#### Abstract

We investigate the vehicle platoon problems, where the actuator saturation and absent velocity measurement are taken into consideration. Firstly, a novel algorithm, where a smooth function is introduced to deal with the sharp corner of the input signals, is proposed for a group of vehicles with actuator saturation by using the consensus theory. Secondly, by applying an auxiliary system for the followers to estimate the velocities, a control strategy for the vehicle platoon with actuator saturation and absent velocity measurement is designed via the adaptive control approach. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed approaches.

#### 1. Introduction

Vehicle platoon is well studied in the field of formation control, which is an effective approach to enhance the traffic safety and efficiency [1]. The main objective of vehicle platoon is to avoid traffic jams and improve the traffic security. Usually, a vehicle platoon contains a group of vehicles, which are moving in a lane with certain velocities. From this definition, we can clearly find that the key point of vehicle platoon control is to adjust the velocities and positions of the vehicles, such that they can reach the desired platoon.

In the past few years, a lot of results, like sliding mode control [2], consensus [3], neural network [4], and so on [5–7], have been proposed to deal with the platoon control problems in some ways. For instance, a robust acceleration tracking control with vehicle longitudinal dynamics is presented in [8] for platoon-level automation. In [9], an adaptive bidirectional platoon control method is proposed by using a coupled sliding mode control algorithm. Furthermore, a novel distributed control architecture is given in [10] for heterogeneous platoons, where a group of linear time-invariant autonomous vehicles are considered.

To further study the vehicle platoon control problems, some limitations are taken into account. For example, in [11], range-limit sensors and delayed actuators are handled by guaranteed-cost platoon control method. Considering the capacity limitation and random packet loss, a closed-form methodology for vehicular platoon control is proposed in [12]. Moreover, a hierarchical platoon control framework is established in [13] to deal with the negative effect of the tracking parameter. The practical string stability of platoon is investigated in [14] for parasitic time delays and lags of the actuators and sensors. In [15], the authors propose control laws in the case of losing communications between the lead vehicle and the other vehicles in a platoon. Besides, disturbances, nonlinear, control input nonlinearities are also taken into consideration in many literatures [16, 17]. However, the fundamental limitations, associated with both actuator saturation and absent velocity measurement, have not been fully studied.

Actuator saturation is one of the most common constraints in practical control systems and it is still a challenge to design the control law with actuator saturation. Considerable attention is paid to the systems with actuator saturation since 1950s [18]. In [19], an online approximation is developed for automation train operation with the presence of actuator saturation. In [20], a position and heading trajectory tracking control law is developed for Swedish wheeled robots with unaccounted actuator saturation. In [21], a modified fault tolerant control strategy is presented in the flexible spacecraft to ensure that the control signal will never incur saturation.

For the vehicle platoon with absent velocity measurement, there are also some results. For instance, in [22], passive filters are employed to design an output feedback coordinated formation controller in multiple spacecrafts with absent velocity measurement. A decentralized control strategy is proposed with the neighboring topology for velocity constraints in double-integrator dynamics in [23]. In [24], an adaptive MIMO-ESO is designed to estimate the unmeasurable linear and angular velocities of underwater robots. Moreover, leader-follower output feedback synchronization scheme is given in [25] for controlling the attitude of two satellites when angular velocity measurements are not available. However, to the best of our knowledge, few results have dealt with both issues, absent velocity measurement and actuator saturation, simultaneously in vehicle platoon control.

Motivated by this fact, we propose a novel control strategy to dispose of both the cases: actuator saturation and absent velocity measurement in vehicle platoon. The main contributions of this paper are twofolds:(i)We solve the “actuator saturation” problem in vehicle platoon, which occurs when the desired force exceeds the maximum output force of the engines. Transient period or even long period of such behavior will increase the loss of engine and reduce its life. A smooth function is introduced to handle this problem such that the control inputs can always be below the maximum inputs.(ii)We solve the “velocity absent measurement” problem in vehicle platoon control, where intensive oscillation may arise without velocity measurement. We apply an auxiliary system into the control law to estimate the velocity values so that the vehicles can relax the information requirements and communication burden.

The remainder of the paper is organized as follows. Section 2 gives the problem formulation. In Section 3, the controller for vehicle platoon control systems with actuator saturation is designed. Section 4 further proposes an algorithm with both actuator saturation and absent velocity measurement. Numerical simulations are presented in Section 5. Finally, Section 6 draws the conclusion.

#### 2. Problem Formulation

Modeling is an important step in understanding the behavior of the system [26]. In this paper, we consider a group of vehicles, which are composed of one leader and followers, moving in one-dimensional plane. The topological structure of this vehicle platoon is shown in Figure 1, where is the desired distance between two consecutive vehicles.

The dynamics of the leader is shown as where and are the position and velocity of the leader, respectively.

The dynamic models of followers are described as where and are the position and velocity of vehicle , respectively. represents the mass of vehicle and denotes the control input.

In practical applications, the engine will work at maximum output status when there is actuator saturation, which will increase the loss of engine and reduce its life. In addition, due to the loss of velocity information, namely, “absent velocity measurement,” the performance of vehicle platoon control will degrade.

Therefore, the main objective of this paper is to control the velocity-measurement-absent vehicles with actuator saturation such that they can move along a platoon with the desired velocities. Meanwhile, they can keep distance from avoiding collision during the process and the safe distances can converge to a stable constant when they reach the desired states.

#### 3. Vehicle Platoon with Actuator Saturation

Due to the physical limitations of actuators, the control inputs can be written as where is the upper bound of .

As there is a sharp corner between and , it is unrealizable to use this kind of input signals in practice. Hence, we use a smooth function to approximate the control inputs, which is defined as [27]

Then, in (3) is expressed as where is a bounded function and its bound can be obtained as

As , the bound increases from 0 to as varies from 0 to . Otherwise, the bound decreases from to 0.

On this basis, an algorithm for the vehicle platoon with actuator saturation is proposed as follows:

Section (a) represents the position control, where and represents the desired distance between each two vehicles. In addition, is the weighted adjacency matrix with and is the strictly positive scalar gain.

Section (b) is the velocity control, where is also the weighted adjacency matrix with and is the strictly positive scalar gain.

Furthermore, the following lemma is introduced for the proposed control system.

Lemma 1. *Suppose , , , and . If is symmetric (), then *

Then, the theorem of platoon control with actuator saturation is presented as follows.

Theorem 2. *Consider a group of vehicles with actuator saturation in a one-dimensional platoon. By using the control law in (7), the vehicles can reach the platoon with desired distances and velocities. Namely, *(i)*;*(ii)*;*(iii)*(iv)*

*Proof. *Based on absolute value inequality, we can easily find that the control input in (7) is bounded as Let . Consider the following Lyapunov function candidate: where is a vector with all elements equal to one.

Note that is positive definite and radially unbound with respect to . Taking the derivative of , we have According to Lemma 1, we obtain that This reveals that , and hence , are globally bounded.

Since is bounded, we can conclude that is bounded. Based on* Barbălat* Lemma, we have and .

As , it leads toThen, we haveEq. (14) satisfiesAs a result, .

*Remark 3. *The purpose of using as a safe distance is to complete the proof by Lyapunov function. With the algorithm (7) proposed in this section, we can obtain that .

#### 4. Vehicle Platoon with Actuator Saturation and Absent Velocity Measurement

In this section, we take both the actuator saturation and absent velocity measurement into consideration and propose a novel platoon control law. The main strategy of this proposed method is to design an auxiliary system for the followers, so that they can estimate the velocities effectively.

Invoking the adaptive control theory, the algorithm for the vehicle platoon with both actuator saturation and absent velocity measurement can be presented as where is the weighted adjacency matrix with . and are positive scalar gains, and is defined as the scalar gain.

The auxiliary system, denoted by and , is defined as follows:where , , and are strictly positive constants, and and have arbitrarily initial values.

Theorem 4. *Consider a group of vehicles with actuator saturation and absent velocity measurement in a platoon. By using the control law in (16), the vehicles can reach the platoon with desired distances and velocities. Namely, *(i)*;*(ii)*;*(iii)*(iv)*

*Proof. *From Theorem 2, we can obtain that the control laws in (16) are bounded as Then denote . Consider the following Lyapunov function candidate where is a vector with all elements equal to one and the functions and are defined elements-wise for a vector.

Invoking (2) and (16), the time derivative of is given by From Lemma 1, we know that and . Then, we obtain thatThis shows that , and , , , and are globally bounded.

From the above derivations, we conclude that is bounded. In addition, due to the boundedness of , and are also bounded. On the basis of the above analysis, we obtain that and . Since , , and are bounded, we know that is also bounded. From* Barbălat* Lemma, we can testify that with the fact that and is bounded.

Then, we have that With , (23) satisfies Therefore, . According to* Barbălat* Lemma, we can obtain that because is bounded.

As the derivative of (18), , is bounded and is also bounded, is proven to be bounded. Then, we can conclude that , and . Furthermore, we know from (18) that .

Based on the above analysis, we can conclude that for all vehicles.

*Remark 5. *As the auxiliary system has the same structure with the actual velocity mode, the proposed approach in this paper provides an* indirect* asymptotic estimation of the velocity. By applying this auxiliary system, the platoon control system can generate the necessary damping for the overall closed stability [28, 29].

#### 5. Simulations

To verify the effectiveness of the proposed algorithms in this paper for handing both the “actuator saturation” and “absent velocity measurement” problem, we design a series of simulations, which consist of 6 followers and 1 leader. The initial conditions and safe distance are shown in Table 1.

Then, the reference velocities of the leader are presented as follows, which includes the processes of acceleration, cruise, and deceleration:

##### 5.1. Vehicle Platoon with Actuator Saturation

In this part, we only consider the control strategy in (7) and select the controller gains such as .

In the simulations, Figure 2 shows the velocities of vehicles, from which we can see that all the vehicles follow the desired velocities based on the present control law. In Figure 3, it is observed that the followers track the positions of leader vehicle and the distance between two consecutive vehicles converges to a desired constant value. In addition, we can see that there is no collision during this process.

Note that, in Figure 4, control laws generate traction force for vehicles during the acceleration phase, while they generate braking force when the desired velocities decline. In particularly, the maximum traction force and maximum braking force are limited. Aforementioned simulations confirm the effectiveness of our proposed approach with actuator saturation.

##### 5.2. Vehicle Platoon with Actuator Saturation and Absent Velocity Measurement

In this part, we consider control law (16)–(18). We set the initial values of signal as and as . Similarly, . The controller gains are chosen as . Figures 5–7 are depicted to validate the effectiveness of control algorithm in (16).

Velocities are illustrated in Figure 5, from which we can see that followers can reach the desired velocity, while oscillations present at the beginning in vehicle platoon. This is caused by the damping from auxiliary in order to handle the absent velocities. Figure 6 depicts the position trajectories of vehicles, which has the similar trend with Figure 3. As shown in Figure 6, position gaps converge to the desired value asymptotically, which indicates that platoon maneuver is achieved.

Control inputs in vehicle platoon with actuator saturation and absent velocity measurement are shown in Figure 7, where active control inputs exist in the first period of acceleration and then they can make a quick and smooth change to accommodate the various desired velocities. Furthermore, we can also see that maximum control input in Figure 7 is 12 m·s^{−2}, which is within the controllable range in (19) due to the limitations of traction behavior and braking behavior. All the simulations in this part demonstrate control strategy in (16) can achieve similar control performance as control law in (7) despite absent velocity measurement.

##### 5.3. Comparative Analysis

In order to illustrate the advantages of the proposed algorithm, we further design two comparative simulations between the proposed method in this paper and the other related methods.

Firstly, we verify the superiority of the algorithm with the smooth function in Section 4. A comparison simulation is designed for the proposed algorithm in Section 4 with an existing algorithm, where the upper bounded input is 15 m·s^{−2}, the lower bounded input is −15 m·s^{−2}, and other parameters are the same. Under the control of the algorithm in Section 4, the vehicle platoon can maintain the desired states without actuator saturation. The results are as follows.

From Figure 8, we can see that the proposed algorithm can deal with the control inputs saturation, while saturation occurs in other related algorithm.

Secondly, we verify the superiority of the algorithm with the auxiliary system in Section 4 for handling the absent velocity measurement problem. The velocity measurement can be estimated by the auxiliary system on the basis of position information. An existing algorithm without considering the absent velocity measurement is performed as the contrast. The results are as follows.

From Figure 9, we can see that, without velocity measurement, other algorithm will lead to intensive oscillation in the control inputs. However, the proposed algorithm in this paper can effectively deal with this problem and the control inputs can converge to zero within a short time. In addition, the velocities in some related literatures are varying during all the simulations.

#### 6. Conclusions

This study proposed a method to investigate vehicle platoon with actuator saturation and absent velocity measurement. Firstly, we applied the consensus theory to the vehicle platoon control strategy with actuator saturation. Based on this achievement, we further proposed vehicle platoon control algorithm for the velocity-measurement-absent vehicles with actuator saturation. The stability of these proposed vehicle platoon systems was proved through the Lyapunov theorem. Finally, numerical simulations were provided to verify the proposed approaches.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work is partially supported by the Fundamental Research Funds for the Central Universities of China (310832163403, 310832171004, and 310832101012), the Key Science and Technology Program of Shaanxi Province (2017JQ6060), and the China Postdoctoral Science Foundation Project (2017M613030).