Research ArticleResearch on Gear Shifting Process without Disengaging Clutchfor a Parallel Hybrid Electric Vehicle Equipped with AMT

Hui-Long Yu12 Jun-Qiang Xi1 Feng-qi Zhang1 and Yu-hui Hu1

1 State Key Laboratory of Vehicle Transmission Beijing Institute of Technology Beijing 100081 China2Department of Mechanical Engineering Politecnico di Milano Campus Bovisa Via La Masa 1 20156 Milano Italy

Correspondence should be addressed to Jun-Qiang Xi xijunqiangbiteducn

Received 31 October 2013 Revised 13 December 2013 Accepted 13 December 2013 Published 11 February 2014

Academic Editor Hamid Reza Karimi

Copyright copy 2014 Hui-Long Yu 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

Dynamicmodels of a single-shaft parallel hybrid electric vehicle (HEV) equipped with automatedmechanical transmission (AMT)were described in different working stages during a gear shifting process without disengaging clutch Parameters affecting the gearshifting time components life and gear shifting jerk in different transient states during a gear shifting process were deeply analyzedThe mathematical models considering the detailed synchronizer working process which can explain the gear shifting failure longtime gear shifting and frequent synchronizer failure phenomenon in HEV were derived Dynamic coordinated control strategyof the engine motor and actuators in different transient states considering the detailed working stages of synchronizer in a gearshifting process of a HEV is for the first time innovatively proposed according to the state of art references Bench test and real roadtest results show that the proposed control strategy can improve the gear shifting quality in all its evaluation indexes significantly

1 Introduction

Auto gearshift can help to improve the driving comfortreduce the friction of clutch and synchronizer and achievea better handling of driving even in a complex environmentFor instance drivers having little driving experience mayhave a bad driving performance or a dangerous accidentdue to their unskilled operating in a manual gear shiftingprocess [1] Compared with other common automatic shifttechnologies including AT CVT and DCT AMT has lowercost and higher transmission efficiency Furthermore it canundertake modification on the traditional transmissionsTherefore it is used more widely in passenger cars especiallyin the hybrid and pure electric vehicles [2]

The main concern of this paper is a single-shaft parallelhybrid electric vehicle equipped with AMT The structurediagram of the HEV powertrain adopted in this paper isshown in Figure 1 The system is composed of a diesel engine(172 kW) a permanent magnetic motor (PMSM 75 kW) aclutch and a 5-speed automated mechanical gearbox Toobtain a smooth quick and successful gearshift the engine

motor and actuator of the gearbox should be coordinatelycontrolled very well

Most of the gearshift control strategies in traditionalvehicles equipped with AMT disengage the clutch duringa gear shifting process [3ndash5] which can introduce long-time power interruption and friction of the clutch A fewresearchers proposed gearshift control strategies withoutdisengaging the clutch using active engine control [6 7]but the conditions have not been significantly improved dueto the slow response of engine The introduction of drivingmotor on the HEV gives chance to achieve a fast and smoothgearshift by its active control due to its good response [8ndash10]

2 Gear Shifting Process without DisengagingClutch in the Hybrid Electric Vehicle

The gear shifting process is shown as in Figure 2 Typicalgear shifting process without disengaging clutch in a parallelhybrid electric vehicle can be divided into the following

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 985652 12 pageshttpdxdoiorg1011552014985652

2 Mathematical Problems in Engineering

Table 1 Parameters of the researched HEV

119869119890

119869119898

119869ci 119869co 119869119894

119869119900

119879119891-static

12 kgsdotm2 024 kgsdotm2 03 kgsdotm2 0028 kgsdotm2 009 kgsdotm2 0015 kgsdotm2 80Nsdotm

Gearbox

SynchronizerClutch Cardan

joint

Main reducer

5 4 3 2 1 R

Wheel

EngineMotor

Figure 1 Configuration of the researched single-shaft parallel hy-brid electric vehicle

detailed stages A unload of the power sources before agearshift B switch to neutral gear C active speed synchro-nization by the motorD mechanical speed synchronizationby the synchronizer E selection and engagement the newgears andF torque recovery

The control in stage A is mainly aimed at reducing thetransmitted torque between the engaged gears close to zerowhich means that the power sources are not output drivingtorque and at themeantime are not driven by the output shaftSeveral problems will come out when trying to disengage thegears transmitting large torque as follows

(a) It is difficult to disengage the gears or damage theteeth surface of the gears due to the contact pressurebetween the meshing teeth

(b) It introduces torsional vibration of the output shaftand affects the comfort of the vehicle due to the stepchange of torque

(c) It increases the difficulty of active synchronizationdue to the introduced vibration of output shaft speed

A gearshift when incompletely unloaded is shown inFigure 3 The main control difficulty in stage A is thatthere is no sensor to measure the transmission torquebetween the meshing teeth so it is easy to unload the powersources incompletely or excessively and introduce reversetorque exerted to the input shaftThis phenomenon becomesmore frequent when the driver needs a snap accelerationor a quick slowdown it seriously affects the gear shiftingquality

Stag C is the active speed synchronization period itneeds to adjust the speed of the input shaft to a desired speedquickly without overshoot and oscillation the response ofactive speed synchronization driven by the PMSM is muchfaster than that driven by engine [11]

In real driving cycles stages D and E which were con-duct at the same timewere also crucial to obtain a high quality

gear shifting process These two stages in HEV are quitedifferent from those in the traditional vehicles Rotationalinertia of input components of the gearbox in a gearshiftwithout disengaging clutch on a HEV is about 30 times largerthan that of those on a traditional vehicle in a gearshift withdisengaging clutch and there is at least a fivefold frictiontorque increase which can be seen from Table 1

We also found that there were frequent phenomenaof gear engaging failure or long time gear engaging aftersynchronization especially when the road condition changedor there was a braking The synchronizer wear was seriousand the obtained gearshift quality was quite poor To thebest of our knowledge there is no relative research on thisproblem

3 Evaluation Indexes of Gear Shifting Quality

31 The Gear Shifting Time The gear shifting time of thesingle-shaft parallel hybrid electric vehicle includes theunload time 119905unload time of disengaging the gear 119905disengtime of active synchronization 119905PMSM time of mechanicalsynchronization and gear engaging 119905eng and time of torquerecovery 119905recv The gear shifting time will affect the dynamicsand comfort performance of the HEV

119905shift = 119905unload + 119905diseng + 119905PMSM + 119905eng + 119905recv (1)

32The FrictionWork The friction work can reflect the wearof the friction pair it is an important index of durability Thesmaller the friction work is the longer the life of synchronizercan be obtained

The friction work 119871 is given as

119871 = int

119905119904

0

119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905 (2)

where 119905119904is the friction time119879

119904is the friction torque120596

119894120596119900are

the angular speed of input part and output part of the frictionpairs respectively [12]

33The Gear Shifting Jerk The jerk is derivative of longitudi-nal acceleration it can reflect the oscillation of driving torqueand can be expressed as

119895 =1198892V1198891199052

=119903

1198940

1198892

120596119900

1198891199052 (3)

where 119905 is time V is the vehicle speed 1198940is the gear ratio of the

main retarder and 119903 is the radius of the driving wheel

Mathematical Problems in Engineering 3

Initial gear

New gear

Motor speed

Engine speed

Engine torque

Motor torque

TimeDriver control

Gear shiftingDriver control

1

Torq

ueSp

eed

Gea

r

Desired speed

2 3 4 65

Figure 2 Gear shifting process without disengaging clutch of the HEV

To calculate the jerk equation (3) can be discretized as

119895 (119899) =119903

1198940

(120596119899+2

minus 120596119899+1) minus (120596

119899+1minus 120596119899)

Δ1199052

=2120587119903

601198940

(119899119899+2

minus 119899119899+1) minus (119899

119899+1minus 119899119899)

Δ1199052

(4)

where 119899 is the angular speed of the output shaft The ridecomfort of the vehicle becomes worse than the increase of thejerk For the quantitative criteria of jerk the recommendedvalue of Germany is |119895| le 10ms3 the one of former Sovietunion is |119895| le 3136ms3 and the one of China is |119895| le1764ms3 [13]

4 Power Transmission System Modeling

The sketch of power transmission system for the researchedsingle-shaft parallel hybrid electric vehicle is shown inFigure 4 It should be able to find out the main factorsaffecting the gear shifting quality in stages A D and EThe rotational inertia of clutch is equaled to the permanentmagnet synchronous motor due to the fact that there is noclutch disengaging in the gear shifting process To analyzethe torsional vibration of shafts between the driving motorand engine the torsion between gearbox and final driveis taken into account The torsion inside the gearbox isignored

When the system is in a certain gear the differential equa-tions are

(119879119890+ 119879119898minus 119879119890119898minus 119879119891) 119894119892minus 119879119900119908minus119879119903

119894119900

= (119869119890119890+ 119869119898+119888119898+ 119869119894119894) 119894119892+ 119869119900119900+ 119869119908119908

(5)

When the system is in neutral gear the differential equa-tions are

119879119890+ 119879119898minus 119879119890119898minus 119879119891= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

minus119879119900119908minus119879119903

119894119900

= 119869119900119900+ 119869119908119908

(6)

When the system transforms from neutral gear to a cer-tain gear the differential equations are

119879119890+ 119879119898minus 119879119890119898minus 119879119891minus119879119904

119894119892

= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

119879119904minus 119879119900V minus

119879119903

119894119900

= 119869119900119900+ 119869119908119908

(7)

where

119879119890119898= 119896119890119898(120579119890minus 120579119898) + 119888 (120596

119890minus 120596119898)

119879119900V = 119896119900V (120579119900 minus 120579V) + 119888 (120596119900 minus 120596V)

120579119909= 120596119909 119909 = (119890 119898 119900 V)

119879119903

119903119908

= 119866 (119891road cos 120579 + sin 120579) +119862119863119860V2119886

2115

(8)

4 Mathematical Problems in Engineering

4

3

2

1

0

minus1

600

540

480

420

360

300

1120

1080

1040

1000

9601840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

Current gear

Displacement of gearshift actuator

Speed of output shaft

Time t (s)

Time t (s)

Time t (s)

139 rpm

Figure 3 A gearshift when incompletely unloaded

TeTsTm

Tem Tow Tr

cem cow

kem kow

Je 120596e Ji 120596iJo 120596o Jw 120596wJm+ c 120596m

Figure 4 Sketch of the power transmission system

where 119879119890is the output torque of engine 119879

119898is the output

torque of driven motor 119879119890119898

is the viscoelastic torque ofthe shaft between motor and engine 119879

119891is the rotating

friction torque of transmission input side (including thefriction torque of engine motor and input shaft) when theclutch is engaged 119879

119900V is the rotating friction torque betweentransmission output shaft and final drive 119879

119903is the resisting

torque of road 119879119904is the friction torque of synchronizer

119903119908is the radius of wheel 119894

119892is the transmission ratio 119894

119900is

10001500

20002500

3000

2040

6080

1000

200

400

600

800

Throttle ()

T(N

m)

N (rpm)

Figure 5 Engine MAP

the final drive ratio 120596119890is the engine speed 120596

119898is the motor

speed 120596119894is the transmission input shaft speed 120596

119900is the

transmission output shaft speed 120579119890is the engine rotation

angle 120579119898

is the motor rotation angle 120579119900is the rotation

angle of transmission output shaft 120579V is the rotation angleof final drive 119866 is the vehicle gravity 119891road is the rollingresistance coefficient 120579 is the road slope angle 119862

119863is the air

resistance coefficient119860 is the face area (m2) V119886is the relative

velocity (kmh) which represents wind-free velocity 119869119894is the

equivalent rotational inertia of first-speed gear and second-speed gear in transmission which is converted to the inputend of synchronizer 119869

119890+119888is the equivalent rotational inertia

of engine crankshaft and clutch pressure plate and 119869119898+119888

isthe equivalent rotational inertia of motor and driven part ofclutch

5 Components Modeling

There are two working states of the transmission in a certaingear or in neutral gear When it is in neutral gear 119879

119904= 0

This paper focuses on the research of the switching processbetween states 1 and 2 It involves the dynamic process of eachcomponent

51 Engine Modeling The engine torque 119879119890can be obtained

from the engine MAP as follows

119879119890= 119891 (120572

119890 119899119890) (9)

where 120572119890is the engine throttle opening and 119899

119890is the engine

speed The engine MAP is shown in Figure 5

52 PMSM Modeling The transient value of three-phasevoltage and current of the motor can be ignored becausethe time constant of motor torque is generally very smallso the driving motor model can be simplified Consideringthe demand torque 119879

119898req the maximum allowable drive andbrake torque of motor with current rotating speed 119879

119898dis and119879119898chg and the maximum allowable torque of motor when

Mathematical Problems in Engineering 5

0 500 1000 1500 2000 2500 30000

10

20

300

400

500

600

N (rpm)

T(N

m)

Figure 6 Motor MAP

the power battery discharges and recharges 119879119887dis and 119879119887chg

the motor torque can be represented as

119879119898=

max (119879119898req 119879119898dis 119879119887dis) when 119879

119898req gt 0

max (119879119898req 119879119898chg 119879119887chg) when 119879

119898req lt 0

(10)

The motor MAP is shown in Figure 6

53 Gear Shifting Actuator Modeling The gear shifting actu-ator is driven by brushless DC motor (BLDCM) In a gearshifting process parameters including the displacement ofactuator and gear shifting force should be under control Thedisplacement should be tuned to a certain position rapidlyand accurately The output force of the BLDCM also shouldbe accurately controlled

The mathematical model and torque characteristics ofBLDCM can be analyzed with the mode of 3-phase 6-stateAt any moment only two phases work and the remainingone shuts down The motor adopts Y connection with non-salient pole Based on the physical structure we assume thatthe three phases of stator are completely symmetrical the air-gapmagnetic field produced by rotor is square wave the backelectromotive force of 3-phase winding is trapezoidal wavethe armature reaction of stator winding and eddy current lossare ignored and the magnetic circuit is not saturated So thebalance equation of the 3-phase stator voltage is given as

119906119896=119877119904119894119896+ 119890119896+ (119871119904minus 119871119898)119889119894119896

119889119905 119896 = 119886 119887 119888 (11)

Leaving out the mechanical loss of rotor and assumingthe electromagnetic power can totally translate to the kineticenergy of rotor the electromagnetic torque of motor is

119879eg =(119890119886119894119886+ 119890119887119894119887+ 119890119888119894119888)

120596mech (12)

The kinetic equation of BLDCM is

119879eg = 119869119889120596119898

119889119905+ 119879119871+ 119861120596mech (13)

The speed of BLDCM 119899 can be represented as

119899 =60120596mech2120587

(14)

The axial force acting on shift sleeve119865119909can be represented

as

119865119909= 119879eg119894119898120578119898 (15)

where 119894119886 119894119887 119894119888are phase currents 119906

119886 119906119887 119906119888are phase voltages

119877119904is the phase resistance of stator 119890

119886 119890119887 119890119888are phase

potential 119871119904is the self-inductance of stator phase 119871

119898is the

mutual-inductor of stator phases 119879119871is the load torque of

motor119861 is the damping coefficient120596mech is the angular speedof rotor 119869 is the rotational inertia of motor 119894

119898is the speed

ratio between motor and sleeve operating mechanism and120578119898

is the mechanical efficiency between motor and sleeveoperating mechanism

54 Synchronizer Modeling Synchronizer is one of the mostimportant parts of transmission The working process ofsynchronizer directly affects the gear shifting quality [14]To derive the optimal coordinated control strategy of eachpart the operating process of synchronizer is divided into fivestages

541 First Free Fly This stage actually includes two pro-cesses First the sleeve moves forward axially without resis-tance and pushes the synchro ring to the target gear Theresistance axial force is very small in this stage Second theforce transfers from sleeve to the sliders then to the conesurfaces of synchro ring The spring of slider here is used tokeep the balance When the force transferred increases to acertain extent the limiting mechanism will lose its balanceSo the sliders are pushed into grooves of the synchro hubby the radial pressure and the sleeve can continue slidingforward Then the spline teeth of sleeve is contacted to thespline teeth of synchro ring and the axial force transfers fromsleeve to the synchro ringThe schematic diagramof this stageis shown as Figure 7

The kinetic equation is

119865119909minus 119865119891minus 119865119903= 119898119904119886119904 (16)

where119898119904is the mass of sleeve 119886

119904is the acceleration of sleeve

119865119891is the sliding resistance when sleeve moves and 119865

119903is the

force acted on sleeve from limiting mechanism

542 Angular Velocity Synchronization The sleeve stopssliding and its axial velocity becomes zero The frictionbetween mesh teeth of sleeve and mesh teeth of synchro ringstarts to increase while the kinetic energy of sleeve and thetarget gear begin to decrease The angular velocity differencebetween them decreases gradually to zero The schematicdiagram of this stage is shown in Figure 8

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

Submit your manuscripts athttpwwwhindawicom

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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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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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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

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Table 1 Parameters of the researched HEV

119869119890

119869119898

119869ci 119869co 119869119894

119869119900

119879119891-static

12 kgsdotm2 024 kgsdotm2 03 kgsdotm2 0028 kgsdotm2 009 kgsdotm2 0015 kgsdotm2 80Nsdotm

Gearbox

SynchronizerClutch Cardan

joint

Main reducer

5 4 3 2 1 R

Wheel

EngineMotor

Figure 1 Configuration of the researched single-shaft parallel hy-brid electric vehicle

detailed stages A unload of the power sources before agearshift B switch to neutral gear C active speed synchro-nization by the motorD mechanical speed synchronizationby the synchronizer E selection and engagement the newgears andF torque recovery

The control in stage A is mainly aimed at reducing thetransmitted torque between the engaged gears close to zerowhich means that the power sources are not output drivingtorque and at themeantime are not driven by the output shaftSeveral problems will come out when trying to disengage thegears transmitting large torque as follows

(a) It is difficult to disengage the gears or damage theteeth surface of the gears due to the contact pressurebetween the meshing teeth

(b) It introduces torsional vibration of the output shaftand affects the comfort of the vehicle due to the stepchange of torque

(c) It increases the difficulty of active synchronizationdue to the introduced vibration of output shaft speed

A gearshift when incompletely unloaded is shown inFigure 3 The main control difficulty in stage A is thatthere is no sensor to measure the transmission torquebetween the meshing teeth so it is easy to unload the powersources incompletely or excessively and introduce reversetorque exerted to the input shaftThis phenomenon becomesmore frequent when the driver needs a snap accelerationor a quick slowdown it seriously affects the gear shiftingquality

Stag C is the active speed synchronization period itneeds to adjust the speed of the input shaft to a desired speedquickly without overshoot and oscillation the response ofactive speed synchronization driven by the PMSM is muchfaster than that driven by engine [11]

In real driving cycles stages D and E which were con-duct at the same timewere also crucial to obtain a high quality

gear shifting process These two stages in HEV are quitedifferent from those in the traditional vehicles Rotationalinertia of input components of the gearbox in a gearshiftwithout disengaging clutch on a HEV is about 30 times largerthan that of those on a traditional vehicle in a gearshift withdisengaging clutch and there is at least a fivefold frictiontorque increase which can be seen from Table 1

We also found that there were frequent phenomenaof gear engaging failure or long time gear engaging aftersynchronization especially when the road condition changedor there was a braking The synchronizer wear was seriousand the obtained gearshift quality was quite poor To thebest of our knowledge there is no relative research on thisproblem

3 Evaluation Indexes of Gear Shifting Quality

31 The Gear Shifting Time The gear shifting time of thesingle-shaft parallel hybrid electric vehicle includes theunload time 119905unload time of disengaging the gear 119905disengtime of active synchronization 119905PMSM time of mechanicalsynchronization and gear engaging 119905eng and time of torquerecovery 119905recv The gear shifting time will affect the dynamicsand comfort performance of the HEV

119905shift = 119905unload + 119905diseng + 119905PMSM + 119905eng + 119905recv (1)

32The FrictionWork The friction work can reflect the wearof the friction pair it is an important index of durability Thesmaller the friction work is the longer the life of synchronizercan be obtained

The friction work 119871 is given as

119871 = int

119905119904

0

119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905 (2)

where 119905119904is the friction time119879

119904is the friction torque120596

119894120596119900are

the angular speed of input part and output part of the frictionpairs respectively [12]

33The Gear Shifting Jerk The jerk is derivative of longitudi-nal acceleration it can reflect the oscillation of driving torqueand can be expressed as

119895 =1198892V1198891199052

=119903

1198940

1198892

120596119900

1198891199052 (3)

where 119905 is time V is the vehicle speed 1198940is the gear ratio of the

main retarder and 119903 is the radius of the driving wheel

Mathematical Problems in Engineering 3

Initial gear

New gear

Motor speed

Engine speed

Engine torque

Motor torque

TimeDriver control

Gear shiftingDriver control

1

Torq

ueSp

eed

Gea

r

Desired speed

2 3 4 65

Figure 2 Gear shifting process without disengaging clutch of the HEV

To calculate the jerk equation (3) can be discretized as

119895 (119899) =119903

1198940

(120596119899+2

minus 120596119899+1) minus (120596

119899+1minus 120596119899)

Δ1199052

=2120587119903

601198940

(119899119899+2

minus 119899119899+1) minus (119899

119899+1minus 119899119899)

Δ1199052

(4)

where 119899 is the angular speed of the output shaft The ridecomfort of the vehicle becomes worse than the increase of thejerk For the quantitative criteria of jerk the recommendedvalue of Germany is |119895| le 10ms3 the one of former Sovietunion is |119895| le 3136ms3 and the one of China is |119895| le1764ms3 [13]

4 Power Transmission System Modeling

The sketch of power transmission system for the researchedsingle-shaft parallel hybrid electric vehicle is shown inFigure 4 It should be able to find out the main factorsaffecting the gear shifting quality in stages A D and EThe rotational inertia of clutch is equaled to the permanentmagnet synchronous motor due to the fact that there is noclutch disengaging in the gear shifting process To analyzethe torsional vibration of shafts between the driving motorand engine the torsion between gearbox and final driveis taken into account The torsion inside the gearbox isignored

When the system is in a certain gear the differential equa-tions are

(119879119890+ 119879119898minus 119879119890119898minus 119879119891) 119894119892minus 119879119900119908minus119879119903

119894119900

= (119869119890119890+ 119869119898+119888119898+ 119869119894119894) 119894119892+ 119869119900119900+ 119869119908119908

(5)

When the system is in neutral gear the differential equa-tions are

119879119890+ 119879119898minus 119879119890119898minus 119879119891= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

minus119879119900119908minus119879119903

119894119900

= 119869119900119900+ 119869119908119908

(6)

When the system transforms from neutral gear to a cer-tain gear the differential equations are

119879119890+ 119879119898minus 119879119890119898minus 119879119891minus119879119904

119894119892

= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

119879119904minus 119879119900V minus

119879119903

119894119900

= 119869119900119900+ 119869119908119908

(7)

where

119879119890119898= 119896119890119898(120579119890minus 120579119898) + 119888 (120596

119890minus 120596119898)

119879119900V = 119896119900V (120579119900 minus 120579V) + 119888 (120596119900 minus 120596V)

120579119909= 120596119909 119909 = (119890 119898 119900 V)

119879119903

119903119908

= 119866 (119891road cos 120579 + sin 120579) +119862119863119860V2119886

2115

(8)

4 Mathematical Problems in Engineering

4

3

2

1

0

minus1

600

540

480

420

360

300

1120

1080

1040

1000

9601840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

Current gear

Displacement of gearshift actuator

Speed of output shaft

Time t (s)

Time t (s)

Time t (s)

139 rpm

Figure 3 A gearshift when incompletely unloaded

TeTsTm

Tem Tow Tr

cem cow

kem kow

Je 120596e Ji 120596iJo 120596o Jw 120596wJm+ c 120596m

Figure 4 Sketch of the power transmission system

where 119879119890is the output torque of engine 119879

119898is the output

torque of driven motor 119879119890119898

is the viscoelastic torque ofthe shaft between motor and engine 119879

119891is the rotating

friction torque of transmission input side (including thefriction torque of engine motor and input shaft) when theclutch is engaged 119879

119900V is the rotating friction torque betweentransmission output shaft and final drive 119879

119903is the resisting

torque of road 119879119904is the friction torque of synchronizer

119903119908is the radius of wheel 119894

119892is the transmission ratio 119894

119900is

10001500

20002500

3000

2040

6080

1000

200

400

600

800

Throttle ()

T(N

m)

N (rpm)

Figure 5 Engine MAP

the final drive ratio 120596119890is the engine speed 120596

119898is the motor

speed 120596119894is the transmission input shaft speed 120596

119900is the

transmission output shaft speed 120579119890is the engine rotation

angle 120579119898

is the motor rotation angle 120579119900is the rotation

angle of transmission output shaft 120579V is the rotation angleof final drive 119866 is the vehicle gravity 119891road is the rollingresistance coefficient 120579 is the road slope angle 119862

119863is the air

resistance coefficient119860 is the face area (m2) V119886is the relative

velocity (kmh) which represents wind-free velocity 119869119894is the

equivalent rotational inertia of first-speed gear and second-speed gear in transmission which is converted to the inputend of synchronizer 119869

119890+119888is the equivalent rotational inertia

of engine crankshaft and clutch pressure plate and 119869119898+119888

isthe equivalent rotational inertia of motor and driven part ofclutch

5 Components Modeling

There are two working states of the transmission in a certaingear or in neutral gear When it is in neutral gear 119879

119904= 0

This paper focuses on the research of the switching processbetween states 1 and 2 It involves the dynamic process of eachcomponent

51 Engine Modeling The engine torque 119879119890can be obtained

from the engine MAP as follows

119879119890= 119891 (120572

119890 119899119890) (9)

where 120572119890is the engine throttle opening and 119899

119890is the engine

speed The engine MAP is shown in Figure 5

52 PMSM Modeling The transient value of three-phasevoltage and current of the motor can be ignored becausethe time constant of motor torque is generally very smallso the driving motor model can be simplified Consideringthe demand torque 119879

119898req the maximum allowable drive andbrake torque of motor with current rotating speed 119879

119898dis and119879119898chg and the maximum allowable torque of motor when

Mathematical Problems in Engineering 5

0 500 1000 1500 2000 2500 30000

10

20

300

400

500

600

N (rpm)

T(N

m)

Figure 6 Motor MAP

the power battery discharges and recharges 119879119887dis and 119879119887chg

the motor torque can be represented as

119879119898=

max (119879119898req 119879119898dis 119879119887dis) when 119879

119898req gt 0

max (119879119898req 119879119898chg 119879119887chg) when 119879

119898req lt 0

(10)

The motor MAP is shown in Figure 6

53 Gear Shifting Actuator Modeling The gear shifting actu-ator is driven by brushless DC motor (BLDCM) In a gearshifting process parameters including the displacement ofactuator and gear shifting force should be under control Thedisplacement should be tuned to a certain position rapidlyand accurately The output force of the BLDCM also shouldbe accurately controlled

The mathematical model and torque characteristics ofBLDCM can be analyzed with the mode of 3-phase 6-stateAt any moment only two phases work and the remainingone shuts down The motor adopts Y connection with non-salient pole Based on the physical structure we assume thatthe three phases of stator are completely symmetrical the air-gapmagnetic field produced by rotor is square wave the backelectromotive force of 3-phase winding is trapezoidal wavethe armature reaction of stator winding and eddy current lossare ignored and the magnetic circuit is not saturated So thebalance equation of the 3-phase stator voltage is given as

119906119896=119877119904119894119896+ 119890119896+ (119871119904minus 119871119898)119889119894119896

119889119905 119896 = 119886 119887 119888 (11)

Leaving out the mechanical loss of rotor and assumingthe electromagnetic power can totally translate to the kineticenergy of rotor the electromagnetic torque of motor is

119879eg =(119890119886119894119886+ 119890119887119894119887+ 119890119888119894119888)

120596mech (12)

The kinetic equation of BLDCM is

119879eg = 119869119889120596119898

119889119905+ 119879119871+ 119861120596mech (13)

The speed of BLDCM 119899 can be represented as

119899 =60120596mech2120587

(14)

The axial force acting on shift sleeve119865119909can be represented

as

119865119909= 119879eg119894119898120578119898 (15)

where 119894119886 119894119887 119894119888are phase currents 119906

119886 119906119887 119906119888are phase voltages

119877119904is the phase resistance of stator 119890

119886 119890119887 119890119888are phase

potential 119871119904is the self-inductance of stator phase 119871

119898is the

mutual-inductor of stator phases 119879119871is the load torque of

motor119861 is the damping coefficient120596mech is the angular speedof rotor 119869 is the rotational inertia of motor 119894

119898is the speed

ratio between motor and sleeve operating mechanism and120578119898

is the mechanical efficiency between motor and sleeveoperating mechanism

54 Synchronizer Modeling Synchronizer is one of the mostimportant parts of transmission The working process ofsynchronizer directly affects the gear shifting quality [14]To derive the optimal coordinated control strategy of eachpart the operating process of synchronizer is divided into fivestages

541 First Free Fly This stage actually includes two pro-cesses First the sleeve moves forward axially without resis-tance and pushes the synchro ring to the target gear Theresistance axial force is very small in this stage Second theforce transfers from sleeve to the sliders then to the conesurfaces of synchro ring The spring of slider here is used tokeep the balance When the force transferred increases to acertain extent the limiting mechanism will lose its balanceSo the sliders are pushed into grooves of the synchro hubby the radial pressure and the sleeve can continue slidingforward Then the spline teeth of sleeve is contacted to thespline teeth of synchro ring and the axial force transfers fromsleeve to the synchro ringThe schematic diagramof this stageis shown as Figure 7

The kinetic equation is

119865119909minus 119865119891minus 119865119903= 119898119904119886119904 (16)

where119898119904is the mass of sleeve 119886

119904is the acceleration of sleeve

119865119891is the sliding resistance when sleeve moves and 119865

119903is the

force acted on sleeve from limiting mechanism

542 Angular Velocity Synchronization The sleeve stopssliding and its axial velocity becomes zero The frictionbetween mesh teeth of sleeve and mesh teeth of synchro ringstarts to increase while the kinetic energy of sleeve and thetarget gear begin to decrease The angular velocity differencebetween them decreases gradually to zero The schematicdiagram of this stage is shown in Figure 8

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Initial gear

New gear

Motor speed

Engine speed

Engine torque

Motor torque

TimeDriver control

Gear shiftingDriver control

1

Torq

ueSp

eed

Gea

r

Desired speed

2 3 4 65

Figure 2 Gear shifting process without disengaging clutch of the HEV

To calculate the jerk equation (3) can be discretized as

119895 (119899) =119903

1198940

(120596119899+2

minus 120596119899+1) minus (120596

119899+1minus 120596119899)

Δ1199052

=2120587119903

601198940

(119899119899+2

minus 119899119899+1) minus (119899

119899+1minus 119899119899)

Δ1199052

(4)

where 119899 is the angular speed of the output shaft The ridecomfort of the vehicle becomes worse than the increase of thejerk For the quantitative criteria of jerk the recommendedvalue of Germany is |119895| le 10ms3 the one of former Sovietunion is |119895| le 3136ms3 and the one of China is |119895| le1764ms3 [13]

4 Power Transmission System Modeling

The sketch of power transmission system for the researchedsingle-shaft parallel hybrid electric vehicle is shown inFigure 4 It should be able to find out the main factorsaffecting the gear shifting quality in stages A D and EThe rotational inertia of clutch is equaled to the permanentmagnet synchronous motor due to the fact that there is noclutch disengaging in the gear shifting process To analyzethe torsional vibration of shafts between the driving motorand engine the torsion between gearbox and final driveis taken into account The torsion inside the gearbox isignored

When the system is in a certain gear the differential equa-tions are

(119879119890+ 119879119898minus 119879119890119898minus 119879119891) 119894119892minus 119879119900119908minus119879119903

119894119900

= (119869119890119890+ 119869119898+119888119898+ 119869119894119894) 119894119892+ 119869119900119900+ 119869119908119908

(5)

When the system is in neutral gear the differential equa-tions are

119879119890+ 119879119898minus 119879119890119898minus 119879119891= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

minus119879119900119908minus119879119903

119894119900

= 119869119900119900+ 119869119908119908

(6)

When the system transforms from neutral gear to a cer-tain gear the differential equations are

119879119890+ 119879119898minus 119879119890119898minus 119879119891minus119879119904

119894119892

= 119869119890119890+ 119869119898+119888119898+ 119869119894119894

119879119904minus 119879119900V minus

119879119903

119894119900

= 119869119900119900+ 119869119908119908

(7)

where

119879119890119898= 119896119890119898(120579119890minus 120579119898) + 119888 (120596

119890minus 120596119898)

119879119900V = 119896119900V (120579119900 minus 120579V) + 119888 (120596119900 minus 120596V)

120579119909= 120596119909 119909 = (119890 119898 119900 V)

119879119903

119903119908

= 119866 (119891road cos 120579 + sin 120579) +119862119863119860V2119886

2115

(8)

4 Mathematical Problems in Engineering

4

3

2

1

0

minus1

600

540

480

420

360

300

1120

1080

1040

1000

9601840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

Current gear

Displacement of gearshift actuator

Speed of output shaft

Time t (s)

Time t (s)

Time t (s)

139 rpm

Figure 3 A gearshift when incompletely unloaded

TeTsTm

Tem Tow Tr

cem cow

kem kow

Je 120596e Ji 120596iJo 120596o Jw 120596wJm+ c 120596m

Figure 4 Sketch of the power transmission system

where 119879119890is the output torque of engine 119879

119898is the output

torque of driven motor 119879119890119898

is the viscoelastic torque ofthe shaft between motor and engine 119879

119891is the rotating

friction torque of transmission input side (including thefriction torque of engine motor and input shaft) when theclutch is engaged 119879

119900V is the rotating friction torque betweentransmission output shaft and final drive 119879

119903is the resisting

torque of road 119879119904is the friction torque of synchronizer

119903119908is the radius of wheel 119894

119892is the transmission ratio 119894

119900is

10001500

20002500

3000

2040

6080

1000

200

400

600

800

Throttle ()

T(N

m)

N (rpm)

Figure 5 Engine MAP

the final drive ratio 120596119890is the engine speed 120596

119898is the motor

speed 120596119894is the transmission input shaft speed 120596

119900is the

transmission output shaft speed 120579119890is the engine rotation

angle 120579119898

is the motor rotation angle 120579119900is the rotation

angle of transmission output shaft 120579V is the rotation angleof final drive 119866 is the vehicle gravity 119891road is the rollingresistance coefficient 120579 is the road slope angle 119862

119863is the air

resistance coefficient119860 is the face area (m2) V119886is the relative

velocity (kmh) which represents wind-free velocity 119869119894is the

equivalent rotational inertia of first-speed gear and second-speed gear in transmission which is converted to the inputend of synchronizer 119869

119890+119888is the equivalent rotational inertia

of engine crankshaft and clutch pressure plate and 119869119898+119888

isthe equivalent rotational inertia of motor and driven part ofclutch

5 Components Modeling

There are two working states of the transmission in a certaingear or in neutral gear When it is in neutral gear 119879

119904= 0

This paper focuses on the research of the switching processbetween states 1 and 2 It involves the dynamic process of eachcomponent

51 Engine Modeling The engine torque 119879119890can be obtained

from the engine MAP as follows

119879119890= 119891 (120572

119890 119899119890) (9)

where 120572119890is the engine throttle opening and 119899

119890is the engine

speed The engine MAP is shown in Figure 5

52 PMSM Modeling The transient value of three-phasevoltage and current of the motor can be ignored becausethe time constant of motor torque is generally very smallso the driving motor model can be simplified Consideringthe demand torque 119879

119898req the maximum allowable drive andbrake torque of motor with current rotating speed 119879

119898dis and119879119898chg and the maximum allowable torque of motor when

Mathematical Problems in Engineering 5

0 500 1000 1500 2000 2500 30000

10

20

300

400

500

600

N (rpm)

T(N

m)

Figure 6 Motor MAP

the power battery discharges and recharges 119879119887dis and 119879119887chg

the motor torque can be represented as

119879119898=

max (119879119898req 119879119898dis 119879119887dis) when 119879

119898req gt 0

max (119879119898req 119879119898chg 119879119887chg) when 119879

119898req lt 0

(10)

The motor MAP is shown in Figure 6

53 Gear Shifting Actuator Modeling The gear shifting actu-ator is driven by brushless DC motor (BLDCM) In a gearshifting process parameters including the displacement ofactuator and gear shifting force should be under control Thedisplacement should be tuned to a certain position rapidlyand accurately The output force of the BLDCM also shouldbe accurately controlled

The mathematical model and torque characteristics ofBLDCM can be analyzed with the mode of 3-phase 6-stateAt any moment only two phases work and the remainingone shuts down The motor adopts Y connection with non-salient pole Based on the physical structure we assume thatthe three phases of stator are completely symmetrical the air-gapmagnetic field produced by rotor is square wave the backelectromotive force of 3-phase winding is trapezoidal wavethe armature reaction of stator winding and eddy current lossare ignored and the magnetic circuit is not saturated So thebalance equation of the 3-phase stator voltage is given as

119906119896=119877119904119894119896+ 119890119896+ (119871119904minus 119871119898)119889119894119896

119889119905 119896 = 119886 119887 119888 (11)

Leaving out the mechanical loss of rotor and assumingthe electromagnetic power can totally translate to the kineticenergy of rotor the electromagnetic torque of motor is

119879eg =(119890119886119894119886+ 119890119887119894119887+ 119890119888119894119888)

120596mech (12)

The kinetic equation of BLDCM is

119879eg = 119869119889120596119898

119889119905+ 119879119871+ 119861120596mech (13)

The speed of BLDCM 119899 can be represented as

119899 =60120596mech2120587

(14)

The axial force acting on shift sleeve119865119909can be represented

as

119865119909= 119879eg119894119898120578119898 (15)

where 119894119886 119894119887 119894119888are phase currents 119906

119886 119906119887 119906119888are phase voltages

119877119904is the phase resistance of stator 119890

119886 119890119887 119890119888are phase

potential 119871119904is the self-inductance of stator phase 119871

119898is the

mutual-inductor of stator phases 119879119871is the load torque of

motor119861 is the damping coefficient120596mech is the angular speedof rotor 119869 is the rotational inertia of motor 119894

119898is the speed

ratio between motor and sleeve operating mechanism and120578119898

is the mechanical efficiency between motor and sleeveoperating mechanism

54 Synchronizer Modeling Synchronizer is one of the mostimportant parts of transmission The working process ofsynchronizer directly affects the gear shifting quality [14]To derive the optimal coordinated control strategy of eachpart the operating process of synchronizer is divided into fivestages

541 First Free Fly This stage actually includes two pro-cesses First the sleeve moves forward axially without resis-tance and pushes the synchro ring to the target gear Theresistance axial force is very small in this stage Second theforce transfers from sleeve to the sliders then to the conesurfaces of synchro ring The spring of slider here is used tokeep the balance When the force transferred increases to acertain extent the limiting mechanism will lose its balanceSo the sliders are pushed into grooves of the synchro hubby the radial pressure and the sleeve can continue slidingforward Then the spline teeth of sleeve is contacted to thespline teeth of synchro ring and the axial force transfers fromsleeve to the synchro ringThe schematic diagramof this stageis shown as Figure 7

The kinetic equation is

119865119909minus 119865119891minus 119865119903= 119898119904119886119904 (16)

where119898119904is the mass of sleeve 119886

119904is the acceleration of sleeve

119865119891is the sliding resistance when sleeve moves and 119865

119903is the

force acted on sleeve from limiting mechanism

542 Angular Velocity Synchronization The sleeve stopssliding and its axial velocity becomes zero The frictionbetween mesh teeth of sleeve and mesh teeth of synchro ringstarts to increase while the kinetic energy of sleeve and thetarget gear begin to decrease The angular velocity differencebetween them decreases gradually to zero The schematicdiagram of this stage is shown in Figure 8

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

4

3

2

1

0

minus1

600

540

480

420

360

300

1120

1080

1040

1000

9601840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

1840 1842 1844 1846 1848 1850

Current gear

Displacement of gearshift actuator

Speed of output shaft

Time t (s)

Time t (s)

Time t (s)

139 rpm

Figure 3 A gearshift when incompletely unloaded

TeTsTm

Tem Tow Tr

cem cow

kem kow

Je 120596e Ji 120596iJo 120596o Jw 120596wJm+ c 120596m

Figure 4 Sketch of the power transmission system

where 119879119890is the output torque of engine 119879

119898is the output

torque of driven motor 119879119890119898

is the viscoelastic torque ofthe shaft between motor and engine 119879

119891is the rotating

friction torque of transmission input side (including thefriction torque of engine motor and input shaft) when theclutch is engaged 119879

119900V is the rotating friction torque betweentransmission output shaft and final drive 119879

119903is the resisting

torque of road 119879119904is the friction torque of synchronizer

119903119908is the radius of wheel 119894

119892is the transmission ratio 119894

119900is

10001500

20002500

3000

2040

6080

1000

200

400

600

800

Throttle ()

T(N

m)

N (rpm)

Figure 5 Engine MAP

the final drive ratio 120596119890is the engine speed 120596

119898is the motor

speed 120596119894is the transmission input shaft speed 120596

119900is the

transmission output shaft speed 120579119890is the engine rotation

angle 120579119898

is the motor rotation angle 120579119900is the rotation

angle of transmission output shaft 120579V is the rotation angleof final drive 119866 is the vehicle gravity 119891road is the rollingresistance coefficient 120579 is the road slope angle 119862

119863is the air

resistance coefficient119860 is the face area (m2) V119886is the relative

velocity (kmh) which represents wind-free velocity 119869119894is the

equivalent rotational inertia of first-speed gear and second-speed gear in transmission which is converted to the inputend of synchronizer 119869

119890+119888is the equivalent rotational inertia

of engine crankshaft and clutch pressure plate and 119869119898+119888

isthe equivalent rotational inertia of motor and driven part ofclutch

5 Components Modeling

There are two working states of the transmission in a certaingear or in neutral gear When it is in neutral gear 119879

119904= 0

This paper focuses on the research of the switching processbetween states 1 and 2 It involves the dynamic process of eachcomponent

51 Engine Modeling The engine torque 119879119890can be obtained

from the engine MAP as follows

119879119890= 119891 (120572

119890 119899119890) (9)

where 120572119890is the engine throttle opening and 119899

119890is the engine

speed The engine MAP is shown in Figure 5

52 PMSM Modeling The transient value of three-phasevoltage and current of the motor can be ignored becausethe time constant of motor torque is generally very smallso the driving motor model can be simplified Consideringthe demand torque 119879

119898req the maximum allowable drive andbrake torque of motor with current rotating speed 119879

119898dis and119879119898chg and the maximum allowable torque of motor when

Mathematical Problems in Engineering 5

0 500 1000 1500 2000 2500 30000

10

20

300

400

500

600

N (rpm)

T(N

m)

Figure 6 Motor MAP

the power battery discharges and recharges 119879119887dis and 119879119887chg

the motor torque can be represented as

119879119898=

max (119879119898req 119879119898dis 119879119887dis) when 119879

119898req gt 0

max (119879119898req 119879119898chg 119879119887chg) when 119879

119898req lt 0

(10)

The motor MAP is shown in Figure 6

53 Gear Shifting Actuator Modeling The gear shifting actu-ator is driven by brushless DC motor (BLDCM) In a gearshifting process parameters including the displacement ofactuator and gear shifting force should be under control Thedisplacement should be tuned to a certain position rapidlyand accurately The output force of the BLDCM also shouldbe accurately controlled

The mathematical model and torque characteristics ofBLDCM can be analyzed with the mode of 3-phase 6-stateAt any moment only two phases work and the remainingone shuts down The motor adopts Y connection with non-salient pole Based on the physical structure we assume thatthe three phases of stator are completely symmetrical the air-gapmagnetic field produced by rotor is square wave the backelectromotive force of 3-phase winding is trapezoidal wavethe armature reaction of stator winding and eddy current lossare ignored and the magnetic circuit is not saturated So thebalance equation of the 3-phase stator voltage is given as

119906119896=119877119904119894119896+ 119890119896+ (119871119904minus 119871119898)119889119894119896

119889119905 119896 = 119886 119887 119888 (11)

Leaving out the mechanical loss of rotor and assumingthe electromagnetic power can totally translate to the kineticenergy of rotor the electromagnetic torque of motor is

119879eg =(119890119886119894119886+ 119890119887119894119887+ 119890119888119894119888)

120596mech (12)

The kinetic equation of BLDCM is

119879eg = 119869119889120596119898

119889119905+ 119879119871+ 119861120596mech (13)

The speed of BLDCM 119899 can be represented as

119899 =60120596mech2120587

(14)

The axial force acting on shift sleeve119865119909can be represented

as

119865119909= 119879eg119894119898120578119898 (15)

where 119894119886 119894119887 119894119888are phase currents 119906

119886 119906119887 119906119888are phase voltages

119877119904is the phase resistance of stator 119890

119886 119890119887 119890119888are phase

potential 119871119904is the self-inductance of stator phase 119871

119898is the

mutual-inductor of stator phases 119879119871is the load torque of

motor119861 is the damping coefficient120596mech is the angular speedof rotor 119869 is the rotational inertia of motor 119894

119898is the speed

ratio between motor and sleeve operating mechanism and120578119898

is the mechanical efficiency between motor and sleeveoperating mechanism

54 Synchronizer Modeling Synchronizer is one of the mostimportant parts of transmission The working process ofsynchronizer directly affects the gear shifting quality [14]To derive the optimal coordinated control strategy of eachpart the operating process of synchronizer is divided into fivestages

541 First Free Fly This stage actually includes two pro-cesses First the sleeve moves forward axially without resis-tance and pushes the synchro ring to the target gear Theresistance axial force is very small in this stage Second theforce transfers from sleeve to the sliders then to the conesurfaces of synchro ring The spring of slider here is used tokeep the balance When the force transferred increases to acertain extent the limiting mechanism will lose its balanceSo the sliders are pushed into grooves of the synchro hubby the radial pressure and the sleeve can continue slidingforward Then the spline teeth of sleeve is contacted to thespline teeth of synchro ring and the axial force transfers fromsleeve to the synchro ringThe schematic diagramof this stageis shown as Figure 7

The kinetic equation is

119865119909minus 119865119891minus 119865119903= 119898119904119886119904 (16)

where119898119904is the mass of sleeve 119886

119904is the acceleration of sleeve

119865119891is the sliding resistance when sleeve moves and 119865

119903is the

force acted on sleeve from limiting mechanism

542 Angular Velocity Synchronization The sleeve stopssliding and its axial velocity becomes zero The frictionbetween mesh teeth of sleeve and mesh teeth of synchro ringstarts to increase while the kinetic energy of sleeve and thetarget gear begin to decrease The angular velocity differencebetween them decreases gradually to zero The schematicdiagram of this stage is shown in Figure 8

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

0 500 1000 1500 2000 2500 30000

10

20

300

400

500

600

N (rpm)

T(N

m)

Figure 6 Motor MAP

the power battery discharges and recharges 119879119887dis and 119879119887chg

the motor torque can be represented as

119879119898=

max (119879119898req 119879119898dis 119879119887dis) when 119879

119898req gt 0

max (119879119898req 119879119898chg 119879119887chg) when 119879

119898req lt 0

(10)

The motor MAP is shown in Figure 6

53 Gear Shifting Actuator Modeling The gear shifting actu-ator is driven by brushless DC motor (BLDCM) In a gearshifting process parameters including the displacement ofactuator and gear shifting force should be under control Thedisplacement should be tuned to a certain position rapidlyand accurately The output force of the BLDCM also shouldbe accurately controlled

The mathematical model and torque characteristics ofBLDCM can be analyzed with the mode of 3-phase 6-stateAt any moment only two phases work and the remainingone shuts down The motor adopts Y connection with non-salient pole Based on the physical structure we assume thatthe three phases of stator are completely symmetrical the air-gapmagnetic field produced by rotor is square wave the backelectromotive force of 3-phase winding is trapezoidal wavethe armature reaction of stator winding and eddy current lossare ignored and the magnetic circuit is not saturated So thebalance equation of the 3-phase stator voltage is given as

119906119896=119877119904119894119896+ 119890119896+ (119871119904minus 119871119898)119889119894119896

119889119905 119896 = 119886 119887 119888 (11)

Leaving out the mechanical loss of rotor and assumingthe electromagnetic power can totally translate to the kineticenergy of rotor the electromagnetic torque of motor is

119879eg =(119890119886119894119886+ 119890119887119894119887+ 119890119888119894119888)

120596mech (12)

The kinetic equation of BLDCM is

119879eg = 119869119889120596119898

119889119905+ 119879119871+ 119861120596mech (13)

The speed of BLDCM 119899 can be represented as

119899 =60120596mech2120587

(14)

The axial force acting on shift sleeve119865119909can be represented

as

119865119909= 119879eg119894119898120578119898 (15)

where 119894119886 119894119887 119894119888are phase currents 119906

119886 119906119887 119906119888are phase voltages

119877119904is the phase resistance of stator 119890

119886 119890119887 119890119888are phase

potential 119871119904is the self-inductance of stator phase 119871

119898is the

mutual-inductor of stator phases 119879119871is the load torque of

motor119861 is the damping coefficient120596mech is the angular speedof rotor 119869 is the rotational inertia of motor 119894

119898is the speed

ratio between motor and sleeve operating mechanism and120578119898

is the mechanical efficiency between motor and sleeveoperating mechanism

54 Synchronizer Modeling Synchronizer is one of the mostimportant parts of transmission The working process ofsynchronizer directly affects the gear shifting quality [14]To derive the optimal coordinated control strategy of eachpart the operating process of synchronizer is divided into fivestages

541 First Free Fly This stage actually includes two pro-cesses First the sleeve moves forward axially without resis-tance and pushes the synchro ring to the target gear Theresistance axial force is very small in this stage Second theforce transfers from sleeve to the sliders then to the conesurfaces of synchro ring The spring of slider here is used tokeep the balance When the force transferred increases to acertain extent the limiting mechanism will lose its balanceSo the sliders are pushed into grooves of the synchro hubby the radial pressure and the sleeve can continue slidingforward Then the spline teeth of sleeve is contacted to thespline teeth of synchro ring and the axial force transfers fromsleeve to the synchro ringThe schematic diagramof this stageis shown as Figure 7

The kinetic equation is

119865119909minus 119865119891minus 119865119903= 119898119904119886119904 (16)

where119898119904is the mass of sleeve 119886

119904is the acceleration of sleeve

119865119891is the sliding resistance when sleeve moves and 119865

119903is the

force acted on sleeve from limiting mechanism

542 Angular Velocity Synchronization The sleeve stopssliding and its axial velocity becomes zero The frictionbetween mesh teeth of sleeve and mesh teeth of synchro ringstarts to increase while the kinetic energy of sleeve and thetarget gear begin to decrease The angular velocity differencebetween them decreases gradually to zero The schematicdiagram of this stage is shown in Figure 8

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

SleeveSynchro ringTarget gear

Centering pinSlider

Spring

Synchro hub

Limiting mechanism

Figure 7 First free fly

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Fa

Ft

Figure 8 Angular velocity synchronization

In the sleeve deceleration stage one has

119865119895119905 = 119898

119904V119904 (17)

where 119865119895is the impact force acted on meshing teeth of

synchro ring from sleeve 119905 is the sleeve deceleration time andV119904is the axial velocity of sleeve under the shifting forceIn the sliding friction stage one has

119879119904= 119865119909lowast 119891 lowast

119903

sin120572

119875119904= 119879119904

1003816100381610038161003816100381610038161003816100381610038161003816

120596119900minus120596119894

119894119892

1003816100381610038161003816100381610038161003816100381610038161003816

(18)

where 119865119909is the axial force of sleeve 119891 is the friction

coefficient 119903 is the mean effective cone radius 120572 is half of thecone angle and 119875

119904is the friction power

543 Turning the Synchro Ring When the two parts of syn-chronizer have been synchronized the driving part rotatesunder the action of the sleeve teeth (the driven part connectsrigidly to the vehicle body so it cannot be turned) A lowangular velocity differential between sleeve and synchro ringoccurs with the turning of synchro ring Then the synchro-nization is broken up and the locking state of synchro ring

Mesh tooth ofsynchro ring

Mesh toothof sleeve

Figure 9 Turning the synchro ring

disappears The schematic diagram of this stage is shown inFigure 9

Before the turning one has

1205961199030= 120596119900 (19)

During the turning process one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = 119869119903119903 (20)

119869119903= 119869119890+119888+ 119869119898+119888

+ 119869119894 (21)

1205961199031= 1205961199030+ 119903119905 (22)

1205961199031

= 120596119900 (23)

where 1205961199030is the initial angular speed of synchronizer driving

part before rotation 120596119900is the speed of output shaft 120596

1199031is the

input speed after rotation 120573 is the angle of roof shape at theinterlock gearing 120583 is the friction coefficient at roof-shapededge 119903

119904is the mean effective radius on interlock gearing

119879loss is the moment of losses 119879119894is the input torque 119869

119903is the

equivalent inertia to the synchronizer input end and 119903is the

angular acceleration of synchronizer input part

544 Turning the Target Gear This stage starts from theseparation of synchro ring and cone surface of the target gearThe driving part of the synchronizer including the synchroring rotates under the action of the sleeve teeth It ends atthe beginning of eventual slideThe schematic diagramof thisstage is shown in Figure 10

During this stage one has

119865119909

1 minus 120583 tan120573120583 + tan120573

119903119904minus 119879loss minus 119879119894119894119892 minus 119879119891 = (119869119903 + 119869119904) 119903

1205961199032= 1205961199031+ 119903119905

(24)

where 119869119904is the rotational inertia of synchro ring

545 Final Free Fly The sleeve engages with the targetgear under the action of the gear shifting force The kinetic

Mathematical Problems in Engineering 7

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

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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

Mesh tooth oftarget gear

Mesh toothof sleeve

Figure 10 Turning the target gear

Engaged

Figure 11 Final free fly

equation is similar to the first stage The schematic diagramof this stage is shown in Figure 11 Consider

119865119909minus 119865119891minus 119865119903= 119898119904119886119904

119865119891=120583119892119879119894119894119892

119903119904

(25)

where 120583119892is the axial dynamic friction coefficient between

sleeve and spline teeth of constant mesh gear and 119879119894is

the transferred torque We can get the conclusion from thederived kinetic equations that in the gear shifting processwithout disengaging clutch for a parallel hybrid vehicleequipped with AMT the inertia of input shaft the speedcontrol precision of driving motor and the input torqueof transmission can directly affect the synchronization timeand friction work The control of shifting force is also veryimportant An ideal control of each component in each stagecan improve the quality of gear shifting and increase the lifeof synchronizer

6 Coordinated Control of Each Component inEach Stage of a Gear Shifting Process

61 Switch to Neutral Gear The necessary condition ofswitching to neutral gear is

119865119909gt 119865119891=120583119892119879119894

119894119892

119903119904

(26)

where the input torque of the power sources is

119879119894= 119879119890+ 119879119898minus 119879119891minus 119879119890119898 (27)

Obviously the reasonable control of gear shifting forceand the input torque is the precondition to disengage thegears smoothly And the basic rules are to increase the gearshifting force 119865

119909and decrease the input torque 119879

119894

But the force to disengage the gears cannot increaseinfinitely and the exceeding force might damage the surfaceof meshing teeth therefore it should be controlled within thestipulated limitThe key to disengage the gears smoothly is tocontrol the input torque close to zero In order to realize thiscontrol wemeasured and got the input torque characteristicscurve of the power sources as well as the friction torquecharacteristics curve of the driving part then established theactual net output of torque model

In this paper we control the engine torque to zero and thefriction torque of the input part of the system is compensatedby the electrical motor to achieve a zero net output torque ofthe power system the control law is represented as

119879119894= 119891 (120572

119890 119879119898 120596119894) (28)

Another problem is the coordinated torque control of thetwo power sources during the unloading phase The controlparameters during the unloading phase involved the enginethrottle opening120572

119890andmotor torque119879

119898 Firstly the torque119879

119894

should be adjusted linearly instead of step change Secondlythe gear shifting time should be as short as possible Finallythe input torque 119879

119894should be controlled close to zero

The engine throttle opening is controlled linearly to zerowithout step change The step change might cause the shafttorsional vibration between the engine and the drivingmotorit also might influence the driving motor

Bench test showed that if the unloading time of theengine is 08 s when the initial throttle opening is 100the shaft torsional vibration between the engine and motorcan be negligible When the hybrid power system starts tounload linearly from the maximum torque 1200Nm and theunload speed is 15Nm10ms the shaft torsional vibration ofthe transmission system can be negligible as well So 08sis the maximum unloading time The PMSM can respondmuch faster than engine it can be controlled to compensatethe resistance torque and adjust the input torque 119879

119894to zero

linearly during the process of the engine unloading

8 Mathematical Problems in Engineering

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

MDesiredposition

PDcomputation Controller Displacement

detect

Feedback position

Figure 12 The diagram block of gearshift BLDCM controller

The target throttle opening during the phase of engineunloading is

120572119894+1= 120572119894minus 1 (119894min = 1 119894max = 120572119890init)

if (120572119894lt 1) 120572

119894= 0

(29)

where 120572119894is the throttle opening percentage of the CAN

protocol its physical values range from 0 to 100 and itsresolution is 01 and 119894max is the total unload steps equal tothe initial physical values of 120572

119894before unloading The time

interval between 120572119894+1

and 120572119894is 10ms which is transmitted to

the engine control unit (ECU) by transmission control unit(TCU) via CAN bus

Without considering the effect of temperature the rela-tionship between friction torque and rotation speed of inputshaft when the clutch is engaged can be given as

119879119891= 119891 (119899

119894) (30)

The target control torque of the motor is

119879119898(119894 + 1) = 119879

119898init+(119879119891minus 119879119898init

)

119894max (31)

where 119894max is the total unload step and its value is equal to theinitial physical value of 120572

119894before unloading

When 119879119894isin (minus15Nm 15Nm) the BLDCM can start

to drive the gearshift actuator to disengage the gears Toget an accurate control position of the gearshift actuatorand a smooth gearshift the displacement of the actuator ischosen as feedback of the BLDCM control The closed loopproportional-derivative (PD) control is chosen as the controlstrategy of gear selecting and shifting motor the relationshipbetween input 119890(119905) and output119906(119905) after discretization is givenas

119906 (119896) = 119870119901119890 (119896) + 119870

119889[119890 (119896) minus 119890 (119896 minus 1)] (32)

where119870119889119870119901are the differential coefficient and proportional

coefficient of the controller respectively 119890(119896) is the deviationof current detecting position and the desired position 119906(119896) isthe output value of the controller after 119896 times sampling andcalculating The diagram block gearshift BLDCM controlleris shown in Figure 12

62 Switch to a Certain Gear

621 Active Speed Synchronous Control Driving by PMSM Inthis stage the gearbox is in neutral gear the driving motoris controlled in speed control mode and its speed should beadjusted to

119899119898aim

= 119899119900sdot 119894119892 (33)

When 119899119898isin (119899119898aim

minus 20 119899119898aim

+ 20) the system startsto execute the next action In addition to reduce the entiregear shifting time further the gear selecting actuator canbe simultaneously controlled during the process of speedadjusting

622 Mechanical Synchronization and Gear Engaging ProcessThePMSM should be controlled to follow the speed of outputshaft after the active speed synchronous control the gearshiftactuator starts to engage the target gears

(1) First Free Fly This stage should be finished quickly andneed a larger gear shifting force but the force cannot beinfinitely increased because excessive shift force may causelarge impact force 119865

119895 and reduce the life of synchro ring and

sleeve

(2) Angular Velocity Synchronization In this stage theBLDCM torque is directly controlled to an allowedmaximumvalue to finish the mechanical synchronization process assoon as possibleThe drivingmotor is still controlled in speedmode following the speed of output shaft in this stage whichcould eliminate the rotation speed difference caused by theroad condition or braking action timely it can help to reducethe sliding friction work and the synchronization time

(3)Turning the Synchro Ring The control of this phase is thekey to engage the gears successfully and quickly we canknow from (20) that to turn the synchro ring and unlockthe synchronizer the gearshift torque should overcome thetorque loss 119879loss the input torque of power sources 119879119894119894119892 andthe friction torque of transmission input side 119879

119891

Real road test also confirmed that it was difficult toovercome the resistance torque 119879

119894(approximately 100Nm)

depending on the gearshift torque which is limited using afixed threshold The target gears cannot be engaged for quitea long time after finishing the rotation speed synchronizationprocess However it is obvious that the input torque of powersources 119879

119894119894119892is also controllable in this stage The engine is

working in the idle state now which can be considered as theload of transmission system The driving motor is in controlphase of following the speed of output shaft Based on thispoint of view we proposed a control method called ldquoactivelyunlock the synchronizer controlrdquo after finishing the speedsynchronization

The specific control method is to control the drivingmotor has a jagged speed fluctuation ranging fromminus10 rpm to10 rpmafter themechanical synchronizationwhich canmakethe sleeve teeth wriggle between teeth of synchro ring so thesynchro ring can be turned with the help of active control of

Mathematical Problems in Engineering 9

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

PMSM and gear shifting force The basic principle is shownin Figure 13

(4) Turning the Target Gear This stage is almost the same asthe last one except that the synchro ring is unlocked Theactual wriggled components are still in the sleeve side thecontrol algorithm can be learned from the previous stage

(5) Final Free Fly In this stage the gear shift actuator iscontrolled to the desired position and the engaging processis finished

63 Torque Recovering After a successful gear shifting thetorque of the engine and driving motor should be controlledto the desired value of the driver separately without largetorsional vibration and long time power interrupt Thecontrol algorithm is shown in Figure 14

7 Bench Test and Road Test

The block diagram of the overall gear shifting control algo-rithm for a single-shaft parallel HEV equipped with AMTwithout disengaging clutch is shown in Figure 15

In the bench the eddy current dynamometer works inthe load following mode that can simulate the actual drivingtest cycle conditions The monitor system of dynamometercan display the output torque of power sources in real timeand it can detect the jerk of output shaft when engagementor disengagement happens The main purpose of the benchtest is to verify and modify the control strategy Figure 16 is apicture of the test bench

After plenty of bench test 1000 kilometers real road testwith a real HEVwas conducted Figure 17 shows the test HEV

Figure 18 represents some exemplary curves of gearshifting process in road test when stage A is the unloadstage the engine is unloading exactly tomaintain its idle stateand the driving motor is unloading to the torque that canjust overcome the friction torque of the power input side thisstage costs about 03 s Then the gears are disengaged within02 s In stage C the driving motor is controlled in speedmode to finish the active synchronization process which costsabout 04 s In stage D the target gears are successfully andquickly engagedwith the small speed oscillation driven by themotor after the mechanical synchronization this stage costsabout 02 s The whole gear shifting process costs about 1 s

Figure 19 shows that the torque of engine and drivingmotor transits to the driverrsquos desired torque smoothly whenthe demand torque is recovering after a gearshift And thisstage costs about 05 s

8 Conclusions

In this paper mathematical models of every componentfor a HEV are described derivation and analysis of thegear shifting dynamics model in every working process ofthe synchronizer is also given Innovative control strategiesincluding dynamic coordinated control in powertrain unload

Y

N

Y

N

Y

N

Start to unlock thesynchro ring

Unlock

Unlock

Unlock

Desired motor speed

+10 rpm for 200ms

Desired motor speed

10 rpm for 200msminus

Finishunlocking

Figure 13 The algorithm to unlock the synchronizer actively

Y

N

Y

N

Y

N

Start to recovertorque

Finish recovering

T lt Req minus

minus

15

T gt Req + 15

ReqT = T+ 10 ReqT = T 10

|T minus Req| lt 15

Figure 14 The algorithm of recovering torque

10 Mathematical Problems in Engineering

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

Table 2 The statistical data of evaluation indexes in road test

Stage 1 2 3 1 2 3 1 2 3Input shaft speed (rmin) 1560 mdash 907 1049 mdash 799 1675 mdash 1279Maximum jerk (ms3) 115 mdash 5 123 mdash 54 9 mdash 54Stage time (ms) 329 488 549 368 414 560 366 480 569Total time (ms) 1353 1321 1412

Y

Y

N

N

N

N

N

Y

Y

N

Y

Y

Start

Shiftschedule

Gearshift

Unloadcontrol

|Ti| lt 10

Disengage thegears

Activesynchronization

|ReqNm minus Nm| lt 10

|ReqNm minus Nm| lt 10

Engage thenew gears

Engage

Engage

Synchro ringunlock control

Torque recoverycontrol

Gearshift fnish

Figure 15 Gear shifting control algorithm for a single-shaft parallelHEW without disengaging clutch

Figure 16 The test bench

Figure 17 The test HEV

process and active unlock control of the synchronizer on aHEV are proposed

By the dynamic coordinated control of the engine motorand actuators in unloading process the net input torqueof the gearbox can be controlled more closely to zero andit is easy to disengage the gears smoothly and quickly Bythe dynamic coordinated control of the engine motor andactuators in every working stage of the synchronizer duringa gear shifting process the gears can be controlled to engagemore quickly and successfully with less friction

According to the statistical data of the evaluation indexesin Table 2 the whole gear shifting time is controlled within15 s And the time consuming for stage 1 (disengaging thegears) and stage 2 (recovering the torque) takes about 50of the whole gear shifting time which is determined by thegearshift points and the driverrsquos demand of torqueThe powerinterrupt time is controlled within 05 s and the dynamicproperty of the HEV is ensured The jerk of gear shiftingwithout disengaging clutch is controlled within 15ms3 andthe ride comfort of the HEV is ensured

Mathematical Problems in Engineering 11

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

Te-

Req

()

Tm

-Req

(Nm

)

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625 630 635 640

620 625

1 2 3 4

630 635 640

620 625 630 635 640

620 625 630 635 640

806040200

806040200

2000

minus200minus400minus600

200

100

0

6

4

2

0

1800

1500

1200

1800

1500

1200

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

Te

()

Tm

(Nm

)G

ear

n2

(rpm

)n1

(rpm

)

Figure 18 Experiment curves of a gear shifting process

The proposed gear shifting control strategy can also beapplied into other kinds of pure electric or hybrid electricvehicles with similar gear box to significantly improve thegear shifting quality in all its evaluation indexes Controlstrategy using more advanced control technologies [15 16]will be proposed to improve the gear shifting quality furtherin the near future

Conflict of Interests

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

610 612 614 616 618 620

800600400200

5004003002001000

0

30

20

10

0

6

4

2

0

60

40

20

0

40

20100

30

400

200

0

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

610 612 614 616 618 620

t (s)

Tm

(Nm

)Te

()

Gea

rTe-

Req

()

Tm

-Req

(Nm

)Teminusd(

)Tmminusd(N

m)

Figure 19 The torque recovering process

Acknowledgment

This work was supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA11A223)

References

[1] M Pettersson and L Nielsen ldquoGear shifting by engine controlrdquoIEEE Transactions on Control Systems Technology vol 8 no 3pp 495ndash507 2000

12 Mathematical Problems in Engineering

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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

[2] L Glielmo L Iannelli V Vacca and F Vasca ldquoGearshift controlfor automatedmanual transmissionsrdquo IEEEASMETransactionson Mechatronics vol 11 no 1 pp 17ndash26 2006

[3] Y Lei M Niu and A Ge ldquoA research on starting controlstrategy of vehicle with AMTrdquo in Proceedings of the FISITAWorld Automotive Congress Seoul Korea 2000

[4] H Lee S Sul H Cho and J Lee ldquoAdvanced gear-shiftingand clutching strategy for a parallel-hybrid vehiclerdquo IEEETransactions on Industry Applications vol 6 no 6 pp 26ndash322000

[5] U Kiencke and L Nielsen Automotive Control Systems ForEngine Driveline and Vehicle Springer 2nd edition 2005

[6] Z Zhong G Kong Z Yu X Xin andX Chen ldquoShifting controlof an automated mechanical transmission without using theclutchrdquo International Journal of Automotive Technology vol 13no 3 pp 487ndash496 2012

[7] X T Lu and G Z Hou ldquoIntroduction of the AMT controlsystem structure and main foreign AMT productsrdquo AutomobileTechnology vol 5 pp 19ndash22 2004 (Chinese)

[8] R C Baraszu and S R Cikanek ldquoTorque fill-in for an auto-mated shift manual transmission in a parallel hybrid electricvehiclerdquo in Proceedings of the American Control Conference vol8ndash10 pp 1431ndash1436 Anchorage Alaska USA May 2002

[9] C L Liao J Z Zhang and Q C Lu ldquoCoordinated powertraincontrol method for shifting process of automated mechanicaltransmission in the hybrid electric vehiclerdquo Journal of Mechan-ical Engineering vol 41 no 12 pp 37ndash41 2005 (Chinese)

[10] M Ye D T Qin and Z J Liu ldquoShift performance control formild hybrid electric vehicle equipped with automatic manualtransmissionrdquo Journal of Mechanical Engineering vol 45 no 5pp 108ndash114 2009 (Chinese)

[11] J Fredriksson and B Egardt ldquoNonlinear control applied togearshifting in automated manual transmissionsrdquo in Proceed-ings of the 39th IEEE Confernce on Decision and Control pp444ndash449 Sydney Australia December 2000

[12] X Dong Dynamic control of engine and motor in a gearshiftfor HEV equipped with AMT [PhD thesis] Beijing Institute ofTechnology 2012

[13] L Guo AGe T Zhang andY Yue ldquoAMT shift process controlrdquoTransactions of the Chinese Society for Agricultural Machineryvol 34 no 2 pp 1ndash10 2004

[14] L Lovas D Play J Marialigeti and J F Rigal ldquoMechanicalbehaviour simulation for synchromesh mechanism improve-mentsrdquo Proceedings of the Institution of Mechanical EngineersPart D vol 220 no 7 pp 919ndash945 2006

[15] H Zhang Y Shi and M X Liu ldquo119867infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 no 1 pp 337ndash345 2013

[16] H Zhang Y Shi and A SMehr ldquoRobust static output feedbackcontrol and remote PID design for networked motor systemsrdquoIEEE Transactions on Industrial Electronics vol 58 no 12 pp5396ndash5405 2011

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