An Optimal Longitudinal Control Strategy of Platoons Using ...downloads.hindawi.com/journals/jat/2020/8822117.pdfCAVplatoonisformed,thestrategycanalsohelpmaintain the platoon operation

Research ArticleAn Optimal Longitudinal Control Strategy of Platoons UsingImproved Particle Swarm Optimization

Zhizhou Wu1 Zhibo Gao 1 Wei Hao 23 and Jiaqi Ma4

113e Key Laboratory of Road and Traffic Engineering Ministry of Education Tongji University 4800 Caorsquoan Road JiadingShanghai 201804 China2College of Transportation Engineering Changsha University of Science and Technology Changsha 410114 China3Hunan Key Laboratory of Smart Roadway and Cooperative Vehicle-Infrastructure SystemChangsha University of Science and Technology Changsha 410004 China4Department of Civil and Architectural Engineering and Construction Management University of Cincinnati CincinnatiOH 45221 USA

Correspondence should be addressed to Zhibo Gao gaozhibotongjieducn

Received 17 June 2020 Revised 3 August 2020 Accepted 29 August 2020 Published 8 September 2020

Academic Editor Alain Lambert

Copyright copy 2020 ZhizhouWu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Most existing longitudinal control strategies for connected and automated vehicles (CAVs) have unclear adaptability withoutscientific analysis regarding the key parameters of the control algorithm is paper presents an optimal longitudinal controlstrategy for a homogeneous CAV platoon First of all the CAV platoon models with constant time-headway gap strategy andconstant spacing gap strategy were respectively established based on the third-order linear vehicle dynamics model en alinear-quadratic optimal controller was designed considering the perspectives of driving safety efficiency and ride comfort withthree performance indicators including vehicle gap error relative speed and desired acceleration An improved particle swarmoptimization algorithm was used to optimize the weighting coefficients for the controller state and control variables Based on theMatlabSimulink experimental simulation the analysis results show that the proposed strategy can significantly reduce the gaperror and relative speed and improve the flexibility and initiative of the platoon control strategy compared with the unoptimizedstrategies Sensitivity analysis was provided for communication lag and actuator lag in order to prove the applicability andeffectiveness of this proposed strategy which will achieve better distribution of system performance

1 Introduction

With the development of the urbanization process trafficdemand and the number of motor vehicles have been greatlyincreased and the contradiction between traffic supply anddemand has become more and more serious leading to aseries of social problems such as traffic congestion [1 2]environmental pollution [3 4] accident frequency [5ndash8]and energy waste [9 10] With the advantages of infor-mation acquisition perception response time and inter-action behavior connected and automated vehicles (CAVs)are expected to improve the traffic flow characteristics fromthe microlevel thus providing an effective method to solveproblems such as traffic congestion and traffic safety [11ndash13]

Adaptive cruise control (ACC) is an important technology inCAVs longitudinal control and ACC vehicles can obtain thereal-time status information (position speed accelerationetc) of the preceding vehicle through on-board detectionequipment with the acceleration optimization algorithm tomaintain the constant gap for the preceding vehicle [14] Asa more complex form of ACC system cooperative adaptivecruise control (CACC) vehicles can actively communicatewith the preceding vehicle through vehicle-to-vehicle (V2V)communication technology to coordinate their speedchanges to form platoons that allow them to follow thepreceding vehicles safely with short gaps [15] A successfulCACC operation strategy can increase the probability ofmaking individual CACC vehicles join the platoons Once a

HindawiJournal of Advanced TransportationVolume 2020 Article ID 8822117 12 pageshttpsdoiorg10115520208822117

CAV platoon is formed the strategy can also help maintainthe platoon operation throughout a highway corridorHowever there are many factors affecting the applicationeffect for the CACC system among which the most criticalones are spacing strategy and control algorithm Before theimplementation of large-scale field real vehicle test it isnecessary to study the influence of different spacing strat-egies and control algorithms on the performance of CACCsystem in order to improve the theory of CACC system andpromote the promotion of its technology

During the process of vehicle longitudinal control thereasonable spacing control strategy is one of the keys toensuring high safety car-following stability and satisfactoryroad capacity In recent years a variety of control strategieshave been put forward which are mainly divided intoconstant spacing gap (CSG) strategy constant time-headwaygap (CTG) strategy and variable time-headway gap (VTG)strategy CSG strategy shows that the vehicle alwaysmaintains a constant intervehicle distance with the pre-ceding vehicle [16 17] Due to its simple structure and lowapplication difficulty it is first applied in highway envi-ronment with relatively little interference [18] HoweverCSG strategy is insensitive to the change of vehicle speed andhas poor adaptability to the platoon stability in the complexand changeable driving environment In order to deal withthese shortcomings some strategies in which the gap varieswith the driving environment are proposed such as CTGstrategy and VTG strategy CTG strategy refers to theconstant time headway between the subject vehicle and thepreceding vehicle by constantly adjusting the speed underthe car-following state [19] With further research theflexibility and stability of CTG strategy like CSG strategycannot meet the requirements of the complex traffic system[20] erefore some VTG strategies that the time headwaycan change with the traffic flow environment have attractedthe attention of researchers With due consideration of theinfluence of vehicle speed acceleration and relative speedchanges by varied time headway Yanakiev and Kanella-kopoulos [21] presented a variable headway strategy andproved that the proposed strategy can guarantee platoonstability In addition some variable time-headway strategieswere proposed for different driving behaviors such asperception judgment and operation [22 23]

In the car-following mode it is greatly necessary to takeinto consideration various factors such as comfort safetycar-following stability traffic capacity and energy con-sumption except considering a stable safety control strategybetween the vehicle and the preceding vehicle From theperspective of optimal control the design of a reasonableand effective control algorithm is another research hotspotof CACC system design e design of a reasonable andeffective control algorithm from the perspective of optimalcontrol is another focus of CACC research field Presentlythe theories and methods applied to the car-following modeof CACC system include classical proportional integraldifferential (PID) control model predictive control (MPC)and artificial neural network and linear-quadratic (LQ)control For the nonlinear characteristics of CACC systemPID is applied to realize the automatic adjustment of the

performance index parameters [24] rough controlstrategies such as prediction model rolling optimizationand feedback correction MPC can take into account mul-tiple control objectives and constraints simultaneouslywhich is conducive to meeting multiple control require-ments during vehicle driving [25] e disadvantage of MPCis that the model structure is too complex e methodsof artificial intelligence mainly include fuzzy control andartificial neural network e idea is to establish the rela-tionship between input (vehicle position velocity acceler-ation etc) and output (desired acceleration angle etc)through the sample training method It has the advantagesof simple design structure mature algorithm applicationconditions and low hardware requirements and thedisadvantages of the model structure are difficult to de-scribe LQ algorithm also known as a linear-quadraticalgorithm is an optimal control law based on quadraticperformance indexes to obtain state linear feedback andrealize closed-loop optimal control Its advantages are notlimited by the concrete model structure and its design isflexible e disadvantage is that the control performancedepends on the weight matrix of state variables and inputvariables and these choices are subjective and lack ob-jective description

To attain optimal gains for PID or LQ algorithm a seriesof intelligent optimization algorithms were used such asGenetic Algorithm (GA) and particle swarm optimization(PSO) methods Compared with GA and other optimizationmethods PSO memorizes satisfactory solutions among allthe particles without considering previous knowledge aftereach evolution so that it attracts increasing attention forsolving complex optimization problems Sowjanya andSrinivas [26] presented the development of an optimal PIDcontroller for a composition control system using PSOtechnique Chen et al [27] used dynamic particle swarmoptimization to develop optimal strategies for two-variableenergy management and gear-shifting in hybrid electricvehicles e results demonstrate that compared with thebaseline case the proposed optimization strategy broughtsuperior performance when applied to hybrid energymanagement and transmission control Rahman et al [28]investigated the use of the accelerated particle swarm op-timization (APSO) method for developing real-time andlarge-scale optimizations for allocating power Comparedwith standard particle swarm optimization consideringcharging time and battery capacity the APSO method canachieve some improvements in terms of best fitness andcomputation timee above studies have revealed that PSOis a reliable tool for designing an optimal strategy and that itcan outperform other evolutionary algorithms [27]

e longitudinal control strategy of CAV platoon iscomplex especially with the uncertainty of key parametersand there is a need to develop computationally efficientalgorithms to find optimal solutions e objective of thispaper is to design an optimal controller considering the keyparameter selection involved in the longitudinal vehicledynamics e optimal control is designed by consideringthe parametric uncertainties in the dynamics model ofplatoons expressed by a third-order linear vehicle model

2 Journal of Advanced Transportation

e optimal platooning is then formulated as a linear-quadratic optimal control problem where optimal desiredaccelerations are generated to minimize the cost functione optimal controller regulates platoon desired accelera-tions to minimize the cost function representing safetyefficiency and comfort subject to speed limits accelerationrange and minimal spacing In addition an improvedparticle swarm optimization (IPSO) algorithm is used tooptimize the weighting coefficients for the state variables andinput variables e designed control strategy is flexible insuch a way that it can be applied to the homogeneousplatooning control with constant time-headway gap andconstant spacing gap

e remainder of this paper is structured as followsSection 2 introduces the system models for constant time-headway and constant spacing gap platoons with CAVsSection 3 presents a linear-quadratic optimal controller withan improved particle swarm optimization algorithm Section4 designs simulation experiments to verify the performanceof the proposed controller and discusses the simulationresults of the CAV platoons under different control strat-egies Finally conclusions and recommendations are de-livered in Section 5

2 Dynamics Models

First of all a single CAV dynamics model is introduced andthen the CAV platoon dynamics models under the strategyof constant time-headway and constant spacing gap areestablished

21 Single-Vehicle Dynamics Model In the dynamicsmodeling process with a single vehicle it is assumed that avehicle travels on the road with ln xn vn and an denotingthe vehicle length location speed and acceleration of thesubject vehicle According to the second law of Newtonrsquoskinematics the dynamic equation of a single vehicle is listedas follows

mn middot euroxn Fn minus Rgn minus Rdn minus Ran Ran Kan middot _x2n1113966 (1)

where mn denotes the mass of vehicle n Fn denotes thedriving force produced by vehiclersquos engine Rgn Rdn andRan denote the grade resistance mechanical drag andaerodynamic drag respectively and Kan denotes theaerodynamic drag coefficient

e vehicle engine model can be considered as a first-order lag system as expressed by [29]

_Fn minusFn

τA+

Un

τA (2)

where τA denotes the vehicle engine time coefficient Un

denotes the throttle command input and Un ge 0 and Un le 0represent power input and brake input respectively

Utilizing the feedback linearization control law [30] thenonlinear model in equation (1) can be linearized as a first-order lag system as expressed by

Un mnun + 2Kan _xn euroxn τA

+ Rgn + Rdn + Ran1113872 1113873

x

n 1τA

middot un minus xn( 1113857(3)

where un denotes the desired acceleration given to vehicle nUnder the strategy of the constant time-headway gap the

control objective is to keep the speed of the subject vehicleconsistent with the preceding and maintain a constant timeheadway that is sd

n vn middot td + s0 where sdn is the desired gap

td is the desired time headway and s0 is the minimum gap atstandstill For a single vehicle n the system state is describedby the gap error relative speed to the preceding vehicle andactual acceleration that is Z (ΔsnΔvn an)T and thecontrol variable is defined as U un Δsn is the deviationbetween the real space gap sn xnminus1 minus xn minus ln and the de-sired gap sd

n to the preceding vehicle that is Δsn sn minus sdn

Δvn is the relative speed to the preceding vehicleAccording to the previous definition of system state and

control variable a third-order model is used to express thelongitudinal dynamics model for vehicle n as shown in [31]

ddt

Z ddtΔsnΔvn an( 1113857

T

ddt

xnminus1 minus xn minus ln minus sdn

vnminus1 minus vn

an

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

vnminus1 minus vn minus an middot td

anminus1 minus an

un minus an( 1113857

τA

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(4)

f(ZU d) A middot Z + B middot U + C middot d (5)

where A 0 1 minust

d

0 0 minus10 0 minus1τA

⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦ B

00

minus1τn

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ C

010

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

d [anminus1] and d is the external disturbance of the system

which refers to the actual acceleration of the precedingvehicle n minus 1 in this paper

Under the strategy of the constant spacing gap thecontrol objective is to maintain a desired space gap between

Journal of Advanced Transportation 3

the subject vehicle n and the preceding vehicle n minus 1 this issd

n Ld where Ld is the desired spacing gap Compared withthe CTG strategy the change of the CSG strategy is theexpression adjustment of gap error that is Δsn xnminus1 minus

xn minus ln minus Ld erefore the single dynamics model of CSGstrategy can still be expressed by equations (4) and (5) and itonly needs to set td 0

22 Platoon Dynamics Models For a homogeneous CAVplatoon system with Nge 2 vehicles the system state

variables can be defined as Zp (Δs1Δv1 a1 Δs2Δv2a2 ΔsNΔvN aN)T the control variable can be definedas Up (u1 u2 uN)T and the disturbance can be de-fined as dp ap where ap is the actual acceleration of thepreceding vehicle n minus 1 For each homogeneous CAV vehiclein this platoon system it is applicable to the single-vehicledynamics model erefore the matrix-form system dy-namics model for homogeneous CAV platoon can beexpressed with

ddtZp

ddtΔs1Δv1 a1Δs2Δv2 a2 ΔsNΔvN aN( 1113857

T g Zp

Up dp

( 1113857

g ZpUp

dp( 1113857 Ap

middot Zp+ Bp

middot Up+ Cp

middot dp

(6)

where

Ap

A3times31 03times3

middot middot middot 03times3

03times3 A3times42 middot middot middot 03times3

⋮ ⋮ ⋱ ⋮

03times2 03times3middot middot middot A3times4

N

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bp

B1 0

⋱

0 BN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

C 0

0 0⎡⎢⎣ ⎤⎥⎦

A3times31

0 1 minustd

0 0 minus1

0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A3times4i

0 0 1 minustd

1 0 0 minus1

0 0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

i 2 3 N

Bk

0

0

1τA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k 1 2 N

(7)

3 Design of Optimal Controllerfor Platoon Operation

In this section a linear-quadratic optimal controller is de-veloped to determine the control command by minimizing arunning cost function

31 PlatooningControl Formulation e centralized controlmethod is adopted in the controller design of the CAVplatoon where all the follower information is collected andcomputed by the leader of a CAV platoon and then thedesired accelerations are broadcast to the followers Since theplatoon dynamics model is linear a linear-quadratic optimalcontroller is designed to optimize the performance of CAVplatoon operation e target is to enable the platoon toobtain better performance in terms of driving safety effi-ciency and ride comfort which is to reduce the gap errorrelative speed and desired acceleration reflecting physicalquantity [31] e controller is used to comprehensivelyconsider each performance indicator through the weightingcoefficients e cost function J is shown in

J minUp

1113946t0+Tp

t0

L Zp(t)Up

(t)( 1113857dt (8)

L c1 1113944

N

i1Δsi( 1113857

2+ c2 1113944

N

i1Δvi( 1113857

2+ c3 1113944

N

i1ui( 1113857

2 (9)

In equation (9) c1 c2 and c3 denote the weightingcoefficients of vehicle gap error relative speed and desiredacceleration respectively and [t0 t0 + Tp] denotes a timehorizon

In addition to the target performance indicator the stateand control variables in the system should also be con-strained e existing constraints are listed as follows

(1) Speed constraint


v le vi le v (10)

(2) Gap constraint

xi minus xi+1 ge s0 + li (11)

(3) Acceleration constraint

a le ui le a (12)

(4) Other constraints since the accurate feedback in-formation on vehicle positions speeds and actualacceleration is obtained via on-board sensors andV2V communication e received state at time t0 isactually measured at time t0 minus τs It is formulated asfollows

1113957Zpt0( 1113857 Zp

t0 minus τs( 1113857 (13)

where v v a and a denote the lower and upper bounds ofthe speed and acceleration respectively and τs is thefeedback delay in communication

In order to facilitate later expression and calculation thecost function is adjusted to matrix forms and the expressionis listed as follows

J minUp t0 t0+Tp1113858 1113859

1113946t0+Tp

t0

Zp( 1113857

TQZp+ Up

( 1113857TRUp

1113876 1113877dt (14)

where

Q

q1 03times3

⋱

03times3 qi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R

r1 0

⋱

0 ri

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

qi

c1 0 0

0 c2 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ri c31113858 11138591times1

i 1 2 N

(15)

e desired acceleration of platoon system can be cal-culated by

Up(t) minusK middot Zp

(t) (16)

where K is the optimal control feedback gain matrix Whenthe vehicle parameter value and the weighting coefficientvalue are determined K can be obtained by solving Riccatiequation

PAp+ Ap

( 1113857TP minus PBpRminus 1 Bp

( 1113857TP + Q 0 (17)

where P is the solution of Riccati equation (17)Lastly the optimal control feedback gain matrix is ob-

tained that is K Rminus 1(Bp)TP

32 ParametersOptimization In equations (9) and (14)ndash(17)it can be seen that the target performance of the systemcompletely depends on the selection of the weighting co-efficients c1 c2 and c3 In previous designs the weightedcoefficient is usually obtained by repeated trials whichmakes the optimal control law lack theory and objectivityIn this section an improved particle swarm optimizationalgorithm is adopted to optimize the weighting coefficientsfor the controller

PSO is an intelligent optimization algorithm based onthe predation behavior of birds which has the characteristicsof swarm optimization However the basic PSO algorithm isprone to fall into local optimization and has a slow con-vergence rate erefore an IPSO algorithm is introduced tooptimize the weight coefficient of the controller e idea isto use the nonlinear function sigmoid to dynamically adjustthe inertia factor and the algorithm can achieve a goodbalance in the global space and local space [32] In theoptimization process the velocity and position of particles inthe search space are determined according to the followingimproved iterative updating formula

ϑt+1 ωtϑt + η1c1 Pt minus χt( 1113857 + η2c2 Gt minus χt( 1113857 (18)

χt+1 χt + ϑt+1 (19)

ωt sigmoid α middotΔJϑt

1113888 1113889 (20)

where χ denotes the position of the particle ϑ denotes thevelocity of the particle ωt denotes the inertia factor η1 andη2 denote accelerated constant c1 and c2 denote a randomnumber between [0 1] Pt denotes the best position that theparticle has searched so far Gt denotes the best location ofthe whole particle swarm so far α denotes the linear changecoefficient and ΔJ denotes the change of fitness functionvalue

e procedure of controller parameter optimizationusing IPSO is shown in Figure 1 and the flow of the IPSOalgorithm is as follows

Step 1 Initialize the particle swarm e positions χt

and velocities ϑt of all particles are generated randomlyand Pt and Gt are determinedStep 2 Calculate the fitness value of each particle andcompare its fitness value with the fitness value of theoptimal position Pt experienced by the particle If it isgood take it as the current PtStep 3 For each particle compare its fitness value withthe fitness value of the optimal position Gt experiencedby the whole particle swarm If it is good take it asthe current Gt


Step 4 Update the velocity and position of particlesaccording to equations (18)ndash(20)Step 5 If the termination condition (the maximumnumber of iterations or the lower limit of the adaptivevalue) is not met then return to step (2) otherwise exitthe algorithm and output the optimal solution

4 Simulation Experimental Design

41 Simulation Scenarios In order to verify the effectivenessof the proposed method a homogeneous CAV platoonsystem with N 4 vehicles is selected for this case studywhich followed an exogenous head vehicle as shown inFigure 2 V2V communication is adopted between eachfollowing CAV For convenience all vehicle lengths thedesired headway and the minimum gap at standstill arethe same

42 Results Analysis To verify the benefits of the optimalcontroller of platoons using IPSO some different strategieswere adopted to conduct the comparison

CTG strategy desired time headway td 1 s [33] andweighting coefficient c1 06 c2 05 and c3 06[31]CSG strategy desired time headway td 0 s desiredspace gap Ld 75m and weighting coefficient c1 06c2 05 and c3 06CTG-IPSO strategy desired time headway td 1 s andthe weighting coefficient value space is [01 100]CSG-IPSO strategy desired time headway td 0 sdesired space gap Ld 75m and the weighting coef-ficient value space is [01 100]

During the simulation experiment the simulation periodis 50 s A step function is used to describe the acceleration forthe exogenous head vehicle which starts with an initialspeed of 25ms decelerates with minus4ms2 from 10 s to 12 sand then accelerates with 1ms2 from 27 s to 35 s For other

time slots its acceleration is 0ms2 For the controllers theparameters are set as follows li 4m s0 2m Ld 75mτA 02s τs 005s v 0ms v 12036ms aminus5ms2and a 2ms2 For the IPSO algorithm the parameters areset as follows η1 η1 2 α 1 particle size is 50 and themaximum iteration is 100

To ensure the stability of the results the simulations ofCTG-IPSO and CSG-IPSO controllers run 10 times and theoptimization procedure is shown in Figure 3e simulationresults of each scheme are shown in Table 1 and Figures 4ndash7In Figure 3 the algorithm has converged when the numberof iterations is less than 15 generations is also showsthat the controller using the improved particle swarm op-timization algorithm has good feasibility At this timethe weighting coefficients of CTG-IPSO controller arec1 60157 c2 68653 and c3 0491 the total cost is1438 the weighting coefficients of CSG-IPSO controller arec1 5276 c2 1000 and c3 0295 and the total cost is1930 As depicted in Figures 4ndash7 and values of costs inTable 1 when the exogenous head vehicle decelerates oraccelerates all controllers give control commands to reducethe relative speeds and gap errors for reducing the costs andsettle to a new equilibrium where the cost is zero after acertain time In addition the behavior of the first vehicle ismore sensitive to the change of the exogenous head vehicleSince the changes of actual accelerations start from the firstvehicle to the last vehicle sequentially the maximum ofabsolute relative speed and gap error of the 4th vehicle issmaller than that of the first vehicle To this end they canensure string stability

In order to describe quantitatively the variation of theperformance indicators the root-mean-square value ofperformance indicator is analyzed for CAV platoon con-sidering the single CAV performance indicator e analysisresults are shown in Table 2 and Figure 8

It can be seen from Table 2 that the total cost of the CTG-IPSO controller is decreased by 100 compared with theunoptimized CTG controller wherein the relative speed isreduced by 83 and the actual acceleration is decreased by33 relative to the gap error which increased by 30 In

Start

Produce swarm of particles

Individual assignment c1 c2 c3

LQR output optimal control feedback gain matrix K

Run simulation models and output performance indicators

Satisfy termina-tion condition

End

Particle swarm update operation

Y

N

IPSO Simulink

Figure 1 e procedure of controller parameters optimization


(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV

Exogenoushead vehicle

V2V

Figure 2 e platooning formations

Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)

Figure 3 Optimization result of CTG-IPSO scheme and CSG-IPSO scheme (a) CTG-IPSO and (b) CSG-IPSO

Table 1 Performance results of platooning controllers

Indicators CTG CSG CTG-IPSO CSG-IPSOmax |Δs+

1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd

4thExogenoushead vehicle

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)

Figure 4 Simulation result of CTG (a) gap error (b) relative speed and (c) acceleration


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)

Figure 5 Simulation result of CSG (a) gap error (b) relative speed and (c) actual acceleration

∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)

Figure 6 Simulation result of CTG-IPSO (a) gap error (b) relative speed and (c) actual acceleration

∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)

Figure 7 Simulation result of CSG-IPSO (a) gap error (b) relative speed and (c) actual acceleration


contrast to the unoptimized CSG controller the total costof the CSG-IPSO controller decreased by 341 whereinthe vehicle gap error decreased by 693 and the relativespeed decreased by 732 combined with the actual ac-celeration which increased by 15 It can be seen from thechanges in total cost and performance that the controllerdesigned in this paper has certain benefits for differentcontrol strategies

As depicted in Figures 8(a) and 8(b) the vehicle gaperror and the relative speed for each controller decrease withthe increase of the vehicle number order e decreasingtrend of CTG and CTG-IPSO is slow while the decreasingtrend of the CSP and CSP-IPSO controllers is dramatic Itcan be seen from Figure 8(c) that the actual acceleration ofthe CTG and CTG-IPSO controllers decreases sharply withthe increase of the number order but the CTG-IPSOcontroller is better than CTG e actual acceleration of theCSG-IPSO and CSG controllers is maintained at 09ms2is is because under the constant time-headway gapstrategy the desired vehicle spacing is no longer fixed butchanges linearly with the change of vehicle speed At thistime the position and speed information of the leading carare independent of the following car e subject vehicleonly needs to maintain a constant time headway with thepreceding vehicle and it can also maintain the safety andstability of the system with less acceleration

43 Sensitivity Analysis of Key Parameters Hysteresis is oneof the three characteristics of the car-following model and isinevitable in the CACC system In CACC system there aremainly two types communication lag andmechanical lag Inthis paper mechanical lag refers to the time coefficient of theautomobile engine namely the actuator lag erefore this

section mainly analyzes the impact of communication lagand actuator lag on the proposed method

431 Communication Lag With the values of other pa-rameters fixed the communication lag value varies between0 s and 02 s with an interval of 005 s e optimized systemperformance indicator and total cost are shown in Tables 3and 4

As can be seen from Table 3 with the increase of thecommunication lag the vehicle gap error of CTG controllerdecreases but the relative speed and actual accelerationincrease Compared with the CTG controller the CTG-IPSOcontroller improves the gap error relative speed and actualacceleration together with the system total cost which de-creases by 81ndash100 It can be seen from Table 4 that thevehicle gap error the relative speed and the actual accel-eration of CSG and CSG-IPSO controllers increase with theincrease of communication lag Compared with the CSGcontroller the CSG-IPSO controller improves the vehiclegap error and the relative speed but the actual acceleration isslightly increased combined with the total cost reduced by222ndash357 e analysis results show that the controllerdesigned in this paper can improve system efficiency withdifferent communication lag and gap control strategies Inaddition when communication lag increases from 0 s to02 s the total cost of CTG and CTG-IPSO controllers will beincreased by 47 and 69 respectively and the totalcost of CSG and CSG-IPSO controllers will be increased by123 and 359 respectively is finding shows that theconstant time-headway gap strategy can overcome the in-terference caused by communication lag which is suitablefor vehicle formation driving under extreme communicationconditions

Table 2 e root-mean-square value of performance indicators

Control strategy Δs (m) Δv (ms) a (ms2) Total costCTG 0166 0626 0612 1597CSG 0735 0317 0898 2927CTG-IPSO 0171 0574 0592 1438CSG-IPSO 0226 0085 0911 1930

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)

2nd 3rd 4th1stVehicle ID

(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)

Figure 8 Performance indicator for single CAV (a) gap error (b) relative speed and (c) actual acceleration


432 Actuator Lag With the values of other parametersfixed the actuator lag value varies between 01 s and 05 swith an interval of 01 s e optimized system performanceindicator and total cost are shown in Tables 5 and 6

It is shown in Table 5 that the performance indicators ofCTG and CTG-IPSO controllers increase with the increaseof actuator lag but the vehicle gap error of CTG-IPSOcontroller decreases Compared with the CTG scheme theCTG-IPSO controller reduces the vehicle gap error relativespeed and actual acceleration together with the total costwhich decreases by 77ndash157 In addition it can be seenfrom Table 6 that the vehicle gap error relative speed andactual acceleration of the CSG and CSG-IPSO controllersincreased with the increase of the actuator lag Comparedwith the CSG controller the CSG-IPSO controller hasimproved by the vehicle gap error and the relative speedhowever the actual acceleration is slightly increased com-bined with the total cost reduced by 331ndash343 e

analysis results also show that the controller designed in thispaper can improve the system efficiency with different ac-tuator lag and gap strategies

In addition when actuator lag increases from 01 s to05 s the total cost of CTG and CTG-IPSO controllers will beincreased by 180 and 78 respectively and the total costof CSG and CSG-IPSO controllers will be increased by 446and 434 respectively is finding also shows that theconstant time-headway gap strategy can overcome the ad-verse effects caused by actuator lag and has excellent flex-ibility and initiative and it is suitable for vehicle formationdriving under extreme mechanical conditions

5 Conclusions

Most studies have used constant time-headway gap or constantspacing gap as the CACC control strategy and the weightingcoefficients of the performance indicators are calculated

Table 3 Sensitivity analysis of communication lag of CTG and CTG-IPSO

τsCTG CTG-IPSO

Δs (m) Δv (ms) a (ms2) Total cost Δs (m) Δv (ms) a (ms2) Total cost0 0167 0624 0608 1583 0177 0572 0587 1425005 0166 0626 0612 1597 0171 0574 0592 1438010 0166 0627 0617 1614 0164 0577 0599 1461015 0164 0629 0623 1634 0157 0581 0608 1489020 0164 0631 0627 1657 0153 0585 0617 1523

Table 4 Sensitivity analysis of communication lag of CSG and CSG-IPSO

τsCSG CSG-IPSO


Table 5 Sensitivity analysis of actuator lag of CTG and CTG-IPSO

τACTG CTG-IPSO


Table 6 Sensitivity analysis of actuator lag of CSG and CSG-IPSO

τACSG CSG-IPSO



according to subjective judgment in order to achieve the op-timal control for the system However the strategy type andweighting coefficient uncertainties will directly affect the op-timal performance of the system under different driving en-vironments In this paper an optimal longitudinal controlstrategy of platoons using improved particle swarm optimi-zation has been proposed to address these issues e con-trolled object in this proposed strategy is the homogeneousplatoon model under the constant time-headway strategy andconstant spacing strategy In order to overcome the interfer-ence caused by changing the acceleration of the exogenoushead vehicle a linear-quadratic optimal controller is designedwith the consideration of the feedback delay and weightingcoefficient uncertainties e controller is optimized by animproved particle swarm optimization algorithm e simu-lation results show that the CTG-IPSO controller can reducethe total cost by 10 vehicle relative speed by 83 and actualacceleration by 33 compared with the unoptimized con-troller e CSG-IPSO controller can reduce the total cost by341 and reduce the vehicle gap error by 693 reducing therelative speed by 732Whether it is a constant time-headwaygap strategy or constant spacing gap strategy the strategyproposed in this paper can obtain good control benefits Inaddition the sensitivity analysis of the two parameters com-munication lag and actuator lag further proves the adaptabilityand effectiveness of the proposed strategy and also reveals thatthe constant time-headway gap strategy has great flexibility andinitiative with the balance of performance indicators which isapplicable to form platoons in extreme conditions

In order to highlight the benefits of the proposedmethod the homogeneous platoon system is chosen as theresearch object which does not consider the design andoptimization of the controller under the heterogeneoustraffic flow mixed with CAVs and human-driven vehiclesDuring the actual complex traffic scenarios the variabletime-headway gap strategy has greater flexibility andadaptability erefore future research should consider thestability analysis and controller design of platoons underthe variable time-headway gap strategy Furthermore thecontroller designed in this paper only considers the objec-tives of safety efficiency and comfort How to introduce theenergy consumption for the original basis is also a potentialtopic which needs to be expanded in the next research

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China under Grant nos 61773288 and51808057 and was supported in part by the National Key

Research and Development Program of China (no2018YFB1600805)

References

[1] J Rios-Torres and A A Malikopoulos ldquoA survey on thecoordination of connected and automated vehicles at inter-sections and merging at highway on-rampsrdquo IEEE Transac-tions on Intelligent Transportation Systems vol 18 no 5pp 1066ndash1077 2017

[2] C Yu Y Feng H X Liu W Ma and X Yang ldquoIntegratedoptimization of traffic signals and vehicle trajectories atisolated urban intersectionsrdquo Transportation Research Part BMethodological vol 112 pp 89ndash112 2018

[3] M Barth and K Boriboonsomsin ldquoReal-world carbon dioxideimpacts of traffic congestionrdquo Transportation Research Re-cord vol 2058 no 1 pp 163ndash171 2008

[4] N Jollands P Waide M Ellis et al ldquoe 25 IEA energyefficiency policy recommendations to the G8 gleneagles planof actionrdquo Energy Policy vol 38 no 11 pp 6409ndash6418 2010

[5] W Hao C Kamga X Yang et al ldquoDriver injury severitystudy for truck involved accidents at highway-rail gradecrossings in the United Statesrdquo Transportation Research PartF Traffic Psychology and Behaviour vol 43 pp 379ndash3862016

[6] Q Zeng W Gu X Zhang et al ldquoAnalyzing freeway crashseverity using a Bayesian spatial generalized ordered logitmodel with conditional autoregressive priorsrdquo AccidentAnalysis amp Prevention vol 127 pp 87ndash95 2019

[7] Q Wen H Wen H Huang and M Abdel-Aty ldquoA bayesianspatial random parameters tobit model for analyzing crashrates on roadway segmentsrdquo Accident Analysis amp Preventionvol 100 pp 37ndash43 2017

[8] F Chen and S Chen ldquoInjury severities of truck drivers insingle- and multi-vehicle accidents on rural highwaysrdquo Ac-cident Analysis amp Prevention vol 43 no 5 pp 1677ndash16882011

[9] X He H X Liu and X Liu ldquoOptimal vehicle speed trajectoryon a signalized arterial with consideration of queuerdquoTransportation Research Part C Emerging Technologiesvol 61 pp 106ndash120 2015

[10] H Yang H Rakha and M V Ala ldquoEco-cooperative adaptivecruise control at signalized intersections considering queueeffectsrdquo IEEE Transactions on Intelligent TransportationSystems vol 18 no 16 pp 1575ndash1585 2016

[11] L Xiao MWang W Schakel and B van Arem ldquoUnravellingeffects of cooperative adaptive cruise control deactivation ontraffic flow characteristics at merging bottlenecksrdquo Trans-portation Research Part C Emerging Technologies vol 96pp 380ndash397 2018

[12] F Chen M Song X Ma and X Zhu ldquoAssess the impacts ofdifferent autonomous trucksrsquo lateral control modes on asphaltpavement performancerdquo Transportation Research Part CEmerging Technologies vol 103 pp 17ndash29 2019

[13] Q Shi and H Zhuang ldquoFault diagnosis of an autonomousvehicle with an improved SVM algorithm subject tounba-lanced datasetsrdquo IEEE Transactions on Industrial Electronics2020

[14] Y Qin H Wang W Wang and D Ni ldquoReview of car-fol-lowing models of adaptive cruise controlrdquo Journal of Trafficand Transportation Engineering vol 17 no 3 pp 121ndash1302017

[15] G Marsden M McDonald and M Brackstone ldquoTowards anunderstanding of adaptive cruise controlrdquo Transportation


Research Part C Emerging Technologies vol 9 no 1pp 33ndash51 2001

[16] S Hoogendoorn R Hoogendoorn M Wang andW Daamen ldquoModeling driver driver support and cooper-ative systems with dynamic optimal controlrdquo TransportationResearch Record Journal of the Transportation ResearchBoard vol 2316 no 1 pp 20ndash30 2012

[17] D Swaroop J K Hedrick C C Chien and P Ioannou ldquoAcomparision of spacing and headway control laws for auto-matically controlled vehiclesrdquo Vehicle System Dynamicsvol 23 no 1 pp 597ndash625 1994

[18] J Zhou and H Peng ldquoRange policy of adaptive cruise controlvehicles for improved flow stability and string stabilityrdquo IEEETransactions on Intelligent Transportation Systems vol 6no 2 pp 229ndash237 2005

[19] P Y Li and A Shrivastava ldquoTraffic flow stability induced byconstant time headway policy for adaptive cruise controlvehiclesrdquo Transportation Research Part C Emerging Tech-nologies vol 10 no 4 pp 275ndash301 2002

[20] Y H Chiang and J C Juang ldquoLongitudinal vehicle controlwith the spacing policy in consideration of brake input limitsrdquoin Proceedings of the 2007 IEEE International Conference onSystems Man and Cybernetics October 2007

[21] D Yanakiev and I Kanellakopoulos ldquoNonlinear spacingpolicies for automated heavy-duty vehiclesrdquo IEEE Transac-tions on Vehicular Technology vol 47 no 4 pp 1365ndash13771998

[22] G N Bifulco L Pariota F Simonelli and R Di PaceldquoDevelopment and testing of a fully adaptive cruise controlsystemrdquo Transportation Research Part C Emerging Technol-ogies vol 29 pp 156ndash170 2013

[23] A Rosenfeld Z Bareket C V Goldman D J LeBlanc andO Tsimhoni ldquoLearning driversrsquo behavior to improve adaptivecruise controlrdquo Journal of Intelligent Transportation Systemsvol 19 no 1 pp 18ndash31 2014

[24] J Villagra B Vinagre and I Tejado ldquoData-driven fractionalPID control application to DC motors in flexible jointsrdquoIFAC Proceedings Volumes vol 45 no 3 pp 709ndash714 2012

[25] L LuoVehicle Adaptive Cruise Control and the CorrespondingMacroscopic Traffic Flow Model Zhejiang University Hang-zhou China 2011 in Chinese

[26] K L Sowjanya and L R Srinivas ldquoTuning of PID controllersusing particle swarm optimizationrdquo International Journal ofIndustrial Electronics and Electrical Engineering vol 3 no 2pp 17ndash22 2015

[27] S-Y Chen C-H Wu Y-H Hung and C-T Chung ldquoOp-timal strategies of energy management integrated withtransmission control for a hybrid electric vehicle using dy-namic particle swarm optimizationrdquo Energy vol 160 no 1pp 154ndash170 2018

[28] I Rahman P M Vasant B S M Singh and M Abdullah-Al-Wadud ldquoOn the performance of accelerated particle swarmoptimization for charging plug-in hybrid electric vehiclesrdquoAlexandria Engineering Journal vol 55 no 1 pp 419ndash4262016

[29] M Wang ldquoInfrastructure assisted adaptive driving to stabiliseheterogeneous vehicle stringsrdquo Transportation Research PartC Emerging Technologies vol 91 pp 276ndash295 2018

[30] I G Jin and G Orosz ldquoDynamics of connected vehiclesystems with delayed acceleration feedbackrdquo TransportationResearch Part C Emerging Technologies vol 46 pp 46ndash642014

[31] N Chen M Wang T Alkim and B van Arem ldquoA robustlongitudinal control strategy of platoons under model

uncertainties and time delaysrdquo Journal of Advanced Trans-portation vol 2018 Article ID 9852721 13 pages 2018

[32] X Zhang Q Wang and W Ji ldquoAn improved particle swarmoptimization algorithm for adaptive inertial weightsrdquo Mi-croelectronics amp Computer vol 36 no 3 pp 66ndash70 2019

[33] L Xiao and F Gao ldquoPractical string stability of platoon ofadaptive cruise control vehiclesrdquo IEEE Transactions on In-telligent Transportation Systems vol 12 no 4 pp 1184ndash11942011


CAV platoon is formed the strategy can also help maintainthe platoon operation throughout a highway corridorHowever there are many factors affecting the applicationeffect for the CACC system among which the most criticalones are spacing strategy and control algorithm Before theimplementation of large-scale field real vehicle test it isnecessary to study the influence of different spacing strat-egies and control algorithms on the performance of CACCsystem in order to improve the theory of CACC system andpromote the promotion of its technology

During the process of vehicle longitudinal control thereasonable spacing control strategy is one of the keys toensuring high safety car-following stability and satisfactoryroad capacity In recent years a variety of control strategieshave been put forward which are mainly divided intoconstant spacing gap (CSG) strategy constant time-headwaygap (CTG) strategy and variable time-headway gap (VTG)strategy CSG strategy shows that the vehicle alwaysmaintains a constant intervehicle distance with the pre-ceding vehicle [16 17] Due to its simple structure and lowapplication difficulty it is first applied in highway envi-ronment with relatively little interference [18] HoweverCSG strategy is insensitive to the change of vehicle speed andhas poor adaptability to the platoon stability in the complexand changeable driving environment In order to deal withthese shortcomings some strategies in which the gap varieswith the driving environment are proposed such as CTGstrategy and VTG strategy CTG strategy refers to theconstant time headway between the subject vehicle and thepreceding vehicle by constantly adjusting the speed underthe car-following state [19] With further research theflexibility and stability of CTG strategy like CSG strategycannot meet the requirements of the complex traffic system[20] erefore some VTG strategies that the time headwaycan change with the traffic flow environment have attractedthe attention of researchers With due consideration of theinfluence of vehicle speed acceleration and relative speedchanges by varied time headway Yanakiev and Kanella-kopoulos [21] presented a variable headway strategy andproved that the proposed strategy can guarantee platoonstability In addition some variable time-headway strategieswere proposed for different driving behaviors such asperception judgment and operation [22 23]

In the car-following mode it is greatly necessary to takeinto consideration various factors such as comfort safetycar-following stability traffic capacity and energy con-sumption except considering a stable safety control strategybetween the vehicle and the preceding vehicle From theperspective of optimal control the design of a reasonableand effective control algorithm is another research hotspotof CACC system design e design of a reasonable andeffective control algorithm from the perspective of optimalcontrol is another focus of CACC research field Presentlythe theories and methods applied to the car-following modeof CACC system include classical proportional integraldifferential (PID) control model predictive control (MPC)and artificial neural network and linear-quadratic (LQ)control For the nonlinear characteristics of CACC systemPID is applied to realize the automatic adjustment of the

performance index parameters [24] rough controlstrategies such as prediction model rolling optimizationand feedback correction MPC can take into account mul-tiple control objectives and constraints simultaneouslywhich is conducive to meeting multiple control require-ments during vehicle driving [25] e disadvantage of MPCis that the model structure is too complex e methodsof artificial intelligence mainly include fuzzy control andartificial neural network e idea is to establish the rela-tionship between input (vehicle position velocity acceler-ation etc) and output (desired acceleration angle etc)through the sample training method It has the advantagesof simple design structure mature algorithm applicationconditions and low hardware requirements and thedisadvantages of the model structure are difficult to de-scribe LQ algorithm also known as a linear-quadraticalgorithm is an optimal control law based on quadraticperformance indexes to obtain state linear feedback andrealize closed-loop optimal control Its advantages are notlimited by the concrete model structure and its design isflexible e disadvantage is that the control performancedepends on the weight matrix of state variables and inputvariables and these choices are subjective and lack ob-jective description

To attain optimal gains for PID or LQ algorithm a seriesof intelligent optimization algorithms were used such asGenetic Algorithm (GA) and particle swarm optimization(PSO) methods Compared with GA and other optimizationmethods PSO memorizes satisfactory solutions among allthe particles without considering previous knowledge aftereach evolution so that it attracts increasing attention forsolving complex optimization problems Sowjanya andSrinivas [26] presented the development of an optimal PIDcontroller for a composition control system using PSOtechnique Chen et al [27] used dynamic particle swarmoptimization to develop optimal strategies for two-variableenergy management and gear-shifting in hybrid electricvehicles e results demonstrate that compared with thebaseline case the proposed optimization strategy broughtsuperior performance when applied to hybrid energymanagement and transmission control Rahman et al [28]investigated the use of the accelerated particle swarm op-timization (APSO) method for developing real-time andlarge-scale optimizations for allocating power Comparedwith standard particle swarm optimization consideringcharging time and battery capacity the APSO method canachieve some improvements in terms of best fitness andcomputation timee above studies have revealed that PSOis a reliable tool for designing an optimal strategy and that itcan outperform other evolutionary algorithms [27]

e longitudinal control strategy of CAV platoon iscomplex especially with the uncertainty of key parametersand there is a need to develop computationally efficientalgorithms to find optimal solutions e objective of thispaper is to design an optimal controller considering the keyparameter selection involved in the longitudinal vehicledynamics e optimal control is designed by consideringthe parametric uncertainties in the dynamics model ofplatoons expressed by a third-order linear vehicle model




2 Dynamics Models






_Fn minusFn

τA+

Un

τA (2)





+ Rgn + Rdn + Ran1113872 1113873

x

n 1τA









ddt


T

ddt


vnminus1 minus vn

an

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


anminus1 minus an


τA

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(4)


where A 0 1 minust

d


⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦ B

00

minus1τn

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ C

010

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦










ddtZp


T g Zp

Up dp

( 1113857

g ZpUp

dp( 1113857 Ap

middot Zp+ Bp

middot Up+ Cp

middot dp

(6)

where

Ap

A3times31 03times3



⋮ ⋮ ⋱ ⋮


N

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bp

B1 0

⋱

0 BN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

C 0

0 0⎡⎢⎣ ⎤⎥⎦

A3times31

0 1 minustd

0 0 minus1

0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A3times4i

0 0 1 minustd

1 0 0 minus1

0 0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

i 2 3 N

Bk

0

0

1τA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k 1 2 N

(7)




J minUp

1113946t0+Tp

t0

L Zp(t)Up

(t)( 1113857dt (8)

L c1 1113944

N

i1Δsi( 1113857

2+ c2 1113944

N

i1Δvi( 1113857

2+ c3 1113944

N

i1ui( 1113857

2 (9)





v le vi le v (10)

(2) Gap constraint



a le ui le a (12)


1113957Zpt0( 1113857 Zp

t0 minus τs( 1113857 (13)



J minUp t0 t0+Tp1113858 1113859

1113946t0+Tp

t0

Zp( 1113857

TQZp+ Up

( 1113857TRUp

1113876 1113877dt (14)

where

Q

q1 03times3

⋱

03times3 qi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R

r1 0

⋱

0 ri

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

qi

c1 0 0

0 c2 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ri c31113858 11138591times1

i 1 2 N

(15)



(t) (16)


PAp+ Ap


( 1113857TP + Q 0 (17)






χt+1 χt + ϑt+1 (19)


1113888 1113889 (20)
















Start






End


Y

N

IPSO Simulink



(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References








































2 Dynamics Models






_Fn minusFn

τA+

Un

τA (2)





+ Rgn + Rdn + Ran1113872 1113873

x

n 1τA









ddt


T

ddt


vnminus1 minus vn

an

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


anminus1 minus an


τA

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(4)


where A 0 1 minust

d


⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦ B

00

minus1τn

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ C

010

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦










ddtZp


T g Zp

Up dp

( 1113857

g ZpUp

dp( 1113857 Ap

middot Zp+ Bp

middot Up+ Cp

middot dp

(6)

where

Ap

A3times31 03times3



⋮ ⋮ ⋱ ⋮


N

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bp

B1 0

⋱

0 BN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

C 0

0 0⎡⎢⎣ ⎤⎥⎦

A3times31

0 1 minustd

0 0 minus1

0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A3times4i

0 0 1 minustd

1 0 0 minus1

0 0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

i 2 3 N

Bk

0

0

1τA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k 1 2 N

(7)




J minUp

1113946t0+Tp

t0

L Zp(t)Up

(t)( 1113857dt (8)

L c1 1113944

N

i1Δsi( 1113857

2+ c2 1113944

N

i1Δvi( 1113857

2+ c3 1113944

N

i1ui( 1113857

2 (9)





v le vi le v (10)

(2) Gap constraint



a le ui le a (12)


1113957Zpt0( 1113857 Zp

t0 minus τs( 1113857 (13)



J minUp t0 t0+Tp1113858 1113859

1113946t0+Tp

t0

Zp( 1113857

TQZp+ Up

( 1113857TRUp

1113876 1113877dt (14)

where

Q

q1 03times3

⋱

03times3 qi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R

r1 0

⋱

0 ri

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

qi

c1 0 0

0 c2 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ri c31113858 11138591times1

i 1 2 N

(15)



(t) (16)


PAp+ Ap


( 1113857TP + Q 0 (17)






χt+1 χt + ϑt+1 (19)


1113888 1113889 (20)
















Start






End


Y

N

IPSO Simulink



(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References











































ddtZp


T g Zp

Up dp

( 1113857

g ZpUp

dp( 1113857 Ap

middot Zp+ Bp

middot Up+ Cp

middot dp

(6)

where

Ap

A3times31 03times3



⋮ ⋮ ⋱ ⋮


N

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Bp

B1 0

⋱

0 BN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

C 0

0 0⎡⎢⎣ ⎤⎥⎦

A3times31

0 1 minustd

0 0 minus1

0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

A3times4i

0 0 1 minustd

1 0 0 minus1

0 0 0 minus1τA

1113888 1113889

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

i 2 3 N

Bk

0

0

1τA

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

k 1 2 N

(7)




J minUp

1113946t0+Tp

t0

L Zp(t)Up

(t)( 1113857dt (8)

L c1 1113944

N

i1Δsi( 1113857

2+ c2 1113944

N

i1Δvi( 1113857

2+ c3 1113944

N

i1ui( 1113857

2 (9)





v le vi le v (10)

(2) Gap constraint



a le ui le a (12)


1113957Zpt0( 1113857 Zp

t0 minus τs( 1113857 (13)



J minUp t0 t0+Tp1113858 1113859

1113946t0+Tp

t0

Zp( 1113857

TQZp+ Up

( 1113857TRUp

1113876 1113877dt (14)

where

Q

q1 03times3

⋱

03times3 qi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R

r1 0

⋱

0 ri

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

qi

c1 0 0

0 c2 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ri c31113858 11138591times1

i 1 2 N

(15)



(t) (16)


PAp+ Ap


( 1113857TP + Q 0 (17)






χt+1 χt + ϑt+1 (19)


1113888 1113889 (20)
















Start






End


Y

N

IPSO Simulink



(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References






































v le vi le v (10)

(2) Gap constraint



a le ui le a (12)


1113957Zpt0( 1113857 Zp

t0 minus τs( 1113857 (13)



J minUp t0 t0+Tp1113858 1113859

1113946t0+Tp

t0

Zp( 1113857

TQZp+ Up

( 1113857TRUp

1113876 1113877dt (14)

where

Q

q1 03times3

⋱

03times3 qi

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

R

r1 0

⋱

0 ri

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

qi

c1 0 0

0 c2 0

0 0 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

ri c31113858 11138591times1

i 1 2 N

(15)



(t) (16)


PAp+ Ap


( 1113857TP + Q 0 (17)






χt+1 χt + ϑt+1 (19)


1113888 1113889 (20)
















Start






End


Y

N

IPSO Simulink



(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References
















































Start






End


Y

N

IPSO Simulink



(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References






































(∆s1 ∆v1)(∆s2 ∆v2)(∆s3 ∆v3)(∆s4 ∆v4)

FollowingCAV


V2V


Tota

l cos

t

143

144

145

146

147

20 40 60 80 1000Iteration

(a)

Tota

l cos

t190

195

200

205

210

215

20 40 60 80 1000Iteration

(b)




1 | (m) 0400 2150 1132 0767max |Δs+

4 | (m) 0240 0263 0247 0059max |Δv+

1 | (ms) 1037 1320 1006 0295max |Δv+

4 | (ms) 0948 0152 0917 0031max |a+

1 | (ms2) 1007 1179 1001 1053max |a+

4 | (ms2) 0954 1265 0994 1101min |Δsminus

1 | (m) 0769 4638 0313 1331min |Δsminus

4 | (m) 0155 0504 0191 0105min |Δvminus

1 | (ms) 3733 2549 3057 0783min |Δvminus

4 | (ms) 1417 0233 1310 0070min |aminus

1 | (ms2) 3553 4625 3324 4205min |aminus

4 | (ms2) 1447 4110 1479 4392

∆s (m

)

1st2nd

3rd4th

ndash1

ndash05

0

05

1

10 20 30 40 500Time (s)

(a)

∆v (m

s)

1st2nd

3rd4th

ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(b)

a (m

s2 )

3rd

1st2nd


ndash4

ndash3

ndash2

ndash1

0

1

2

10 20 30 40 500Time (s)

(c)



∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References






































∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

4

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash05

0

05

1

15

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

(c)


∆s (m

)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash15

ndash1

ndash05

0

05

1

(a)

∆v (m

s)

1st2nd

3rd4th

10 20 30 40 500Time (s)

ndash1

ndash05

0

05

(b)

a (m

s2 )

3rd

1st2nd


10 20 30 40 500Time (s)

ndash6

ndash4

ndash2

0

2

(c)











CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References














































CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

15

∆s (m

)


(a)

CTGCSG

CTG-IPSOCSG-IPSO

0

05

1

∆v (m

s)


(b)

CTGCSG

CTG-IPSOCSG-IPSO

04

06

08

1

a (m

s2 )


(c)







5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References










































5 Conclusions



τsCTG CTG-IPSO



τsCSG CSG-IPSO



τACTG CTG-IPSO



τACSG CSG-IPSO





Data Availability




Acknowledgments



References








































Data Availability




Acknowledgments



References



























































Documents

An Optimal Longitudinal Control Strategy of Platoons Using ...downloads.hindawi.com/journals/jat/2020/8822117.pdfCAVplatoonisformed,thestrategycanalsohelpmaintain the platoon operation