# Distributed Adaptive Sliding Mode Control for Vehicle Platoon with Uncertain Driving Resistance and Actuator Saturation

Authors: Jia-cheng Song; Yong-feng Ju
Journal: Complexity
 (2020)

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## Abstract

Vehicle platoon has been demonstrated to be a promising driving pattern for its prominent advantages in enhancing traffic safety, improving highway capacity, and increasing fuel economy as well as reducing carbon emissions. However, the uncertain driving resistance and saturated actuator output decay the control performance and may even lead to the instability of a vehicle platoon. Therefore, a distributed adaptive sliding mode control algorithm for vehicle platoon with uncertain driving resistance and actuator saturation is proposed in this paper. First of all, sliding mode control technique, together with the coupled sliding surface (CSS) method, is adopted to design the vehicle platoon control algorithm and an adaptive updating law is proposed to estimate the unknown driving resistance coefficients. Then, for the problem of actuator saturation, an antiwindup compensation-based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in the case of actuator saturation. In addition, considering the chattering problem inherent in sliding mode control, a sigmoid-like functionsgn⋅ is deployed to weaken the influence of chattering, which is expected to enhance the driving comfortableness. Both theoretical analysis and numerical simulation verify the feasibility and effectiveness of the proposed vehicle platoon algorithm.


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## Body

## 1. Introduction

In recent years, the automated highway system (AHS) has gained considerable attentions from governments, automobile manufactures, and academia because of the increasing traffic congestion problem in large cities [1–3]. As an effective measure to relieve congestions, platoon control [4, 5], which requires vehicles to move in a string with predefined intervehicle distance and with the same velocity, has demonstrated its unique advantage in enhancing traffic safety, improving highway capacity, increasing fuel economy, and reducing carbon emissions [6–8].During the past few years, many achievements have been developed for vehicle platoon control, such as backstepping approach [9], information consensus approach [10, 11], adaptive sliding mode approach [12], etc. However, most of the above works treat the vehicle platoon control problem in a simplified fashion without considering the driving resistance inherent in the vehicle dynamics. As a matter of fact, driving resistance, which includes rolling resistance, air resistance, and grade resistance [13], has significant influence on vehicle platoon such as degrading the controller performance and leading to the instability of a vehicle string [14]. In addition, driving resistance is virtually influenced by many factors such as vehicle mass, motorcycle type, and road and weather conditions [15, 16], some of which may even vary with the driving conditions [17]. Therefore, it is a great challenge in obtaining the driving resistance parameters in an accurate way.For the problem of unknown driving resistance in vehicle dynamics, Altmannshofer and Endisch [18] proposed a robust parameter estimation algorithm for identifying the vehicle driving resistance. Tannoury et al. [16] designed a nonlinear observer for the estimation of tire radius and rolling resistance to compensate for the unknown parameters. Guo et al. [14] deployed a quadratic function to represent the unknown and time-varying coefficients of driving resistance. However, as for the unknown driving resistance in vehicle platoon control, existing literatures mainly treat it as time-varying external disturbance without explicitly obtaining its true value [12]. In [9], the uncertain driving resistance in vehicle dynamics is modeled as an unknown time-varying disturbance and an adaptive method is adopted to estimate this value. Similarly, the uncertain driving resistance is described as a bounded disturbance and also estimated by adaptive approach in [14]. In [13], the vehicle resistance, which is relevant to vehicle mass, weather conditions, deadweight, motorcycle type, etc., is analyzed and the vehicle platoon model is established.On the other hand, the mechanical constraints, especially the actuator saturation [19], are important concern in vehicle dynamics. The actuator saturation of vehicle, whether it is the servo motor for an electric vehicle or engine for a gasoline vehicle, has been proved to be the source of performance degradation in vehicle driving procedure [20], which therefore deserves further investigation in vehicle platoon control. However, to the best of our knowledge, literatures that specifically address this issue seem very few. As a ubiquitous phenomenon in mechanical systems, actuator saturation, which is usually referred to as input saturation, has been intensively investigated in various control domains. In [20], by explicitly considering the actuator saturation, a novel robust adaptive control law is proposed to ensure the stability of the closed-loop system for high-speed train. In [21], the Nussbaum function is introduced to compensate for the nonlinear term arising from input saturation. In [22], a saturated adaptive robust control strategy, by adding an anti-windup block, is designed for vehicle active suspension systems, which is beneficial for the stability and performance preservation in the presence of saturation. In [23], an adaptive coordinated control algorithm of multiple high-speed trains with input saturation is proposed, where an antiwindup compensation block is used to optimize the control algorithm such that it is more resilient to input saturation. In [24], a smooth function tanh⋅ is introduced to handle the “actuator saturation” problem in vehicle platoon such that the control input can always be below the maximum inputs. Therefore, the above work provides fruitful inspirations for the actuator saturation problem of vehicle platoon.In this paper, we are trying to investigate the vehicle-platoon problem with uncertain driving resistance and actuator saturation via adaptive sliding mode control approach. Firstly, coupled sliding surface (CSS) is deployed to link interconnected vehicles and an adaptive control method is adopted to identify and estimate the variations of resistance coefficients. After that, a distributed adaptive sliding mode control algorithm for vehicle platoon with uncertain driving resistance is proposed. Then, an antiwindup compensation based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in case of actuator saturation. Moreover, the chattering phenomena inherent in sliding mode control are relieved by using a sigmoid-like function. Finally, various numerical simulations are performed to demonstrate the feasibility and effectiveness of the proposed control algorithm.The rest of this paper is organized as follows. In Section2, the vehicle dynamic model and the problems considered in this paper are described. In Section 3, the adaptive sliding mode controller is designed to realize the vehicle platoon in the presence of uncertain driving resistance and actuator saturation. Simulations are performed in Section 4 to demonstrate the feasibility and effectiveness of our algorithm. Concluding remarks are given in Section 5.


## 2. Vehicle Dynamics and Problem Formulation

Assume a vehicle platoon, which consists of a string of autonomous vehicles, includes a leader vehicle andn followers. As is shown in Figure 1, each follower regulates its motion according to the received information (e.g., position, velocity, and acceleration) from its front and back vehicles via wireless communication technique.Figure 1
Topological structure of vehicle platoon.


### 2.1. Vehicle Dynamics

Consider a vehicle platoon moving in a string with the following longitudinal dynamics:(1)r˙it=vit,Miv˙it=Fit−fi,i=1,2,…,n,where Mi is the mass of the ith vehicle, rit and vit denote the ith vehicle’s position and velocity, respectively, and Fit, which is taken as the control input, represents the traction or braking force of the ith vehicle. In addition, fi denotes its driving resistance.Generally,fi is influenced by the rolling resistance fr, air resistance fw, and grade resistance fg, etc. [25], which can be described as(2)fi=fr+fw+fg.The explicit form offr, fw, and fg can be written as follows:(i)
Rolling resistancefr:(3)fr=Wk,whereW is the normal load and k is the rolling resistance coefficient following such experience values as(4)k=0.0076+0.000056v,wherev is the velocity of vehicle.(ii)
Air resistancefw:(5)fw=12CDAρv2,whereCD is the coefficient of air resistance, A is windward area of vehicle, and ρ is the air density.(iii)
Grade resistancefg:If the vehicle is running on the hill, the component of gravity along the slope is defined as the grade resistance(6)fg=Gsinα,whereG and α are, respectively, the gravity and road-grade.Ask, CD, and ρ are uncertain values and heterogeneous with respect to different vehicles, therefore, fi is time-varying in different driving speed and road conditions. To facilitate the research, we write fi as(7)fi=At+Btvi+Ctvi2,where At,Bt,Ct, describing the uncertainty of driving resistance, are unknown and time-varying values.In order to simplify the protocol design and stability analysis, we rewrite (1) as(8)r˙it=vit,v˙it=Fit−ait−bitvit−citvit2,where uit:=Fit/Mi denotes the acceleration or deceleration of vehicle i (it is designed as the control input in this paper) and ait:=Ait/Mi, bit:=Bit/Mi, cit:=Cit/Mi are the vehicle driving resistance coefficients by considering the influence of vehicle mass Mi.


### 2.2. Problem Formulation

Generally speaking, the main purpose of vehicle platoon is to enhance the highway capacity and relieve traffic congestion by maintaining the desired safety distance between two consecutive vehicles and reaching the velocity consensus among vehicles [4].In particular, the time-varying driving resistance will inevitably influence the vehicle dynamics and decay the vehicle platoon performance. Therefore, one needs to specifically design the distributed control inputuit for a single vehicle such that the vehicle platoon is achieved under the disturbance of time-varying driving resistance.In addition, due to the physical and mechanical limitations of actuators, the control inputuit for vehicles will be under some constraints. The control input uit with saturation is given in the following form:(9)uit=satumin,uit,umax=umin,ifuit<umin,uit,ifumin≤uit≤umax,umax,ifuit>umax,where umin and umax are known constants, which represent the bounds of the control force.Based on above description, the main control objective of this paper can be concluded as follows:(1)
The vehicle platoon is achieved such that the follower’s velocity can converge to the velocity of the leader and each vehicle can maintain a safe intervehicle distance to avoid collision with each other(2)
The unknown time-varying coefficientsait, bit, and cit can be estimated such that the vehicle platoon is achieved by handling the parameter uncertainties via an adaptive control approach(3)
When the control input exceeds the maximum output of the vehicle actuator (servo motor for an electric vehicle or engine for a gasoline vehicle), the proposed algorithm can regulate the actuator output autonomously such that the actuator life as well as the vehicle platoon performance is guaranteed


## 2.1. Vehicle Dynamics

Consider a vehicle platoon moving in a string with the following longitudinal dynamics:(1)r˙it=vit,Miv˙it=Fit−fi,i=1,2,…,n,where Mi is the mass of the ith vehicle, rit and vit denote the ith vehicle’s position and velocity, respectively, and Fit, which is taken as the control input, represents the traction or braking force of the ith vehicle. In addition, fi denotes its driving resistance.Generally,fi is influenced by the rolling resistance fr, air resistance fw, and grade resistance fg, etc. [25], which can be described as(2)fi=fr+fw+fg.The explicit form offr, fw, and fg can be written as follows:(i)
Rolling resistancefr:(3)fr=Wk,whereW is the normal load and k is the rolling resistance coefficient following such experience values as(4)k=0.0076+0.000056v,wherev is the velocity of vehicle.(ii)
Air resistancefw:(5)fw=12CDAρv2,whereCD is the coefficient of air resistance, A is windward area of vehicle, and ρ is the air density.(iii)
Grade resistancefg:If the vehicle is running on the hill, the component of gravity along the slope is defined as the grade resistance(6)fg=Gsinα,whereG and α are, respectively, the gravity and road-grade.Ask, CD, and ρ are uncertain values and heterogeneous with respect to different vehicles, therefore, fi is time-varying in different driving speed and road conditions. To facilitate the research, we write fi as(7)fi=At+Btvi+Ctvi2,where At,Bt,Ct, describing the uncertainty of driving resistance, are unknown and time-varying values.In order to simplify the protocol design and stability analysis, we rewrite (1) as(8)r˙it=vit,v˙it=Fit−ait−bitvit−citvit2,where uit:=Fit/Mi denotes the acceleration or deceleration of vehicle i (it is designed as the control input in this paper) and ait:=Ait/Mi, bit:=Bit/Mi, cit:=Cit/Mi are the vehicle driving resistance coefficients by considering the influence of vehicle mass Mi.


## 2.2. Problem Formulation

Generally speaking, the main purpose of vehicle platoon is to enhance the highway capacity and relieve traffic congestion by maintaining the desired safety distance between two consecutive vehicles and reaching the velocity consensus among vehicles [4].In particular, the time-varying driving resistance will inevitably influence the vehicle dynamics and decay the vehicle platoon performance. Therefore, one needs to specifically design the distributed control inputuit for a single vehicle such that the vehicle platoon is achieved under the disturbance of time-varying driving resistance.In addition, due to the physical and mechanical limitations of actuators, the control inputuit for vehicles will be under some constraints. The control input uit with saturation is given in the following form:(9)uit=satumin,uit,umax=umin,ifuit<umin,uit,ifumin≤uit≤umax,umax,ifuit>umax,where umin and umax are known constants, which represent the bounds of the control force.Based on above description, the main control objective of this paper can be concluded as follows:(1)
The vehicle platoon is achieved such that the follower’s velocity can converge to the velocity of the leader and each vehicle can maintain a safe intervehicle distance to avoid collision with each other(2)
The unknown time-varying coefficientsait, bit, and cit can be estimated such that the vehicle platoon is achieved by handling the parameter uncertainties via an adaptive control approach(3)
When the control input exceeds the maximum output of the vehicle actuator (servo motor for an electric vehicle or engine for a gasoline vehicle), the proposed algorithm can regulate the actuator output autonomously such that the actuator life as well as the vehicle platoon performance is guaranteed


## 3. Distributed Adaptive Sliding Mode Control Algorithm for Vehicle Platoon

Firstly, the position tracking error for theith vehicle is defined as(10)ei=ri−1−ri−d,where d>0 is a constant value, representing the required distance between two consecutive vehicles.We also denote the velocity error by(11)v˜i=vi−1−vi.Here, the following assumptions are made for facilitating the control protocol design and theoretical analysis.Assumption 1.
ait, bit, and cit are bounded variables; i.e., ait≤ai+, bit≤bi+, and cit≤ci+.


### 3.1. Vehicle Platoon Control Algorithm with Uncertain Driving Resistance

We first consider the case of vehicle platoon with uncertain driving resistance. For the dynamics of vehicles with errorei, the control objective is to make ei converge to zero and to guarantee string stability.Hence, sliding mode control technique is employed to develop the vehicle platoon controller; we choose each sliding surface as(12)ni=e˙i+αiei,where αi>0 is a positive constant.The solution of (12) is(13)eit=niαi+exp−αitei0−niαiexp−αit−t0.From (13), one can see that when t⟶∞, ei⟶ni/αi. As αi is a positive constant, ei⇔ni. Furthermore, combined with (12), we have e˙i⟶0 when ni⟶0.Taking the time derivative of (12), one obtains(14)n˙i=e¨i+αie˙i=r¨i−1−r¨i+αie˙i.It is worth noting that (14) only describes the characteristic of a single vehicle. In order to describe the stability of the whole platoon, we adopt the coupled sliding surface (CSS) [7] of the ith vehicle for the control of the whole platoon system(15)Ni=βini−ni+1,where βi>0i=1,2,…,n is a weighting factor.Sinceni+1 does not exist for the last vehicle (i.e., i=n), we set ni+1=0. Then, we have N1:=n1,n2,…,nnT and N2:=N1,N2,…,NnT; the relationship between N1 and N2 can be described as(16)N2=BN1,where(17)B=β1−10⋯00β2−1⋯0⋮⋮⋮⋱⋮000⋯−1000⋯βn,with βi>0i=1,2,…,n being the parameters to be designed.In order to illustrate the same convergence ofN1 and N2, the following lemma is given.Lemma 1 (see [7]).
Equivalence of the convergence of the CSS and each sliding surface toward zero:N2 becomes zero if and only if N1 becomes zero at the same time.Therefore, the problem of makingni and ei converge to zero is converted into making Ni converge to zero. The time derivative of Ni in (15) can be written as(18)N˙i=βin˙i−n˙i+1=βir¨i−1−r¨i+αie˙i−r¨i−r¨i+1+αi+1e¨i+1=−βi+1ui−ai−bivi−civi2+Di,where Di=βir¨i−1+r¨i+1+αiβie˙i−αi+1e˙i+1.Accordingly, the novel adaptive platoon control law for theith vehicle is designed as(19)ui=k1βi+1Ni+1βi+1Di+sgnNia^i++b^i+vi+c^i+vi2,where k1 is a positive parameter that needs to be designed and a^i+, b^i+, and c^i+ are the estimated values of unknown constant coefficients ai+, bi+, and ci+.The adaptive estimation law for unknown coefficients is determined by(20)a^˙i+=λ1βi+1Ni,b^˙i+=λ2βi+1Nivi,c^˙i+=λ3βi+1Nivi2,where λ1,λ2,λ3>0 are positive constants.Particularly, wheni=n, we know Nn=βnnn from the definition of (15); the time derivative of Nn can be described as(21)N˙n=βnn˙n=βnr¨n−1−r¨n+αne˙n=−βnun−an−bnvn−cnvn2+Dn,where Dn=βnr¨n−1+αne˙n for i=n.The adaptive platoon control law for thenth vehicle is therefore formulated as(22)un=k2βnNn+1βnDn+sgnNna^n++b^n+vn+c^n+vn2,where k2 is a positive parameter that needs to be designed.Thus, the coefficients adaptation law can be designed as(23)a^˙n+=λ1βnNn,b^˙n+=λ2βnNnvn,c^˙n+=λ3βnNnvn2.Then, the following theorem, which guarantees the stability of each vehicle and string stability of the whole vehicle platoon, can be obtained.Theorem 1.
Consider a vehicle platoon described by (8), the proposed control algorithm of vehicle platoons (19) and (22) and the adaptive control law of coefficients (20) and (23) can ensure that the sliding surfaces Ni and ni and the distance error ei converge to zero.Proof.
First, we define the estimation error of coefficientsa˜i+, b˜i+, and c˜i+ as(24)a˜i+=a^i+−ai+,b˜i+=b^i+−bi+,c˜i+=c^i+−ci+.
Choose the following Lyapunov function candidates:(25)V=∑i=1nVi,where(26)Vi=12Ni2+12λ1a˜i+2+12λ2b˜i+2+12λ3c˜i+2.
Then, the time derivative ofVi can be calculated as(27)V˙i=NiN˙i+1λ1a˜i+a˜˙i++1λ2b˜i+b˜˙i++1λ3c˜i+c˜˙i+.
Becauseai+, bi+, and ci+ are constants, we know a˜˙i+=a^˙i+, b˜˙i+=b^˙i+, and c˜˙i+=c^˙i+ according to (24). Combining (18) and (20) with (27) yields(28)V˙i=−k1Ni2−βi+1Nia^i+−ai+b^i+−bivi+c^i+−civi2+βi+1Nia˜i++b˜i+vi+c˜i+vi2.
According to Young’s inequality [14], we have(29)ai≤ai+,bivi≤bi+vi,civi2≤ci+vi2.
Substituting (29) into (28) gives(30)V˙i≤−k1Ni2−βi+1Nia^i+−ai++b^i+−bi+vi+c^i+−ci+vi2+βi+1Nia˜i++b˜i+vi+c˜i+vi2≤−k1Ni2.
Thus,(31)V˙≤−k1∑i=1nNi2≤0.
From (30) and (31), we know that V˙i and V˙ are negative semidefinite; therefore, Vi and V are not monotonic increasing and boundedness. In addition, we can also obtain that Ni is boundedness and N˙i is boundedness.
Taking the derivative of (30) and (31), we have V¨i≤−k1/2NiN˙i, V¨≤k1/2∑i=1nNiN˙i. As V¨i and V¨ are boundedness, V˙i and V˙ are uniformly continuous. According to the Barbalat lemma, we know that limt⟶∞V˙i=0 and limt⟶∞V˙=0. Since k1 is positive, limt⟶∞Ni=0 for all i. This implies that N2, N1, and ei in Lemma 1 converge to zero.Remark 1.
Cconstrued sliding surface (12) facilitates the control design and stability analysis. Open-loop dynamics (12) contains ei and e˙i; the closed-loop dynamics n˙i will contain system dynamics (8) and the control input ui. Inspired by the adaptive technique, the control inputs (19) and (22) and the adaptive laws (19) and (23) can be designed by choosing the suitable Lyapunov function.Remark 2.
The main objective of the designed controller is to makeNi, a˜i+, b˜i+, and c˜i+ converge to zero; thus the Lyapunov function Vi is chosen as (26). The principle of controller design is to make V˙i≤0.


### 3.2. Vehicle Platoon Control Algorithm with Uncertain Driving Resistance and Actuator Saturation

In practice, the actuator saturation of vehicles has proved to be a source of performance degradation [26]. To handle this problem, an antiwindup compensation block is used to modify the control input such that it is more resilient to actuator saturation. An antiwindup compensator is used to generate a signal ϕit for each vehicle as the output of a differential equation:(32)ϕ˙it=−χiϕit+βi+1ui0t−uit,ϕ˙nt=−χnϕnt+βnun0t−unt.Letδit=Nit−ϕit. We have the following error dynamic system:(33)δ˙it=−βi+1uit−ai−bivi−civi2−βi+1ui0t−uit+Di+χiϕit,δ˙nt=−βnunt−an−bnvn−cnvn2−βnun0t−unt+Dn+χnϕnt.By considering explicitly the actuator saturation of vehicles, the modified adaptive control inputuit for each vehicle can be written as(34)uit=satumin,ui0t,umax,(35)ui0t=k1βi+1δi+1βi+1Di−1βi+1χiϕit+sgnδia^i++b^i+vi+c^i+vi2.The adaptive estimation laws for unknown coefficients are determined by(36)a^˙i+=λ1βi+1δi,b^˙i+=λ2βi+1δivi,c^˙i+=λ3βi+1δivi2.The adaptive platoon control law of thenth vehicle is therefore formulated as(37)un0t=knβnδn+1βnDn−1βnχnϕnt+sgnδna^n++b^n+vn+c^n+vn2.The coefficients adaptation law is designed as(38)a^˙n+=λ1βnδn,b^˙n+=λ2βnδnvn,c^˙n+=λ3βnδnvn2.The signalϕit is used for attenuating the integral windup of the adaptive coordinated control laws in case of actuator saturation. When actuator saturation happens, there will be a rise of error ϕit, and meanwhile the signal Ni will also increase, which in turn ensures that no sudden rise will happen to the newly defined tracking error δit. When the actuator saturation stops, the term ui0t−uit=0, and the signal ϕit will converge to zero.In addition, for the signalϕit, we have the following theorem.Theorem 2.
For the signalϕit, there exists a positive ϖi such that, for any t>0, it holds that(39)ϕit≤ui0t−uitβi+1χi+ϖi.Proof.
From (32), we can obtain that(40)ϕit=ϕi0e−χit+ui0t−uitβi+1χie−χit∫0te−χiϖdϖ=ϕi0e−χit+ui0t−uitβi+1χi1−e−χit,which implies that ϕit≤ui0t−uitβi+1/χi as t⟶∞. Thus, there exists a positive scaler ϖi, such that, for any t>0, it holds that ϕit≤ui0t−uitβi+1/χi; i.e., ϕit is bounded.
The following theorem will provide our result on the vehicle platoon with actuator saturation.Theorem 3.
Consider a vehicle platoon described by (8), the proposed control algorithm of vehicle platoon (35) and (37) and the adaptive control law of coefficients (36) and (38) can ensure the desired distances between two consecutive vehicles. Then, we have the following results:(1)
When the saturation does not occur, i.e.,ui−ui0=0, all the results in Theorems 1 and 2 will hold automatically.(2)
When the saturation occurs, i.e.,ui−ui0≠0, the modified δit will converge to zero, and Ni will also converge to zero by adjusting variable χi in Theorem 2. This implies that N2, N1, and ei in Lemma 1 converge to zero.Proof.
For the case that the actuator is not saturated,ui−ui0=0, the results in Theorem 1 hold automatically.
When saturation occurs, for the error dynamic (33) of the vehicles movement with control input (35) and (37), construct the following Lyapunov-like function candidate:(41)V=∑i=1nVi,where(42)Vi=12δi2+12λ1a˜i+2+12λ2b˜i+2+12λ3c˜i+2.
Then, the time derivative ofVi can be calculated as(43)V˙i=δiδ˙i+1λ1a˜i+a˜˙i++1λ2b˜i+b˜˙i++1λ3c˜i+c˜˙i+.
Becauseai+, bi+, and ci+ are constants, we know that a˜˙i+=a^˙i+, b˜˙i+=b^˙i+, and c˜˙i+=c^˙i+ according to (24). Therefore,(44)V˙i≤−k1δi2−βi+1δia^i+−ai++b^i+−bi+vi+c^i+−ci+vi2+βi+1δia˜i++b˜i+vi+c˜i+vi2≤−k1δi2.
Thus,(45)V˙≤−k1∑i=1nδi2≤0.
Using the same analysis method as in Theorem1, it can be shown that δi will converge to zero eventually. From Theorem 2, we know that variable χi can be adjusted, and ϕit can also be adjusted to an arbitrarily small value. Therefore, we have Ni≈δi. Combining Theorem 2 and equations (12) and (15), we know that ni and ei will converge to zero eventually.


### 3.3. Reduction of Chattering

It is well known that the sliding mode control has inherently the phenomena of chattering, which is detrimental for the system performance [27]. In this paper, a sigmoid-like function δi/δi+ς is used to replace the sgn⋅ function, where ς is a small positive constant.Then, the control inputuit of the ith vehicle can be rewritten as(46)uit=satumin,ui0t,umax,where ui0t is the calculated control input. It is determined by(47)ui0t=k1βi+1δi+1βi+1Di−1βi+1χiϕit+δiδi+ςa^i++b^i+vi+c^i+vi2.The adaptive estimation law ofith vehicle for unknown coefficients is determined by(48)a^˙i+=λ1βi+1δi2δi+ς,b^˙i+=λ2βi+1δi2viδi+ς,c^˙i+=λ3βi+1δi2vi2δi+ς.Similar to the previous case,un0t of the nth vehicle can be described as(49)ui0t=k1βnδn+1βnDn−1βnχnϕnt+δnδn+ςa^n++b^n+vn+c^n+vn2.The adaptive estimation law of thenth vehicle for unknown coefficients is determined by(50)a^˙n+=λ1βnδn2δn+ς,b^˙n+=λ2βnδn2vnδn+ς,c^˙n+=λ3βnδn2vn2δn+ς.


## 3.1. Vehicle Platoon Control Algorithm with Uncertain Driving Resistance

We first consider the case of vehicle platoon with uncertain driving resistance. For the dynamics of vehicles with errorei, the control objective is to make ei converge to zero and to guarantee string stability.Hence, sliding mode control technique is employed to develop the vehicle platoon controller; we choose each sliding surface as(12)ni=e˙i+αiei,where αi>0 is a positive constant.The solution of (12) is(13)eit=niαi+exp−αitei0−niαiexp−αit−t0.From (13), one can see that when t⟶∞, ei⟶ni/αi. As αi is a positive constant, ei⇔ni. Furthermore, combined with (12), we have e˙i⟶0 when ni⟶0.Taking the time derivative of (12), one obtains(14)n˙i=e¨i+αie˙i=r¨i−1−r¨i+αie˙i.It is worth noting that (14) only describes the characteristic of a single vehicle. In order to describe the stability of the whole platoon, we adopt the coupled sliding surface (CSS) [7] of the ith vehicle for the control of the whole platoon system(15)Ni=βini−ni+1,where βi>0i=1,2,…,n is a weighting factor.Sinceni+1 does not exist for the last vehicle (i.e., i=n), we set ni+1=0. Then, we have N1:=n1,n2,…,nnT and N2:=N1,N2,…,NnT; the relationship between N1 and N2 can be described as(16)N2=BN1,where(17)B=β1−10⋯00β2−1⋯0⋮⋮⋮⋱⋮000⋯−1000⋯βn,with βi>0i=1,2,…,n being the parameters to be designed.In order to illustrate the same convergence ofN1 and N2, the following lemma is given.Lemma 1 (see [7]).
Equivalence of the convergence of the CSS and each sliding surface toward zero:N2 becomes zero if and only if N1 becomes zero at the same time.Therefore, the problem of makingni and ei converge to zero is converted into making Ni converge to zero. The time derivative of Ni in (15) can be written as(18)N˙i=βin˙i−n˙i+1=βir¨i−1−r¨i+αie˙i−r¨i−r¨i+1+αi+1e¨i+1=−βi+1ui−ai−bivi−civi2+Di,where Di=βir¨i−1+r¨i+1+αiβie˙i−αi+1e˙i+1.Accordingly, the novel adaptive platoon control law for theith vehicle is designed as(19)ui=k1βi+1Ni+1βi+1Di+sgnNia^i++b^i+vi+c^i+vi2,where k1 is a positive parameter that needs to be designed and a^i+, b^i+, and c^i+ are the estimated values of unknown constant coefficients ai+, bi+, and ci+.The adaptive estimation law for unknown coefficients is determined by(20)a^˙i+=λ1βi+1Ni,b^˙i+=λ2βi+1Nivi,c^˙i+=λ3βi+1Nivi2,where λ1,λ2,λ3>0 are positive constants.Particularly, wheni=n, we know Nn=βnnn from the definition of (15); the time derivative of Nn can be described as(21)N˙n=βnn˙n=βnr¨n−1−r¨n+αne˙n=−βnun−an−bnvn−cnvn2+Dn,where Dn=βnr¨n−1+αne˙n for i=n.The adaptive platoon control law for thenth vehicle is therefore formulated as(22)un=k2βnNn+1βnDn+sgnNna^n++b^n+vn+c^n+vn2,where k2 is a positive parameter that needs to be designed.Thus, the coefficients adaptation law can be designed as(23)a^˙n+=λ1βnNn,b^˙n+=λ2βnNnvn,c^˙n+=λ3βnNnvn2.Then, the following theorem, which guarantees the stability of each vehicle and string stability of the whole vehicle platoon, can be obtained.Theorem 1.
Consider a vehicle platoon described by (8), the proposed control algorithm of vehicle platoons (19) and (22) and the adaptive control law of coefficients (20) and (23) can ensure that the sliding surfaces Ni and ni and the distance error ei converge to zero.Proof.
First, we define the estimation error of coefficientsa˜i+, b˜i+, and c˜i+ as(24)a˜i+=a^i+−ai+,b˜i+=b^i+−bi+,c˜i+=c^i+−ci+.
Choose the following Lyapunov function candidates:(25)V=∑i=1nVi,where(26)Vi=12Ni2+12λ1a˜i+2+12λ2b˜i+2+12λ3c˜i+2.
Then, the time derivative ofVi can be calculated as(27)V˙i=NiN˙i+1λ1a˜i+a˜˙i++1λ2b˜i+b˜˙i++1λ3c˜i+c˜˙i+.
Becauseai+, bi+, and ci+ are constants, we know a˜˙i+=a^˙i+, b˜˙i+=b^˙i+, and c˜˙i+=c^˙i+ according to (24). Combining (18) and (20) with (27) yields(28)V˙i=−k1Ni2−βi+1Nia^i+−ai+b^i+−bivi+c^i+−civi2+βi+1Nia˜i++b˜i+vi+c˜i+vi2.
According to Young’s inequality [14], we have(29)ai≤ai+,bivi≤bi+vi,civi2≤ci+vi2.
Substituting (29) into (28) gives(30)V˙i≤−k1Ni2−βi+1Nia^i+−ai++b^i+−bi+vi+c^i+−ci+vi2+βi+1Nia˜i++b˜i+vi+c˜i+vi2≤−k1Ni2.
Thus,(31)V˙≤−k1∑i=1nNi2≤0.
From (30) and (31), we know that V˙i and V˙ are negative semidefinite; therefore, Vi and V are not monotonic increasing and boundedness. In addition, we can also obtain that Ni is boundedness and N˙i is boundedness.
Taking the derivative of (30) and (31), we have V¨i≤−k1/2NiN˙i, V¨≤k1/2∑i=1nNiN˙i. As V¨i and V¨ are boundedness, V˙i and V˙ are uniformly continuous. According to the Barbalat lemma, we know that limt⟶∞V˙i=0 and limt⟶∞V˙=0. Since k1 is positive, limt⟶∞Ni=0 for all i. This implies that N2, N1, and ei in Lemma 1 converge to zero.Remark 1.
Cconstrued sliding surface (12) facilitates the control design and stability analysis. Open-loop dynamics (12) contains ei and e˙i; the closed-loop dynamics n˙i will contain system dynamics (8) and the control input ui. Inspired by the adaptive technique, the control inputs (19) and (22) and the adaptive laws (19) and (23) can be designed by choosing the suitable Lyapunov function.Remark 2.
The main objective of the designed controller is to makeNi, a˜i+, b˜i+, and c˜i+ converge to zero; thus the Lyapunov function Vi is chosen as (26). The principle of controller design is to make V˙i≤0.


## 3.2. Vehicle Platoon Control Algorithm with Uncertain Driving Resistance and Actuator Saturation

In practice, the actuator saturation of vehicles has proved to be a source of performance degradation [26]. To handle this problem, an antiwindup compensation block is used to modify the control input such that it is more resilient to actuator saturation. An antiwindup compensator is used to generate a signal ϕit for each vehicle as the output of a differential equation:(32)ϕ˙it=−χiϕit+βi+1ui0t−uit,ϕ˙nt=−χnϕnt+βnun0t−unt.Letδit=Nit−ϕit. We have the following error dynamic system:(33)δ˙it=−βi+1uit−ai−bivi−civi2−βi+1ui0t−uit+Di+χiϕit,δ˙nt=−βnunt−an−bnvn−cnvn2−βnun0t−unt+Dn+χnϕnt.By considering explicitly the actuator saturation of vehicles, the modified adaptive control inputuit for each vehicle can be written as(34)uit=satumin,ui0t,umax,(35)ui0t=k1βi+1δi+1βi+1Di−1βi+1χiϕit+sgnδia^i++b^i+vi+c^i+vi2.The adaptive estimation laws for unknown coefficients are determined by(36)a^˙i+=λ1βi+1δi,b^˙i+=λ2βi+1δivi,c^˙i+=λ3βi+1δivi2.The adaptive platoon control law of thenth vehicle is therefore formulated as(37)un0t=knβnδn+1βnDn−1βnχnϕnt+sgnδna^n++b^n+vn+c^n+vn2.The coefficients adaptation law is designed as(38)a^˙n+=λ1βnδn,b^˙n+=λ2βnδnvn,c^˙n+=λ3βnδnvn2.The signalϕit is used for attenuating the integral windup of the adaptive coordinated control laws in case of actuator saturation. When actuator saturation happens, there will be a rise of error ϕit, and meanwhile the signal Ni will also increase, which in turn ensures that no sudden rise will happen to the newly defined tracking error δit. When the actuator saturation stops, the term ui0t−uit=0, and the signal ϕit will converge to zero.In addition, for the signalϕit, we have the following theorem.Theorem 2.
For the signalϕit, there exists a positive ϖi such that, for any t>0, it holds that(39)ϕit≤ui0t−uitβi+1χi+ϖi.Proof.
From (32), we can obtain that(40)ϕit=ϕi0e−χit+ui0t−uitβi+1χie−χit∫0te−χiϖdϖ=ϕi0e−χit+ui0t−uitβi+1χi1−e−χit,which implies that ϕit≤ui0t−uitβi+1/χi as t⟶∞. Thus, there exists a positive scaler ϖi, such that, for any t>0, it holds that ϕit≤ui0t−uitβi+1/χi; i.e., ϕit is bounded.
The following theorem will provide our result on the vehicle platoon with actuator saturation.Theorem 3.
Consider a vehicle platoon described by (8), the proposed control algorithm of vehicle platoon (35) and (37) and the adaptive control law of coefficients (36) and (38) can ensure the desired distances between two consecutive vehicles. Then, we have the following results:(1)
When the saturation does not occur, i.e.,ui−ui0=0, all the results in Theorems 1 and 2 will hold automatically.(2)
When the saturation occurs, i.e.,ui−ui0≠0, the modified δit will converge to zero, and Ni will also converge to zero by adjusting variable χi in Theorem 2. This implies that N2, N1, and ei in Lemma 1 converge to zero.Proof.
For the case that the actuator is not saturated,ui−ui0=0, the results in Theorem 1 hold automatically.
When saturation occurs, for the error dynamic (33) of the vehicles movement with control input (35) and (37), construct the following Lyapunov-like function candidate:(41)V=∑i=1nVi,where(42)Vi=12δi2+12λ1a˜i+2+12λ2b˜i+2+12λ3c˜i+2.
Then, the time derivative ofVi can be calculated as(43)V˙i=δiδ˙i+1λ1a˜i+a˜˙i++1λ2b˜i+b˜˙i++1λ3c˜i+c˜˙i+.
Becauseai+, bi+, and ci+ are constants, we know that a˜˙i+=a^˙i+, b˜˙i+=b^˙i+, and c˜˙i+=c^˙i+ according to (24). Therefore,(44)V˙i≤−k1δi2−βi+1δia^i+−ai++b^i+−bi+vi+c^i+−ci+vi2+βi+1δia˜i++b˜i+vi+c˜i+vi2≤−k1δi2.
Thus,(45)V˙≤−k1∑i=1nδi2≤0.
Using the same analysis method as in Theorem1, it can be shown that δi will converge to zero eventually. From Theorem 2, we know that variable χi can be adjusted, and ϕit can also be adjusted to an arbitrarily small value. Therefore, we have Ni≈δi. Combining Theorem 2 and equations (12) and (15), we know that ni and ei will converge to zero eventually.


## 3.3. Reduction of Chattering

It is well known that the sliding mode control has inherently the phenomena of chattering, which is detrimental for the system performance [27]. In this paper, a sigmoid-like function δi/δi+ς is used to replace the sgn⋅ function, where ς is a small positive constant.Then, the control inputuit of the ith vehicle can be rewritten as(46)uit=satumin,ui0t,umax,where ui0t is the calculated control input. It is determined by(47)ui0t=k1βi+1δi+1βi+1Di−1βi+1χiϕit+δiδi+ςa^i++b^i+vi+c^i+vi2.The adaptive estimation law ofith vehicle for unknown coefficients is determined by(48)a^˙i+=λ1βi+1δi2δi+ς,b^˙i+=λ2βi+1δi2viδi+ς,c^˙i+=λ3βi+1δi2vi2δi+ς.Similar to the previous case,un0t of the nth vehicle can be described as(49)ui0t=k1βnδn+1βnDn−1βnχnϕnt+δnδn+ςa^n++b^n+vn+c^n+vn2.The adaptive estimation law of thenth vehicle for unknown coefficients is determined by(50)a^˙n+=λ1βnδn2δn+ς,b^˙n+=λ2βnδn2vnδn+ς,c^˙n+=λ3βnδn2vn2δn+ς.


## 4. Numerical Simulations

To verify the feasibility and effectiveness of the proposed platoon control algorithm, numerical simulations are performed with 8 vehicles (1 leader and 7 followers).


### 4.1. Simulation Setup

Without loss of generality, we suppose the vehicle platoon drives in various driving conditions, such as acceleration, cruising, and braking. In addition, we also consider the influence of time-varying driving resistance as a disturbance on the vehicle velocity in the form of sine wave.Therefore, the desired velocity of the leading vehicle is specifically designed as(51)vLt=20sinπ200t,ift<100,vLt=20,if100≤t<150,vLt=20+3sinπ10t,if150≤t<170,vLt=20,if170≤t<230,vLt=20+3sinπ10t,if230≤t<250,vLt=20,if250≤t<320,vLt=20+3sinπ10t,if320≤t<340,vLt=20,if340≤t<400,vLt=20sinπ200t,if400≤t<500,vLt=0,if500≤t<520.Inspired by our previous work on adaptive control for vehicle platoon [24], the variation of driving resistance coefficients ai, bi, ci is assumed to be determined by ai=a11+sint, bi=b11+cos3t, ci=c11+cos4t, respectively. It should be pointed out that ai, bi, ci are time-varying and bounded, so they can model the uncertainty and external noise of the vehicle. The other parameters such as a1, b1, c1, initial states of vehicles, and control parameters are listed in Tables 1 and 2.Table 1
Initial states of each vehicle and initial estimated values of driving resistance coefficients.
VehicleLeaderV1V2V3V4V5V6V7v000000000v˙000000000r068634638302180a^+000.80.90.850.850.80.850.8b^+000.0040.0040.00450.0040.00450.0040.0045c^+000.000160.000160.000160.000160.000160.000160.00016a100.30.40.40.40.50.40.5b100.0040.0030.0050.0040.0050.0040.005c100.000180.000160.000140.000170.000150.000170.00015Table 2
Control parameters of vehicles.
dαiβiλik1k2uminumax10100510−100.50.5−22


### 4.2. Simulation Results

To better illustrate the effectiveness of the proposed vehicle platoon control algorithm, three simulation cases are performed.Case 1.
Platoon with uncertain driving resistance.
In this case, we only consider the influence of uncertain driving resistance for vehicle platoon, where (19), (20), (22), and (23) are applied to the vehicle.
Figure2(a) shows the vehicle velocities and velocity errors (between followers and leader) in the simulation, from which one can clearly see that the velocity of followers will approach the leader’s velocity and the velocity errors will decline to an extremely small value around zero. Figure 2(b) gives the position curves of all vehicles in the platoon, from which we can see that vehicles will not collide during the whole driving cycle. Figure 2(c) also demonstrates that all vehicles will keep a given distance (about 10 m) under the proposed platoon algorithms (19) and (22). Figure 2(d) shows the control input (acceleration or deceleration) of all followers, from which we can see that the control input decreases sharply from 400 m/s2 to 20 m/s2 within about 0.03 s. Then, the control input will converge to a relatively steady state (around 20 m/s2 and −20 m/s2). However, such control input is infeasible in practical applications, as it exceeds the maximum output of vehicle actuator and causes the phenomena of actuator saturation. Therefore, we will consider the problem of actuator saturation in Case 2.Figure 2
The simulation results of each vehicle with uncertain driving resistance: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Case 2.
Platoon with actuator saturation.
In this case, the control laws (34), (35), and (37) and coefficients adaptive laws (36) and (38) are applied to the vehicle platoon. We set the initial values of signal ϕi as ϕi=0 and ϕ˙i=0 for i=1,2,…,n. The parameter of χi is chosen as χi=100 for i=1,2,…,n.
Figure3(a) shows the vehicle velocities and velocity errors (between followers and leader) by considering the influence of actuator saturation in the simulation. Compared with Figure 2(a), the control algorithms (34), (35), and (37) can also guarantee the performance of vehicle velocities and velocity errors. From Figures 3(b) and 3(c), we can also see that the proposed algorithms (34), (35), and (37) can maintain a desired intervehicle distance with their neighbors to ensure driving safety. Figure 3(d) gives the control input of all followers; it is obvious that the input (acceleration or deceleration) can be limited to 2 m/s by adopting the proposed control algorithms (34), (35), and (37), which satisfies the maximum output of vehicle actuators.
However, it triggers the chattering phenomenon, which may deteriorate the vehicle actuator and lead to the driving uncomfortableness for passengers.
To avoid chattering in practical implementation, the modified control algorithms (47) and (49) are proposed and simulations are performed in Case 3.Figure 3
The simulation results of each vehicle with uncertain driving resistance and actuator saturation: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Case 3.
Platoon with reduced chattering
In this case, the control laws (47) and (49) and coefficients adaptive laws (48) and (50) are applied to the vehicle platoon. The sigmoid-like function δi/δi+ς is used to replace the sgn⋅, where(52)ς=1∗10−5.
Figure4(a) shows the vehicle velocities and velocity errors (between followers and leader). Compared with Figure 3(a), the amendment control algorithms (47) and (49) have not greatly influenced the performance of vehicle velocities and velocity errors. From Figures 4(b) and 4(c), one can see that the proposed amendment algorithms (47) and (49) can also maintain a desired intervehicle distance with their neighbors, which ensures the driving safety of vehicle platoon. Figure 4(d) illustrates the control input of all followers; it is obvious that influence of chatting is significantly weakened compared with Figure 3(d). Therefore, better vehicle platoon performance can be obtained and the driving comfortableness can be enhanced.Figure 4
The simulation results of each vehicle with uncertain driving resistance, actuator saturation, and reduced chattering: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Remark 3.
In Case1, we only consider the influence of uncertain driving resistance for vehicle platoon, where (19), (20), (22), and (23) are applied to the vehicle. However, it exceeds the maximum output of vehicle actuator and causes the phenomena of actuator saturation. Therefore, we consider the problem of actuator saturation in Case 2. And the control laws (34), (35), and (37) and coefficients adaptive laws (36) and (38) are applied to the vehicle platoon. However, it triggers the chattering phenomenon in simulation, which may deteriorate the vehicle actuator and lead to the driving uncomfortableness for passengers. To avoid chattering in practical implementation, the modified control algorithm (47), (49) is proposed and simulations are performed in Case 3. Finally, better vehicle platoon performance is obtained and the driving comfortableness is enhanced.


## 4.1. Simulation Setup

Without loss of generality, we suppose the vehicle platoon drives in various driving conditions, such as acceleration, cruising, and braking. In addition, we also consider the influence of time-varying driving resistance as a disturbance on the vehicle velocity in the form of sine wave.Therefore, the desired velocity of the leading vehicle is specifically designed as(51)vLt=20sinπ200t,ift<100,vLt=20,if100≤t<150,vLt=20+3sinπ10t,if150≤t<170,vLt=20,if170≤t<230,vLt=20+3sinπ10t,if230≤t<250,vLt=20,if250≤t<320,vLt=20+3sinπ10t,if320≤t<340,vLt=20,if340≤t<400,vLt=20sinπ200t,if400≤t<500,vLt=0,if500≤t<520.Inspired by our previous work on adaptive control for vehicle platoon [24], the variation of driving resistance coefficients ai, bi, ci is assumed to be determined by ai=a11+sint, bi=b11+cos3t, ci=c11+cos4t, respectively. It should be pointed out that ai, bi, ci are time-varying and bounded, so they can model the uncertainty and external noise of the vehicle. The other parameters such as a1, b1, c1, initial states of vehicles, and control parameters are listed in Tables 1 and 2.Table 1
Initial states of each vehicle and initial estimated values of driving resistance coefficients.
VehicleLeaderV1V2V3V4V5V6V7v000000000v˙000000000r068634638302180a^+000.80.90.850.850.80.850.8b^+000.0040.0040.00450.0040.00450.0040.0045c^+000.000160.000160.000160.000160.000160.000160.00016a100.30.40.40.40.50.40.5b100.0040.0030.0050.0040.0050.0040.005c100.000180.000160.000140.000170.000150.000170.00015Table 2
Control parameters of vehicles.
dαiβiλik1k2uminumax10100510−100.50.5−22


## 4.2. Simulation Results

To better illustrate the effectiveness of the proposed vehicle platoon control algorithm, three simulation cases are performed.Case 1.
Platoon with uncertain driving resistance.
In this case, we only consider the influence of uncertain driving resistance for vehicle platoon, where (19), (20), (22), and (23) are applied to the vehicle.
Figure2(a) shows the vehicle velocities and velocity errors (between followers and leader) in the simulation, from which one can clearly see that the velocity of followers will approach the leader’s velocity and the velocity errors will decline to an extremely small value around zero. Figure 2(b) gives the position curves of all vehicles in the platoon, from which we can see that vehicles will not collide during the whole driving cycle. Figure 2(c) also demonstrates that all vehicles will keep a given distance (about 10 m) under the proposed platoon algorithms (19) and (22). Figure 2(d) shows the control input (acceleration or deceleration) of all followers, from which we can see that the control input decreases sharply from 400 m/s2 to 20 m/s2 within about 0.03 s. Then, the control input will converge to a relatively steady state (around 20 m/s2 and −20 m/s2). However, such control input is infeasible in practical applications, as it exceeds the maximum output of vehicle actuator and causes the phenomena of actuator saturation. Therefore, we will consider the problem of actuator saturation in Case 2.Figure 2
The simulation results of each vehicle with uncertain driving resistance: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Case 2.
Platoon with actuator saturation.
In this case, the control laws (34), (35), and (37) and coefficients adaptive laws (36) and (38) are applied to the vehicle platoon. We set the initial values of signal ϕi as ϕi=0 and ϕ˙i=0 for i=1,2,…,n. The parameter of χi is chosen as χi=100 for i=1,2,…,n.
Figure3(a) shows the vehicle velocities and velocity errors (between followers and leader) by considering the influence of actuator saturation in the simulation. Compared with Figure 2(a), the control algorithms (34), (35), and (37) can also guarantee the performance of vehicle velocities and velocity errors. From Figures 3(b) and 3(c), we can also see that the proposed algorithms (34), (35), and (37) can maintain a desired intervehicle distance with their neighbors to ensure driving safety. Figure 3(d) gives the control input of all followers; it is obvious that the input (acceleration or deceleration) can be limited to 2 m/s by adopting the proposed control algorithms (34), (35), and (37), which satisfies the maximum output of vehicle actuators.
However, it triggers the chattering phenomenon, which may deteriorate the vehicle actuator and lead to the driving uncomfortableness for passengers.
To avoid chattering in practical implementation, the modified control algorithms (47) and (49) are proposed and simulations are performed in Case 3.Figure 3
The simulation results of each vehicle with uncertain driving resistance and actuator saturation: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Case 3.
Platoon with reduced chattering
In this case, the control laws (47) and (49) and coefficients adaptive laws (48) and (50) are applied to the vehicle platoon. The sigmoid-like function δi/δi+ς is used to replace the sgn⋅, where(52)ς=1∗10−5.
Figure4(a) shows the vehicle velocities and velocity errors (between followers and leader). Compared with Figure 3(a), the amendment control algorithms (47) and (49) have not greatly influenced the performance of vehicle velocities and velocity errors. From Figures 4(b) and 4(c), one can see that the proposed amendment algorithms (47) and (49) can also maintain a desired intervehicle distance with their neighbors, which ensures the driving safety of vehicle platoon. Figure 4(d) illustrates the control input of all followers; it is obvious that influence of chatting is significantly weakened compared with Figure 3(d). Therefore, better vehicle platoon performance can be obtained and the driving comfortableness can be enhanced.Figure 4
The simulation results of each vehicle with uncertain driving resistance, actuator saturation, and reduced chattering: (a) the velocity curves; (b) the position curves; (c) the distances curves between two neighbor vehicles; (d) the control input curves.
(a)(b)(c)(d)Remark 3.
In Case1, we only consider the influence of uncertain driving resistance for vehicle platoon, where (19), (20), (22), and (23) are applied to the vehicle. However, it exceeds the maximum output of vehicle actuator and causes the phenomena of actuator saturation. Therefore, we consider the problem of actuator saturation in Case 2. And the control laws (34), (35), and (37) and coefficients adaptive laws (36) and (38) are applied to the vehicle platoon. However, it triggers the chattering phenomenon in simulation, which may deteriorate the vehicle actuator and lead to the driving uncomfortableness for passengers. To avoid chattering in practical implementation, the modified control algorithm (47), (49) is proposed and simulations are performed in Case 3. Finally, better vehicle platoon performance is obtained and the driving comfortableness is enhanced.


## 5. Conclusions

In this paper, we discuss the distributed adaptive control problem for vehicle platoon with uncertain driving resistance and actuator saturation. Coupled sliding surface (CSS) is deployed to link interconnected vehicles and an adaptive control method is adopted to estimate the variations of resistance coefficients. An antiwindup compensation based approach is utilized to attenuate the integral windup of the adaptive platoon control laws in case of actuator saturation. Theoretical results are verified via numerical simulations, which demonstrate that the proposed control algorithm can make every vehicle keep the desired distance with the preceding vehicle and all followers’ velocity will gradually converge to the velocity of the leader even in the presence of uncertain driving resistance coefficients and actuator saturation.



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*Source: 7581517-2020-07-13.xml*