#### Abstract

In city traffic, it is important to improve transportation efficiency and the spacing of platoon should be shortened when crossing the street. The best method to deal with this problem is automatic control of vehicles. In this paper, a mathematical model is established for the platoon’s longitudinal movement. A systematic analysis of longitudinal control law is presented for the platoon of vehicles. However, the parameter calibration for the platoon model is relatively difficult because the platoon model is complex and the parameters are coupled with each other. In this paper, the particle swarm optimization method is introduced to effectively optimize the parameters of platoon. The proposed method effectively finds the optimal parameters based on simulations and makes the spacing of platoon shorter.

#### 1. Introduction

With the rapid development of economy and growth of population, the number of vehicles becomes more and more in China and the increase in vehicle number is unprecedented. As a result, the traffic congestion has been becoming a serious problem. Therefore, it attracts many researchers’ attention to improve the platoon’s control which can shorten the spacing of platoon and increase traffic flow.

The best method to deal with this problem is automatic control of vehicles. One of the transportation automatic control systems is the Automated Highway System (AHS) which includes the longitudinal control, the lateral control, and the comprehensive control. An AHS is a proposed intelligent transportation system technology designed to provide driverless cars with specific rights-of-way. It is most often touted as a means of traffic congestion relief, as it would drastically reduce following distances and headway, thus allowing more cars to occupy a given stretch of road [1]. It belongs to longitudinal control to cross the traffic intersection at a fast speed. The aim of longitudinal control is to ensure a safe distance and maintain a relatively stable speed between vehicles.

The idea of longitudinal vehicle control has developed very quickly and become very attractive with the increasing issues of traffic congestion and road safety. The researchers showed many technical considerations in the design of longitudinal control systems, such as external forces, process and measurement noise, and sampling and quantization of measurements [2]. Many models which include linear model [3, 4] and nonlinear model [5, 6] had been established and the researchers had presented systematic analysis of longitudinal control for the platoon. The scholars applied many controllers, such as classical proportional integral controller [7, 8] and the fuzzy controller [9] in longitudinal control of platoon. Wang et al. [10] analyzed the local stability of platoon and studied the relationship between the safe headway and the global stability of platoon. However, there are many parameters to be set for the platoon. When the global optimization method is applied to find the optimum parameters of platoon, the method evaluates the fitness of application and numerous fitness evaluations are needed. As such, the convergence is also an important factor in the selection of a method for preventing a platoon from numerous iterations of the method.

The particle swarm optimization (PSO) method is a population based stochastic optimization method proposed by Kennedy and Eberhart in 1995 and is inspired by social behavior such as flocks of birds or schools of fish [11]. The main advantages of PSO method are simple to understand, easy to implement, and quick in convergence [12]. The PSO method has been widely studied. Pan et al. [13] analyzed the kinetic characteristic of three models of PSO method. Ren and Wang [14] proposed an accelerated convergence particle swarm optimization algorithm based on analyzing the convergence of basal PSO method. Coelho [15] presented a novel quantum-behaved PSO method using chaotic mutation operator. Chen and Zhao [16] proposed a particle swarm optimization method that uses an adaptive variable population size and periodic partial increasing or declining individuals in the form of ladder function. The PSO method has been successfully applied in the CPG parameter optimization [17], continuous nonlinear function optimization [11], reactive power and voltage control [18], and parameter tuning of controller for a power system [19].

The remainder of this paper is organized as follows. Section 2 builds a dynamic platoon model and presents a systematic analysis of a longitudinal control law for the platoon. Section 3 shows comprehensive and detailed analysis on parameters optimization based on the PSO method. Section 4 shows the simulation. The conclusion and some ideas about further research are discussed in Section 5.

#### 2. Dynamic Model of Platoon

Various models for vehicles dynamics have been used in the study of longitudinal control of platoon. For a platoon travelling at a constant speed in a fixed direction, we adopt the following third-order model [2]. The state equation of the vehicle is given by where , , and denote the absolute position, velocity, and acceleration of the vehicle, respectively. The parameter is the time constant of the vehicle propulsion system and the parameter is the control input. The assumed configuration for a platoon of five vehicles is shown in Figure 1.

The parameter is assigned position of vehicle along the road and the parameter is the deviation of the th vehicle position from its assigned position. And it is described by

Given the direction of platoon from right to left, the platoon variables are the velocity of the lead vehicle, the velocity of the th vehicle, the acceleration of the lead vehicle and the acceleration of the th vehicle.

From the input/output point of view of the th linearized model [3, 4], we can obtain

The objective of longitudinal control is to maintain the spacing error below a predetermined level or, if possible, at zero. As to the first vehicle, the controller [6] can be written as where , , , , and are design constants [6]. The parameter denotes the steady-state velocity of the lead vehicle and the parameter is the steady-state acceleration of the lead vehicle.

When and , we can obtain

Differentiating both sides of (2) three times with respect to the time variable, we obtain

Taking Laplace transforms, we obtain

For the th vehicle, we can also obtain where , , , , and are design constants [6]. Differentiating both sides of (2) three times with respect to the time variable, we obtain

Taking Laplace transforms, we obtain

For the second vehicle, the transfer function is

Therefore, the main design objective for the longitudinal control law is shown in Figure 2, where .

We use the block diagram in Figure 2 to analyze the platoon. Some considerations suggest the main design objective for the longitudinal control law.

In order to keep the platoon stable, all poles of the transfer function are required to be in the left half plane. According to Routh-Hurwitz stability criterion, we obtain , , , and .

Since the perturbations in the lead vehicle’s velocity from its steady-state value should not get magnified from one vehicle to the next as one goes down the platoon, we require that for all .

Therefore, these parameters , , , , , , , , , and are needed to be selected correctly.

#### 3. Particle Swarm Optimization Method

The particle swarm optimization method uses the concept called particle and swarm [17, 20]. The particles correspond to an animal, bird, and insect in a herd, flock, and swarm, respectively. Let us consider a swarm including particles which are seeking the optimum value of the objective function in an -dimensional search space, each particle having a vector of position , which is associated with a solution, a vector of velocity , which determines the movement value of a particle in each dimension to improve its current position, and a vector of particle best position which is associated with most fitted positions of a particle from the first step of the algorithm. It is notable that the fitness of a position can easily be calculated considering the objective function of the optimization problem. A vector of the global best particle is reserved for knowledge sharing among all particles of a swarm [21]. Using the aforementioned notations, each argument of the velocity and position vector for each particle in the swarm is updated through the multiple iterations of the algorithm using the following model: where is a velocity and is a position of the particle at iteration. is a previous best position and is a global best position of each particle obtained so far. and are determined as and . is a cognition learning factor and is a social learning factor. is the inertia weight which determines the particle speed prior to the current velocity and thus functions as a balancing algorithm between global search and local search capacities.

#### 4. Simulations

##### 4.1. Simulation Scheme

In this paper, the PSO method is used to optimize the parameters and . In order to reduce the particle dimension and avoid complicated analysis, we choose , , and . Then (7), (10), and (11) have the same poles. The values of parameters are set as , , , , , , , and [6]. The spacing between these vehicles is selected as fitness function. To examine the behavior of the platoon with optimized parameters, the simulation for platoon consisting of 5 vehicles is run in MATLAB. In the simulation, all the vehicles are assumed to be initially travelling at the velocity of 0 m/s. When s, the lead vehicle’s velocity increases from initial value and the acceleration is 2 m/s^{2}. Finally it reaches the value of 20 m/s. Taking inverse Laplace transforms for (7), the fitness function is obtained:
where s.

The size of the swarm is 50 and the number of iterations is 300. Parameters and are varied in the interval in step of 0.01. The parameters and are set as and . The inertia weight [22] is described by where and are the maximal and minimum values of the inertia weight, and they are set as and . is the maximal iteration time and is current iteration time.

The optimization procedure of the PSO method is given by the following steps.

*Step 1. *Initialize a population of particles with random positions and velocities.

*Step 2. *Evaluate each particle’s fitness value.

*Step 3. *Compare each particle’s fitness with the particle’s . If the current value is better than , then set the value equal to the current value and the location equal to the current location.

*Step 4. *Compare the fitness with the population’s overall previous best . If the current value is better than , then reset the value to the current particle’s array index and value.

*Step 5. *Update each particle’s velocity and position according to (12).

*Step 6. *Return to Step 2 until the maximum number of iterations is reached.

##### 4.2. Simulations

Through the particle swarm optimization method, we obtain the values of parameters which are and . The curve of is obtained, as shown in Figure 3.

Then transfer functions are described by

The Bode Diagrams of two transfer functions are shown in Figure 4. We can see that the frequence response meets the second condition of the platoon.

**(a)**

**(b)**

According to the longitudinal control law, the diagram of platoon is shown in Figure 5.

**(a) Spacing of the platoon**

**(b) Position of vehicles**

**(c) Velocity of vehicles**

**(d) Acceleration of vehicles**

The simulation results show that the deviations of vehicles from their preassigned positions do not exceed 0.06 m. The accelerations of vehicles in the platoon are within the range of acceptable comfort limits.

#### 5. Conclusion

In this paper, a mathematical model is built for the platoon’s longitudinal movement and a longitudinal control law is analyzed in detail. It is well known that the parameter calibration of the platoon is a difficult problem. However, the PSO method effectively finds the parameters by storing the previous knowledge of particles and estimating the best positions and achieves the computation time of 2.7 s. The Automated Highway System is seen as a better way to deal with the problem of the traffic congestion. In general, the road conditions are complex. A number of studies should be done to analyze the effects of disturbances and modeling errors which may need further investigation.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was supported by the National Natural Science Foundation of China under Grants 61040056 and 61070127 and Shanghai Key Project of Foundation funding under Grant 09JC1414600, China.