Research ArticleDistributed Model Predictive Control over Multiple Groups ofVehicles in Highway Intelligent Space for Large Scale System

Tang Xiaofeng Gao Feng Xu Guoyan Ding Nenggen Cai Yao and Liu Jian Xing

School of Transportation Science and Engineering Beihang University No 37 Xueyuan Road Haidian DistrictBeijing 100191 China

Correspondence should be addressed to Gao Feng gaofbuaaeducn

Received 29 May 2014 Revised 18 June 2014 Accepted 11 July 2014 Published 23 July 2014

Academic Editor Wuhong Wang

Copyright copy 2014 Tang Xiaofeng et al This 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

The paper presents the three time warning distances for solving the large scale system of multiple groups of vehicles safety drivingcharacteristics towards highway tunnel environment based on distributed model prediction control approach Generally speakingthe system includes two parts First multiple vehicles are divided intomultiple groupsMeanwhile the distributedmodel predictivecontrol approach is proposed to calculate the information framework of each group Each group of optimization performanceconsiders the local optimization and the neighboring subgroup of optimization characteristics which could ensure the globaloptimization performance Second the three time warning distances are studied based on the basic principles used for highwayintelligent space (HIS) and the information framework concept is proposed according to the multiple groups of vehicles The mathmodel is built to avoid the chain avoidance of vehiclesThe results demonstrate that the proposed highway intelligent space methodcould effectively ensure driving safety of multiple groups of vehicles under the environment of fog rain or snow

1 Introduction

A new research concept about highway intelligent space(HIS) is proposed in the literature [1] The main objective ofthe HIS system is to create an intelligent driving spacewhere sensors are arranged in some necessary places such ashighway tunnel road sections that are prone to accidentsor poor visibility under adverse weather conditions Mean-while advanced vehicle-to-vehicle (V2V) and vehicle-to-server (V2S) are adopted to inform drivers of road sectionsituation ahead to avoid collision under the environment ofspace communication mode The HIS system is introducedto provide vehicles with useful driving state to ensure thatvehicles could have better access to road information aheadespeciallymultiple groups of vehicles of driving environmentHence studying vehicle driving state is important for the HISsystem to build an intelligent space according to the abovementioned situations This paper mainly focuses on multi-vehicle of driving state when passing through the highwaytunnel

The highway tunnel has its own complex driving featuresfor example the channel tunnel fire resulting in multiplefatalities and injuries [2] several traffic accidents appearingin some tunnels [3] and vehicle crash caused by visualdysfunction of drivers [4] When accidents happened insome highway tunnel the effective vehicle state should betransmitted to vehicles towards highway tunnel in real timeby the communication between vehicles and HIS systemespecially multivehicle driving towards highway tunnel Mul-tivehicle has different driving modes such as single vehicleform groups of vehicle forms and vehicle organizing formsamong which groups of vehicle driving forms are the com-mon While HIS server could store large amounts of datainformation for vehicles multivehicle can be decomposed todifferent groups of vehicles driving towards highway tunnelsand different groups of vehicles need different driving statesaccording to the specific driving environment Each groupof vehicles driving states is defined as information frame-work Information frameworkmainly shows that the effectivedriving state obtained from HIS server can ensure different

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 809124 12 pageshttpdxdoiorg1011552014809124

2 Mathematical Problems in Engineering

groups of vehicles of safety characteristics Therefore study-ing multiple groups of vehicles driving characteristics as wellas accurate evaluations of the risk rear-end collisions is thekey technologies under the environment of highway tunnel

For the large scale of multi-intelligent vehicle systemsthere are three MPC schemes for designing such large scalesystems namely distributed MPC centralized MPC anddecentralizedMPCThe distributedMPC strategy is arguablythe most promising one because it beats the centralizedMPCone in terms of computational load and outperforms thedecentralized MPC one in control performance [5] In aDMPC design different model predictive controllers com-municate through a communication network to cooperatetheir actions in order to achieve optimal performance [6] forexample collision avoidance constraints [7] The distributedframework of MPC is also gradually developing for thecontrol of large scale system There are two methods ofdistributed MPCs that appeared in the literature for thelarge scale system One method is that each local controllerexchanges estimation states with its neighbors and thereforeimproves the performance of closed-loop subsystem How-ever the performance of other subsystems is not considered inthis optimization The second method could achieve a goodperformance close to the centralized MPC However thisstrategy requires much more communication resources andthe structure of controller is relatively complex [8] In thispaper the neighbor optimizationmethod is used for the largescale system inwhich each subsystem interacts in sequence bystate

Based on the above state a new vehicle safety distanceamong groups based on the HIS system is proposed todevelop a warning system Three time warning distanceswill be studied considering the complex tunnel space Innature human beings suffer from perception limitations witha typical reactive time of 075 seconds to 15 seconds on emer-gency events Therefore it is highly important that vehiclesshould work in the three warning distances to guaranteeits immediate and effective stopping to avoid collisions Thefirst warning distance is calculated by the time the driverreaction time and the message propagation delay As to thesecond warning distance the compensated safety distance isproposed as the buffer distance to reduce its acceleration tothe range of the vehicle near the tunnel The third warningdistance is the tracking safety distance The proposed HISsystem is adapted for each group and can effectively achievevehiclersquos safety Figure 1 shows the work principle among thegroups Distributed model prediction control approach isused to solve groups of vehicles collision avoidance Mean-while the method based on neighbor optimization is usedto solve the large scale system Therefore by predicting thestates based on the prediction model over the finite horizonthe third warning distance which is collision-free at discretetime steps is planned In other words since the collisionavoidance is considered only at the discrete prediction timesteps a collision may occur in the intervals between theprediction time steps In particular since it is difficult tokeep the sampling steps small enough to ensure the con-vergence of the trajectory the collisions between prediction

Groups of vehicles

HIS serverDCP DMC

MPC1 only for theleading vehicle

Group-1

MPC2 only for theleading vehicle

Group-2

MPCn only for theleading vehicle

Group-nmiddot middot middot

Figure 1 The work principle among groups

time steps become quite significant [9] Therefore as to themultiple groups of vehicles collision avoidance problemadopting the above mentioned could solve vehicle environ-ment perception

The organization of the paper is as follows System des-cription and objective are presented in Section 2 In Section 3distributed model predictive control approach is studied InSection 4 the stability of vehicles in each group is studiedIn Section 5 simulation and analysis are presented Finallyconclusions are given

2 System Description and Objective

In this section we define the system dynamics and pose anintegrated cost function for every group to ensure systemstabilization and vehicles safety drivingMeanwhile the threetime warning distances are studied addressing the systemdynamics

21 System Description Figure 2 shows the basic workprinciple of overall systemThe set 119895 isin 120588 = 1 119873V1 1 119873V2 1 119873V119899 is used to denote themultivehicle drivingtowards highway tunnel Autonomous vehicles organize dif-ferent groups according to the specific driving environmentor driving state Therefore the symbol 119894 isin Γ = 1 119899 isused to denote group-1 group-2 group-119899 For each group-119894 the leading vehicle 119896 isin Ω = 119881

11 11988121 119881

1198991 should

track the three time warning distances The last vehicle 120575 isinΘ = 1119873V1 2119873V2 119899119873V acts on the intervehicle dynamicsmodel between it and the next group For each followingvehicle in each group the reference trajectory should trackthe leading vehicles Therefore distributed model predictivecontrol approaches used for the leading vehicles of each groupcan achieve the vehicles safety driving characteristics Thisstudy is divided into several groups This class of seriallyconnected subgroup is composed of many similar subgroupsplaced after one another in such a way that each subgroup isconnected with dynamic state between its neighbors Somealgorithms and assumptions are made as follows

Assumption 1 The vehicles in each group are self-organiza-tion form the following vehicles in each group can favorablytrack the leading vehicle The time delay communication ofthe vehicle driving state transmission between vehicles is notconsidered

Mathematical Problems in Engineering 3

∆s

MPC1 MPC2 MPC-n

O

Vehicle accidentzone ahead

Multiple groups of vehicles

s1 the first time warning distance

s2 the second time warning distance

s3 the third time warning distance

s1

s1

s0

V0 V11 V21V12 V1n V2n Vn1 Vnn

s2

s2

s3

sn

sdes

(i minus 1)N(iminus1)n iNi1

HIS serverDCP DMC

Group-1 Group-2 Group-nmiddot middot middot

middot middot middot middot middot middotmiddot middot middotmiddot middot middot

Figure 2 Overall system work principle

Assumption 2 Based on highway tunnel characteristics theovertaking situation is not considered in the research rangeand the velocity change of vehicles is small

Algorithm 3 Distributed model predictive controller for theleading vehicle 119896 isin Ω initialization the driving state of thevehicle 119881

0is sent to the first group by HIS server starting

at time 1199050= 0 In addition other groups will adjust to the

driving states by the communication between the HIS systemand the groups

22 Analysis of theThree TimeWarning Distances The safetyof the driving behavior for each group is typically related tovehicle driving environment Hence regarding each group ofsafety aspect the information framework includes interve-hicle distance relative velocity and acceleration The basicprinciple of the three time warning distances can be seenfrom Figure 2 When an accident has happened in a highwaytunnel the vehicle 119881

0near the tunnel has obtained the

message and lowers its velocity subconsciously The leadingvehicle 119881

11received the road information ahead by the HIS

server and passed through the three time warning distancesThe latter groups will adjust to the driving states based onthe vehicle 119881

0and the former groups The overall groups of

distance could be shown as

119904 = V0sdot 1199051+1

2119886 sdot 1199052

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199041

+ min2

(1199042(1199052) 1199092(1199052)) max2

(1199042(1199052) 1199092(1199052))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199042

+ (120585119890(1199053) + 119889des (1199053))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199043

(1)

The first time warning distance for each leading vehicle ofsafety distance is calculated before arriving at its secondwarning and mainly includes two parts of time the reactiontime of the driver 119905

1and message propagation delay 119905

2

Message propagation delay is composed of two parts 11990521and

11990522 11990521

is the time when the message about an accident intunnel is sent to the HIS server and 119905

22is the time when a

message is sent to related leading vehicles of the group Withthis notation we can define the first warning distance

1199041(119905) = V

0119905 +1

21198861199052 (119905 le 119905

2) (2)

Furthermore we can solve the first time warning distance foreach group denoted by

119904119894= 1199041(119905) (119905 le 119905

2 119894 = 1 119899) (3)

The second time warning distance is calculated before arriv-ing at its third time warning distanceThe goal for the secondtime warning distance is to lower the leading vehicle 119896 isin Ωof velocity after arriving at the third time warning distancethe leading vehicle of driving states could satisfy the trackingmode

Definition 4 Compensated safety distance in our work isdefined as the buffer distance among groups to avoid colli-sions

Based on the above assumptions and definition themodel for compensated safety distance for each group 119894 isin Γis denoted as follows

= 119866119909 + 119867119906 119909 = [1199042119894(119905)

V1(119905)] 119866 = [

0 10 0]

119867 = [01] 119906 = 119886des (119905)

(4)

4 Mathematical Problems in Engineering

where 1199042119894(119905) is the compensated safety distance and 119906 is the

desired deceleration for the leading vehicle

Definition 5 The vehicle is in the safety driving when thedesired acceleration is defined in the range |119886

119897| le 11988611 11988611

isa special state that denotes the leading vehicles entering thetracking mode after arriving at the third warning distance

The second time warning distance is a buffer range so weintroduce the virtual vehicle concept to calculate the distanceAdditional advantage of the virtual vehicle scheme is that themotion of the vehicle can be smoothly controlled when a newleading vehicle cuts in or the current vehicle cuts out [10]

Definition 6 The virtual vehicle distance 1199092(119905) is defined as

a constant safety distance For every group 119894 isin 1 2 119899the compensated safety distance should be set in the range asfollows

min119894

1199042119894(119905) 1199092(119905) lt 119904

2119894(119905) le max

119894

1199042119894(119905) 1199092(119905) (5)

The constant safety distance of virtual vehicle distance can beshown

1199092(119905) = ℎ sdot V + 119863min (6)

The second time warning distance is built to reduce vehiclersquosacceleration during some adverse weather or road sectionsthat are prone to accidents

The third warning time is the tracking mode beforearriving at its permitted minimum distanceThe vehicle119881

0is

uncontrollable so the leading vehicle 119896 isin 11988111 11988121 119881

1198991

in each group should be controlled to ensure a suitable rela-tive distance As can be seen from Figure 2 the intervehicledynamics model is designed among groups as follows

V119891119894= 119886119891119894

120585119889119894= 119897119897119894minus 119897119891119894

ΔV119894= V119897119894minus V119891119894

120585119890119894= 120585119889119894minus 119889des 119894

(7)

The longitudinal dynamics of the leading vehicle 119896 isin11988111 11988121 119881

1198991 are nonlinear According to the vehicle

dynamics in [11] the longitudinal dynamics is transferred asfollows

119886119891119894= 119892119891119894(V119891119894 119886119891119894) + ℎ119891119894(V119891119894) 120575119891119894

119892119891119894(V119891119894 119886119891119894) = minus

2119870119886119889

119898minus1

120591119891

[119886119891119894+119870119886119889

119898119891119894

V2119891119894+119870119898119889

119898119891119894

]

ℎ119891119894(V119891119894) =

1

119898119891119894

120591119891119894

(8)

If the parameters in (7) are exactly known the followingfeedback linearizing control law could be adopted

120575119891119894= 119898119891119894120583119891119888119894+ 119870119886119889V2119891119894+ 119870119898119889+ 2120591119891119894119870119886119889V119891119894119886119891119894 (9)

where 120585119889119894

is intervehicle distance ΔV119894is the error among

groups and 120585119890119894is the error between actual intervehicle dis-

tance and desired vehicle safety distance V119897119894 V119891119894are vehicle3rsquos

speed and vehicle1rsquos speed 120583119891119888119894

is the input signal that makesthe closed-loop system satisfy certain performance criteriaIn controller (8) we achieve the following objectives (1) thefeedback linearization results in a linear system as discussedabove however uncertainties in parameters can potentiallymake the linearization process inexact the study of sucha case would be an interesting topic to be considered infuture research (2) the simplification of the system model byexcluding some characteristic parameters (eg the mechani-cal drag mass and air resistance) from the vehicle dynamicsManipulating (6) through (8) the equation becomes

119891119894=1

120591119891119894

(120583119891119888119894

minus 120572119891119894) (10)

where 120591119891119894is the engine time constant and its value is 025 a

single parameter that describes the dynamics of the propul-sion system and internal disturbances 120583

119891119888119894

can be viewed asthe throttlebrake input causing accelerationdeceleration inthe controlled vehicle The system thus takes the form whichcan be described by the following standard equations

[[[

[

V119891119894

120585119890119894

ΔV119894

119891119894

]]]

]

=

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

+

[[[[[

[

0001

120591119891119894

]]]]]

]

sdot [119906 (119905)] +[[[

[

0010

]]]

]

sdot [120572119897119894]

119910 (119905)119894=[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

(11)

Writing the above equation as standard state-space equationswe have

119909 (119905)119894= 119860119909 (119905)

119894+ 119861119906 (119905)

119894+ 119862119908(119905)

119894

119910 (119905)119894= 119863119909 (119905)

119894

119860 =

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

119861 =

[[[[[

[

0001

120591119891119894

]]]]]

]

119862 =[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

119863 =[[[

[

0010

]]]

]

(12)

where state variables are119909(119905)119894= [V119891119894 120585119890119894 ΔV119894 120572119891119894]

119879 and119908(119905)119894

is the acceleration of the last vehicle 120575

Mathematical Problems in Engineering 5

By (12) the overall system of vehicle dynamics can beexpressed as

(

1199091

1199092

119909119899+1

) =(

119860104times4

sdot sdot sdot 04times4

04times4

1198602sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 119860119899

) sdot(

1199091

1199092

119909119899

)

+(

119861104times1

sdot sdot sdot 04times1

04times1

1198612sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 119861119899

) sdot(

1199061

1199062

119906119899

)

+(

11989610 sdot sdot sdot 0

0 1198962sdot sdot sdot 0

0 0 sdot sdot sdot 00 0 sdot sdot sdot 119896

119899

) sdot(

1199081

1199082

119908119899

)

(13)

Simplifying the above equation

119883119899+1= 119860 sdot 119883

119899+ 119861 sdot 119880

119899+ 119870 sdot 119882

119899 (14)

where 119883119899+1= (1199091 1199092 sdot sdot sdot 119909119899+1)

119879 119883119899= (1199091 1199092 sdot sdot sdot 119909119899)

119879119880119899= (1199061 1199062 sdot sdot sdot 119906119899)

119879 and 119908119899= (1199081 1199082 sdot sdot sdot 119908119899)

119879

Criterion 1 The dynamics model (12) is stable controllableand observability

The stability system is analyzed by eigenvalue criteriaThe controllability is verified by using rank (119860 119861) and theobservability is studied by using rank (119862 119860)

As we have defined it the three time warning distancesare designed to imply that combinedwithHIS space vehiclessafety driving could be guaranteed under the environment ofsomeuncertainweather conditions such as fog rain or snow

3 Distributed Model PredictiveControl Approach

31 Induction Process In this section we introduce notationand define the optimal control problem and the distributedmodel predictive control approach (DMPC) for the leadingvehicles in each group Combined with the above schematicsderivation theDMPConly is developed in the leading vehicle119896 isin Ω the following vehicles in each group can maintain thesafety driving distance in the form of the platoonTheDMPCalgorithm captures an important class of practical problemsincluding for examplemaneuvering a group of vehicles fromone point to another while maintaining relative formationandor avoiding collisions [12] Figure 3 shows the principleof the optimal control application for each leading vehicle

As stated above each group exchanges information withthe HIS server system Each group sequentially achievesperformance development in distributed MPC algorithmsunder the environment of the highway tunnel

Because the latter groups irregularly drive on the highwaytunnel the interconnections between different subsystemsare assumed to be weak and are considered as disturbances

HIS server

DPC center DMC

Relative distancerelative velocity

longitudinal velocity and acceleration

Predictive modelintervehicle dynamics

Receding optimizationDesired distance

Vehicle-to-HIS

communication

Optimal control

state

Vehicledynamic

model

Figure 3 The principle for the optimal control application

which can send the relative driving information to the lattergroup via HIS server system In addition the driving statesof the last vehicle in each group are uncontrollable butcan be measured by sensors so the acceleration 119908(119905)

119894of

the last vehicle 120575 isin Θ can be regarded as the measureddisturbance signal when calculating the latter groups Theoverall of longitudinal dynamics system that is composedof 119898 interconnected subsystems can be descried by (14)The global optimization problem can be decomposed intoa number of local optimization subproblems and the wholecontrol performance can be efficiently improved [13] Thecooperation between subsystems is achieved by exchanginginformation between each subsystem and its neighbors ina distributed structure via network between HIS server andvehicles

By (12) according to the vehicle driving characteristicsin highway intelligent space subsystem 119878

119894interacts with 119878

119895

and the output state acceleration of subsystem 119878119894is affected by

subgroup 119878119895 In this case 119878

119895is called input neighboring sub-

group of 119878119894 119878119894is called the output neighboring subsystem In

(12) the state acceleration of the 119878119894is regarded as disturbance

In every group of distributed optimal control strategywe assume that the same constant prediction horizon 119873

119901isin

(0infin) and constant update period 120574 isin (0119873119901] are used In

practice the update period 120574 isin (0119873119901] is the sample interval

The common update times are denoted by 119905119896= 1199050+120593119896 where

1199050= 0 and 119896 isin 119873 = 0 1 2 The leading vehicle in each

group sequentially solves an optimal control problem at theupdate period 120575 isin (0119873

119901] and applies the optimal control

trajectory until its next update time We have that 119911119894(119905) and

119906119894(119905) are the actual error state and control input respectively

For each leading vehicle 119896 isin Ω at any time 119905 ge 1199050 over any

prediction interval [119905119896 119905119896+119879] 119896 isin 119873 associatedwith updated

time 119905119896 we denote two trajectories

119906lowast119894(120591 119905119896) the optimal control trajectory

119894(120591 119905119896) the predicted state trajectory

where 120591 isin [119905119896 119905119896+ 119879]

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

groups of vehicles of safety characteristics Therefore study-ing multiple groups of vehicles driving characteristics as wellas accurate evaluations of the risk rear-end collisions is thekey technologies under the environment of highway tunnel

For the large scale of multi-intelligent vehicle systemsthere are three MPC schemes for designing such large scalesystems namely distributed MPC centralized MPC anddecentralizedMPCThe distributedMPC strategy is arguablythe most promising one because it beats the centralizedMPCone in terms of computational load and outperforms thedecentralized MPC one in control performance [5] In aDMPC design different model predictive controllers com-municate through a communication network to cooperatetheir actions in order to achieve optimal performance [6] forexample collision avoidance constraints [7] The distributedframework of MPC is also gradually developing for thecontrol of large scale system There are two methods ofdistributed MPCs that appeared in the literature for thelarge scale system One method is that each local controllerexchanges estimation states with its neighbors and thereforeimproves the performance of closed-loop subsystem How-ever the performance of other subsystems is not considered inthis optimization The second method could achieve a goodperformance close to the centralized MPC However thisstrategy requires much more communication resources andthe structure of controller is relatively complex [8] In thispaper the neighbor optimizationmethod is used for the largescale system inwhich each subsystem interacts in sequence bystate

Based on the above state a new vehicle safety distanceamong groups based on the HIS system is proposed todevelop a warning system Three time warning distanceswill be studied considering the complex tunnel space Innature human beings suffer from perception limitations witha typical reactive time of 075 seconds to 15 seconds on emer-gency events Therefore it is highly important that vehiclesshould work in the three warning distances to guaranteeits immediate and effective stopping to avoid collisions Thefirst warning distance is calculated by the time the driverreaction time and the message propagation delay As to thesecond warning distance the compensated safety distance isproposed as the buffer distance to reduce its acceleration tothe range of the vehicle near the tunnel The third warningdistance is the tracking safety distance The proposed HISsystem is adapted for each group and can effectively achievevehiclersquos safety Figure 1 shows the work principle among thegroups Distributed model prediction control approach isused to solve groups of vehicles collision avoidance Mean-while the method based on neighbor optimization is usedto solve the large scale system Therefore by predicting thestates based on the prediction model over the finite horizonthe third warning distance which is collision-free at discretetime steps is planned In other words since the collisionavoidance is considered only at the discrete prediction timesteps a collision may occur in the intervals between theprediction time steps In particular since it is difficult tokeep the sampling steps small enough to ensure the con-vergence of the trajectory the collisions between prediction

Groups of vehicles

HIS serverDCP DMC

MPC1 only for theleading vehicle

Group-1

MPC2 only for theleading vehicle

Group-2

MPCn only for theleading vehicle

Group-nmiddot middot middot

Figure 1 The work principle among groups

time steps become quite significant [9] Therefore as to themultiple groups of vehicles collision avoidance problemadopting the above mentioned could solve vehicle environ-ment perception

The organization of the paper is as follows System des-cription and objective are presented in Section 2 In Section 3distributed model predictive control approach is studied InSection 4 the stability of vehicles in each group is studiedIn Section 5 simulation and analysis are presented Finallyconclusions are given

2 System Description and Objective

In this section we define the system dynamics and pose anintegrated cost function for every group to ensure systemstabilization and vehicles safety drivingMeanwhile the threetime warning distances are studied addressing the systemdynamics

21 System Description Figure 2 shows the basic workprinciple of overall systemThe set 119895 isin 120588 = 1 119873V1 1 119873V2 1 119873V119899 is used to denote themultivehicle drivingtowards highway tunnel Autonomous vehicles organize dif-ferent groups according to the specific driving environmentor driving state Therefore the symbol 119894 isin Γ = 1 119899 isused to denote group-1 group-2 group-119899 For each group-119894 the leading vehicle 119896 isin Ω = 119881

11 11988121 119881

1198991 should

track the three time warning distances The last vehicle 120575 isinΘ = 1119873V1 2119873V2 119899119873V acts on the intervehicle dynamicsmodel between it and the next group For each followingvehicle in each group the reference trajectory should trackthe leading vehicles Therefore distributed model predictivecontrol approaches used for the leading vehicles of each groupcan achieve the vehicles safety driving characteristics Thisstudy is divided into several groups This class of seriallyconnected subgroup is composed of many similar subgroupsplaced after one another in such a way that each subgroup isconnected with dynamic state between its neighbors Somealgorithms and assumptions are made as follows

Assumption 1 The vehicles in each group are self-organiza-tion form the following vehicles in each group can favorablytrack the leading vehicle The time delay communication ofthe vehicle driving state transmission between vehicles is notconsidered

Mathematical Problems in Engineering 3

∆s

MPC1 MPC2 MPC-n

O

Vehicle accidentzone ahead

Multiple groups of vehicles

s1 the first time warning distance

s2 the second time warning distance

s3 the third time warning distance

s1

s1

s0

V0 V11 V21V12 V1n V2n Vn1 Vnn

s2

s2

s3

sn

sdes

(i minus 1)N(iminus1)n iNi1

HIS serverDCP DMC

Group-1 Group-2 Group-nmiddot middot middot

middot middot middot middot middot middotmiddot middot middotmiddot middot middot

Figure 2 Overall system work principle

Assumption 2 Based on highway tunnel characteristics theovertaking situation is not considered in the research rangeand the velocity change of vehicles is small

Algorithm 3 Distributed model predictive controller for theleading vehicle 119896 isin Ω initialization the driving state of thevehicle 119881

0is sent to the first group by HIS server starting

at time 1199050= 0 In addition other groups will adjust to the

driving states by the communication between the HIS systemand the groups

22 Analysis of theThree TimeWarning Distances The safetyof the driving behavior for each group is typically related tovehicle driving environment Hence regarding each group ofsafety aspect the information framework includes interve-hicle distance relative velocity and acceleration The basicprinciple of the three time warning distances can be seenfrom Figure 2 When an accident has happened in a highwaytunnel the vehicle 119881

0near the tunnel has obtained the

message and lowers its velocity subconsciously The leadingvehicle 119881

11received the road information ahead by the HIS

server and passed through the three time warning distancesThe latter groups will adjust to the driving states based onthe vehicle 119881

0and the former groups The overall groups of

distance could be shown as

119904 = V0sdot 1199051+1

2119886 sdot 1199052

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199041

+ min2

(1199042(1199052) 1199092(1199052)) max2

(1199042(1199052) 1199092(1199052))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199042

+ (120585119890(1199053) + 119889des (1199053))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199043

(1)

The first time warning distance for each leading vehicle ofsafety distance is calculated before arriving at its secondwarning and mainly includes two parts of time the reactiontime of the driver 119905

1and message propagation delay 119905

2

Message propagation delay is composed of two parts 11990521and

11990522 11990521

is the time when the message about an accident intunnel is sent to the HIS server and 119905

22is the time when a

message is sent to related leading vehicles of the group Withthis notation we can define the first warning distance

1199041(119905) = V

0119905 +1

21198861199052 (119905 le 119905

2) (2)

Furthermore we can solve the first time warning distance foreach group denoted by

119904119894= 1199041(119905) (119905 le 119905

2 119894 = 1 119899) (3)

The second time warning distance is calculated before arriv-ing at its third time warning distanceThe goal for the secondtime warning distance is to lower the leading vehicle 119896 isin Ωof velocity after arriving at the third time warning distancethe leading vehicle of driving states could satisfy the trackingmode

Definition 4 Compensated safety distance in our work isdefined as the buffer distance among groups to avoid colli-sions

Based on the above assumptions and definition themodel for compensated safety distance for each group 119894 isin Γis denoted as follows

= 119866119909 + 119867119906 119909 = [1199042119894(119905)

V1(119905)] 119866 = [

0 10 0]

119867 = [01] 119906 = 119886des (119905)

(4)

4 Mathematical Problems in Engineering

where 1199042119894(119905) is the compensated safety distance and 119906 is the

desired deceleration for the leading vehicle

Definition 5 The vehicle is in the safety driving when thedesired acceleration is defined in the range |119886

119897| le 11988611 11988611

isa special state that denotes the leading vehicles entering thetracking mode after arriving at the third warning distance

The second time warning distance is a buffer range so weintroduce the virtual vehicle concept to calculate the distanceAdditional advantage of the virtual vehicle scheme is that themotion of the vehicle can be smoothly controlled when a newleading vehicle cuts in or the current vehicle cuts out [10]

Definition 6 The virtual vehicle distance 1199092(119905) is defined as

a constant safety distance For every group 119894 isin 1 2 119899the compensated safety distance should be set in the range asfollows

min119894

1199042119894(119905) 1199092(119905) lt 119904

2119894(119905) le max

119894

1199042119894(119905) 1199092(119905) (5)

The constant safety distance of virtual vehicle distance can beshown

1199092(119905) = ℎ sdot V + 119863min (6)

The second time warning distance is built to reduce vehiclersquosacceleration during some adverse weather or road sectionsthat are prone to accidents

The third warning time is the tracking mode beforearriving at its permitted minimum distanceThe vehicle119881

0is

uncontrollable so the leading vehicle 119896 isin 11988111 11988121 119881

1198991

in each group should be controlled to ensure a suitable rela-tive distance As can be seen from Figure 2 the intervehicledynamics model is designed among groups as follows

V119891119894= 119886119891119894

120585119889119894= 119897119897119894minus 119897119891119894

ΔV119894= V119897119894minus V119891119894

120585119890119894= 120585119889119894minus 119889des 119894

(7)

The longitudinal dynamics of the leading vehicle 119896 isin11988111 11988121 119881

1198991 are nonlinear According to the vehicle

dynamics in [11] the longitudinal dynamics is transferred asfollows

119886119891119894= 119892119891119894(V119891119894 119886119891119894) + ℎ119891119894(V119891119894) 120575119891119894

119892119891119894(V119891119894 119886119891119894) = minus

2119870119886119889

119898minus1

120591119891

[119886119891119894+119870119886119889

119898119891119894

V2119891119894+119870119898119889

119898119891119894

]

ℎ119891119894(V119891119894) =

1

119898119891119894

120591119891119894

(8)

If the parameters in (7) are exactly known the followingfeedback linearizing control law could be adopted

120575119891119894= 119898119891119894120583119891119888119894+ 119870119886119889V2119891119894+ 119870119898119889+ 2120591119891119894119870119886119889V119891119894119886119891119894 (9)

where 120585119889119894

is intervehicle distance ΔV119894is the error among

groups and 120585119890119894is the error between actual intervehicle dis-

tance and desired vehicle safety distance V119897119894 V119891119894are vehicle3rsquos

speed and vehicle1rsquos speed 120583119891119888119894

is the input signal that makesthe closed-loop system satisfy certain performance criteriaIn controller (8) we achieve the following objectives (1) thefeedback linearization results in a linear system as discussedabove however uncertainties in parameters can potentiallymake the linearization process inexact the study of sucha case would be an interesting topic to be considered infuture research (2) the simplification of the system model byexcluding some characteristic parameters (eg the mechani-cal drag mass and air resistance) from the vehicle dynamicsManipulating (6) through (8) the equation becomes

119891119894=1

120591119891119894

(120583119891119888119894

minus 120572119891119894) (10)

where 120591119891119894is the engine time constant and its value is 025 a

single parameter that describes the dynamics of the propul-sion system and internal disturbances 120583

119891119888119894

can be viewed asthe throttlebrake input causing accelerationdeceleration inthe controlled vehicle The system thus takes the form whichcan be described by the following standard equations

[[[

[

V119891119894

120585119890119894

ΔV119894

119891119894

]]]

]

=

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

+

[[[[[

[

0001

120591119891119894

]]]]]

]

sdot [119906 (119905)] +[[[

[

0010

]]]

]

sdot [120572119897119894]

119910 (119905)119894=[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

(11)

Writing the above equation as standard state-space equationswe have

119909 (119905)119894= 119860119909 (119905)

119894+ 119861119906 (119905)

119894+ 119862119908(119905)

119894

119910 (119905)119894= 119863119909 (119905)

119894

119860 =

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

119861 =

[[[[[

[

0001

120591119891119894

]]]]]

]

119862 =[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

119863 =[[[

[

0010

]]]

]

(12)

where state variables are119909(119905)119894= [V119891119894 120585119890119894 ΔV119894 120572119891119894]

119879 and119908(119905)119894

is the acceleration of the last vehicle 120575

Mathematical Problems in Engineering 5

By (12) the overall system of vehicle dynamics can beexpressed as

(

1199091

1199092

119909119899+1

) =(

119860104times4

sdot sdot sdot 04times4

04times4

1198602sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 119860119899

) sdot(

1199091

1199092

119909119899

)

+(

119861104times1

sdot sdot sdot 04times1

04times1

1198612sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 119861119899

) sdot(

1199061

1199062

119906119899

)

+(

11989610 sdot sdot sdot 0

0 1198962sdot sdot sdot 0

0 0 sdot sdot sdot 00 0 sdot sdot sdot 119896

119899

) sdot(

1199081

1199082

119908119899

)

(13)

Simplifying the above equation

119883119899+1= 119860 sdot 119883

119899+ 119861 sdot 119880

119899+ 119870 sdot 119882

119899 (14)

where 119883119899+1= (1199091 1199092 sdot sdot sdot 119909119899+1)

119879 119883119899= (1199091 1199092 sdot sdot sdot 119909119899)

119879119880119899= (1199061 1199062 sdot sdot sdot 119906119899)

119879 and 119908119899= (1199081 1199082 sdot sdot sdot 119908119899)

119879

Criterion 1 The dynamics model (12) is stable controllableand observability

The stability system is analyzed by eigenvalue criteriaThe controllability is verified by using rank (119860 119861) and theobservability is studied by using rank (119862 119860)

As we have defined it the three time warning distancesare designed to imply that combinedwithHIS space vehiclessafety driving could be guaranteed under the environment ofsomeuncertainweather conditions such as fog rain or snow

3 Distributed Model PredictiveControl Approach

31 Induction Process In this section we introduce notationand define the optimal control problem and the distributedmodel predictive control approach (DMPC) for the leadingvehicles in each group Combined with the above schematicsderivation theDMPConly is developed in the leading vehicle119896 isin Ω the following vehicles in each group can maintain thesafety driving distance in the form of the platoonTheDMPCalgorithm captures an important class of practical problemsincluding for examplemaneuvering a group of vehicles fromone point to another while maintaining relative formationandor avoiding collisions [12] Figure 3 shows the principleof the optimal control application for each leading vehicle

As stated above each group exchanges information withthe HIS server system Each group sequentially achievesperformance development in distributed MPC algorithmsunder the environment of the highway tunnel

Because the latter groups irregularly drive on the highwaytunnel the interconnections between different subsystemsare assumed to be weak and are considered as disturbances

HIS server

DPC center DMC

Relative distancerelative velocity

longitudinal velocity and acceleration

Predictive modelintervehicle dynamics

Receding optimizationDesired distance

Vehicle-to-HIS

communication

Optimal control

state

Vehicledynamic

model

Figure 3 The principle for the optimal control application

which can send the relative driving information to the lattergroup via HIS server system In addition the driving statesof the last vehicle in each group are uncontrollable butcan be measured by sensors so the acceleration 119908(119905)

119894of

the last vehicle 120575 isin Θ can be regarded as the measureddisturbance signal when calculating the latter groups Theoverall of longitudinal dynamics system that is composedof 119898 interconnected subsystems can be descried by (14)The global optimization problem can be decomposed intoa number of local optimization subproblems and the wholecontrol performance can be efficiently improved [13] Thecooperation between subsystems is achieved by exchanginginformation between each subsystem and its neighbors ina distributed structure via network between HIS server andvehicles

By (12) according to the vehicle driving characteristicsin highway intelligent space subsystem 119878

119894interacts with 119878

119895

and the output state acceleration of subsystem 119878119894is affected by

subgroup 119878119895 In this case 119878

119895is called input neighboring sub-

group of 119878119894 119878119894is called the output neighboring subsystem In

(12) the state acceleration of the 119878119894is regarded as disturbance

In every group of distributed optimal control strategywe assume that the same constant prediction horizon 119873

119901isin

(0infin) and constant update period 120574 isin (0119873119901] are used In

practice the update period 120574 isin (0119873119901] is the sample interval

The common update times are denoted by 119905119896= 1199050+120593119896 where

1199050= 0 and 119896 isin 119873 = 0 1 2 The leading vehicle in each

group sequentially solves an optimal control problem at theupdate period 120575 isin (0119873

119901] and applies the optimal control

trajectory until its next update time We have that 119911119894(119905) and

119906119894(119905) are the actual error state and control input respectively

For each leading vehicle 119896 isin Ω at any time 119905 ge 1199050 over any

prediction interval [119905119896 119905119896+119879] 119896 isin 119873 associatedwith updated

time 119905119896 we denote two trajectories

119906lowast119894(120591 119905119896) the optimal control trajectory

119894(120591 119905119896) the predicted state trajectory

where 120591 isin [119905119896 119905119896+ 119879]

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

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Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

∆s

MPC1 MPC2 MPC-n

O

Vehicle accidentzone ahead

Multiple groups of vehicles

s1 the first time warning distance

s2 the second time warning distance

s3 the third time warning distance

s1

s1

s0

V0 V11 V21V12 V1n V2n Vn1 Vnn

s2

s2

s3

sn

sdes

(i minus 1)N(iminus1)n iNi1

HIS serverDCP DMC

Group-1 Group-2 Group-nmiddot middot middot

middot middot middot middot middot middotmiddot middot middotmiddot middot middot

Figure 2 Overall system work principle

Assumption 2 Based on highway tunnel characteristics theovertaking situation is not considered in the research rangeand the velocity change of vehicles is small

Algorithm 3 Distributed model predictive controller for theleading vehicle 119896 isin Ω initialization the driving state of thevehicle 119881

0is sent to the first group by HIS server starting

at time 1199050= 0 In addition other groups will adjust to the

driving states by the communication between the HIS systemand the groups

22 Analysis of theThree TimeWarning Distances The safetyof the driving behavior for each group is typically related tovehicle driving environment Hence regarding each group ofsafety aspect the information framework includes interve-hicle distance relative velocity and acceleration The basicprinciple of the three time warning distances can be seenfrom Figure 2 When an accident has happened in a highwaytunnel the vehicle 119881

0near the tunnel has obtained the

message and lowers its velocity subconsciously The leadingvehicle 119881

11received the road information ahead by the HIS

server and passed through the three time warning distancesThe latter groups will adjust to the driving states based onthe vehicle 119881

0and the former groups The overall groups of

distance could be shown as

119904 = V0sdot 1199051+1

2119886 sdot 1199052

1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199041

+ min2

(1199042(1199052) 1199092(1199052)) max2

(1199042(1199052) 1199092(1199052))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199042

+ (120585119890(1199053) + 119889des (1199053))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1199043

(1)

The first time warning distance for each leading vehicle ofsafety distance is calculated before arriving at its secondwarning and mainly includes two parts of time the reactiontime of the driver 119905

1and message propagation delay 119905

2

Message propagation delay is composed of two parts 11990521and

11990522 11990521

is the time when the message about an accident intunnel is sent to the HIS server and 119905

22is the time when a

message is sent to related leading vehicles of the group Withthis notation we can define the first warning distance

1199041(119905) = V

0119905 +1

21198861199052 (119905 le 119905

2) (2)

Furthermore we can solve the first time warning distance foreach group denoted by

119904119894= 1199041(119905) (119905 le 119905

2 119894 = 1 119899) (3)

The second time warning distance is calculated before arriv-ing at its third time warning distanceThe goal for the secondtime warning distance is to lower the leading vehicle 119896 isin Ωof velocity after arriving at the third time warning distancethe leading vehicle of driving states could satisfy the trackingmode

Definition 4 Compensated safety distance in our work isdefined as the buffer distance among groups to avoid colli-sions

Based on the above assumptions and definition themodel for compensated safety distance for each group 119894 isin Γis denoted as follows

= 119866119909 + 119867119906 119909 = [1199042119894(119905)

V1(119905)] 119866 = [

0 10 0]

119867 = [01] 119906 = 119886des (119905)

(4)

4 Mathematical Problems in Engineering

where 1199042119894(119905) is the compensated safety distance and 119906 is the

desired deceleration for the leading vehicle

Definition 5 The vehicle is in the safety driving when thedesired acceleration is defined in the range |119886

119897| le 11988611 11988611

isa special state that denotes the leading vehicles entering thetracking mode after arriving at the third warning distance

The second time warning distance is a buffer range so weintroduce the virtual vehicle concept to calculate the distanceAdditional advantage of the virtual vehicle scheme is that themotion of the vehicle can be smoothly controlled when a newleading vehicle cuts in or the current vehicle cuts out [10]

Definition 6 The virtual vehicle distance 1199092(119905) is defined as

a constant safety distance For every group 119894 isin 1 2 119899the compensated safety distance should be set in the range asfollows

min119894

1199042119894(119905) 1199092(119905) lt 119904

2119894(119905) le max

119894

1199042119894(119905) 1199092(119905) (5)

The constant safety distance of virtual vehicle distance can beshown

1199092(119905) = ℎ sdot V + 119863min (6)

The second time warning distance is built to reduce vehiclersquosacceleration during some adverse weather or road sectionsthat are prone to accidents

The third warning time is the tracking mode beforearriving at its permitted minimum distanceThe vehicle119881

0is

uncontrollable so the leading vehicle 119896 isin 11988111 11988121 119881

1198991

in each group should be controlled to ensure a suitable rela-tive distance As can be seen from Figure 2 the intervehicledynamics model is designed among groups as follows

V119891119894= 119886119891119894

120585119889119894= 119897119897119894minus 119897119891119894

ΔV119894= V119897119894minus V119891119894

120585119890119894= 120585119889119894minus 119889des 119894

(7)

The longitudinal dynamics of the leading vehicle 119896 isin11988111 11988121 119881

1198991 are nonlinear According to the vehicle

dynamics in [11] the longitudinal dynamics is transferred asfollows

119886119891119894= 119892119891119894(V119891119894 119886119891119894) + ℎ119891119894(V119891119894) 120575119891119894

119892119891119894(V119891119894 119886119891119894) = minus

2119870119886119889

119898minus1

120591119891

[119886119891119894+119870119886119889

119898119891119894

V2119891119894+119870119898119889

119898119891119894

]

ℎ119891119894(V119891119894) =

1

119898119891119894

120591119891119894

(8)

If the parameters in (7) are exactly known the followingfeedback linearizing control law could be adopted

120575119891119894= 119898119891119894120583119891119888119894+ 119870119886119889V2119891119894+ 119870119898119889+ 2120591119891119894119870119886119889V119891119894119886119891119894 (9)

where 120585119889119894

is intervehicle distance ΔV119894is the error among

groups and 120585119890119894is the error between actual intervehicle dis-

tance and desired vehicle safety distance V119897119894 V119891119894are vehicle3rsquos

speed and vehicle1rsquos speed 120583119891119888119894

is the input signal that makesthe closed-loop system satisfy certain performance criteriaIn controller (8) we achieve the following objectives (1) thefeedback linearization results in a linear system as discussedabove however uncertainties in parameters can potentiallymake the linearization process inexact the study of sucha case would be an interesting topic to be considered infuture research (2) the simplification of the system model byexcluding some characteristic parameters (eg the mechani-cal drag mass and air resistance) from the vehicle dynamicsManipulating (6) through (8) the equation becomes

119891119894=1

120591119891119894

(120583119891119888119894

minus 120572119891119894) (10)

where 120591119891119894is the engine time constant and its value is 025 a

single parameter that describes the dynamics of the propul-sion system and internal disturbances 120583

119891119888119894

can be viewed asthe throttlebrake input causing accelerationdeceleration inthe controlled vehicle The system thus takes the form whichcan be described by the following standard equations

[[[

[

V119891119894

120585119890119894

ΔV119894

119891119894

]]]

]

=

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

+

[[[[[

[

0001

120591119891119894

]]]]]

]

sdot [119906 (119905)] +[[[

[

0010

]]]

]

sdot [120572119897119894]

119910 (119905)119894=[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

(11)

Writing the above equation as standard state-space equationswe have

119909 (119905)119894= 119860119909 (119905)

119894+ 119861119906 (119905)

119894+ 119862119908(119905)

119894

119910 (119905)119894= 119863119909 (119905)

119894

119860 =

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

119861 =

[[[[[

[

0001

120591119891119894

]]]]]

]

119862 =[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

119863 =[[[

[

0010

]]]

]

(12)

where state variables are119909(119905)119894= [V119891119894 120585119890119894 ΔV119894 120572119891119894]

119879 and119908(119905)119894

is the acceleration of the last vehicle 120575

Mathematical Problems in Engineering 5

By (12) the overall system of vehicle dynamics can beexpressed as

(

1199091

1199092

119909119899+1

) =(

119860104times4

sdot sdot sdot 04times4

04times4

1198602sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 119860119899

) sdot(

1199091

1199092

119909119899

)

+(

119861104times1

sdot sdot sdot 04times1

04times1

1198612sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 119861119899

) sdot(

1199061

1199062

119906119899

)

+(

11989610 sdot sdot sdot 0

0 1198962sdot sdot sdot 0

0 0 sdot sdot sdot 00 0 sdot sdot sdot 119896

119899

) sdot(

1199081

1199082

119908119899

)

(13)

Simplifying the above equation

119883119899+1= 119860 sdot 119883

119899+ 119861 sdot 119880

119899+ 119870 sdot 119882

119899 (14)

where 119883119899+1= (1199091 1199092 sdot sdot sdot 119909119899+1)

119879 119883119899= (1199091 1199092 sdot sdot sdot 119909119899)

119879119880119899= (1199061 1199062 sdot sdot sdot 119906119899)

119879 and 119908119899= (1199081 1199082 sdot sdot sdot 119908119899)

119879

Criterion 1 The dynamics model (12) is stable controllableand observability

The stability system is analyzed by eigenvalue criteriaThe controllability is verified by using rank (119860 119861) and theobservability is studied by using rank (119862 119860)

As we have defined it the three time warning distancesare designed to imply that combinedwithHIS space vehiclessafety driving could be guaranteed under the environment ofsomeuncertainweather conditions such as fog rain or snow

3 Distributed Model PredictiveControl Approach

31 Induction Process In this section we introduce notationand define the optimal control problem and the distributedmodel predictive control approach (DMPC) for the leadingvehicles in each group Combined with the above schematicsderivation theDMPConly is developed in the leading vehicle119896 isin Ω the following vehicles in each group can maintain thesafety driving distance in the form of the platoonTheDMPCalgorithm captures an important class of practical problemsincluding for examplemaneuvering a group of vehicles fromone point to another while maintaining relative formationandor avoiding collisions [12] Figure 3 shows the principleof the optimal control application for each leading vehicle

As stated above each group exchanges information withthe HIS server system Each group sequentially achievesperformance development in distributed MPC algorithmsunder the environment of the highway tunnel

Because the latter groups irregularly drive on the highwaytunnel the interconnections between different subsystemsare assumed to be weak and are considered as disturbances

HIS server

DPC center DMC

Relative distancerelative velocity

longitudinal velocity and acceleration

Predictive modelintervehicle dynamics

Receding optimizationDesired distance

Vehicle-to-HIS

communication

Optimal control

state

Vehicledynamic

model

Figure 3 The principle for the optimal control application

which can send the relative driving information to the lattergroup via HIS server system In addition the driving statesof the last vehicle in each group are uncontrollable butcan be measured by sensors so the acceleration 119908(119905)

119894of

the last vehicle 120575 isin Θ can be regarded as the measureddisturbance signal when calculating the latter groups Theoverall of longitudinal dynamics system that is composedof 119898 interconnected subsystems can be descried by (14)The global optimization problem can be decomposed intoa number of local optimization subproblems and the wholecontrol performance can be efficiently improved [13] Thecooperation between subsystems is achieved by exchanginginformation between each subsystem and its neighbors ina distributed structure via network between HIS server andvehicles

By (12) according to the vehicle driving characteristicsin highway intelligent space subsystem 119878

119894interacts with 119878

119895

and the output state acceleration of subsystem 119878119894is affected by

subgroup 119878119895 In this case 119878

119895is called input neighboring sub-

group of 119878119894 119878119894is called the output neighboring subsystem In

(12) the state acceleration of the 119878119894is regarded as disturbance

In every group of distributed optimal control strategywe assume that the same constant prediction horizon 119873

119901isin

(0infin) and constant update period 120574 isin (0119873119901] are used In

practice the update period 120574 isin (0119873119901] is the sample interval

The common update times are denoted by 119905119896= 1199050+120593119896 where

1199050= 0 and 119896 isin 119873 = 0 1 2 The leading vehicle in each

group sequentially solves an optimal control problem at theupdate period 120575 isin (0119873

119901] and applies the optimal control

trajectory until its next update time We have that 119911119894(119905) and

119906119894(119905) are the actual error state and control input respectively

For each leading vehicle 119896 isin Ω at any time 119905 ge 1199050 over any

prediction interval [119905119896 119905119896+119879] 119896 isin 119873 associatedwith updated

time 119905119896 we denote two trajectories

119906lowast119894(120591 119905119896) the optimal control trajectory

119894(120591 119905119896) the predicted state trajectory

where 120591 isin [119905119896 119905119896+ 119879]

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 1199042119894(119905) is the compensated safety distance and 119906 is the

desired deceleration for the leading vehicle

Definition 5 The vehicle is in the safety driving when thedesired acceleration is defined in the range |119886

119897| le 11988611 11988611

isa special state that denotes the leading vehicles entering thetracking mode after arriving at the third warning distance

The second time warning distance is a buffer range so weintroduce the virtual vehicle concept to calculate the distanceAdditional advantage of the virtual vehicle scheme is that themotion of the vehicle can be smoothly controlled when a newleading vehicle cuts in or the current vehicle cuts out [10]

Definition 6 The virtual vehicle distance 1199092(119905) is defined as

a constant safety distance For every group 119894 isin 1 2 119899the compensated safety distance should be set in the range asfollows

min119894

1199042119894(119905) 1199092(119905) lt 119904

2119894(119905) le max

119894

1199042119894(119905) 1199092(119905) (5)

The constant safety distance of virtual vehicle distance can beshown

1199092(119905) = ℎ sdot V + 119863min (6)

The second time warning distance is built to reduce vehiclersquosacceleration during some adverse weather or road sectionsthat are prone to accidents

The third warning time is the tracking mode beforearriving at its permitted minimum distanceThe vehicle119881

0is

uncontrollable so the leading vehicle 119896 isin 11988111 11988121 119881

1198991

in each group should be controlled to ensure a suitable rela-tive distance As can be seen from Figure 2 the intervehicledynamics model is designed among groups as follows

V119891119894= 119886119891119894

120585119889119894= 119897119897119894minus 119897119891119894

ΔV119894= V119897119894minus V119891119894

120585119890119894= 120585119889119894minus 119889des 119894

(7)

The longitudinal dynamics of the leading vehicle 119896 isin11988111 11988121 119881

1198991 are nonlinear According to the vehicle

dynamics in [11] the longitudinal dynamics is transferred asfollows

119886119891119894= 119892119891119894(V119891119894 119886119891119894) + ℎ119891119894(V119891119894) 120575119891119894

119892119891119894(V119891119894 119886119891119894) = minus

2119870119886119889

119898minus1

120591119891

[119886119891119894+119870119886119889

119898119891119894

V2119891119894+119870119898119889

119898119891119894

]

ℎ119891119894(V119891119894) =

1

119898119891119894

120591119891119894

(8)

If the parameters in (7) are exactly known the followingfeedback linearizing control law could be adopted

120575119891119894= 119898119891119894120583119891119888119894+ 119870119886119889V2119891119894+ 119870119898119889+ 2120591119891119894119870119886119889V119891119894119886119891119894 (9)

where 120585119889119894

is intervehicle distance ΔV119894is the error among

groups and 120585119890119894is the error between actual intervehicle dis-

tance and desired vehicle safety distance V119897119894 V119891119894are vehicle3rsquos

speed and vehicle1rsquos speed 120583119891119888119894

is the input signal that makesthe closed-loop system satisfy certain performance criteriaIn controller (8) we achieve the following objectives (1) thefeedback linearization results in a linear system as discussedabove however uncertainties in parameters can potentiallymake the linearization process inexact the study of sucha case would be an interesting topic to be considered infuture research (2) the simplification of the system model byexcluding some characteristic parameters (eg the mechani-cal drag mass and air resistance) from the vehicle dynamicsManipulating (6) through (8) the equation becomes

119891119894=1

120591119891119894

(120583119891119888119894

minus 120572119891119894) (10)

where 120591119891119894is the engine time constant and its value is 025 a

single parameter that describes the dynamics of the propul-sion system and internal disturbances 120583

119891119888119894

can be viewed asthe throttlebrake input causing accelerationdeceleration inthe controlled vehicle The system thus takes the form whichcan be described by the following standard equations

[[[

[

V119891119894

120585119890119894

ΔV119894

119891119894

]]]

]

=

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

+

[[[[[

[

0001

120591119891119894

]]]]]

]

sdot [119906 (119905)] +[[[

[

0010

]]]

]

sdot [120572119897119894]

119910 (119905)119894=[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

sdot[[[

[

V119891119894

120585119890119894

ΔV119894

120572119891119894

]]]

]

(11)

Writing the above equation as standard state-space equationswe have

119909 (119905)119894= 119860119909 (119905)

119894+ 119861119906 (119905)

119894+ 119862119908(119905)

119894

119910 (119905)119894= 119863119909 (119905)

119894

119860 =

[[[[[

[

0 0 0 10 0 1 minusℎ

119894

0 0 0 minus1

0 0 0minus1

120591119891119894

]]]]]

]

119861 =

[[[[[

[

0001

120591119891119894

]]]]]

]

119862 =[[[

[

1 0 0 00 1 0 00 0 1 00 0 0 1

]]]

]

119863 =[[[

[

0010

]]]

]

(12)

where state variables are119909(119905)119894= [V119891119894 120585119890119894 ΔV119894 120572119891119894]

119879 and119908(119905)119894

is the acceleration of the last vehicle 120575

Mathematical Problems in Engineering 5

By (12) the overall system of vehicle dynamics can beexpressed as

(

1199091

1199092

119909119899+1

) =(

119860104times4

sdot sdot sdot 04times4

04times4

1198602sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 119860119899

) sdot(

1199091

1199092

119909119899

)

+(

119861104times1

sdot sdot sdot 04times1

04times1

1198612sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 119861119899

) sdot(

1199061

1199062

119906119899

)

+(

11989610 sdot sdot sdot 0

0 1198962sdot sdot sdot 0

0 0 sdot sdot sdot 00 0 sdot sdot sdot 119896

119899

) sdot(

1199081

1199082

119908119899

)

(13)

Simplifying the above equation

119883119899+1= 119860 sdot 119883

119899+ 119861 sdot 119880

119899+ 119870 sdot 119882

119899 (14)

where 119883119899+1= (1199091 1199092 sdot sdot sdot 119909119899+1)

119879 119883119899= (1199091 1199092 sdot sdot sdot 119909119899)

119879119880119899= (1199061 1199062 sdot sdot sdot 119906119899)

119879 and 119908119899= (1199081 1199082 sdot sdot sdot 119908119899)

119879

Criterion 1 The dynamics model (12) is stable controllableand observability

The stability system is analyzed by eigenvalue criteriaThe controllability is verified by using rank (119860 119861) and theobservability is studied by using rank (119862 119860)

As we have defined it the three time warning distancesare designed to imply that combinedwithHIS space vehiclessafety driving could be guaranteed under the environment ofsomeuncertainweather conditions such as fog rain or snow

3 Distributed Model PredictiveControl Approach

31 Induction Process In this section we introduce notationand define the optimal control problem and the distributedmodel predictive control approach (DMPC) for the leadingvehicles in each group Combined with the above schematicsderivation theDMPConly is developed in the leading vehicle119896 isin Ω the following vehicles in each group can maintain thesafety driving distance in the form of the platoonTheDMPCalgorithm captures an important class of practical problemsincluding for examplemaneuvering a group of vehicles fromone point to another while maintaining relative formationandor avoiding collisions [12] Figure 3 shows the principleof the optimal control application for each leading vehicle

As stated above each group exchanges information withthe HIS server system Each group sequentially achievesperformance development in distributed MPC algorithmsunder the environment of the highway tunnel

Because the latter groups irregularly drive on the highwaytunnel the interconnections between different subsystemsare assumed to be weak and are considered as disturbances

HIS server

DPC center DMC

Relative distancerelative velocity

longitudinal velocity and acceleration

Predictive modelintervehicle dynamics

Receding optimizationDesired distance

Vehicle-to-HIS

communication

Optimal control

state

Vehicledynamic

model

Figure 3 The principle for the optimal control application

which can send the relative driving information to the lattergroup via HIS server system In addition the driving statesof the last vehicle in each group are uncontrollable butcan be measured by sensors so the acceleration 119908(119905)

119894of

the last vehicle 120575 isin Θ can be regarded as the measureddisturbance signal when calculating the latter groups Theoverall of longitudinal dynamics system that is composedof 119898 interconnected subsystems can be descried by (14)The global optimization problem can be decomposed intoa number of local optimization subproblems and the wholecontrol performance can be efficiently improved [13] Thecooperation between subsystems is achieved by exchanginginformation between each subsystem and its neighbors ina distributed structure via network between HIS server andvehicles

By (12) according to the vehicle driving characteristicsin highway intelligent space subsystem 119878

119894interacts with 119878

119895

and the output state acceleration of subsystem 119878119894is affected by

subgroup 119878119895 In this case 119878

119895is called input neighboring sub-

group of 119878119894 119878119894is called the output neighboring subsystem In

(12) the state acceleration of the 119878119894is regarded as disturbance

In every group of distributed optimal control strategywe assume that the same constant prediction horizon 119873

119901isin

(0infin) and constant update period 120574 isin (0119873119901] are used In

practice the update period 120574 isin (0119873119901] is the sample interval

The common update times are denoted by 119905119896= 1199050+120593119896 where

1199050= 0 and 119896 isin 119873 = 0 1 2 The leading vehicle in each

group sequentially solves an optimal control problem at theupdate period 120575 isin (0119873

119901] and applies the optimal control

trajectory until its next update time We have that 119911119894(119905) and

119906119894(119905) are the actual error state and control input respectively

For each leading vehicle 119896 isin Ω at any time 119905 ge 1199050 over any

prediction interval [119905119896 119905119896+119879] 119896 isin 119873 associatedwith updated

time 119905119896 we denote two trajectories

119906lowast119894(120591 119905119896) the optimal control trajectory

119894(120591 119905119896) the predicted state trajectory

where 120591 isin [119905119896 119905119896+ 119879]

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

By (12) the overall system of vehicle dynamics can beexpressed as

(

1199091

1199092

119909119899+1

) =(

119860104times4

sdot sdot sdot 04times4

04times4

1198602sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 04times4

04times4

04times4

sdot sdot sdot 119860119899

) sdot(

1199091

1199092

119909119899

)

+(

119861104times1

sdot sdot sdot 04times1

04times1

1198612sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 04times1

04times1

04times1

sdot sdot sdot 119861119899

) sdot(

1199061

1199062

119906119899

)

+(

11989610 sdot sdot sdot 0

0 1198962sdot sdot sdot 0

0 0 sdot sdot sdot 00 0 sdot sdot sdot 119896

119899

) sdot(

1199081

1199082

119908119899

)

(13)

Simplifying the above equation

119883119899+1= 119860 sdot 119883

119899+ 119861 sdot 119880

119899+ 119870 sdot 119882

119899 (14)

where 119883119899+1= (1199091 1199092 sdot sdot sdot 119909119899+1)

119879 119883119899= (1199091 1199092 sdot sdot sdot 119909119899)

119879119880119899= (1199061 1199062 sdot sdot sdot 119906119899)

119879 and 119908119899= (1199081 1199082 sdot sdot sdot 119908119899)

119879

Criterion 1 The dynamics model (12) is stable controllableand observability

The stability system is analyzed by eigenvalue criteriaThe controllability is verified by using rank (119860 119861) and theobservability is studied by using rank (119862 119860)

As we have defined it the three time warning distancesare designed to imply that combinedwithHIS space vehiclessafety driving could be guaranteed under the environment ofsomeuncertainweather conditions such as fog rain or snow

3 Distributed Model PredictiveControl Approach

31 Induction Process In this section we introduce notationand define the optimal control problem and the distributedmodel predictive control approach (DMPC) for the leadingvehicles in each group Combined with the above schematicsderivation theDMPConly is developed in the leading vehicle119896 isin Ω the following vehicles in each group can maintain thesafety driving distance in the form of the platoonTheDMPCalgorithm captures an important class of practical problemsincluding for examplemaneuvering a group of vehicles fromone point to another while maintaining relative formationandor avoiding collisions [12] Figure 3 shows the principleof the optimal control application for each leading vehicle

As stated above each group exchanges information withthe HIS server system Each group sequentially achievesperformance development in distributed MPC algorithmsunder the environment of the highway tunnel

Because the latter groups irregularly drive on the highwaytunnel the interconnections between different subsystemsare assumed to be weak and are considered as disturbances

HIS server

DPC center DMC

Relative distancerelative velocity

longitudinal velocity and acceleration

Predictive modelintervehicle dynamics

Receding optimizationDesired distance

Vehicle-to-HIS

communication

Optimal control

state

Vehicledynamic

model

Figure 3 The principle for the optimal control application

which can send the relative driving information to the lattergroup via HIS server system In addition the driving statesof the last vehicle in each group are uncontrollable butcan be measured by sensors so the acceleration 119908(119905)

119894of

the last vehicle 120575 isin Θ can be regarded as the measureddisturbance signal when calculating the latter groups Theoverall of longitudinal dynamics system that is composedof 119898 interconnected subsystems can be descried by (14)The global optimization problem can be decomposed intoa number of local optimization subproblems and the wholecontrol performance can be efficiently improved [13] Thecooperation between subsystems is achieved by exchanginginformation between each subsystem and its neighbors ina distributed structure via network between HIS server andvehicles

By (12) according to the vehicle driving characteristicsin highway intelligent space subsystem 119878

119894interacts with 119878

119895

and the output state acceleration of subsystem 119878119894is affected by

subgroup 119878119895 In this case 119878

119895is called input neighboring sub-

group of 119878119894 119878119894is called the output neighboring subsystem In

(12) the state acceleration of the 119878119894is regarded as disturbance

In every group of distributed optimal control strategywe assume that the same constant prediction horizon 119873

119901isin

(0infin) and constant update period 120574 isin (0119873119901] are used In

practice the update period 120574 isin (0119873119901] is the sample interval

The common update times are denoted by 119905119896= 1199050+120593119896 where

1199050= 0 and 119896 isin 119873 = 0 1 2 The leading vehicle in each

group sequentially solves an optimal control problem at theupdate period 120575 isin (0119873

119901] and applies the optimal control

trajectory until its next update time We have that 119911119894(119905) and

119906119894(119905) are the actual error state and control input respectively

For each leading vehicle 119896 isin Ω at any time 119905 ge 1199050 over any

prediction interval [119905119896 119905119896+119879] 119896 isin 119873 associatedwith updated

time 119905119896 we denote two trajectories

119906lowast119894(120591 119905119896) the optimal control trajectory

119894(120591 119905119896) the predicted state trajectory

where 120591 isin [119905119896 119905119896+ 119879]

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The predicted value 119894(120591 119905119896) is transmitted to all other

followers as soon as the optimal control problem at 119905 = 119896120575is solved to take account of collision avoidance The overallsystem running process is as follows

Step 1 At the time 119905 = 119896120575 the leading vehicle11988111building the

model predictive controller is defined

Input State 119894(120591 119905119896) = 119906lowast119894(120591 119905119896)

Disturbance Variance119908(119905)119908(119905) = 120572119897119894 Because of the uncon-

trollable vehicle 1198810 the V(119905) can be regarded as a disturbance

variance

Step 2 The DMPC used for the leading vehicle 119896 isin Ω obeysthe following implementation strategy

(1) At the time 119905 all the controllers receive the full statemeasurement 119909(119905) from the sensors

(2) Each controller evaluates its own future input tra-jectory based on 119909(119905) and 119908(119905) Based on the inputtrajectories each controller calculates the currentdecided set of inputs trajectories

(3) Each controller updates its entire future input tra-jectory and sends the future state to the followingvehicles in each group

(4) When a new measurement is received go to Step 1(119905 larr 119905 + 1)

At each iteration for 119901 = 0 119873 minus 1 each controllersolves the following optimization problem

min119906119897(119905)119906119897(119905+119873minus1)

119869 (119905)

119906119897(119905 + 119901) isin 119880

119897 119901 ge 0

1199061198971015840 (119905 + 119901) = 119906

1198971015840(119905 + 119901)

119888minus1

forall1198971015840= 119894

119909 (119905 + 119901) isin 119883 119895 gt 0

119909 (119905 + 119873) isin 119883119891

(15)

with119869 (119905) = sum

119897

119869119897(119905)

119869119894(119905) =119873minus1

sum119901=0

[1003817100381710038171003817119909119897 (119905 + 119901)

10038171003817100381710038172

119876119897

+1003817100381710038171003817119906119897 (119905 + 119901)

10038171003817100381710038172

119877119897

] + 119909 (119905 + 119873)2

119901119894

(16)

where 119876 and 119875 are the weighting matrices that tune therelative importance of the output vectorrsquos elements as well asthe magnitude of the control effort 119876119875 formulation is usedto solve the model predictive control problem

32 Control Objectives and Constraints Typically the pri-mary control objective of the HIS server system is to keepvehicles at a safety distance So there exit some constraintson the information framework for each group and theinformation unit for each vehicle

Step 1 The first parameter is the intervehicle distance Theintervehicle distance includes two parts One part is theintervehicle distance in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ and theleading vehicle 119896 isin Ω About the first part of the intervehicledistance the so-called safety distance can be defined as (6)As to constraints of relative distance error when the vehicle1198810runs uniformly larger or smaller intervehicle distance

may occur in real time The inequality for subobjective is asfollows

120585min119890119894le 120585119890119894le 120585

max119890119894 (17)

where 120585min119890119894

= minus5m is the lower boundary which is againobtained from the driver experimental data and 120585max

119890119894le 6m

is the higher boundary in the literature [14]The intervehicle distance between the last vehicle 120575 isin Θ

and the leading vehicle 119896 isin Ω can be studied on the threetime warning distances The second time warning distanceconstraint can be shown in Definition 5

The third time warning distance is the tracking modeand the constraint is the same as the first part of intervehicledistance

Step 2 The second parameter is the intervehicle velocityThe intervehicle velocity also includes two parts One is theintervehicle velocity in each group and the other is theintervehicle distance between the last vehicle 120575 isin Θ andthe leading vehicle 119896 isin Ω About the first part of theintervehicle velocity we define the initial vehicle velocityas 0 le V

119891le 80 kms The relative velocity in each group

should be minimized in the range minus1 le ΔV le 09msThe second intervehicle velocity is researched based on thethird time warning distance because during the first and thesecond time warning distances the vehicle is in accelerationor decelerationwithout any ruleThe constraint is in the rangeminus1 le ΔV le 12ms

Step 3 The third parameter is the acceleration of the leadingvehicle 119896 isin Ω in each group First the vehicle 119881

11of acceler-

ation 11988611constraint is defined in the range 119886

119891min= minus30msminus2

119886ℎmax isin [20 30]msminus2 The absolute value of 120572

119891minbeing bigger than that of 120572

119891maxcan accommodate larger brak-

ing degree to prevent rear-end collisions [15]

Step 4 The fourth parameter is the weight 119876119890and the

control input The weight 119876119890which is the weight of the error

120585119890between the desired and the actual distance has to be

considered The larger 119876119890is the smaller the time reaches

a steady-state situation Although the focus is on safety ithas to be remarked that for increasing 119876

119890 the acceleration

will increase as well The control input constraint includesthrottle input or brake input So we could define the controlformulation as 119906(119896) isin [minus1 1]

4 The Dynamics Model in Each Group

We consider 119899 groups of vehicles that travel in a straightline towards a highway tunnel In this section we focus on

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 5 10 15 20 25 30minus05

0

05

1

15

2

25

3

Time t

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired interdistance for gouple-1 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

(a)

0 5 10 15 20 25 30minus09

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV11 and V0 (ms)

Time t

(b)

0 5 10 15 20 25 30minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Plan

t out

put

acce

lera

tion

(ms

2)

Acceleration of the leading vehicle V11 (ms2)Reference acceleration (ms2)

(c)

Figure 4 The first group the intervehicle distance the intervelocity and the acceleration

the stability analysis for vehicles in each group 119894 isin 1 119899Meanwhile we suppose the intervehicle distance following aconstant time headway policy to be (6) In this way the actualdistance between vehicle 119894119895 and vehicle 119894119895minus1 should be definedas follows

120576119894119895(119905) = 119909

119894119895minus1(119905) minus 119909

119894119895(119905) minus 119871

119894119895minus1 (18)

where 119871119894119895minus1

is vehicle 119894119895minus1 length and 119909119894119895minus1(119905) 119909119894119895(119905) stand for

the position for vehicle 119894119895 and vehicle 119894119895 minus 1 So respectively

the continuous vehicle of vehicle distance error is calculatedby the actual distance and desired distance as follows

120575119894119895(119905) = 120576

119894119895(119905) minus 119909

2(119905) (19)

From the view of vehicle platoon stability the objective for thevehicle-to-HIS framework is that vehicle interdistance shouldtend to 0 which holds the desired interdistance between theleading vehicle and the following vehicle Equation (19) canbe defined as follows

119878119894119895equiv 120575119894119895(119905) (20)

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus1

0

1

2

3

4

5

Interdistance3-3 for vehicle3-3 of simulationInterdistance3-2 for vehicle3-2 of simulationInterdistance3-1 for vehicle3-1 of simulationDesired interdistance for gouple-2 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus14

minus12

minus1

minus08

minus06

minus04

minus02

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV21and V15 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V21 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 5 The second group the intervehicle distance the intervelocity and the acceleration

According to slid mode control method principle when thevariable 120575

119894(119905) satisfies the following expression

119878119894119895= minus120582119894119895sdot 119878119894119895 (21)

If the equation 119878119894119895rarr 0 is established 120575

119894119895(119905) rarr 0means that

the 119894119895th vehicle is said to provide individual vehicle stabilityThe spacing error derivation process for vehicle platoon isshown in [16] which is used in the paper as follows

119867(119904) =120575119894119895(119904)

120575119894119895minus1(119904)=

119904 + 120582

119905119892sdot 120591 sdot 1199043 + 119905

119892sdot 1199042 + (1 + 120582119905

119892) 119904 + 120582

(22)

where 119905119892= 2 s and 120591 = 05 s By using the above transfer

function string stability of vehicle platoon can be analyzed Itis shown that condition 119867(119904)

infinle 1 ensures system stability

5 Simulation and Analysis

In this section we mainly research the simulation processof the groups The three time warning distances have beenused to ensure multivehicle safety The first and the secondtime warning distances mainly lower the deceleration ofthe leading vehicle in each group The third time warningdistance mainly provides the safety driving distance withgroups for example deceleration state of safety distance

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

Interdistance4-4 for vehicle4-4 of simulationInterdistance4-3 for vehicle4-3 of simulationInterdistance4-2 for vehicle4-2 of simulationInterdistance4-1 for vehicle4-1 of simulationDesired interdistance for gouple3 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between the vehiclesV31and V23 (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading vehicle V31 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 6 The third group the intervehicle distance the intervelocity and the acceleration

or track mode of safety distance When the leading vehiclein each group is in the deceleration state during the thirdwarning distance other vehicles in each group are in a normalsafety driving Therefore we mainly study the tracking modefor each group during the third warning distance usingDMPC It is assumed that groups Γ = 4 119873V1 = 5 119873V2 = 3119873V3 = 4 and 119873V4 = 5 The prediction horizon and controlhorizon for the DPMC are 300 sampling times At time 119905

0

the vehicle 1198810acceleration is defined

1198860= 0 119905 lt 4

minus04 119905 ge 4(23)

Meanwhile we assume that after the first and the second timewarning distance vehicle119881

11of velocity is 105msWhen the

leading vehicle in each group lowers its acceleration |119886119897| le

11988611in the third time warning distance the relative distances

between the following vehicles in each group and the distancebetween the last vehicle and the leading vehicle are 3m

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 5 10 15 20 25 30minus2

0

2

4

6

8

10

12

14

16

Interdistance5-5 for vehicle5-5 of simulationInterdistance5-4 for vehicle5-4 of simulationInterdistance5-3 for vehicle5-3 of simulationInterdistance5-2 for vehicle5-2 of simulationInterdistance5-1 for vehicle5-1 of simulationDesired inter-distance for gouple-4 of vehicles

Plan

t out

put

inte

rdist

ance

am

ong

vehi

cles

(m

)

Time t

(a)

0 5 10 15 20 25 30minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

Plan

t out

put

rela

tive

spee

d (m

s)

Reference speed (ms)Relative speed between vehicles (ms)

Time t

(b)

0 5 10 15 20 25 30minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Time

Acceleration of the leading V41 (ms2)Reference acceleration (ms2)

Plan

t out

put

acce

lera

tion

(ms2)

(c)

Figure 7 The fourth group the intervehicle distance the intervelocity and the acceleration

5m 10m and 15m The following vehicles in each groupare tracking the reference trajectories Hence the drivingstate of the leading vehicles can represent each group ofinformation framework The leading vehicles can only bestudied Four groups of simulation results are shown as shownin Figures 4 5 6 and 7

From Figure 4 it can be seen that the intervehicle5-1distance between the leading vehicle 119881

11and the vehicle 119881

0

can maintain stability until the time 15 s When the vehicle1198810decelerated irregularly the leading vehicle 119881

11adjusts to

its velocity subsequently Meanwhile the following vehicles

have received the road information ahead from the HISserver system they still retain the constant safety distanceAs to the acceleration curve the reference acceleration 119886

11=

minus04ms2 by using Definition 5 we can know that only if|119886119897| le 119886

11 the system is controllable Figure 5 shows the

intervehicle distance between vehicles can retain stabilityuntil the time 10 s The change range of the intervelocityis larger than the former group Generally speaking thefirst group and the second group are stability The thirdand the fourth group signify that the intervehicle distancebetween vehicles can retain stability until the time 10 s to 15 s

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 5 10 15 20 25 30minus4

minus2

0

2

4

6

8

10

12

14

16

t (s)

Plan

t out

put

inte

rdist

ance

s am

ong

the

grou

ps (m

)

e interdistance5-1 between V51and V0 in the first groupe interdistance3-1 between V31 and V55 in the second groupe interdistance4-1 between V41 and V33 in the third groupe interdistance5-1 between V51 and V44 in the fourth group

Figure 8 The intervehicle distance among groups

These groups show that the intervelocity between vehicleschanges in real timeThe acceleration also adapts to the envi-ronment tracking characteristics The simulated informationframework demonstrated that by the first and the secondtime warning distances to groups of vehicles decelerationsystem can realize effectively vehicle safety characteristicsFigure 8 shows the intervehicle distances among groupsThrough comparative analysis of the intervehicle5-1 distancein the first group the intervehicle3-1 distance in the secondgroup the intervehicle4-1 distance in the third group and theintervehicle5-1 distance in the fourth group we can concludethat the distributed optimization basedDMPC controller andthe three time warning distances are carried out to verifythe performance of the vehicles passing the highway tunnelenvironment

6 Conclusion

The paper presents the distributed model predictive controlapproach for groups of vehicles of information frameworkanalysis on the highway tunnel environment The three timewarning distances are proposed to ensure vehicles of safetydriving The framework can be adapted for a highway tunneland some adverse weather such as fog rain and snow whendrivers cannot distinguish the road ahead because of weatherand road reasons The system is different from traditionalvehicle system of active electric functions its goal is toprovide a warning system based on the HIS system fordrivers to obtain the road situation in advance especiallymultiple groups of vehicles towards some road section proneto accidents or some adverse places First the three timewarning distances are introduced to guide the mechanismSecond the distributed model predictive control approachis studied to develop the optimal control sequence Last in

order to verify the feasibility for the system four differentgroups are simulatedThe simulation results demonstrate thatthe proposed three time warning distances could achievevehicles of safety and stability Furthermore it is shown thatthe local optimization of distributedmodel predictive controlapproach could ensure that the whole control performance iseffective

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The present work is supported by the National Foundationof Beijing (3133040) and China Postdoctoral Science Foun-dation (124609)

References

[1] F Gao J Liu G Xu K Guo and Y Cui ldquoVehicle state detectionin highway intelligent space based on information fusionrdquoin Proceedings of the Twelfth COTA International Conferenceof Transportation Professionals (CICTP rsquo12) pp 2307ndash2318Beijing China August 2012

[2] J R Mawhinney ldquoFixed fire protection systems in tunnelsissues and directionsrdquo Fire Technology vol 49 no 2 pp 477ndash508 2013

[3] T-T Chen and Y T Hsu ldquoThe research on the operating perfor-mance assessment for the automatic incident detection systemof the highway tunnel-as an example of ba-guah mountaintunnel in Taiwanrdquo in Proceedings of the GeoHunan InternationalConferencemdashTunnel Management Emerging Technologies andInnovation pp 40ndash47 Hunan China June 2011

[4] B Yan H Chen and L Wang ldquoVisual characteristics of thedriver to tunnel group traffic safetyrdquo in Proceedings of the 3rdInternational Conference on Transportation Engineering (ICTErsquo11) pp 3009ndash3014 July 2011

[5] H Li and Y Shi ldquoDistributed model predictive control ofconstrained nonlinear systems with communication delaysrdquoSystems and Control Letters vol 62 no 10 pp 819ndash826 2013

[6] J Zhang and J Liu ldquoDistributed moving horizon state estima-tion for nonlinear systems with bounded uncertaintiesrdquo Journalof Process Control vol 23 pp 1281ndash1295 2013

[7] W B Dunbar and R M Murray ldquoDistributed receding horizoncontrol for multi-vehicle formation stabilizationrdquo Automaticavol 42 no 4 pp 549ndash558 2006

[8] Y Zheng S Li andN Li ldquoDistributedmodel predictive controlover network information exchange for large-scale systemsrdquoControl Engineering Practice vol 19 no 7 pp 757ndash769 2011

[9] K Kon S Habasaki H Fukushima and F Matsuno ldquoModelpredictive based multi-vehicle formation control with collisionavoidance and localization uncertaintyrdquo in Proceedings of theIEEESICE International Symposium on System Integration (SIIrsquo12) pp 212ndash217 Kyushu University Fukuoka Japan December2012

[10] S G Kim M Tomizuka and K H Cheng ldquoSmooth motioncontrol of the adaptive cruise control system by a virtual lead

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

vehiclerdquo International Journal of Automotive Technology vol 13no 1 pp 77ndash85 2012

[11] Y F Peng ldquoAdaptive intelligent backstepping longitudinalcontrol of vehicleplatoons using output recurrent cerebellarmodel articulation controllerrdquoExpert Systemswith Applicationsvol 37 no 3 pp 2016ndash2027 2010

[12] S Li K Li R Rajamani and J Wang ldquoModel predictive multi-objective vehicular adaptive cruise controlrdquo IEEE Transactionson Control Systems Technology vol 19 no 3 pp 556ndash566 2011

[13] Y Zhang and S Li ldquoNetworked model predictive control basedon neighbourhood optimization for serially connected large-scale processesrdquo Journal of Process Control vol 17 no 1 pp 37ndash50 2007

[14] G J L Naus J Ploeg M J G van de Molengraft P MH Heemels and M Steinbuch ldquoDesign and implementationof parameterized adaptive cruise control an explicit modelpredictive control approachrdquo Control Engineering Practice vol18 no 8 pp 882ndash892 2010

[15] T Petrinic and I Petrovic ldquoLongitudinal spacing control ofvehicles in a platoon for stable and increased traffic flowrdquo inProceedings of the IEEE International Conference on ControlApplications (CCA rsquo12) pp 178ndash183 Dubrovnik Croatia Octo-ber 2012

[16] P D Christofides R Scattolini D Munoz de la Pena and JLiu ldquoDistributedmodel predictive control a tutorial review andfuture research directionsrdquo Computers and Chemical Engineer-ing vol 51 pp 21ndash41 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of