An urban expressway is divided into N cells and there are three types of cells, as follows: mainline cell, on-ramp cell, and off-ramp cell. To describe the actual traffic flow processing of the studied object, we established an improved CTM,^[28,29] where the length of each cell that may be different and the length of cell m is l_m. Considering that there are three types of segments on the urban expressway, three types of connections were defined in the CTM, as shown in Fig. 1, where simple connection exists in basic segment, fused connection exists in confluence segment, and separate connection exists in distributary segment. and denote the numbers of vehicles that can move from mainline cell and on-ramp cell p to mainline cell m at discrete time step t, respectively. and denote the numbers of vehicles moving from mainline cell and on-ramp cell p to mainline cell m at step t, respectively. denotes the number of vehicles moving from mainline cell to off-ramp cell o at step t. The numbers of vehicles for mainline cell m and off-ramp o that they can be accepted at step t are and , respectively. denotes the capacity of cell m. v is the free flow speed and ω is the backward propagation of the crowded wave velocity. represents the density of mainline cell m at step t. The congestion density is K_J. α denotes the fuse ratio of on-ramp p to mainline cell and β denotes the separation ratio between off-ramp cell and mainline cell.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. Transmission relation in three types of connections.

Flow conservation for simple connection is expressed by the following equations: 1 2 3 where denotes the flow of vehicles into cell m at step t. Equation (1) indicates that the number of vehicles for a mainline cell moving from its upstream cell is equal to the minimal value between the number of vehicles that can move from its upstream cell and the number of vehicles that can be accepted.^[29] Equation (2) indicates that the number of vehicles for a mainline cell that can move to its downstream cell is not larger than the capacity of the cell. Equation (3) indicates that the number of vehicles that a cell can be accepted is equal to the minimal value between the capacity of the cell and the number of its remaining vehicles.

Flow conservation for fused connection is expressed by the following equations: 4 where function med denotes take median of three items.^[29] Equation (4) indicates that all vehicles can move from cell and on-ramp cell p to cell m if cell m has sufficient space for acceptance, otherwise there will be a certain proportion for and ,

Flow conservation for separate connection is expressed by the following equations: 5 6 Equations (5) and (6) indicate that vehicles can move from cell to cell m and off-ramp cell o if they have sufficient spaces for acceptance, otherwise there will be a certain proportion for and .^[29]

For mainline cell m, which has three types of connections, the state transfer equation of flux is described as follow: 7 Equation (7) indicates that the number of vehicles in a cell is equal to its number at the previous simulation step plus the number of vehicles from its upstream cell and on-ramp cell, and minus the number of vehicles that move to its downstream cell and off-ramp cell. For a mainline cell with only one or two types of connections, we set the number of moving vehicles be zero at the direction without adjacent cell.

Considering the particularity of urban expressway, we use a normal subsystem as shown in Fig. 2 as a node, which consists of a mainline cell, an on-ramp cell, and an off-ramp cell. Each node is only connected with two adjacent nodes, just as each subsystem of urban expressway is only connected to its upstream and downstream. For other types of nodes, we set the flux of on-ramp to be zero for the subsystem without on-ramp, the flux of off-ramp to be zero for the subsystem without off-ramp, and the flux of on-ramp and off-ramp to be zero for the only mainline subsystem. The urban expressway consists of N nodes, where an overpass is divided into four types of nodes according to the concrete component.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. Sketch of normal node.

Obviously, the state transfer equation of flux for mainline cell m of node i can be described by Eq. (7). For node i, the number of accepted vehicles for cell m of node i at step t is . The number of vehicles moving from cell m is . The number of vehicles moving from on-ramp cell p of node i is . Here we assume that the number of vehicles moving from on-ramp p to mainline cell m is within the permissible range , the number of vehicles moving from on-ramp p to mainline cell m is , . In addition, we assume that the reception capacity of off-ramp cell o is infinite; that is, the number of vehicles moving from the mainline cell m to off-ramp o is . According to Eq. (7), the state transfer equation of flux for mainline cell m of node i is described as follows: 8 where and represent the numbers of vehicles moving from node to i and from node i to at step t, respectively. Equation (8) indicates that the number of vehicles for a cell at the next step is equal to the number at the present step plus the number of vehicles from its upstream cell and on-ramp cell, and minus the number of vehicles that move to its downstream cell and off-ramp cell.

Here we use , which represents the traffic density of mainline cell m of node i, as the state variable of node i. l_i represents the product of the mainline cell length of node i and the number of lanes. The state transfer equation of density for node i called node coupling model of urban expressway can be derived as follows if equation (8) is divided by l_i on both sides: 9 where denotes the internal state vector of node i at t and ε denotes the simulation step.

Considering that many of the urban expressways in China are ring networks, we used a ring network as the concrete studied objective, the state transfer equation of Eq. (9) for node i called node coupling model of urban expressway can be derived as follows: 10 11 12 where represents the coupling relationship between nodes i and j. The matrix in Eq. (11) represents the adjacency configuration of the networks. If there is a connection between nodes i and j ( ), then , otherwise, ( ). Diagonal elements are , , where c_i denotes the degree of node i. Transmission relations of three connections are considered among mainline cell, on-ramp cell, and off-ramp cell. The coupling strength is a, where a = 1 in Eq. (10). As the nearest neighbor complex networks, each node of urban expressway is only connected to its downstream node and upstream node. So, except for the main diagonal elements, the matrix only has two nonzero elements.

To simplify the problem, only the mode of multi-on-ramp coordinated control was studied. By using the information gathering facilities and relating data processing, the parameters of traffic flow for each node (e.g., traffic volume, traffic density, average velocity, time headway, and on-ramp queue length, etc.) are gathered and computed. After having extracted the features vector of time series data for traffic flow as the input variables, we can compute the maximum Lyapunov exponent (λ_max) of each node.^[15,16] If λ_max of each node is not larger than zero, the system is not in a chaotic state, meaning that the system is in ordered motion. No on-ramp metering signal is inputted, otherwise the system is in a chaotic state. The coordinated chaos control signals will be input in the next step. After the control signals have been input many steps, λ_max of each node possesses negative values continuously. The traffic flow resumes ordered motion; that is, it is not in a chaotic state. The coordinated on-ramp metering control signals will be input continuously in the next 5 or 10 steps to maximize the effectiveness of control. We then cancel the coordinated on-ramp metering signals.

The DFC mechanism proposed by Pyragas is the most important method of chaos control,^[30,31] in which the unknown unstable periodic orbits (UPO) are estimated by a time-delayed state. The feedback used in the control strategy is based on the difference between the current state of the system and a time-delayed one. The delay constant is chosen to be equal to the period of the target UPO, which is assumed to be known a priori. As such, the DFC method is also referred to as the time-delayed feedback control or time-delayed auto-synchronization method. From the viewpoint of experimental physics, DFC does not need to model the internal dynamics of the object to be controlled. Therefore, it can be easily implemented in the chaos control of many complex uncertain systems, particularly to social and economic systems; for example, the traffic system. So we use DFC as the coordinated chaos control method of an urban expressway. To save control cost discussed in section 1, only partial nodes are inputted as coordinated control signals or added as control variables. The controlled nodes are called pinning nodes. The state transfer equation of the urban expressway in Eq. (10) is revised as follows: 13 where 14

Here the control variable represents the adjusted value of density for node i. This can be realized by the inflow regulating of on-ramp. The feedback gain if and if , where g represents the number of pinning control nodes, the method is called pinning chaos control if , otherwise it is called global chaos control. In particular, the method is called an on-ramp control if g = 1, where the ALINEA algorithm is a classic one.

Define the error of node i as 15 Then, the errors can be described as 16 If the errors tend to zero, then the urban expressway described in Subsection 2.2 will stabilize and achieve synchronization.

Before stating the main results of this paper, some preliminaries need to be given for convenient analysis.

Assumption 1 There is a constant θ that makes the nonlinear function in the network of urban expressway satisfy 17

Lemma 1 (Schur complement)^[32] The following linear matrix inequality (LMI) 18 is equivalent to one of the following conditions: 19 where , .

Theorem 1 If the matrix satisfies the following conditions: 20 the network of urban expressway described in Subsection 2.2 will stabilize and achieve synchronization, where .

Proof Construct a Lyapunov function , and use formulas (13)–(16), then 21 By using Assumption 1, we have 22

We can then obtain Eq. (20) by using Lemma 1.

Remark 1 According to Assumption 1 and Lemma 1, if a network is given, then the stable condition is determined by Eq. (20), which is determined by the coupling strength a, the adjacency matrix , and the feedback gain matrix of the network. An appropriate θ and may make Theorem 1 hold. Considering the particularity of urban expressway, we choose to input the control signal on the on-ramp of the upstream nodes. We set the number of control nodes from 1 to N until the feasible solution in Eq. (20) can be calculated by using LMI toolbox in Matlab. We then obtain the pinning nodes and , where the pinning number is the smallest. It is concluded that such a weighted complex dynamical network can be pinned to its equilibrium by using some controllers. Therefore, the goal of locally asymptotically stable for urban expressway can be achieved.

The clockwise direction of the northwest half-ring Tianjin City Expressway in China, which is shown in Fig. 3, was used as the studied objective, where the length of the one-way four-lane mainline is 20 km. The numbers of on-ramp and off-ramp in the urban expressway are 15 and 17, respectively. Each ramp is numbered, where p and o represent on-ramp and off-ramp, respectively. According to traffic survey and design information of the urban expressway, the maximal flow of urban expressway Q = 1800 veh/( ), free flow velocity v = 60 km/h, backward wave velocity ω =20 km/h, critical density , and congestion density . The vehicles consist mainly of cars and large vehicles, where the proportion of cars is 0.88. Traffic congestion is particularly severe during rush hour.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. Topology diagram of the northwest semicircular Tianjin City Expressway.

Using the rules of section 2 and considering the concrete geometric alignment of the expressway, the studied objective in Fig. 3 is divided into 16 mainline cells, 15 on-ramp cells, and 17 off-ramp cells. Each ramp corresponds to a ramp cell and its length is that of its ramp cell. Each mainline cell starts from the weaving segment between one ramp and the mainline and ends at the starting point of its downstream mainline cell. The length of a mainline cell is the distance from its starting point to the ending point. The system consists of 16 nodes and each node from upstream to downstream is marked to the number from 1 to 16 in the clockwise direction, where each node only includes a mainline cell. Table 1 shows the length of mainline cell of each node and the concrete ramps that the node contains and the starting and ending points of each node, where the concrete ramps number that a node contains may be different due to the uneven distribution of the ramps. In the simulation, the desired density and the duration of each time step is 10 s. To display the control effect, we assume that queue in each on-ramp is out of constrained. The first 360 time steps are discarded to avoid transient behavior. The results described in this paper are obtained after 360 time steps.

Table 1.

Table 1.

Table 1. Mainline cell lengths of nodes and their containing ramps. . Node 1 2 3 4 5 6 7 8 Mainline cell length/km 1.00 1.08 0.68 0.75 0.93 2.10 1.90 1.00 Containing ramps p1,o1 p2,o2,o3 p3,o4 p4,o5 p5,o6 p6,o7 p7 p8,o8 Starting and ending points p1,o2 o2,o4 o4,o5 o5,o6 o6,p6 p6,p7 p7,p8 p8,p9 Node 9 10 11 12 13 14 15 16 Mainline cell length/km 1.20 1.60 1.20 1.30 1.25 0.93 1.60 0.67 Containing ramps p9,o9 p10,o10 p11,o11 p12,o12 o13,o14 p13,o15 p14,o16 p15,o17 Starting and ending points p9,p10 p10,p11 p11,o12 o12,o13 o13,p13 p13,p14 p14,o17 o17,p15

Table 1. Mainline cell lengths of nodes and their containing ramps. .

To test the validities of the proposed models and identify the relevant parameters, some comparisons of the phase diagrams of the actual statistics data and the simulation results for the CTMs of all nodes and the node coupling model of the studied objective were made. Figure 4 shows a comparison of flow-density in macroscopic phase diagram for the studied section of the urban expressway. Obviously, the two fitting curves are basically the same. There is a similar comparison diagram between the actual statistics data of the node and the simulation result for each node. Therefore, the validity is preliminarily proved.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. Comparison of phase diagram between the actual statistics and the simulation result.

The clockwise direction of the northwest half-ring Tianjin City Expressway from 7:00 am to 8:00 am was used as our experiment site and time period. According to the traffic survey, we can obtain the traffic flow with time-dependent characteristics. Table 2 shows the intervals of on-ramp demand and off-ramp flow for partial nodes during the time period. Through the statistics and the related calculation, the combined ratio α and the separation ratio β are set to 0.4 and 0.3, respectively. Figure 5 shows the fluctuations of density and λ_max for nodes 6–8 in a case from step 50 to 150, where no-control signal (NCS) and the proposed pinning chaos control signal (PCC) are inputted, respectively. In the case, λ_max of node 8 at step 53 is 0.000076, meaning that it is larger than 0 and the coordinated chaos control signal needs to be inputted. By using the PCC method and setting θ =0.94, we obtain the pining number of 3, meaning that the inflows of on-ramps from nodes 6–8 need to be regulated. The obtained feedback gains are , , . The pinning chaos control signals were then inputted after two steps. Table 3 shows the comparison of flux and total travel time (TTT) by using NCS and PCC, respectively, where the flux represents the total number of vehicle moving from the studied objective per hour, and TTT is defined as follow:^[33] 23

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) The simulation result of (a) density profile and (b) λmax profile using NCS and PCC in a case: (a1) and (b1) node 8, (a2) and (b2) node 7, and (a3) and (b3) node 6.

Table 2.

Table 2.

Table 2. Intervals of on-ramp demand and off-ramp flow for partial nodes. . Node 15 13 11 9 8 6 5 On-ramp [720,840] [900,1080] [1380,1680] [1500,1800] [720,800] [840,960] [1680,1740] demand/(veh/h) Off-ramp [480,540] [600,660] [300,360] [540,600] [600,660] [1260,1320] [1260,1300] flow/(veh/h)

Table 2. Intervals of on-ramp demand and off-ramp flow for partial nodes. .

Table 3.

Table 3.

Table 3. Comparison of TTT and flux by using NCS and PCC. . Control mode TTT/s Flux/ NCS 923110 24753 PCC 809371 26934

Table 3. Comparison of TTT and flux by using NCS and PCC. .

The findings are as follows.

1) Without coordinated on-ramp metering signal, the traffic jams phenomenon shown in Fig. 5(a1) appears in node 8. The density of node 8 increases from 45 veh/( ) at step 55 to 65 veh/( ) at step 95, and then increases rapidly to 95 veh/( ) at step 150, which eventually causes a jam phenomenon. λ_max of node 8 shown in Fig. 5(b1) keeps positive values after step 55. At the same time, the traffic jams phenomenon shown in Fig. 5 spreads rapidly toward upstream nodes especially nodes 7 and 6. For node 7, the density increases from 45 veh/( ) at step 65 to 51 veh/( ) at step 87, and then increases rapidly to 90 veh/( ) at step 150. λ_max keeps positive values after step 65. For node 6, the density increases from 46 veh/( ) at step 76 to 51 veh/( ) at step 85, and then increases rapidly to 86 veh/( ) at step 150. λ_max keeps positive values after step 76. This means that the operating efficiency of expressway declines and TTT increases. The conclusion is tested again that a wide-moving jam may occur If the traffic flow is in a chaotic state and no chaos control signal is implemented.

2) Using the PCC method, as can be seen from Fig. 5, after 40 steps, 32 steps, and 30 steps of inputting coordinated pining chaos control signal, λ_max of each node remains negative. In other words, we can control the traffic flow chaos using 40 steps or 400 s. The validity of chaos control by using PCC for urban expressway is tested. At the same time, the densities of nodes 6–8 keep vary within the range 40–45 veh/( ). Although the unknown unstable fixed point or periodic orbit has not been set, the traffic flow remains ordered motion if DFC is applied once. The validity of suppressing a wide-moving traffic jam by the proposed chaos control method is tested.

3) Using the PCC method, TTT declines from 923110 s to 809371 s, which decreases 12.6% compared to NCS. The flux increases from 24753 veh/h to 26934 veh/h, which is an improvement of 8.8% compared to NCS. The operating efficiency of expressway is enhanced because the characteristics of traffic flow and the flux for urban expressway could be varied by appropriately regulating the inflow from the on-ramp to the mainline. The system will vary from a chaotic state to ordered motion and a wide-moving traffic jam will be suppressed. The vehicles keep moving at a free flow speed. This makes the flux increase and TTT decrease.

Four control methods were used: PCC, global chaos control (GCC), pinning control signal (PCS), and CORDIN,^[8] were used to make a comparison. In the PCS method, a small negative neighborhood of critical density is used as the desired density for the condition of inputting a coordinated control signal. in Eq. (15) is replaced by the desired density with the same principle and method as PCC, where . In the GCC method, all nodes are used as control nodes with the same principle and method as PCC. We only select nodes 5–8 to be the control ones in CORDIN, where ALINEA is used as the on-ramp metering algorithm of node 8 and the parameter to be calibrated is α₁=0.7 for node 7, α₂=0.8 for nodes 6 and 5. Figure 6 shows the simulation results of density and λ_max using four control methods for partial nodes. Figure 7 and 8 show the changes of congestion range for the studied objective and the changes of flux for node 8 using five control methods, respectively. Figure 9 shows the on-ramp queue lengths of partial nodes. Table 4 shows the simulation results of the four indices, where the average queue length of the on-ramps (AQL) is defined as the queue length accumulation of all on-ramps in all simulation steps divided by the number of the on-ramps and the number of simulation steps, and the maximal queue length of the on-ramps (MQL) is defined as the maximum queue length in all on-ramps for all simulation steps. The findings are as follows.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (color online) The simulation results of (a) density profile and (b) λmax profile using four control methods in a case: (a1) and (b1) node 8, and (a2) and (b2) node 7.

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. (color online) Changes of congestion range.

Fig. 8.

	Figure Option ViewDownloadNew Window
	Fig. 8. (color online) Flow changes of node 8.

Fig. 9.

	Figure Option ViewDownloadNew Window
	Fig. 9. (color online) On-ramp queue length profiles of partial nodes using coordination control of (a) NCS, (b) GCC, (c) PCC, (d) PCS, and (e) CORDIN.

Table 4.

Table 4.

Table 4. Comparison of four indices among five methods. . Control mode NCS PCC PCS GCC CORDIN TTT/s 923110 806400 806505 877531 809371 Flux/ 24753 26934 26915 26052 26656 AQL/veh 25 15 17 23 18 MQL/veh 150 190 191 184 193

Table 4. Comparison of four indices among five methods. .

1) As can be seen from Fig. 6, the density of node 8 varies within the range 38–46 veh/( ) from step 55 to 125 after multi-on-ramp metering signals have been input by using GCC method, PCS method, and CORDIN, respectively. The density of node 8 then varies below 42 veh/( ) after step 126. λ_max of node 8 varies at first between positive value and negative value, and then remains negative value after multi-on-ramp metering signals have been input 25 steps, 36 steps, and 33 steps, respectively. The same change trend is seen for node 7. The congestion range and duration of the system shown in Fig. 7 decrease greatly and the flux of node 8 shown in Fig. 8 increases. As shown in Table 4, TTT decreases and the flux increases for each control method compared with NCS. A wide-moving jamming phenomenon can be suppressed using each control method. The operating efficiency of urban expressway can be enhanced. At the same time, AQL decreases and MQL does not decrease and perhaps increases for each control method compared with NCS. This happens because the coordinated control signal makes many vehicles that want to travel on the expressway have to travel on the urban streets or queue on the on-ramps. The queue length of some metering on-ramps for each control method may become larger than that with NCS. This creates a problem of social inequity for the ramp users but the average queue length of all on-ramps can be smaller than that with NCS. In addition, the operating efficiency of the urban expressway for all users can be enhanced. The validities of PCS method and GCC method are also proven and the validity of CORDIN is tested again.

2) Using GCC method, the density of node 8 increases at first, it then decreases rapidly, and ultimately keeps varying within the range 37–39 veh/( ). The same change trend is seen for node 7. The TTT for each node is the largest and the flux is the smallest among the four control methods. Assuming that the queue in each on-ramp is out of constraint, the AQL is the largest but the MQL is the smallest among the four control methods. This happens because the expanded control sections or increasing on-ramp metering number make many vehicles that want to travel on the expressway travel instead on ordinary urban streets or queue on the on-ramps. The flux drops due to the restrained traffic demand, although the travel time of most vehicles traveling on the mainline may decline. At the same time, this makes the problem of social inequity for some ramp users more serious. Therefore, the control cost by using GCC method cannot be saved and the operating efficiency is the lowest of the four control methods.

3) Using PCS method, the densities of nodes 8 and 7 increase at first, they then decrease slowly, and ultimately reach 42 veh/( ). Using CORDIN, the densities of nodes 8 and 7 increase at first, they then decrease slowly, and ultimately keep varying within the range 40–42 veh/( ). At the same time, the four control methods have the same change trends of on-ramp queue length, congestion range, and flux for each node, such as nodes 5–8. This happens because PCS is a special method of chaos control, where the desired density is chosen to be the target UPO. The essence of CORDIN, especially in the control goal, is also equal to that of PCS method and PCC method. Therefore, the control effects of the three methods are similar.

4) For the PCC method, TTT is the smallest, the flux is the largest, AQL is the smallest, and MQL is not the best among the four methods. This happens because the PCC method does not need any preliminary knowledge about the target UPO, except for its period. Compared with the PCS method, it can easily be implemented in the chaos control of an uncertain expressway. For CORDIN, its parameters of controllers and the number of control nodes are only determined by the traffic managers rather than by more accurate models and their calculation. The control effect may not be optimal, especially in multi-jamming nodes. Therefore, TTT, the flux, and AQL using the PCC method are the best among the four control methods. Although MQL by using PCC method is not the best and is perhaps the worst among the four control methods, and it makes the problem of social inequity for some ramp users more serious, the operating efficiency of urban expressway for all users can be enhanced. Consequently, the control effect of PCC method is the best among all four control methods.