2893	0.4866	5.1421	30.7006
70	15.6686	2.0597	2.3442	23.0034
75	22.6839	4.8839	0.5945	16.3878
80	32.0428	9.1180	0.0000	10.9167

TABLE 8. Value of $$J_v$$, evenly distribution

Speed limit
Initial speed	40	60	80	100
60	0.5844	0.0000	1.1524	7.2498
65	1.0123	0.0551	0.7108	5.8826
70	1.5824	0.2388	0.3410	4.4955
75	2.3953	0.6083	0.0928	3.3497
80	3.5640	1.1893	0.0000	2.3750

TABLE 9. Value of $$J_a$$, random distribution

Speed limit
Initial speed	40	60	80	100
60	0.0188	0.0000	0.0192	0.0838
65	0.0292	0.0014	0.0104	0.0642
70	0.0471	0.0047	0.0024	0.0640
75	0.0584	0.0098	0.0010	0.1520
80	0.0806	0.0197	0.0000	0.2629

TABLE 10. Value of $$J_a$$, all CAVs in the middle lane

Speed limit
Initial speed	40	60	80	100
60	0.0605	0.0000	0.0174	0.0538
65	0.0857	0.0018	0.0088	0.0408
70	0.1139	0.0067	0.0035	0.0287
75	0.1384	0.0129	0.0007	0.0193
80	0.1548	0.0189	0.0000	0.0141

TABLE 11. Value of $$J_a$$, evenly distribution

Speed limit
Initial speed	40	60	80	100
60	0.0076	0.0000	0.0144	0.0661
65	0.0121	0.0007	0.0082	0.0490
70	0.0191	0.0028	0.0036	0.0343
75	0.0251	0.0063	0.0009	0.0231
80	0.0404	0.0130	0.0000	0.0152

6.3 Influence of platoon length

The purpose of pinning control is to keep all vehicles in the platoon in the same state. This includes: (1) In a given initial state, the vehicle state can achieve the same; (2) When a disturbance occurs in the traffic, the propagation of the disturbance can be suppressed, and then the vehicle state can be the same. In this part, we will compare the effectiveness of the pinning strategy under different platoon lengths, hence to illustrate the string stability of the method we proposed. In the first group of experiments, the initial speed of the vehicle is 60 km/h, and the speed limit is 80 km/h; in the second group of experiments, the initial speed of the vehicle is 80 km/h, and the speed limit is 60 km/h. Platoon length increased from 1 vehicle to 20 vehicles. $$\delta ={1\over 3}$$. The figures of the relevant indicators changing with the length of the platoon are as following:

Figures 13 and 14 illustrate that when the number of vehicles in the platoon is small, the difference between random distribution and evenly distribution is not large. This conclusion is also compatible with existing conclusions[23]. That is, when the length of the platoon is small, the selection of different distribution of CAVs has little effect on traffic. As the length of the platoon increases, the effect of evenly distribution on traffic optimization gradually gets better, while the optimization effect of random distribution on traffic fluctuates around a fixed value. Thus, because the scale of actual traffic is generally very large, the method proposed in this paper has practical value.

FIGURE 12

Three-lanes scenario

FIGURE 13

Initial speed=60 km/h, speed limit=80 km/h

FIGURE 14

Initial speed=80 km/h, speed limit=60 km/h

6.4 Influence of penetration rate

In 4.3, we proposed theorem 3 to illustrate that the controllability of mixed traffic could be improved by increasing the amount of CAVs. In this part, we demonstrated through experiments that the increase in CAVs (i.e. the increase in penetration) can make the system perform better on the two indicators of $$J_v$$ and $$J_a$$. In the first group of experiments, the initial speed of the vehicle is 60 km/h, and the speed limit is 40 km/h; in the second group of experiments, the initial speed of the vehicle is 60 km/h, and the speed limit is 80 km/h. There are 20 vehicles in each lane. Penetration rate is denoted by $$\delta ={p\over 20}$$, where $$p$$ increased from 1 to 19. The figures of the relevant indicators changing with the length of the platoon are as following:

Figures 15 and 16 illustrate that when $$\delta$$ is very low (less than or equal to 0.1), the effect of evenly distribution is even worse than that of random distribution. And when $$\delta$$ is sufficiently high (greater than or equal to 0.6), the difference between the two distributions is not significant. In other cases, the effect of evenly distribution is significantly better than random distribution. This is in agreement with the conclusion drawn in Section 3.

FIGURE 15

Initial speed=60 km/h, speed limit=40 km/h

FIGURE 16

Initial speed=60 km/h, speed limit=80 km/h

6.5 Influence of lane changing

Lane changing is a common vehicle behavior in actual traffic, and it will have a non-negligible impact on traffic. In this part, the initial speed of the vehicle and the speed limit are 60 km/h. Different from previous parts, every second, one vehicle is randomly generated at 0m and the end is at 1000 m. The type and lane of the vehicle are randomly assigned. The probability that it is CAV is $$\delta$$, and the probability of it appearing in each lane is equal. Let $$P_c$$ be the propotion of HVs that will change lanes in all vehicles. Assume that CAV will not change lanes. HVs will only change lanes at most once after entering the road, and the target is a random adjacent lane. The simulation duration is set to 1800 s (30 min). The tables are:

It could be seen that both $$J_a$$ and $$J_v$$ have a positive correlation with $$P_c$$, while they are not always have a negative correlation with $$\delta$$. The increase of $$\delta$$ might have a negative impact on $$J_a$$ and $$J_v$$, this conclusion is consistent with the conclusions drawn by [57]-[58]: the mixing of CAV does not necessarily have a positive impact on traffic. Comparing Table 12 with Table 13 and Table 14 with Table 15, it could be concluded that evenly distribution is better than random when considering lane changing behavior. When $$P_c=0.5$$, $$0.1\le \delta \le 0.3,J_a$$ and $$J_v$$ have a negative correlation with $$\delta$$. This shows that when the penetration rate is appropriate, the increase in it has a positive impact on traffic.

TABLE 12. Value of $$J_v$$, random distribution

δ
$${P_c}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0447	0.0508	0.0535	0.0542	0.0475
0.3	0.0726	0.0631	0.0716	0.0757	0.0593
0.5	0.1985	0.1538	0.1071	0.0810	0.0784

TABLE 13. Value of $$J_v$$, evenly distribution

$${\delta }$$
$${P_c}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0248	0.0238	0.0355	0.0369	0.0417
0.3	0.0352	0.0396	0.0265	0.0355	0.0405
0.5	0.0564	0.0589	0.0375	0.0455	0.0454

TABLE 14. Value of $$J_a$$, random distribution

$${\delta }$$
$${P_c}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0142	0.0147	0.0151	0.0156	0.0145
0.3	0.0215	0.0201	0.0211	0.0212	0.0177
0.5	0.0411	0.0387	0.0292	0.0238	0.0237

TABLE 15. Value of $$J_a$$, evenly distribution

$${\delta }$$
$${P_c}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0073	0.0070	0.0095	0.0099	0.0108
0.3	0.0100	0.0115	0.0104	0.0106	0.0121
0.5	0.0164	0.0180	0.0114	0.0132	0.0147

6.6 Influence of information delay

In this subsection, other conditions remain unchanged, let $$P_c=0.2$$ and applying evenly distribution, then the tables are (Tables 16 and 17):

TABLE 16. Value of $$J_v$$, different information delay

$${\delta }$$
$${\tau (s)}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0357	0.0416	0.0351	0.0432	0.0461
0.2	0.0399	0.0391	0.0387	0.0448	0.0494
0.3	0.0319	0.0365	0.0312	0.0497	0.0504
0.4	0.0417	0.0419	0.0383	0.0454	0.0581
0.5	0.0473	0.0549	0.0583	0.0583	0.0632
$$\infty$$	0.0467	0.0450	0.0491	0.0575	0.0594

TABLE 17. Value of $$J_a$$, evenly distribution

$${\delta }$$
$${\tau (s)}$$	0.1	0.2	0.3	0.4	0.5
0.1	0.0114	0.0128	0.0107	0.0130	0.0145
0.2	0.0117	0.0122	0.0115	0.0140	0.0166
0.3	0.0106	0.0120	0.0094	0.0136	0.0170
0.4	0.0121	0.0122	0.0114	0.0143	0.0191
0.5	0.0212	0.0209	0.0216	0.0219	0.0299
$$\infty$$	0.0208	0.0193	0.0204	0.0216	0.0249

$$\infty$$ means that the communication between CAVs is interrupted or the delay is very large, and the pining input is 0 at this time. When $$\delta \le 0.3$$, increase in $$\tau$$ does not always cause traffic deterioration. When $$\delta \ge 0.4$$, both $$J_v$$ and $$J_a$$ are positively correlated with $$\tau$$. Moreover, when $$\tau \ge 0.4$$, the increase in $$\delta$$ will cause traffic deterioration instead. When there are no communications between CAVs, the relevant indicators are better than the case of $$\tau =0.5$$ s. It can be seen that the influence of information delay on traffic cannot be ignored. Only in a good communication environment, the pinning method can play the effect of optimizing traffic.

6.7 Results

Through the above experiments, we can obtain the following conclusions:

• a. The conclusions obtained from the indicators are the same or similar to the known ones, which can illustrate the effectiveness of the indicators used in this paper;
• b. When $$\delta$$ is sufficiently high (greater than or equal to 0.6), the difference between the two distributions are not significant;
• c. When the number of vehicles is large and there is disturbance in the platoon, the evenly distribution can effectively suppress the negative impact of the disturbance on the traffic.
• d. When considering lane changing behavior of HVs, evenly distribution is better than random.

Due to the uncertainty of HVs, disturbances are likely to occur. And in the foreseeable future, the proportion of CAVs in traffic will not be too high. Therefore, the pinning method mentioned in this paper can effectively optimize mixed traffic.

However, from the experimental results, the pinning method has its scope of application:

• a. When the traffic conditions are ideal, the impact of this strategy on traffic can be ignored;
• b. When the length of the platoon is small, the selection of different distribution of CAVs has negligible effect on traffic;
• c. The pinning method can only have significant effect when the scale of traffic is large enough.

7 CONCLUSION

In this paper, a novel cyber-physical description of traffic and pinning approach of mixed traffic are proposed. First, the homogeneous traffic and mixed traffic in the cyber space are discussed, and the small-world and scale-free characteristics of mixed traffic are revealed. Then we proposed the pinning approach. The stability and consistency analysis is proposed then. The influence of adding CAVs is studied, and the pinning node selection strategy for mixed traffic is briefly discussed.

Numerical simulations show that the mixing of CAVs will have a positive effect on traffic, and results under three-lane scenario illustrated specific pinning selection is better random pinning selection. In this way, the effectiveness of the pinning nodes control strategy and the pinning nodes selection strategy can be explained. Besides, impact of speed limit, platoon length, and penetration rate are also researched.

The method proposed in this paper has shortcomings. Both HVs and CAVs could influence the pinning method. On the one hand, since the main idea of pinning is by controlling CAVs to indirectly control other HVs, it is foreseeable that when HV's behavior is unpredictable (e.g. frequent lane changes without a reason etc.), the effect of this indirect control will be greatly reduced. On the other hand, because the pinning method relies on the cooperation between CAVs, when the communication environment is not good, these CAVs will degenerate into AVs or even HVs, which will affect the control efficiency. At this time, it may be more appropriate to apply the existing CACC algorithm. In our future research, more realistic assumptions will be considered, and we will explore the feasibility of pinning control in more application scenarios.

ACKNOWLEDGEMENTS

This work was supported by the National Key R&D Program of China (Grant No. 2018YFB1600600) and the National Natural Science Foundation of China (Grant No. 62073049).

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

Open Research

DATA AVAILABILITY STATEMENT

Research data are not shared.