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Car-Following Model-Based Stochastic Distance Regulation Between Vehicles⁎
5th IFAC Conference on 5th IFACand Conference onControl, Simulation and Modeling Engine Powertrain 5th IFAC Conference on Engine and Powertrain Control, Simulation Modeling Available online at www.sciencedirect.com Changchun, China, September 2018 and Engine Powertrain Simulation and Modeling 5th IFACand Conference onControl,20-22, Changchun, China, September 20-22, 2018 Changchun, China, September 2018 and Modeling Engine and Powertrain Control,20-22, Simulation Changchun, China, September 20-22, 2018
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IFAC PapersOnLine 51-31 (2018) 859–862
Car-Following Model-Based Stochastic Car-Following Model-Based Stochastic Car-Following Model-Based Stochastic Distance Regulation Between Vehicles Car-Following Model-Based Stochastic Distance Regulation Between Vehicles Distance Regulation Between Vehicles Distance Between Jingwei Li, Regulation Changli Zhao, Hongwei Yue, Vehicles Wenjun Fu
Jingwei Jingwei Li, Li, Changli Changli Zhao, Zhao, Hongwei Hongwei Yue, Yue, Wenjun Wenjun Fu Fu Jingwei Li, ofChangli Zhao, Hongwei Shandong Yue, Wenjun Fu Department Automotive Engineering, Jiaotong Department of Engineering, Shandong Department of Automotive Automotive Engineering, Shandong Jiaotong Jiaotong University, Jinan, China(e-mail: [email protected], University, Jinan, China(e-mail: [email protected], Department of Automotive Engineering, Shandong Jiaotong University, Jinan, China(e-mail: [email protected], [email protected], [email protected], [email protected].) [email protected], [email protected], [email protected].) University, Jinan, China(e-mail: [email protected], [email protected], [email protected], [email protected].) [email protected], [email protected], [email protected].) Abstract: The problem of car-following model-based stochastic distance regulation between Abstract: The problem model-based stochastic distance regulation between Abstract: problem of ofofcar-following car-following model-based stochastic distance between vehicles withThe consideration the disturbance caused by the sensitivity of the regulation driver is researched vehicles with consideration of the disturbance caused by the sensitivity of the driver Abstract: ofofdistance car-following model-based stochastic distance regulation between vehicles withThe consideration the disturbance caused by the sensitivity of the driver is is researched researched in this paper. Anproblem stochastic regulator is designed based on the car-following dynamic in this An stochastic regulator is based on the car-following dynamic vehicles with ofdistance the disturbance caused byvehicles. the sensitivity of the driver is researched in this paper. paper. An stochastic distance regulator is designed designed basedThe on validation the car-following dynamic model, for theconsideration purpose of distance regulation between of the proposed model, for the purpose of regulation vehicles. The of the in this paper. stochastic distance regulatorbetween is designed based on validation the car-following dynamic model, for regulator the An purpose of distance distance between vehicles. The validation the proposed proposed stochastic is given by the regulation numerical simulation. The simulation resultsofshow that the stochastic regulator is given by the numerical simulation. The simulation results show that the model, the purpose of can distance regulation vehicles. The validation ofshow the proposed stochastic regulator is given by numerical simulation. The of simulation results that the distancefor between vehicles bethe regulated to abetween neighbourhood its desired value under various distance between vehicles can be regulated to a neighbourhood of its desired value under various stochastic regulator is given numerical The of simulation results the distance between vehicles canby bethe regulated to asimulation. neighbourhood its desired value show underthat various working conditions. working distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. conditions. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. working conditions. Keywords: Car-Following Model, Stochastic Control, Distance Regulation Between Vehicles, Keywords: Car-Following Model, Stochastic Keywords:Model, Car-Following Model, Stochastic Control, Control, Distance Distance Regulation Regulation Between Between Vehicles, Vehicles, Dynamic Numerical Simulation. Dynamic Model, Numerical Simulation. Keywords: Car-Following Model, Stochastic Control, Distance Regulation Between Vehicles, Dynamic Model, Numerical Simulation. Dynamic Model, Numerical Simulation. 1. INTRODUCTION on the base of the control theory. An improved adaptive 1. INTRODUCTION on the base of theory. An improved adaptive 1. INTRODUCTION on thefuzzy base inference of the the control control improved neuro systemtheory. model An is proposed byadaptive A. Khoneuro fuzzy inference system model is proposed by A. INTRODUCTION thefuzzy base inference of to thesimulate control An improved neuro systemtheory. model is proposed byadaptive A. KhoKhodayari (2014) and predict the car-following The increasing of 1. traffic demand has led to serious traffic on dayari (2014) to simulate and predict the car-following The increasing of traffic demand has led to serious traffic neuro fuzzy inference system model is proposed by A. Khodayari (2014) to simulate and predict the car-following based on the reaction delay of driver-vehicle The increasing of traffic demand led to serious traffic behaviour congestion problem, and results inhas environmental pollution behaviour based on the delay of the driver-vehicle congestion problem, and in environmental pollution (2014) and predict car-following behaviour based onsimulate the reaction reaction delay the driver-vehicle unit, where theto reaction delay is used asofan input and other The increasing ofproblems. traffic demand has ledresearch to serious traffic congestion problem, and results results incrucial environmental pollution and traffic safety As an area, the dayari unit, where the reaction delay is used as an input and and traffic safety problems. As an crucial research area, the behaviour based on the reaction delay of the driver-vehicle unit, where the reaction delay is used as an input andtoother other inputs-outputs of the model are chosen with respect this congestion problem, and results in environmental pollution and traffic safety problems. As an crucial research area, the dynamic analysis and control problems of traffic systems inputs-outputs of the model are chosen with respect this dynamic analysis and control problems of traffic systems unit, where In theorder reaction delay is chosen used aswith an input andto inputs-outputs of thetomodel are respect toother this suppress the traffic jam effectively, and safety As an crucial years. research the parameter. dynamic and control problems of trafficarea, systems havetraffic beenanalysis widelyproblems. investigated in recent parameter. In order to suppress the traffic jam effectively, have been widely investigated in recent years. inputs-outputs of the model are chosen with respect to this parameter. In order to suppress the traffic jam effectively, is dynamic and control problems of traffic systems a new congestion feedback control car-following model have beenanalysis widely investigated in recent years. aparameter. new congestion feedback control car-following model is A coupled map traffic flow model is introduced based In order to suppress the traffic jam effectively, a new congestion feedback control car-following model is presented in T. Zhou (2015), by considering the effect of have been widely investigated in recent years. A coupled map traffic flow model is introduced based in T. Zhou (2015), by considering the effect of A traffic model is introduced based presented on coupled optimal map velocity in S.flow Tadaki (1998), and the model a new congestion feedback control car-following model is presented in T. Zhou (2015), by considering the effect of the safe headway to the traffic system. Due to the fact that on optimal velocity in Tadaki (1998), and model safe headway to traffic system. Due fact that A map traffic flow model is conditions. introduced based on optimal velocity in S. S. Tadaki (1998), and the the A model cancoupled be simulated under open boundary car- the presented in T.car-following Zhou by considering thecombined effect of safe headway to the the(2015), trafficand system. Due to to the the fact that the traditional lane-changing can be simulated under open boundary conditions. A cartraditional car-following lane-changing on optimal velocity in S. Tadaki (1998), and the control model can be simulated open boundary A car- the following model isunder established based onconditions. optimal safe headway tothe the driver’s trafficand system. Due tointo thecombined fact that the traditional car-following and lane-changing combined models fail to take anticipation considerfollowing model is established based on optimal control models fail to take the driver’s anticipation into considercan be simulated under open boundary conditions. A carfollowing model is established based on optimal control by analyzing the information perception and operation the traditional car-following and lane-changing combined models fail to take the driver’s anticipation into considerationbased on full velocity difference car-following model by analyzing the perception and operation on full velocity difference car-following model following established based on optimal control by analyzing theisinformation information perception and the operation decision ofmodel drivers by Y. Y. Zhang (2011), and results ationbased models fail to take the driver’s anticipation into considerationbased on full velocity difference car-following model and probabilistic lane-changing model, a double-head cardecision of by Zhang (2011), and the results and probabilistic lane-changing model, a double-head carby analyzing information perception and operation decision of drivers drivers by Y. Y. Y. Y. variable Zhang (2011), and the results show that thethe perceptual relative velocity has ationbased on full velocity difference car-following model and probabilistic lane-changing model,model a double-head carfollowing and lane-changing combined was proposed show that the perceptual variable relative velocity has and lane-changing combined model was proposed decision of drivers Y. Y. variable Zhang (2011), andvelocity the results show the perceptual relative has great that influence onbydriver’s decision in car-following. A following and probabilistic lane-changing model, a double-head carfollowing and lane-changing combined model was proposed in H. Wang (2015), and the method of model calibration great influence on driver’s decision in car-following. A H. Wang (2015), and the method of model calibration show themethod perceptual variable velocity has great influence on driver’s decision in car-following. A in simplethat control to suppress therelative traffic congestion in following and lane-changing combined model was proposed in H. Wang (2015), and the method of model calibration was presented as well. The gap between traffic flow modelsimple control method to the traffic congestion in presented as The gap between traffic flow modelgreat influence on driver’s decision A was simple control method to suppress suppress theinin traffic in the coupled map car-following model X.car-following. M.congestion Zhao (2006), in Wang (2015), and the method modelby calibration was presented as well. well. The gap between traffic flow ingH. and communication approaches is of bridged D.modelY. Jia the coupled map car-following model in X. M. Zhao and communication approaches is bridged by D. Y. Jia simple method to suppress the traffic congestion in the map car-following model X. M.congestions Zhao (2006), (2006), andcoupled thecontrol effect of the control signal onintraffic is ing was presented as well. The gap between traffic flow ing and in communication approaches is bridged by D.modelY. Jia (2016) order to build up better cooperative traffic and the effect of the control signal on traffic congestions is (2016) in order to build up better cooperative traffic the coupled map car-following model in X. M. Zhao (2006), and the effect of the control signal on traffic congestions is investigated with the CM car-following model. The prop- (2016) ing and in communication approaches is bridged by D. Y. Jia order to build up better cooperative traffic systems, and an enhanced cooperative microscopic investigated with the CM car-following model. The propand an enhanced cooperative microscopic traffic and of the signal traffic congestions is systems, investigated with thecontrol CM car-following model. The propertiesthe of effect the General Motors based on car-following models (2016) in order to build up better cooperative systems, and an enhanced cooperative microscopic traffic model is developed by considering V2V and V2I comerties of General based car-following models is is developed by V2V and cominvestigated with theMotors CM car-following model. The properties of the the Motors based car-following models is model evaluated byGeneral P. Chakroborty (1999), and identifies their systems, enhanced cooperative microscopic model is and developed by considering considering V2V and V2I V2I traffic communication. Aanparameter which is newly introduced into evaluated by P. Chakroborty (1999), and identifies their A parameter which is newly introduced into erties of the Motors based car-following models is munication. evaluated byGeneral P.and Chakroborty and identifies their characteristics, proposes a(1999), fuzzy inference logic based model is developed by considering V2V and V2I munication. A parameter which is newly introduced into Newell’s car-following model is presented by X. D. comHua characteristics, and proposes a fuzzy inference logic based Newell’s car-following model is presented by X. D. Hua evaluated by P. Chakroborty (1999), and identifies their characteristics, and proposes a fuzzy inference logic based model that can overcome some of the shortcomings of the Newell’s munication. A parameter which is newly introduced into car-following model is presented by X. D. Hua (2016) to represent the pre-reaction of drivers to traffic model that can overcome some of shortcomings the to represent the pre-reaction of drivers to traffic characteristics, fuzzy inference logic of based model thatmodels. can and overcome of the thethe shortcomings of the (2016) GM based Inproposes ordersome to areflect traffic flow’s charNewell’s car-following model is presented by X. D. Hua (2016) to represent the pre-reaction of drivers to traffic flow information provided by V2V devices. Formulations GM based In to reflect the traffic flow’s information provided by devices. Formulations model thatmodels. can overcome of shortcomings the flow GM based models. In order ordersome toon reflect the traffic flow’sofcharcharacteristics more realistically, thethe basis of CM(Coupled(2016) tooperational represent the pre-reaction of traffic drivers traffic flow information provided by V2V V2V devices. Formulations of traffic capacity in mixed is to provided acteristics more realistically, on the basis of CM(Coupledtraffic operational capacity in mixed traffic is GM models. Inmodel order proposed toon reflect the flow’s characteristics more realistically, the basis of CM(CoupledMap)based car following by traffic Konishia new ex- of flow information provided by V2V devices. Formulations of traffic operational capacity in mixed traffic is provided provided in D. J. Chen (2017), which consists of automated vehicles Map) car following model proposed by Konishia new exJ. (2017), which consists of automated vehicles acteristics more on the basis ofC.CM(CoupledMap) caroffollowing model isproposed by in Konishia ex- in pression safelyrealistically, distance established Zhainew (2017), of D. traffic operational capacity in ismixed traffic is provided in D. J. Chen Chen (2017),when whichtraffic consists automated vehicles and regular vehicles, inofequilibrium. In order pression safely isproposed established C. (2017), regular vehicles, when traffic is in equilibrium. In order Map) carof following model by in Konishia new expression ofconsideration safely distance distance established in C.ofZhai Zhai (2017), with the of is the safe distance vehicles at and in D. J. Chen (2017), which consists of automated vehicles and regular vehicles, when traffic is in equilibrium. In order to solve the problem of time variability, randomness and with the consideration of the safe distance of vehicles at solve the problem of time variability, randomness and pression safelyAdistance established inisC. (2017), with the of consideration of is the safe which distance ofZhai vehicles at to different speeds. dynamic model established by and regular vehicles, when traffic is in equilibrium. In order to solve the problem of time variability, randomness and uncertainty in the traffic flow forecasting, a new combined different speeds. A model is by in the traffic flow forecasting, aa new combined with consideration oftwo the safe which distance of vehicles at uncertainty different A dynamic dynamic model which is established established by H. X.the Ge speeds. (2012) contains successive vehicles’ headway to solve the problem of time variability, randomness and uncertainty in the traffic flow forecasting, new combined control predictive algorithm was proposed in Dong (2018), H. X. Ge (2012) contains two successive vehicles’ headway predictive algorithm was proposed in Dong (2018), different Aofdynamic which isincorporated established by H. X. Ge speeds. (2012) twomodel successive vehicles’ headway distances in frontcontains the considered one are in control uncertainty in the traffic flow forecasting, a new combined control predictive algorithm was proposed in Dong (2018), and the results show that the average relative error and distances in of considered one are incorporated in thepredictive results show that the relative error and H. Ge (2012) contains two successive vehicles’ headway distances in front front of the the considered incorporated in and theX. optimal velocity function, and one the are stability condition control algorithm wasaverage proposed in Dong (2018), and results show that the average error and meanthe square deviation obtained with therelative proposed method the optimal velocity function, and the stability condition square deviation obtained with the proposed method distances front of the one are incorporated in mean the optimal velocity function, and of the stability condition is given forinthe change ofconsidered the speed the preceding vehicle and the results show that the average relative error and mean square deviation obtained with the proposed method are 74% and 85% lower than those of other methods. is for change of the preceding vehicle and 85% than other methods. thegiven optimal velocity and of condition is given for the the changefunction, of the the speed speed ofthe thestability preceding vehicle are mean square withof proposed method are 74% 74% and deviation 85% lower lowerobtained than those those ofthe other methods. In this thelower disturbance caused by the sensitivity is givenwork for the change ofbythe speed of the preceding vehicle is supported Scientific Research Fund Project of are 74%paper, and 85% than those of other methods. This In this paper, disturbance caused the This work is supported by Scientific Research Fund Project of In this driver paper,isthe the disturbance caused by by the sensitivity sensitivity of the modeled as a standard wiener processes. Shandongjiaotong University Science Class), NO:Project Z201704, This work is supported by(Natural Scientific Research Fund of of the driver modeled as wiener processes. Shandongjiaotong University (Natural Science Class), NO: Z201704, In this paper,is disturbance caused by thetosensitivity of driver isthe modeled as aa standard standard wiener processes. Thethe stochastic distance regulator is designed overcome Z201635; Shandong Provincial Higher School Science and Technology Shandongjiaotong University Science Class), NO: Z201704, This work is supported by(Natural Scientific Research Fund Project of The stochastic distance regulator is designed to overcome Z201635; Shandong Provincial Higher School Science and Technology of the driver isdistance modeledregulator as a standard wiener processes. Project ofShandong China under Grant(Natural J17KA036. The stochastic is designed to overcome Z201635; Provincial Higher School Science and Technology Shandongjiaotong University Science Class), NO: Z201704, Project of China under Grant J17KA036. Project ofShandong China under Grant Higher J17KA036. The stochastic distance regulator is designed to overcome Z201635; Provincial School Science and Technology Project of © China under J17KA036. 2405-8963 IFAC Grant (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 903 Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2018 responsibility IFAC 903Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 903 10.1016/j.ifacol.2018.10.097 Copyright © 2018 IFAC 903
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the influence of the disturbance. The numerical simulation shows that the stochastic distance regulator is availably when the accelerated velocity of the leading vehicle both sine curve and step curve.
7.5
2. STOCHASTIC CONTROL PROBLEM
6.5
2.1 Control Problem Description
where x is distance between the vehicles, v is the relative velocity, al is accelerated velocity of the leading vehicle, af is the accelerated velocity of the following vehicle, the standard wiener processes aδ denotes the disturbance caused by the sensitivity of the driver, and δ is the disturbance intensity. For the system (1), given an stochastic regulator as (2)
where x∗ is the desired value of x, such that the distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. 2.2 Stochastic Regulator Design In this section, the process of the stochastic regulator design process will be expressed based on the Lyapunov method. Step 1. Introducing the two error variables e1 = x − x∗ , e2 = v − x∗(1) − α(x, x∗ ). de1 = (v − x )dt = (e2 + α(x, x∗ ))dt. Choose the Lyapunov function 1 V1 = e41 . 4 From (4) and (5), we have LV1 = e31 (e2 + α(x, x∗ )) 3 1 ≤ e42 + e31 ( e1 + α(x, x∗ )). 4 4 By choosing
3 α(x, x∗ ) = −c1 e1 − e1 , 4
where c1 is the design parameter, we can obtain 1 LV1 ≤ −c1 e41 + e42 . 4
(3)
(4)
(5)
5
0
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 1. Distance between the vehicles of W1 Let us consider the Lyapunov candidate 1 V2 = V1 + e42 . 4 In view of (8) and (9), we have 1 LV2 = −c1 e41 + e42 + e32 (al − af − x∗(2) 4 ∂α(x, x∗ ) ∗(1) ∂α(x, x∗ ) 3 v− − x ) + e22 δ 2 ∂x ∂x∗ 2 1 ≤ −c1 e41 + e32 ( e2 + al − af − x∗(2) 4 ∂α(x, x∗ ) ∗(1) ∂α(x, x∗ ) v− − x ∂x ∂x∗ 3 3 4 + e2 ) + δ . 4 4
(10)
(11)
By choosing af as 1 ∂α(x, x∗ ) e2 + al − x∗(2) − v 4 ∂x ∗ ∂α(x, x ) ∗(1) 3 − x + e2 − c2 e2 , ∂x∗ 4
(12)
where c2 is the design parameter. From (11), denote V = V2 , we get 3 (13) LV ≤ −cV + δ 4 , 4 where the design parameter c = min{c1 , c2 }. 3. NUMERICAL SIMULATION
(6)
(7)
(8)
Step 2.From (3), we can obtain ∂α(x, x∗ ) v de2 = (al − af − x∗(2) − ∂x ∗ ∂α(x, x ) ∗(1) − x )dt + δdaδ . ∂x∗
5.5
af =
From (1) and (3), one has ∗(1)
Distance between the vehicles/(m)
6
In this section, the description of the control problem is given. The car-following model with consideration of the sensitivity of the driver is provided as follows: dx = vdt, (1) dv = (al − af )dt + δdaδ ,
af = f (x, v, x∗ , x∗(1) , x∗(2) ),
7
(9)
904
In a principle phase to test the effectiveness of the control scheme, the numerical simulation is established based on (1) and (12). The numerical simulation is run under working conditions W1 and W2 . For working condition W1 , the accelerated velocity of the leading vehicle is a sine curve with disturbance, the noise intensity of the disturbance is 0.001, the the initial value and the desired value of the distance between the vehicles are 5m and 7m, respectively. For working condition W2 , the accelerated velocity of the leading vehicle is a step curve with disturbance, the noise intensity of the disturbance is 0.001, the the initial value and the desired value of the distance between the vehicles are 9m and 6m, respectively. The signals of the distance between the vehicles, the relative velocity between vehicles, the accelerated velocity of the leading vehicle and the accelerated velocity of
IFAC E-CoSM 2018 Changchun, China, September 20-22, 2018 Jingwei Li et al. / IFAC PapersOnLine 51-31 (2018) 859–862
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2
861
0.5 0 Relative velocity between vehicles/(m/s)
−0.5 Relative velocity between vehicles/(m/s)
−1 −1.5 −2 −2.5 0
50
−3
100 150 200 250 300 350 400 450 500 Time/(s)
100 150 200 250 300 350 400 450 500 Time/(s)
1.5
Accelerated velocity of the leading vehicle/(m/s2)
1
1
0. 5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5 0
50
Fig. 6. Relative velocity between vehicles of W2
Fig. 2. Relative velocity between vehicles of W1 1. 5
0
50
−1.5 0
100 150 200 250 300 350 400 450 500 Time/(s)
Accelerated velocity of the leading vehicle/(m/s2)
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 7. Accelerated velocity of the leading vehicle of W2
Fig. 3. Accelerated velocity of the leading vehicle of W1
10
40 35 30 Accelerated velocity of the following vehicle/(m/s2) 25 20 15 10 5 0 −5 0 50 100 150 200 250 300 350 400 450 500 Time/(s)
0 −10 −20
−40 −50 −60 0
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 8. Accelerated velocity of the following vehicle of W2
Fig. 4. Accelerated velocity of the following vehicle of W1 9 8.5 8 Distance between the vehicles/(m)
7.5
Accelerated velocity of the following vehicle/(m/s2)
−30
7 6.5
the following vehicle of working condition W1 are given in Figures 1-4. From Figures 1-4 we can observed that the distance between the vehicles can be regulated to a neighbourhood of its desired value, when the accelerated velocity of the leading vehicle is varied as a sine curve with disturbance. Figures 5-8 shows the signals of the distance between the vehicles, the relative velocity between vehicles, the accelerated velocity of the leading vehicle and the accelerated velocity of the following vehicle of working condition W2 from which we can obtain that the proposed stochastic regulator is still valid.
6 5.5 0
4. CONCLUSION 50
100 150 200 250 300 350 400 450 500 Time/(s)
A car-following model in which the sensitivity of the driver is modeled as a stochastic disturbance is provided in this paper. An stochastic distance regulator is designed
Fig. 5. Distance between the vehicles of W2 905
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to achieves the distance regulation between vehicles. By the proposed stochastic regulator, the distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. REFERENCES A. Khodayari, A. Ghaffari, R.K.e.a. (2014). Improved adaptive neuro fuzzy inference system car-following behavior model based on the driver-vehicle delay. Intelligent Transport Systems, 8, 323–332. C. Zhai, W. Y. Wu, L.H.e.a. (2017). Feedback control of car following model on the basis of time-varying safety distance. Journal of South China University of Technology (Natural Science Edition), 45, 126–134, in Chinese. D. J. Chen, S. AHN, M.C.e.a. (2017). Towards vehicle automation: roadway capacity formulation for traffic mixed with regular and automated vehicles. Transportation Research Part B: Methodological, 100, 196–221. D. Y. Jia, D.N. (2016). Enhanced cooperative carfollowing traffic model with the combination of v2v and v2i communication. Transportation Research Part B: Methodological, 90, 172–191. Dong, W. (2018). Combined control predictive model and method for stochastic traffic flow. Journal of Shenyang University of Technology, 40, 88–93, in Chinese. H. Wang, Z. Q. Liu, Z.X.Z.e.a. (2015). Doubel-head carfollowing and lane-changing combined model. Journal of Southeast University: Natural Science Edition, 45, 985– 989, in Chinese. H. X. Ge, Y. X. Liu, R.J.C.e.a. (2012). A modified coupled map car following model and its traffic congestion analysis. Commun Nonlinear Sci Numer Simulat, 17, 4439–4445. P. Chakroborty, S.K. (1999). Evaluation of the general motors based car-following models and a proposed fuzzy inference model. Transportation Research Part C: Emerging Technologies, 7, 209–235. S. Tadaki, M. Kikuchi, Y.S.e.a. (1998). Coupled map traffic flow simulator based on optimal velocity functions. Physica A: Statistical Journal of the Physical Society of Japan, 67, 2270–2276. T. Zhou, Y. X. Li, Z.Y.Y.e.a. (2015). Congestion feedback control carfollowing model and simulation based on safe headway. Computer Simulation, 32, 192–196, in Chinese. X. D. Hua, W. Wang, H.W. (2016). Traffic car-following model based on car communication technology. Acta Physica Sinica, 65, 52–63, in Chinese. X. M. Zhao, Z.Y.G. (2006). A control method for congested traffic induced by bottlenecks in the coupled map car-following model. Physica A: Statistical Mechanics and its Applications, 366, 513–522. Y. Y. Zhang, X. Y. Wang, D.R.T.e.a. (2011). Car-following model based on optimal control. Journal of Shandong University of Technology (Natural Science Edition), 25, 6–10, in Chinese.
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IFAC PapersOnLine 51-31 (2018) 859–862
Car-Following Model-Based Stochastic Car-Following Model-Based Stochastic Car-Following Model-Based Stochastic Distance Regulation Between Vehicles Car-Following Model-Based Stochastic Distance Regulation Between Vehicles Distance Regulation Between Vehicles Distance Between Jingwei Li, Regulation Changli Zhao, Hongwei Yue, Vehicles Wenjun Fu
Jingwei Jingwei Li, Li, Changli Changli Zhao, Zhao, Hongwei Hongwei Yue, Yue, Wenjun Wenjun Fu Fu Jingwei Li, ofChangli Zhao, Hongwei Shandong Yue, Wenjun Fu Department Automotive Engineering, Jiaotong Department of Engineering, Shandong Department of Automotive Automotive Engineering, Shandong Jiaotong Jiaotong University, Jinan, China(e-mail: [email protected], University, Jinan, China(e-mail: [email protected], Department of Automotive Engineering, Shandong Jiaotong University, Jinan, China(e-mail: [email protected], [email protected], [email protected], [email protected].) [email protected], [email protected], [email protected].) University, Jinan, China(e-mail: [email protected], [email protected], [email protected], [email protected].) [email protected], [email protected], [email protected].) Abstract: The problem of car-following model-based stochastic distance regulation between Abstract: The problem model-based stochastic distance regulation between Abstract: problem of ofofcar-following car-following model-based stochastic distance between vehicles withThe consideration the disturbance caused by the sensitivity of the regulation driver is researched vehicles with consideration of the disturbance caused by the sensitivity of the driver Abstract: ofofdistance car-following model-based stochastic distance regulation between vehicles withThe consideration the disturbance caused by the sensitivity of the driver is is researched researched in this paper. Anproblem stochastic regulator is designed based on the car-following dynamic in this An stochastic regulator is based on the car-following dynamic vehicles with ofdistance the disturbance caused byvehicles. the sensitivity of the driver is researched in this paper. paper. An stochastic distance regulator is designed designed basedThe on validation the car-following dynamic model, for theconsideration purpose of distance regulation between of the proposed model, for the purpose of regulation vehicles. The of the in this paper. stochastic distance regulatorbetween is designed based on validation the car-following dynamic model, for regulator the An purpose of distance distance between vehicles. The validation the proposed proposed stochastic is given by the regulation numerical simulation. The simulation resultsofshow that the stochastic regulator is given by the numerical simulation. The simulation results show that the model, the purpose of can distance regulation vehicles. The validation ofshow the proposed stochastic regulator is given by numerical simulation. The of simulation results that the distancefor between vehicles bethe regulated to abetween neighbourhood its desired value under various distance between vehicles can be regulated to a neighbourhood of its desired value under various stochastic regulator is given numerical The of simulation results the distance between vehicles canby bethe regulated to asimulation. neighbourhood its desired value show underthat various working conditions. working distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. conditions. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. working conditions. Keywords: Car-Following Model, Stochastic Control, Distance Regulation Between Vehicles, Keywords: Car-Following Model, Stochastic Keywords:Model, Car-Following Model, Stochastic Control, Control, Distance Distance Regulation Regulation Between Between Vehicles, Vehicles, Dynamic Numerical Simulation. Dynamic Model, Numerical Simulation. Keywords: Car-Following Model, Stochastic Control, Distance Regulation Between Vehicles, Dynamic Model, Numerical Simulation. Dynamic Model, Numerical Simulation. 1. INTRODUCTION on the base of the control theory. An improved adaptive 1. INTRODUCTION on the base of theory. An improved adaptive 1. INTRODUCTION on thefuzzy base inference of the the control control improved neuro systemtheory. model An is proposed byadaptive A. Khoneuro fuzzy inference system model is proposed by A. INTRODUCTION thefuzzy base inference of to thesimulate control An improved neuro systemtheory. model is proposed byadaptive A. KhoKhodayari (2014) and predict the car-following The increasing of 1. traffic demand has led to serious traffic on dayari (2014) to simulate and predict the car-following The increasing of traffic demand has led to serious traffic neuro fuzzy inference system model is proposed by A. Khodayari (2014) to simulate and predict the car-following based on the reaction delay of driver-vehicle The increasing of traffic demand led to serious traffic behaviour congestion problem, and results inhas environmental pollution behaviour based on the delay of the driver-vehicle congestion problem, and in environmental pollution (2014) and predict car-following behaviour based onsimulate the reaction reaction delay the driver-vehicle unit, where theto reaction delay is used asofan input and other The increasing ofproblems. traffic demand has ledresearch to serious traffic congestion problem, and results results incrucial environmental pollution and traffic safety As an area, the dayari unit, where the reaction delay is used as an input and and traffic safety problems. As an crucial research area, the behaviour based on the reaction delay of the driver-vehicle unit, where the reaction delay is used as an input andtoother other inputs-outputs of the model are chosen with respect this congestion problem, and results in environmental pollution and traffic safety problems. As an crucial research area, the dynamic analysis and control problems of traffic systems inputs-outputs of the model are chosen with respect this dynamic analysis and control problems of traffic systems unit, where In theorder reaction delay is chosen used aswith an input andto inputs-outputs of thetomodel are respect toother this suppress the traffic jam effectively, and safety As an crucial years. research the parameter. dynamic and control problems of trafficarea, systems havetraffic beenanalysis widelyproblems. investigated in recent parameter. In order to suppress the traffic jam effectively, have been widely investigated in recent years. inputs-outputs of the model are chosen with respect to this parameter. In order to suppress the traffic jam effectively, is dynamic and control problems of traffic systems a new congestion feedback control car-following model have beenanalysis widely investigated in recent years. aparameter. new congestion feedback control car-following model is A coupled map traffic flow model is introduced based In order to suppress the traffic jam effectively, a new congestion feedback control car-following model is presented in T. Zhou (2015), by considering the effect of have been widely investigated in recent years. A coupled map traffic flow model is introduced based in T. Zhou (2015), by considering the effect of A traffic model is introduced based presented on coupled optimal map velocity in S.flow Tadaki (1998), and the model a new congestion feedback control car-following model is presented in T. Zhou (2015), by considering the effect of the safe headway to the traffic system. Due to the fact that on optimal velocity in Tadaki (1998), and model safe headway to traffic system. Due fact that A map traffic flow model is conditions. introduced based on optimal velocity in S. S. Tadaki (1998), and the the A model cancoupled be simulated under open boundary car- the presented in T.car-following Zhou by considering thecombined effect of safe headway to the the(2015), trafficand system. Due to to the the fact that the traditional lane-changing can be simulated under open boundary conditions. A cartraditional car-following lane-changing on optimal velocity in S. Tadaki (1998), and the control model can be simulated open boundary A car- the following model isunder established based onconditions. optimal safe headway tothe the driver’s trafficand system. Due tointo thecombined fact that the traditional car-following and lane-changing combined models fail to take anticipation considerfollowing model is established based on optimal control models fail to take the driver’s anticipation into considercan be simulated under open boundary conditions. A carfollowing model is established based on optimal control by analyzing the information perception and operation the traditional car-following and lane-changing combined models fail to take the driver’s anticipation into considerationbased on full velocity difference car-following model by analyzing the perception and operation on full velocity difference car-following model following established based on optimal control by analyzing theisinformation information perception and the operation decision ofmodel drivers by Y. Y. Zhang (2011), and results ationbased models fail to take the driver’s anticipation into considerationbased on full velocity difference car-following model and probabilistic lane-changing model, a double-head cardecision of by Zhang (2011), and the results and probabilistic lane-changing model, a double-head carby analyzing information perception and operation decision of drivers drivers by Y. Y. Y. Y. variable Zhang (2011), and the results show that thethe perceptual relative velocity has ationbased on full velocity difference car-following model and probabilistic lane-changing model,model a double-head carfollowing and lane-changing combined was proposed show that the perceptual variable relative velocity has and lane-changing combined model was proposed decision of drivers Y. Y. variable Zhang (2011), andvelocity the results show the perceptual relative has great that influence onbydriver’s decision in car-following. A following and probabilistic lane-changing model, a double-head carfollowing and lane-changing combined model was proposed in H. Wang (2015), and the method of model calibration great influence on driver’s decision in car-following. A H. Wang (2015), and the method of model calibration show themethod perceptual variable velocity has great influence on driver’s decision in car-following. A in simplethat control to suppress therelative traffic congestion in following and lane-changing combined model was proposed in H. Wang (2015), and the method of model calibration was presented as well. The gap between traffic flow modelsimple control method to the traffic congestion in presented as The gap between traffic flow modelgreat influence on driver’s decision A was simple control method to suppress suppress theinin traffic in the coupled map car-following model X.car-following. M.congestion Zhao (2006), in Wang (2015), and the method modelby calibration was presented as well. well. The gap between traffic flow ingH. and communication approaches is of bridged D.modelY. Jia the coupled map car-following model in X. M. Zhao and communication approaches is bridged by D. Y. Jia simple method to suppress the traffic congestion in the map car-following model X. M.congestions Zhao (2006), (2006), andcoupled thecontrol effect of the control signal onintraffic is ing was presented as well. The gap between traffic flow ing and in communication approaches is bridged by D.modelY. Jia (2016) order to build up better cooperative traffic and the effect of the control signal on traffic congestions is (2016) in order to build up better cooperative traffic the coupled map car-following model in X. M. Zhao (2006), and the effect of the control signal on traffic congestions is investigated with the CM car-following model. The prop- (2016) ing and in communication approaches is bridged by D. Y. Jia order to build up better cooperative traffic systems, and an enhanced cooperative microscopic investigated with the CM car-following model. The propand an enhanced cooperative microscopic traffic and of the signal traffic congestions is systems, investigated with thecontrol CM car-following model. The propertiesthe of effect the General Motors based on car-following models (2016) in order to build up better cooperative systems, and an enhanced cooperative microscopic traffic model is developed by considering V2V and V2I comerties of General based car-following models is is developed by V2V and cominvestigated with theMotors CM car-following model. The properties of the the Motors based car-following models is model evaluated byGeneral P. Chakroborty (1999), and identifies their systems, enhanced cooperative microscopic model is and developed by considering considering V2V and V2I V2I traffic communication. Aanparameter which is newly introduced into evaluated by P. Chakroborty (1999), and identifies their A parameter which is newly introduced into erties of the Motors based car-following models is munication. evaluated byGeneral P.and Chakroborty and identifies their characteristics, proposes a(1999), fuzzy inference logic based model is developed by considering V2V and V2I munication. A parameter which is newly introduced into Newell’s car-following model is presented by X. D. comHua characteristics, and proposes a fuzzy inference logic based Newell’s car-following model is presented by X. D. Hua evaluated by P. Chakroborty (1999), and identifies their characteristics, and proposes a fuzzy inference logic based model that can overcome some of the shortcomings of the Newell’s munication. A parameter which is newly introduced into car-following model is presented by X. D. Hua (2016) to represent the pre-reaction of drivers to traffic model that can overcome some of shortcomings the to represent the pre-reaction of drivers to traffic characteristics, fuzzy inference logic of based model thatmodels. can and overcome of the thethe shortcomings of the (2016) GM based Inproposes ordersome to areflect traffic flow’s charNewell’s car-following model is presented by X. D. Hua (2016) to represent the pre-reaction of drivers to traffic flow information provided by V2V devices. Formulations GM based In to reflect the traffic flow’s information provided by devices. Formulations model thatmodels. can overcome of shortcomings the flow GM based models. In order ordersome toon reflect the traffic flow’sofcharcharacteristics more realistically, thethe basis of CM(Coupled(2016) tooperational represent the pre-reaction of traffic drivers traffic flow information provided by V2V V2V devices. Formulations of traffic capacity in mixed is to provided acteristics more realistically, on the basis of CM(Coupledtraffic operational capacity in mixed traffic is GM models. Inmodel order proposed toon reflect the flow’s characteristics more realistically, the basis of CM(CoupledMap)based car following by traffic Konishia new ex- of flow information provided by V2V devices. Formulations of traffic operational capacity in mixed traffic is provided provided in D. J. Chen (2017), which consists of automated vehicles Map) car following model proposed by Konishia new exJ. (2017), which consists of automated vehicles acteristics more on the basis ofC.CM(CoupledMap) caroffollowing model isproposed by in Konishia ex- in pression safelyrealistically, distance established Zhainew (2017), of D. traffic operational capacity in ismixed traffic is provided in D. J. Chen Chen (2017),when whichtraffic consists automated vehicles and regular vehicles, inofequilibrium. In order pression safely isproposed established C. (2017), regular vehicles, when traffic is in equilibrium. In order Map) carof following model by in Konishia new expression ofconsideration safely distance distance established in C.ofZhai Zhai (2017), with the of is the safe distance vehicles at and in D. J. Chen (2017), which consists of automated vehicles and regular vehicles, when traffic is in equilibrium. In order to solve the problem of time variability, randomness and with the consideration of the safe distance of vehicles at solve the problem of time variability, randomness and pression safelyAdistance established inisC. (2017), with the of consideration of is the safe which distance ofZhai vehicles at to different speeds. dynamic model established by and regular vehicles, when traffic is in equilibrium. In order to solve the problem of time variability, randomness and uncertainty in the traffic flow forecasting, a new combined different speeds. A model is by in the traffic flow forecasting, aa new combined with consideration oftwo the safe which distance of vehicles at uncertainty different A dynamic dynamic model which is established established by H. X.the Ge speeds. (2012) contains successive vehicles’ headway to solve the problem of time variability, randomness and uncertainty in the traffic flow forecasting, new combined control predictive algorithm was proposed in Dong (2018), H. X. Ge (2012) contains two successive vehicles’ headway predictive algorithm was proposed in Dong (2018), different Aofdynamic which isincorporated established by H. X. Ge speeds. (2012) twomodel successive vehicles’ headway distances in frontcontains the considered one are in control uncertainty in the traffic flow forecasting, a new combined control predictive algorithm was proposed in Dong (2018), and the results show that the average relative error and distances in of considered one are incorporated in thepredictive results show that the relative error and H. Ge (2012) contains two successive vehicles’ headway distances in front front of the the considered incorporated in and theX. optimal velocity function, and one the are stability condition control algorithm wasaverage proposed in Dong (2018), and results show that the average error and meanthe square deviation obtained with therelative proposed method the optimal velocity function, and the stability condition square deviation obtained with the proposed method distances front of the one are incorporated in mean the optimal velocity function, and of the stability condition is given forinthe change ofconsidered the speed the preceding vehicle and the results show that the average relative error and mean square deviation obtained with the proposed method are 74% and 85% lower than those of other methods. is for change of the preceding vehicle and 85% than other methods. thegiven optimal velocity and of condition is given for the the changefunction, of the the speed speed ofthe thestability preceding vehicle are mean square withof proposed method are 74% 74% and deviation 85% lower lowerobtained than those those ofthe other methods. In this thelower disturbance caused by the sensitivity is givenwork for the change ofbythe speed of the preceding vehicle is supported Scientific Research Fund Project of are 74%paper, and 85% than those of other methods. This In this paper, disturbance caused the This work is supported by Scientific Research Fund Project of In this driver paper,isthe the disturbance caused by by the sensitivity sensitivity of the modeled as a standard wiener processes. Shandongjiaotong University Science Class), NO:Project Z201704, This work is supported by(Natural Scientific Research Fund of of the driver modeled as wiener processes. Shandongjiaotong University (Natural Science Class), NO: Z201704, In this paper,is disturbance caused by thetosensitivity of driver isthe modeled as aa standard standard wiener processes. Thethe stochastic distance regulator is designed overcome Z201635; Shandong Provincial Higher School Science and Technology Shandongjiaotong University Science Class), NO: Z201704, This work is supported by(Natural Scientific Research Fund Project of The stochastic distance regulator is designed to overcome Z201635; Shandong Provincial Higher School Science and Technology of the driver isdistance modeledregulator as a standard wiener processes. Project ofShandong China under Grant(Natural J17KA036. The stochastic is designed to overcome Z201635; Provincial Higher School Science and Technology Shandongjiaotong University Science Class), NO: Z201704, Project of China under Grant J17KA036. Project ofShandong China under Grant Higher J17KA036. The stochastic distance regulator is designed to overcome Z201635; Provincial School Science and Technology Project of © China under J17KA036. 2405-8963 IFAC Grant (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 903 Hosting by Elsevier Ltd. All rights reserved. Copyright ©under 2018 responsibility IFAC 903Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 903 10.1016/j.ifacol.2018.10.097 Copyright © 2018 IFAC 903
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the influence of the disturbance. The numerical simulation shows that the stochastic distance regulator is availably when the accelerated velocity of the leading vehicle both sine curve and step curve.
7.5
2. STOCHASTIC CONTROL PROBLEM
6.5
2.1 Control Problem Description
where x is distance between the vehicles, v is the relative velocity, al is accelerated velocity of the leading vehicle, af is the accelerated velocity of the following vehicle, the standard wiener processes aδ denotes the disturbance caused by the sensitivity of the driver, and δ is the disturbance intensity. For the system (1), given an stochastic regulator as (2)
where x∗ is the desired value of x, such that the distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. 2.2 Stochastic Regulator Design In this section, the process of the stochastic regulator design process will be expressed based on the Lyapunov method. Step 1. Introducing the two error variables e1 = x − x∗ , e2 = v − x∗(1) − α(x, x∗ ). de1 = (v − x )dt = (e2 + α(x, x∗ ))dt. Choose the Lyapunov function 1 V1 = e41 . 4 From (4) and (5), we have LV1 = e31 (e2 + α(x, x∗ )) 3 1 ≤ e42 + e31 ( e1 + α(x, x∗ )). 4 4 By choosing
3 α(x, x∗ ) = −c1 e1 − e1 , 4
where c1 is the design parameter, we can obtain 1 LV1 ≤ −c1 e41 + e42 . 4
(3)
(4)
(5)
5
0
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 1. Distance between the vehicles of W1 Let us consider the Lyapunov candidate 1 V2 = V1 + e42 . 4 In view of (8) and (9), we have 1 LV2 = −c1 e41 + e42 + e32 (al − af − x∗(2) 4 ∂α(x, x∗ ) ∗(1) ∂α(x, x∗ ) 3 v− − x ) + e22 δ 2 ∂x ∂x∗ 2 1 ≤ −c1 e41 + e32 ( e2 + al − af − x∗(2) 4 ∂α(x, x∗ ) ∗(1) ∂α(x, x∗ ) v− − x ∂x ∂x∗ 3 3 4 + e2 ) + δ . 4 4
(10)
(11)
By choosing af as 1 ∂α(x, x∗ ) e2 + al − x∗(2) − v 4 ∂x ∗ ∂α(x, x ) ∗(1) 3 − x + e2 − c2 e2 , ∂x∗ 4
(12)
where c2 is the design parameter. From (11), denote V = V2 , we get 3 (13) LV ≤ −cV + δ 4 , 4 where the design parameter c = min{c1 , c2 }. 3. NUMERICAL SIMULATION
(6)
(7)
(8)
Step 2.From (3), we can obtain ∂α(x, x∗ ) v de2 = (al − af − x∗(2) − ∂x ∗ ∂α(x, x ) ∗(1) − x )dt + δdaδ . ∂x∗
5.5
af =
From (1) and (3), one has ∗(1)
Distance between the vehicles/(m)
6
In this section, the description of the control problem is given. The car-following model with consideration of the sensitivity of the driver is provided as follows: dx = vdt, (1) dv = (al − af )dt + δdaδ ,
af = f (x, v, x∗ , x∗(1) , x∗(2) ),
7
(9)
904
In a principle phase to test the effectiveness of the control scheme, the numerical simulation is established based on (1) and (12). The numerical simulation is run under working conditions W1 and W2 . For working condition W1 , the accelerated velocity of the leading vehicle is a sine curve with disturbance, the noise intensity of the disturbance is 0.001, the the initial value and the desired value of the distance between the vehicles are 5m and 7m, respectively. For working condition W2 , the accelerated velocity of the leading vehicle is a step curve with disturbance, the noise intensity of the disturbance is 0.001, the the initial value and the desired value of the distance between the vehicles are 9m and 6m, respectively. The signals of the distance between the vehicles, the relative velocity between vehicles, the accelerated velocity of the leading vehicle and the accelerated velocity of
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1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2
861
0.5 0 Relative velocity between vehicles/(m/s)
−0.5 Relative velocity between vehicles/(m/s)
−1 −1.5 −2 −2.5 0
50
−3
100 150 200 250 300 350 400 450 500 Time/(s)
100 150 200 250 300 350 400 450 500 Time/(s)
1.5
Accelerated velocity of the leading vehicle/(m/s2)
1
1
0. 5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5 0
50
Fig. 6. Relative velocity between vehicles of W2
Fig. 2. Relative velocity between vehicles of W1 1. 5
0
50
−1.5 0
100 150 200 250 300 350 400 450 500 Time/(s)
Accelerated velocity of the leading vehicle/(m/s2)
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 7. Accelerated velocity of the leading vehicle of W2
Fig. 3. Accelerated velocity of the leading vehicle of W1
10
40 35 30 Accelerated velocity of the following vehicle/(m/s2) 25 20 15 10 5 0 −5 0 50 100 150 200 250 300 350 400 450 500 Time/(s)
0 −10 −20
−40 −50 −60 0
50
100 150 200 250 300 350 400 450 500 Time/(s)
Fig. 8. Accelerated velocity of the following vehicle of W2
Fig. 4. Accelerated velocity of the following vehicle of W1 9 8.5 8 Distance between the vehicles/(m)
7.5
Accelerated velocity of the following vehicle/(m/s2)
−30
7 6.5
the following vehicle of working condition W1 are given in Figures 1-4. From Figures 1-4 we can observed that the distance between the vehicles can be regulated to a neighbourhood of its desired value, when the accelerated velocity of the leading vehicle is varied as a sine curve with disturbance. Figures 5-8 shows the signals of the distance between the vehicles, the relative velocity between vehicles, the accelerated velocity of the leading vehicle and the accelerated velocity of the following vehicle of working condition W2 from which we can obtain that the proposed stochastic regulator is still valid.
6 5.5 0
4. CONCLUSION 50
100 150 200 250 300 350 400 450 500 Time/(s)
A car-following model in which the sensitivity of the driver is modeled as a stochastic disturbance is provided in this paper. An stochastic distance regulator is designed
Fig. 5. Distance between the vehicles of W2 905
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to achieves the distance regulation between vehicles. By the proposed stochastic regulator, the distance between vehicles can be regulated to a neighbourhood of its desired value under various working conditions. REFERENCES A. Khodayari, A. Ghaffari, R.K.e.a. (2014). Improved adaptive neuro fuzzy inference system car-following behavior model based on the driver-vehicle delay. Intelligent Transport Systems, 8, 323–332. C. Zhai, W. Y. Wu, L.H.e.a. (2017). Feedback control of car following model on the basis of time-varying safety distance. Journal of South China University of Technology (Natural Science Edition), 45, 126–134, in Chinese. D. J. Chen, S. AHN, M.C.e.a. (2017). Towards vehicle automation: roadway capacity formulation for traffic mixed with regular and automated vehicles. Transportation Research Part B: Methodological, 100, 196–221. D. Y. Jia, D.N. (2016). Enhanced cooperative carfollowing traffic model with the combination of v2v and v2i communication. Transportation Research Part B: Methodological, 90, 172–191. Dong, W. (2018). Combined control predictive model and method for stochastic traffic flow. Journal of Shenyang University of Technology, 40, 88–93, in Chinese. H. Wang, Z. Q. Liu, Z.X.Z.e.a. (2015). Doubel-head carfollowing and lane-changing combined model. Journal of Southeast University: Natural Science Edition, 45, 985– 989, in Chinese. H. X. Ge, Y. X. Liu, R.J.C.e.a. (2012). A modified coupled map car following model and its traffic congestion analysis. Commun Nonlinear Sci Numer Simulat, 17, 4439–4445. P. Chakroborty, S.K. (1999). Evaluation of the general motors based car-following models and a proposed fuzzy inference model. Transportation Research Part C: Emerging Technologies, 7, 209–235. S. Tadaki, M. Kikuchi, Y.S.e.a. (1998). Coupled map traffic flow simulator based on optimal velocity functions. Physica A: Statistical Journal of the Physical Society of Japan, 67, 2270–2276. T. Zhou, Y. X. Li, Z.Y.Y.e.a. (2015). Congestion feedback control carfollowing model and simulation based on safe headway. Computer Simulation, 32, 192–196, in Chinese. X. D. Hua, W. Wang, H.W. (2016). Traffic car-following model based on car communication technology. Acta Physica Sinica, 65, 52–63, in Chinese. X. M. Zhao, Z.Y.G. (2006). A control method for congested traffic induced by bottlenecks in the coupled map car-following model. Physica A: Statistical Mechanics and its Applications, 366, 513–522. Y. Y. Zhang, X. Y. Wang, D.R.T.e.a. (2011). Car-following model based on optimal control. Journal of Shandong University of Technology (Natural Science Edition), 25, 6–10, in Chinese.
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