# Neural Adaptive Sliding-Mode Control of a Bidirectional Vehicle Platoon with Velocity Constraints and Input Saturation

Authors: Maode Yan; Jiacheng Song; Panpan Yang; Lei Zuo
Journal: Complexity
 (2018)

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## Abstract

This paper investigates the vehicle platoon control problems with both velocity constraints and input saturation. Firstly, radial basis function neural networks (RBF NNs) are employed to approximate the unknown driving resistance of a vehicle’s dynamic model. Then, a bidirectional topology, where vehicles can only communicate with their direct preceding and following neighbors, is used to depict the relationship among the vehicles in the platoon. On this basis, a neural adaptive sliding-mode control algorithm with an anti-windup compensation technique is proposed to maintain the vehicle platoon with desired distance. Moreover, the string stability and the strong string stability of the whole vehicle platoon are proven through the stability theorem. Finally, numerical simulations verify the feasibility and effectiveness of the proposed control method.


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## Body

## 1. Introduction

The vehicle platoon, which means that all vehicles in a group communicate with each other and regulate their motion to reach the desired intervehicle distance, has received considerable attention in recent years [1, 2]. Its prominent advantages in enhancing traffic safety and improving highway capacity as well as reducing carbon emission have been well studied [3, 4].Vehicle platoon control can be traced back to the PATH (Partners for Advanced Transit and Highways) program in the 1980s, in which many well-known issues are studied, such as control architecture, collision avoidance, and string stability [5–7]. Since then, many other topics on the vehicle platoon control have been discussed in terms of the range policy [8, 9], the communication topology [10–12], and the dynamic heterogeneity [13]. Moreover, some advanced platoon control algorithms have been proposed for the vehicle dynamics model under the framework of multiagent consensus control [3, 4] and the framework of sliding-mode control [14, 15]. However, most of them do not consider the realistic problems in terms of input saturation, velocity constraints, and model nonlinearities.Actually, input saturation and velocity constraints are two important realistic problems in vehicle platoon systems, which may lead to performance degradation and instability of vehicles or even instability of the whole vehicle platoon system [16, 17]. Input saturation in the vehicle denotes that the throttle output characteristics of the vehicle are restricted subject to the physical limitations of actuators. And velocity constraints represent the discrepancy of velocity between actual reachability values and theoretical values, which are caused by throttle output characteristics, limitations of gearbox, and so on [18]. Actually, in many known algorithms, simulation results of velocity appear to have positive and negative shocks at the initial phase because of the random initial errors. However, this is the very last thing we needed or wanted. Thus, it is necessary to develop the study about vehicle platoon control with input saturation and velocity constraints. In fact, input saturation and velocity constraints have been studied independently in some literatures [19–22]. However, to the best of our knowledge, few studies take into account the influence of input saturation and velocity constraints simultaneously.To solve the input saturation problem, a feasible approach is to add a smooth functiontanh⋅ to the algorithm such that the proposed algorithm can exert great control effect on a single target [23–25]. For the platoon system, a similar method is used to design the distributed control algorithm such that the control inputs are limited in a specified range [26, 27]. However, the bounds of control inputs are associated with the number of vehicles in the platoon and the value of control gains in the above works. There is another method to deal with the input saturation considering needlessly the influence of the number of vehicles [28, 29]. In [28], a modified control design framework is presented to prevent the destroyed learning capabilities in the presence of input saturation and is extended for the control of an aircraft model. In [29], an adaptive coordinated control algorithm is proposed on the basis of [28] to control a multi-high-speed train system with input saturation. Moreover, literature that studies particularly the problem of velocity constraints in vehicle platoon control seems few. However, studies about state constraints of a cooperative control system in theory are a hot study area [19, 30, 31]. In [19], considering the influence of input or state constraints, a unified approach is proposed to analyze and design consensus control laws for a multiagent system. Discarded consensus algorithms are also effective methods to handle the problem of state or input constraints [32, 33].Leveraging the above works, vehicles in the platoon are designed by only communicating with their direct preceding and following neighbors in this paper. And a neural adaptive sliding-mode control algorithm is proposed to handle the problems of vehicle platoon control with both input saturation and velocity constraints, simultaneously and systematically. The main features of this paper can be summarized as(i)
RBF NNs are used to learn online the unknown driving resistance such that it can be described in mathematics accurately. And the adaptive technique is used to estimate the optimal weight vectors in RBF NNs. In order to improve the robustness of the system, a sliding surface, which can reduce the disturbance from unknown driving resistance, is constructed
(ii)
Considering the problems of vehicle platoon control with velocity constraints and input saturation, an anti-windup compensation method is utilized to attenuate the integral windup of the adaptive control laws and velocity. Furthermore, a neural adaptive sliding-mode control algorithm is proposed to guarantee the string stability (Definition1) and the strong string stability (Definition 2) of the whole vehicle platoonThe remainder of this paper is organized as follows. The preliminaries and the problem formulation are described in Section2. In Section 3, the control algorithms of vehicle platoon systems are proposed. Simulations are performed in Section 4 to demonstrate the effectiveness of our algorithms. The conclusion is given in Section 5.


## 2. Problem Formulation and Preliminaries

Consider a vehicle platoon with following longitudinal dynamics:(1)ṙit=vit,Miv̇it=Fit−firi,vi,t,i=1,2,…,n,where Mi is the mass of the ith vehicle. rit and vit denote the position and velocity, respectively, of the ith vehicle. Fit represents the actuator force of the ith vehicle. In addition, firi,vi,t describes the unknown driving resistance.In order to simplify the design process of the controller, the dynamics of the vehicle can be rewritten as(2)ṙit=vit,v̇it=uit−giri,vi,t,i=1,2,…,n,where uit≔Fit/Mi, giri,vi,t≔firi,vi,t/Mi.In practice, the control inputs and velocities of vehicles are curbed because of the physical constraints of the actuator. In order to show the saturation characteristic in vehicle dynamics, the control inputuit and the velocity vit with saturation are described in the following form:
(3a)ui0t=satumin,uit,umax=umin,ifuit<umin,uit,ifumin≤uit≤umax,umax,ifuit>umax,(3b)vi0t=satvmin,vit,vmax=vmin,ifvit<vmin,vit,ifvmin≤vit≤vmax,vmax,ifvit>vmax,where ui0t, the saturated control input, is the real control input used in the controller. vi0t denotes the constrained velocity. umin and umax are known constants, which represent the bounds of the control force. vmin and vmax denote the bounds of the vehicle’s velocity.Then, the dynamics of vehicle can be rewritten as(4)ṙit=vit,v̇it=uit−giri,vi,t,uit=ui0t,vit=vi0t,i=1,2,…,n.The structure of the vehicle platoon is described by Figure1 which consists of a leader and n followers. Bidirectional topology is adopted to ensure that every vehicle in the platoon communicates with its neighboring preceding and following vehicles in this paper.Remark 1.
Bidirectional topology means that only the information of the forward and backward vehicles are transferred to the target vehicle by communication facilities. In other words, bidirectional topology can reduce the complexity of communication among the vehicles in the platoon compared with other topologies used in algorithms under the framework of multiagent consensus control (the information of every vehicle in the platoon is needed). Consequently, the communication load of the proposed algorithm is reduced.Figure 1
Bidirectional topology of the vehicle platoon.The main control objectives of this paper can be described as follows:(i)
The string stability of the vehicle platoon system can be guaranteed, which means that (1) the position tracking error of each vehicle in the platoon is kept bounded, i.e.,er,i=ri−1−ri−d≤ρ, where ρ is a small positive constant, d indicates the prespecified distance between two intervehicles, and er,i represents the position tracking error and (2) the Lyapunov functions of individual vehicle and whole vehicle platoon system satisfy the stability proof
(ii)
The strong string stability of the whole platoon system can be guaranteed, i.e.,er,n≤er,n−1≤⋯≤er,1
(iii)
The control input and velocity of each vehicle are limited in the reachable rangeTo this end, the following definitions, assumptions, and lemmas are used throughout the paper.Assumption 1.
The desired velocityvL and its derivative v̇L are known and bounded.Assumption 2 [34].
Thegiri,vi,t is an unknown bounded smooth function.Remark 2.
Assumptions1 and 2 are standard conditions required in many control schemes, which can be commonly found in the existing literature [35].Definition 1 [36] (string stability).
Originer,i=0 with vehicle dynamics modeled by (10) is string stable, if given any ι>0. There exists σ>0 such that
(5)er,i0∞<σ⇒supier,i·∞<ι,i=1,2,…,n.Definition 2 [37] (strong string stability).
Originer,i=0 with each vehicle’s dynamics modeled by (10) is strong string stable if error propagation transfer function Gis≔Ei+1s/Eis satisfies ∣Gis∣<1 for all i=1,2,…,n.Lemma 1 [38].
There is a continuous functionVt≥0, and V0 is bounded. Then, Vt is bounded if the following inequality holds:
(6)V̇t≤−p1Vt+p2,where p1>0 and p2 are constants.Lemma 2 [39, 40].
RBF NNs can approximate online an unknown smooth functionQz in the form of Qz=WTΨz, where z∈Rq denotes the input of the neural network and q represents the dimension of the neural network input. W=w1,w2,…,wmT, where wl is the parameter vector and m indicates the number of neurons. Ψz=φ1z⋯φmzT, where φlz is Gaussian functions
(7)φlz=exp−z−μlTz−μlηl2,l=1,2,…,m,where μl and ηl are the centers and widths, respectively, of the Gaussian functions. RBF NNs can approximate Qz to arbitrary accuracy by setting numerous hidden neurons
(8)Qz=W∗TΨz+εz.
The approximation errorεz can be arbitrarily adjusted to be small by choosing the ideal bounded weight vector. And εz≤ε¯≤∞ is a small positive constant
(9)W∗≔argminW⊂Rqsupz∈ΩzQz−WTΨz.


## 3. Main Results

In this section, we will give the detail steps of the controller design. First step: RBF NNs are adopted to establish the model of the vehicle, and the sliding surface is further constructed; second step: the platoon system is constructed based on bidirectional topology; third step: a neural adaptive sliding-mode algorithm is proposed to handle the velocity constraints and input saturation for a vehicle platoon.


### 3.1. First Step: Establish the Vehicle Model Based on RBF NNs and Construct the Sliding Surface

In order to estimate the unknown driving resistance, RBF NNs are adopted to approximate online thegiri,vi,t, and we further establish the model
(10)v̇it=uit−Wi∗Ψiz−εiz,where z denotes the input of RBF NNs and z=er,i,ėr,i. er,i represents the position tracking error of the ith vehicle and is defined as
(11)er,i=ri−1−ri−d,where d indicates the prespecified distance between two intervehicles.To overcome the degradation of system transient performance caused by large nonzero initial position tracking error, a modified tracking error is defined as(12)e¯r,it=er,it−χit,with
(13)χit=er,i0+ζier,i0+ėr,i0te−ζit,where er,i0=er,it∣t=0,ėr,i0=ėr,it∣t=0 and ζi is a positive constant. Thus, we have
(14)e¯r,it∣t=0=0,e¯̇r,it∣t=0=0.It is obvious thate¯r,it converges to er,it when χit converges to zero, where the rate of convergence is determined by ζi. In addition, in order to improve the robustness of the proposed controller, a sliding surface is constructed as
(15)si=e¯̇r,i+αie¯r,i,where αi is a positive constant. It is known that the convergence and boundness of the sliding surface si and e¯r,it are equivalent.Remark 3.
The importance ofχit is that it can transform the nonzero initial position tracking error problem to a zero initial position tracking error problem. And it can make that all the system states are on the sliding hyperplane at the initial instant, i.e., si0=0. As a result, the reaching phase of the sliding-mode control is eliminated and the transient response of the system is improved.


### 3.2. Second Step: Construct the Bidirectional Communication Topology

It is worth noting that (15) only describes the dynamic characteristics of a single vehicle. To describe the stability of the whole platoon, we adopt the coupled sliding surface (CSS) to establish the relationship between the ith and the i+1th vehicle:
(16)Si=βisi−si+1.It is clear thatsn+1 does not exist, so we set sn+1=0, then, Sn=βnsn. Furthermore, we define matrices S1=s1,s2,…,snT and S2=S1,S2,…,SnT to depict the total vehicle platoon system. The relationship between S1 and S2 can be described as
(17)S2=BS1,where
(18)B=β1−10⋯00β2−1⋯0⋮⋮⋮⋱⋮000⋯−1000⋯βn.It is necessary to know that the parameterβi, i=1,2,…,n, can be designed. If we choose βi>0, the matrix B is nonsingular. Thus, the asymptotic convergence and the ultimate boundedness of Si are equivalent to si.Therefore, taking the time derivative ofSi in (16) yields
(19)Ṡi=βiṡi−ṡi+1=βie¯¨r,i+αie¯̇r,i−e¯¨r,i+1+αi+1e¯̇r,i+1=βir¨i−1−r¨i−χ¨i+αiṙi−1−ṙi−χ̇i−r¨i−r¨i+1−χ¨i+1+αi+1ṙi−ṙi+1−χ̇i+1=−βi+1uit−Wi∗Ψiz−εiz+Di,where Di=βir¨i−1−χ¨i+αiṙi−1−ṙi−χ̇i−−r¨i+1−χ¨i+1+αi+1ṙi−ṙi+1−χ̇i+1.Particularly, we know thatSn=βnsn when i=n. The time derivative of Sn can be written as
(20)Ṡn=βnṡn=βne¯¨r,n+αne¯̇r,n=βnr¨n−1−r¨n−χ¨n+αnṙn−1−ṙn−χ̇n=−βnunt−Wn∗Ψnz−εnz+Dn,where Dn=βnr¨n−1−χ¨n+αnṙn−1−ṙn−χ̇n.Remark 4.
It is clear that CSS includes the information from the neighbor vehicles (direct predecessor and direct follower) of (19) and (20). It means that the CSS achieves the bidirectional topology for vehicle platoon.


### 3.3. Third Step: Design the Control Algorithm by Considering the Velocity Constraints and Input Saturation

Actually, the velocity constraints and input saturation appear frequently in application and have been proven to be the main source of performance degradation [41]. To overcome the degradation of system transient performance caused by saturation, an anti-windup compensator method is utilized to modify the tracking error. The signal ϕit is generated as the output of the following equation:
(21a)ϕ̇it=−ψiϕit+βi+1ui0t−uit,(21b)ϕ̇nt=−ψnϕnt+βnun0t−unt.Letδit=Sit−ϕit. The new tracking error can be written as
(22a)δ̇i=−βi+1ui−Wi∗Ψiz−εiz−βi+1ui0−ui+Di+ψiϕi=−βi+1ui0−ΘiTHi+Di+ψiϕi,(22b)δ̇n=−βnun−Wn∗Ψnz−εnz+Dn−βnun0−un+ψnϕn=−βnun0−ΘnTHn+Dn+ψnϕn,where Hi=Ψiz,1T, ΘiT=Wi∗,εiz, Hn=Ψnz,1T, and ΘnT=Wn∗,εnz.Remark 5.
The function of the signalϕit is to attenuate the integral windup of the control law and velocity when velocity constraints or input saturation occur. In case of velocity constraints or input saturation (vi0−vi≠0 or ui0−ui≠0), the signal Si will rise and the constructed signal ϕit will also rise, which, in turn, ensures that the newly defined tracking error δit cannot rise abruptly.In addition, for the signalϕit, we have the following lemma.Lemma 3
For the signalϕit, it holds for the following equation for any t>0:
(23)ϕit≤Zi+ϖi,i=1,2,…,n,where Zi=βi+1ui0−ui/ψi. ϖi is a positive constant.Proof.
The output of (21a) and (21b) can be described as
(24)ϕit=ϕi0e−ψit+Zie−ψit∫0teψiϖdϖ=ϕi0e−ψit+Zi1−e−ψit.
From (24), we know that ϕit≤Zi as t→∞. Thus, for any t>0, if there exists a positive scalar ϖi, ϕit≤Zi+ϖi can be ensured, i.e., ϕit is bounded.Furthermore, the control law of theith vehicle is formulated as
(25)uit=k1βi+1δi+1βi+1Di+Θ^i∗THi+1βi+1ψiϕi,i=1,2,…,n−1,where Θi∗T=Wi∗,ε¯i and Θ^i∗T=W^i∗,ε¯^i. W^i∗ and ε¯^i represent the estimated value of Wi∗ and ε¯i, respectively. It is necessary to point out that εiz≤ε¯i and ε¯i is a small positive constant.The control law of thenth vehicle is designed as
(26)unt=k2βnδn+1βnDn+Θ^n∗THn+1βnψnϕn,where Θn∗T=Wn∗,ε¯n and Θ^n∗T=W^n∗,ε¯^n. W^n∗ and ε¯^n represent the estimated value of Wn∗ and ε¯n, respectively. εnz≤ε¯n and ε¯n is a small positive constant.The results about the vehicle platoon with velocity constraints and input saturation are as follows.Theorem 1
Consider a vehicle platoon modeled as (10) with velocity constraints and input saturation. The proposed control laws (3a) and (3b), (25), and (26) are used to control each vehicle. In addition, we design the following adaptive estimation laws:
(27a)Θ^̇i∗=Γ1βi+1Hiδi−ξ1iΘ^i∗,i=1,2,…,n−1,(27b)Θ^̇n∗=Γ2βnHnδn−ξ1nΘ^n∗,where ξ1i, ξ1n, Γ1=λi1,λi2T, and Γ2=λn1,λn2T are the parameters to be designed. Then, we have the following results:
(i)
The new tracking errorδit, coupled sliding surface Si, and position tracking error er,i will converge to an arbitrarily small value
(ii)
Meanwhile, the stability of each vehicle in the platoon and the string stability of the total vehicle platoon system can be guaranteed
(iii)
The strong string stability of the vehicle platoon can be ensured, i.e.,er,n≤er,n−1≤⋯≤er,1Proof.
For the caseui−ui0≠0 or vi−vi0≠0, the vehicles move under the control inputs (25) and (26). We design the following Lyapunov function candidate:
(28a)V=∑i=1nVi,(28b)Vi=12δi2+12Θ~i∗TΓ1−1Θ~i∗,where Θ~i∗T=W^i∗−Wi∗,ε¯^i−ε¯i, i=1,2,…,n.
Taking the time derivative of (28b), it follows
(29)V̇i=δiδ̇i+Θ~i∗TΓ1−1Θ~̇i∗T=δi−βi+1k1βi+1δi+Θ^i∗THi−ΘiTHi+1βi+1Di+1βi+1ψiϕi+Di+ψiϕi+Θ~i∗TΓ1−1Θ~̇i∗T≤δi−βi+1k1βi+1δi+Θ^i∗THi−Θi∗THi+1βi+1Di+1βi+1ψiϕi+Di+ψiϕi+Θ~i∗TΓ1−1Θ~̇i∗T=−k1δi2−βi+1δiΘ~i∗THi+Θ~i∗TΓ1−1Θ~̇i∗T=−k1δi2−ξ1iΘ~i∗TΘ^i∗,i=1,2,…,n−1.
Particularly, wheni=n, the time derivative of (28b) can be written as
(30)V̇n=δnδ̇n+Θ~n∗TΓ2−1Θ~̇n∗T=δn−βnk2βnδn+Θ^n∗THn−ΘnTHn+1βnDn+1βnψnϕn+Dn+ψnϕn+Θ~n∗TΓ2−1Θ~̇n∗T≤δn−βnk2βnδn+Θ^n∗THn−Θn∗THn+1βnDn+1βnψnϕn+Dn+ψnϕn+Θ~n∗TΓ2−1Θ~̇n∗T=−k2δn2−ξ1nΘ~n∗TΘ^n∗.
According to the Young’s inequality, we have(31)−ξ1iΘ~i∗TΘ^i∗=−ξ1iΘ~i∗TΘi∗+Θ~i∗≤ξ1iΘi∗TΘi∗2−ξ1iΘ~i∗TΘ~i∗2.
Then, (29) and (30) can be written as
(32a)V̇i≤−k1δi2+ξ1iΘi∗TΘi∗2−ξ1iΘ~i∗TΘ~i∗2,i=1,2,…,n−1,(32b)V̇n≤−k2δn2+ξ1nΘn∗TΘn∗2−ξ1nΘ~n∗TΘ~n∗2,i=n.
Meanwhile, chooseΓ1>2 and Γ2>2 and further define
(33a)φi=ξ1iΘi∗TΘi∗2,φn=ξ1nΘn∗TΘn∗2,(33b)γ1i=mink1,ξ1i2,γ1n=mink2,ξ1n2,(33c)γ2i=max12,Γ1−12,γ2n=max12,Γ2−12.
Then(34a)V̇i≤−γ1iγ2iVi+φi=−ω1Vi+φi,(34b)V̇n≤−γ1nγ2nVn+φn=−ω2Vn+φn,where ω1=γ1i/γ2i and ω2=γ1n/γ2n.
According to Lemma1, we know that Vi is bounded. And Vi≤Vi0+φi/ω (when i=1,2,…,n−1, ω=ω1; i=n, ω=ω2) with Vi0 being the initial value of Vi when t→∞. Furthermore, we can know that the signal δi and the coefficient estimation error Θ~i∗ converge to the following compact set:
(35a)12δi2≤Vi0+φiω⇒δi≤2Vi0+2φiω,(35b)Θ~i∗j≤2λijVi0+2λijφiω,j=1,2.
This implies that the tracking performance and stability of each vehicle can be guaranteed. Meanwhile, the estimation error of driving resistance can be limited to a bounded value. Then, for the string stability of the vehicle platoon, the time derivative of global Lyapunov candidates is written as(36)V̇≤−ω∑i=1nVi+∑i=1nφi=−ωV+∑i=1nφi.
Therefore,(37)0≤V≤∑i=1nφiω+exp−ωtV0−∑i=1nφiω.
Furthermore,(38)δi≤2∑i=1nφiω+2exp−ωtV0−∑i=1nφiω.
Thus,(39)δi≤2∑i=1nφiω+2V0=B+,where B+ is a positive constant.
From the above analysis, it can be seen thatδit will be limited in the range of 0 to B+. Therefore, we know that Si will be limited in the range of 0 to B++ϕi, which can be adjusted to an arbitrary small value by designing the ideal parameters χi and ξ1i. It means that the string stability of the vehicle platoon is guaranteed.
Next, we will prove the strong string stability of the total vehicle platoon system. SinceSi=βisi−si+1=B++Zi∗+ϖi, we can get
(40)βiėr,i+αier,i=ėr,i1+αi+1er,i+1+B++Zi∗+ϖi.
Taking the Laplace transform of (40), it follows that
(41)βisEr,is+αiEr,is=B++Zi∗+ϖi+sEr,i1+1s+αi+1Er,i+1s.
Lettingαi=αi+1, we have
(42)Gis=Er,i+1sEr,is≤s+αi+1Er,i+1s+B++Zi∗+ϖis+αiEr,is=βi,where B++∣Zi∗∣+ϖi>0.
Thus, the strong string stability can be guaranteed ifβi satisfies 0<∣βi∣<1.


## 3.1. First Step: Establish the Vehicle Model Based on RBF NNs and Construct the Sliding Surface

In order to estimate the unknown driving resistance, RBF NNs are adopted to approximate online thegiri,vi,t, and we further establish the model
(10)v̇it=uit−Wi∗Ψiz−εiz,where z denotes the input of RBF NNs and z=er,i,ėr,i. er,i represents the position tracking error of the ith vehicle and is defined as
(11)er,i=ri−1−ri−d,where d indicates the prespecified distance between two intervehicles.To overcome the degradation of system transient performance caused by large nonzero initial position tracking error, a modified tracking error is defined as(12)e¯r,it=er,it−χit,with
(13)χit=er,i0+ζier,i0+ėr,i0te−ζit,where er,i0=er,it∣t=0,ėr,i0=ėr,it∣t=0 and ζi is a positive constant. Thus, we have
(14)e¯r,it∣t=0=0,e¯̇r,it∣t=0=0.It is obvious thate¯r,it converges to er,it when χit converges to zero, where the rate of convergence is determined by ζi. In addition, in order to improve the robustness of the proposed controller, a sliding surface is constructed as
(15)si=e¯̇r,i+αie¯r,i,where αi is a positive constant. It is known that the convergence and boundness of the sliding surface si and e¯r,it are equivalent.Remark 3.
The importance ofχit is that it can transform the nonzero initial position tracking error problem to a zero initial position tracking error problem. And it can make that all the system states are on the sliding hyperplane at the initial instant, i.e., si0=0. As a result, the reaching phase of the sliding-mode control is eliminated and the transient response of the system is improved.


## 3.2. Second Step: Construct the Bidirectional Communication Topology

It is worth noting that (15) only describes the dynamic characteristics of a single vehicle. To describe the stability of the whole platoon, we adopt the coupled sliding surface (CSS) to establish the relationship between the ith and the i+1th vehicle:
(16)Si=βisi−si+1.It is clear thatsn+1 does not exist, so we set sn+1=0, then, Sn=βnsn. Furthermore, we define matrices S1=s1,s2,…,snT and S2=S1,S2,…,SnT to depict the total vehicle platoon system. The relationship between S1 and S2 can be described as
(17)S2=BS1,where
(18)B=β1−10⋯00β2−1⋯0⋮⋮⋮⋱⋮000⋯−1000⋯βn.It is necessary to know that the parameterβi, i=1,2,…,n, can be designed. If we choose βi>0, the matrix B is nonsingular. Thus, the asymptotic convergence and the ultimate boundedness of Si are equivalent to si.Therefore, taking the time derivative ofSi in (16) yields
(19)Ṡi=βiṡi−ṡi+1=βie¯¨r,i+αie¯̇r,i−e¯¨r,i+1+αi+1e¯̇r,i+1=βir¨i−1−r¨i−χ¨i+αiṙi−1−ṙi−χ̇i−r¨i−r¨i+1−χ¨i+1+αi+1ṙi−ṙi+1−χ̇i+1=−βi+1uit−Wi∗Ψiz−εiz+Di,where Di=βir¨i−1−χ¨i+αiṙi−1−ṙi−χ̇i−−r¨i+1−χ¨i+1+αi+1ṙi−ṙi+1−χ̇i+1.Particularly, we know thatSn=βnsn when i=n. The time derivative of Sn can be written as
(20)Ṡn=βnṡn=βne¯¨r,n+αne¯̇r,n=βnr¨n−1−r¨n−χ¨n+αnṙn−1−ṙn−χ̇n=−βnunt−Wn∗Ψnz−εnz+Dn,where Dn=βnr¨n−1−χ¨n+αnṙn−1−ṙn−χ̇n.Remark 4.
It is clear that CSS includes the information from the neighbor vehicles (direct predecessor and direct follower) of (19) and (20). It means that the CSS achieves the bidirectional topology for vehicle platoon.


## 3.3. Third Step: Design the Control Algorithm by Considering the Velocity Constraints and Input Saturation

Actually, the velocity constraints and input saturation appear frequently in application and have been proven to be the main source of performance degradation [41]. To overcome the degradation of system transient performance caused by saturation, an anti-windup compensator method is utilized to modify the tracking error. The signal ϕit is generated as the output of the following equation:
(21a)ϕ̇it=−ψiϕit+βi+1ui0t−uit,(21b)ϕ̇nt=−ψnϕnt+βnun0t−unt.Letδit=Sit−ϕit. The new tracking error can be written as
(22a)δ̇i=−βi+1ui−Wi∗Ψiz−εiz−βi+1ui0−ui+Di+ψiϕi=−βi+1ui0−ΘiTHi+Di+ψiϕi,(22b)δ̇n=−βnun−Wn∗Ψnz−εnz+Dn−βnun0−un+ψnϕn=−βnun0−ΘnTHn+Dn+ψnϕn,where Hi=Ψiz,1T, ΘiT=Wi∗,εiz, Hn=Ψnz,1T, and ΘnT=Wn∗,εnz.Remark 5.
The function of the signalϕit is to attenuate the integral windup of the control law and velocity when velocity constraints or input saturation occur. In case of velocity constraints or input saturation (vi0−vi≠0 or ui0−ui≠0), the signal Si will rise and the constructed signal ϕit will also rise, which, in turn, ensures that the newly defined tracking error δit cannot rise abruptly.In addition, for the signalϕit, we have the following lemma.Lemma 3
For the signalϕit, it holds for the following equation for any t>0:
(23)ϕit≤Zi+ϖi,i=1,2,…,n,where Zi=βi+1ui0−ui/ψi. ϖi is a positive constant.Proof.
The output of (21a) and (21b) can be described as
(24)ϕit=ϕi0e−ψit+Zie−ψit∫0teψiϖdϖ=ϕi0e−ψit+Zi1−e−ψit.
From (24), we know that ϕit≤Zi as t→∞. Thus, for any t>0, if there exists a positive scalar ϖi, ϕit≤Zi+ϖi can be ensured, i.e., ϕit is bounded.Furthermore, the control law of theith vehicle is formulated as
(25)uit=k1βi+1δi+1βi+1Di+Θ^i∗THi+1βi+1ψiϕi,i=1,2,…,n−1,where Θi∗T=Wi∗,ε¯i and Θ^i∗T=W^i∗,ε¯^i. W^i∗ and ε¯^i represent the estimated value of Wi∗ and ε¯i, respectively. It is necessary to point out that εiz≤ε¯i and ε¯i is a small positive constant.The control law of thenth vehicle is designed as
(26)unt=k2βnδn+1βnDn+Θ^n∗THn+1βnψnϕn,where Θn∗T=Wn∗,ε¯n and Θ^n∗T=W^n∗,ε¯^n. W^n∗ and ε¯^n represent the estimated value of Wn∗ and ε¯n, respectively. εnz≤ε¯n and ε¯n is a small positive constant.The results about the vehicle platoon with velocity constraints and input saturation are as follows.Theorem 1
Consider a vehicle platoon modeled as (10) with velocity constraints and input saturation. The proposed control laws (3a) and (3b), (25), and (26) are used to control each vehicle. In addition, we design the following adaptive estimation laws:
(27a)Θ^̇i∗=Γ1βi+1Hiδi−ξ1iΘ^i∗,i=1,2,…,n−1,(27b)Θ^̇n∗=Γ2βnHnδn−ξ1nΘ^n∗,where ξ1i, ξ1n, Γ1=λi1,λi2T, and Γ2=λn1,λn2T are the parameters to be designed. Then, we have the following results:
(i)
The new tracking errorδit, coupled sliding surface Si, and position tracking error er,i will converge to an arbitrarily small value
(ii)
Meanwhile, the stability of each vehicle in the platoon and the string stability of the total vehicle platoon system can be guaranteed
(iii)
The strong string stability of the vehicle platoon can be ensured, i.e.,er,n≤er,n−1≤⋯≤er,1Proof.
For the caseui−ui0≠0 or vi−vi0≠0, the vehicles move under the control inputs (25) and (26). We design the following Lyapunov function candidate:
(28a)V=∑i=1nVi,(28b)Vi=12δi2+12Θ~i∗TΓ1−1Θ~i∗,where Θ~i∗T=W^i∗−Wi∗,ε¯^i−ε¯i, i=1,2,…,n.
Taking the time derivative of (28b), it follows
(29)V̇i=δiδ̇i+Θ~i∗TΓ1−1Θ~̇i∗T=δi−βi+1k1βi+1δi+Θ^i∗THi−ΘiTHi+1βi+1Di+1βi+1ψiϕi+Di+ψiϕi+Θ~i∗TΓ1−1Θ~̇i∗T≤δi−βi+1k1βi+1δi+Θ^i∗THi−Θi∗THi+1βi+1Di+1βi+1ψiϕi+Di+ψiϕi+Θ~i∗TΓ1−1Θ~̇i∗T=−k1δi2−βi+1δiΘ~i∗THi+Θ~i∗TΓ1−1Θ~̇i∗T=−k1δi2−ξ1iΘ~i∗TΘ^i∗,i=1,2,…,n−1.
Particularly, wheni=n, the time derivative of (28b) can be written as
(30)V̇n=δnδ̇n+Θ~n∗TΓ2−1Θ~̇n∗T=δn−βnk2βnδn+Θ^n∗THn−ΘnTHn+1βnDn+1βnψnϕn+Dn+ψnϕn+Θ~n∗TΓ2−1Θ~̇n∗T≤δn−βnk2βnδn+Θ^n∗THn−Θn∗THn+1βnDn+1βnψnϕn+Dn+ψnϕn+Θ~n∗TΓ2−1Θ~̇n∗T=−k2δn2−ξ1nΘ~n∗TΘ^n∗.
According to the Young’s inequality, we have(31)−ξ1iΘ~i∗TΘ^i∗=−ξ1iΘ~i∗TΘi∗+Θ~i∗≤ξ1iΘi∗TΘi∗2−ξ1iΘ~i∗TΘ~i∗2.
Then, (29) and (30) can be written as
(32a)V̇i≤−k1δi2+ξ1iΘi∗TΘi∗2−ξ1iΘ~i∗TΘ~i∗2,i=1,2,…,n−1,(32b)V̇n≤−k2δn2+ξ1nΘn∗TΘn∗2−ξ1nΘ~n∗TΘ~n∗2,i=n.
Meanwhile, chooseΓ1>2 and Γ2>2 and further define
(33a)φi=ξ1iΘi∗TΘi∗2,φn=ξ1nΘn∗TΘn∗2,(33b)γ1i=mink1,ξ1i2,γ1n=mink2,ξ1n2,(33c)γ2i=max12,Γ1−12,γ2n=max12,Γ2−12.
Then(34a)V̇i≤−γ1iγ2iVi+φi=−ω1Vi+φi,(34b)V̇n≤−γ1nγ2nVn+φn=−ω2Vn+φn,where ω1=γ1i/γ2i and ω2=γ1n/γ2n.
According to Lemma1, we know that Vi is bounded. And Vi≤Vi0+φi/ω (when i=1,2,…,n−1, ω=ω1; i=n, ω=ω2) with Vi0 being the initial value of Vi when t→∞. Furthermore, we can know that the signal δi and the coefficient estimation error Θ~i∗ converge to the following compact set:
(35a)12δi2≤Vi0+φiω⇒δi≤2Vi0+2φiω,(35b)Θ~i∗j≤2λijVi0+2λijφiω,j=1,2.
This implies that the tracking performance and stability of each vehicle can be guaranteed. Meanwhile, the estimation error of driving resistance can be limited to a bounded value. Then, for the string stability of the vehicle platoon, the time derivative of global Lyapunov candidates is written as(36)V̇≤−ω∑i=1nVi+∑i=1nφi=−ωV+∑i=1nφi.
Therefore,(37)0≤V≤∑i=1nφiω+exp−ωtV0−∑i=1nφiω.
Furthermore,(38)δi≤2∑i=1nφiω+2exp−ωtV0−∑i=1nφiω.
Thus,(39)δi≤2∑i=1nφiω+2V0=B+,where B+ is a positive constant.
From the above analysis, it can be seen thatδit will be limited in the range of 0 to B+. Therefore, we know that Si will be limited in the range of 0 to B++ϕi, which can be adjusted to an arbitrary small value by designing the ideal parameters χi and ξ1i. It means that the string stability of the vehicle platoon is guaranteed.
Next, we will prove the strong string stability of the total vehicle platoon system. SinceSi=βisi−si+1=B++Zi∗+ϖi, we can get
(40)βiėr,i+αier,i=ėr,i1+αi+1er,i+1+B++Zi∗+ϖi.
Taking the Laplace transform of (40), it follows that
(41)βisEr,is+αiEr,is=B++Zi∗+ϖi+sEr,i1+1s+αi+1Er,i+1s.
Lettingαi=αi+1, we have
(42)Gis=Er,i+1sEr,is≤s+αi+1Er,i+1s+B++Zi∗+ϖis+αiEr,is=βi,where B++∣Zi∗∣+ϖi>0.
Thus, the strong string stability can be guaranteed ifβi satisfies 0<∣βi∣<1.


## 4. Numerical Simulations

To evaluate the effectiveness and feasibility of the developed platoon control approaches, numerical simulations are performed in this section. We apply the designed algorithms to an 8-vehicle platoon with one leader and 7 followers.


### 4.1. Simulation Setup

The prespecified velocity profiles are described as(43)vLt=10sin0.025t,ift≤20,vLt=10,if20<t≤40,vLt=10+3sin0.1t,if40<t≤60,vLt=10,if60<t≤80,vLt=10sin0.025t,if80<t≤100.The desired distance between two consecutive vehicles is set to bed=10 m (which can be set to be arbitrary values according to actual requirements). Each vehicle’s initial values of velocity and position are set as vi=0,0,0,0,0,0,0 and ri=60,48,40,28,20,8,0, respectively. The leader’s initial values of velocity and position are set as vL=0 and rL=71, respectively. Meanwhile, the driving resistance of each vehicle is designed as gi=0.1sin0.1πt. Furthermore, the neural networks used in this simulation contain 5 neurons, where ηl=0.5 and μl=−1,−0.5,0,0.5,1. The other values are set to be zero. The control parameters are listed in Table 1.Table 1
Control parameters of the vehicles (i=1,2,…,7).
αi
βi
ψi
umin
umax
vmin
vmax
ζi
ε1i
Γ1=Γ2
k1=k2
0.2
0.9999
100
−1.3
1.3
0
13
0.5
0.005
[18,2]
8


### 4.2. Simulation Results and Discussion

Case 1.
Vehicle platoon control with velocity constraints and input saturation.The simulation results using the proposed algorithms (3a) and (3b), (25), and (26) are shown in Figure 2. The tracking performance of velocity in Figures 2(a) and 2(b) is excellent, and each vehicle in the platoon moves stably with acceptable velocity error (the biggest velocity error is 0.4 m/s in the adjustment phase and is adjusted to a small number close to zero, eventually). Meanwhile, from the local enlarged drawing in Figure 2(a), the values of velocity of the vehicles are positive, which satisfy the actual operation conditions.Figure 2
Vehicle platoon control with velocity constraints and input saturation.
(a)
Velocity curves
(b)
Velocity error curves
(c)
The distance between two vehicles
(d)
The position tracking error curves
(e)
Control input curves
(f)
Estimated driving resistanceFurthermore, it can be seen from Figures2(c) and 2(d) that the distance between two vehicles converges to the desired values of 10 m and the position tracking error is adjusted to a small number close to zero. What is more, from the local enlarged drawing in Figure 2(d), we know that the strong string stability of the vehicle platoon can be achieved, i.e., ∣er,7∣≤∣er,6∣≤⋯≤∣er,1∣, and this advantage benefits from the CSS with βi<1.Figure2(e) shows the control input curves. It is clear that the control input can be limited in a desired range (−1.3 m/s2–1.3 m/s2). Figure 2(f) shows the estimated uncertain driving resistance using RBF NNs, and the performance of estimated uncertain driving resistance is excellent.Actually, the main sources to solve the problems of velocity constraints and input saturation are the signalχit and δit. The function of the signal χit is that it can make all the system states on the sliding hyperplane at the initial instant. As a result, the reaching phase of the sliding-mode control is eliminated as well as the transient response of the system is improved (reduce adjustment error in the beginning). The function of the signal δit is that it can attenuate the integral windup of the control law when input saturation or velocity constraints occur. In case of input saturation or velocity constraints (vi0−vi≠0 or ui0−ui≠0), the signal Si will rise and the constructed signal ϕit will also rise, which, in turn, ensures that the newly defined tracking error δit cannot rise abruptly.Case 2.
Vehicle platoon control without input saturation and velocity constraints.In this part, we further track the same desired velocity curves in (43) by adopting the other related method, which has not taken the input saturation and velocity constraints into consideration. Comparing the numerical simulation results using the method in this case, the control algorithm proposed in this paper can effectively solve the problems of velocity constraints and input saturation.Figure3(a) shows the velocity tracking performance by adopting the control algorithm, which has not taken the input saturation and velocity constraints into consideration. The velocities of the vehicles reach negative values in Figure 3(a), which is unreasonable in practical applications. In addition, as shown in Figure 3(b), the control input goes up to −20 m/s2 and 20 m/s2, which cannot be used in application due to the physical structure limitations.Figure 3
Vehicle platoon control without input saturation and velocity constrains.
(a)
Velocity curves
(b)
Control input curvesTherefore, the vehicle platoon control algorithm proposed in this paper is more practical and pragmatic.


## 4.1. Simulation Setup

The prespecified velocity profiles are described as(43)vLt=10sin0.025t,ift≤20,vLt=10,if20<t≤40,vLt=10+3sin0.1t,if40<t≤60,vLt=10,if60<t≤80,vLt=10sin0.025t,if80<t≤100.The desired distance between two consecutive vehicles is set to bed=10 m (which can be set to be arbitrary values according to actual requirements). Each vehicle’s initial values of velocity and position are set as vi=0,0,0,0,0,0,0 and ri=60,48,40,28,20,8,0, respectively. The leader’s initial values of velocity and position are set as vL=0 and rL=71, respectively. Meanwhile, the driving resistance of each vehicle is designed as gi=0.1sin0.1πt. Furthermore, the neural networks used in this simulation contain 5 neurons, where ηl=0.5 and μl=−1,−0.5,0,0.5,1. The other values are set to be zero. The control parameters are listed in Table 1.Table 1
Control parameters of the vehicles (i=1,2,…,7).
αi
βi
ψi
umin
umax
vmin
vmax
ζi
ε1i
Γ1=Γ2
k1=k2
0.2
0.9999
100
−1.3
1.3
0
13
0.5
0.005
[18,2]
8


## 4.2. Simulation Results and Discussion

Case 1.
Vehicle platoon control with velocity constraints and input saturation.The simulation results using the proposed algorithms (3a) and (3b), (25), and (26) are shown in Figure 2. The tracking performance of velocity in Figures 2(a) and 2(b) is excellent, and each vehicle in the platoon moves stably with acceptable velocity error (the biggest velocity error is 0.4 m/s in the adjustment phase and is adjusted to a small number close to zero, eventually). Meanwhile, from the local enlarged drawing in Figure 2(a), the values of velocity of the vehicles are positive, which satisfy the actual operation conditions.Figure 2
Vehicle platoon control with velocity constraints and input saturation.
(a)
Velocity curves
(b)
Velocity error curves
(c)
The distance between two vehicles
(d)
The position tracking error curves
(e)
Control input curves
(f)
Estimated driving resistanceFurthermore, it can be seen from Figures2(c) and 2(d) that the distance between two vehicles converges to the desired values of 10 m and the position tracking error is adjusted to a small number close to zero. What is more, from the local enlarged drawing in Figure 2(d), we know that the strong string stability of the vehicle platoon can be achieved, i.e., ∣er,7∣≤∣er,6∣≤⋯≤∣er,1∣, and this advantage benefits from the CSS with βi<1.Figure2(e) shows the control input curves. It is clear that the control input can be limited in a desired range (−1.3 m/s2–1.3 m/s2). Figure 2(f) shows the estimated uncertain driving resistance using RBF NNs, and the performance of estimated uncertain driving resistance is excellent.Actually, the main sources to solve the problems of velocity constraints and input saturation are the signalχit and δit. The function of the signal χit is that it can make all the system states on the sliding hyperplane at the initial instant. As a result, the reaching phase of the sliding-mode control is eliminated as well as the transient response of the system is improved (reduce adjustment error in the beginning). The function of the signal δit is that it can attenuate the integral windup of the control law when input saturation or velocity constraints occur. In case of input saturation or velocity constraints (vi0−vi≠0 or ui0−ui≠0), the signal Si will rise and the constructed signal ϕit will also rise, which, in turn, ensures that the newly defined tracking error δit cannot rise abruptly.Case 2.
Vehicle platoon control without input saturation and velocity constraints.In this part, we further track the same desired velocity curves in (43) by adopting the other related method, which has not taken the input saturation and velocity constraints into consideration. Comparing the numerical simulation results using the method in this case, the control algorithm proposed in this paper can effectively solve the problems of velocity constraints and input saturation.Figure3(a) shows the velocity tracking performance by adopting the control algorithm, which has not taken the input saturation and velocity constraints into consideration. The velocities of the vehicles reach negative values in Figure 3(a), which is unreasonable in practical applications. In addition, as shown in Figure 3(b), the control input goes up to −20 m/s2 and 20 m/s2, which cannot be used in application due to the physical structure limitations.Figure 3
Vehicle platoon control without input saturation and velocity constrains.
(a)
Velocity curves
(b)
Control input curvesTherefore, the vehicle platoon control algorithm proposed in this paper is more practical and pragmatic.


## 5. Conclusion

This paper considered the vehicle platoon control problems with velocity constraints and input saturation. Firstly, RBF NNs were introduced to learn the unknown driving resistance. Next, an anti-windup compensation method was designed to avoid the tracking error rising abruptly when velocity constraints or input saturation occurred. Then, a neural adaptive sliding-mode control algorithm was proposed based on bidirectional topology. The string stability and the strong string stability of the whole platoon system were guaranteed. Numerical simulations were performed to verify the feasibility and effectiveness of the proposed control methods.



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*Source: 1696851-2018-12-02.xml*