Three-parameter transmission gear-shifting schedule for ... · Traditionally the transmission gear-shifting schedule is based upon the throttle position and the vehicle (or engine)

Original article

Proc IMechE Part DJ Automobile Engineering2018 Vol 232(4) 521ndash533 IMechE 2017Reprints and permissionssagepubcoukjournalsPermissionsnavDOI 1011770954407017703234journalssagepubcomhomepid

Three-parameter transmissiongear-shifting schedule for improved fueleconomy

Chengsheng Miao12 Haiou Liu1 and Guoming G Zhu2

AbstractTraditionally the transmission gear-shifting schedule is based upon the throttle position and the vehicle (or engine) speedThis paper proposes to add a third parameter called the terrain coefficient to form a three-parameter gear-shiftingschedule for improving the fuel economy of a vehicle The terrain coefficient is a compound parameter consisting of theroad grade and the rolling resistance coefficient It can be estimated in real time by the proposed multi-step recursiveleast-squares method The dynamic programming and the moving least-squares method are adopted to optimize thegear sequences and to generate the three-parameter gear-shifting schedule The proposed gear-shifting schedule is evalu-ated against the traditional two-parameter gear-shifting schedule via Simulink simulations and on-road experiments usinga heavy-duty vehicle The simulation results for the Urban Dynamometer Driving Schedule and the US06 SupplementalFederal Test Procedure driving cycles show that the fuel economies of the proposed gear-shifting schedule are improvedby 33 and 27 respectively over that of the traditional two-parameter schedule The experimental results indicatethat the three-parameter gear-shifting schedule improves the fuel economy by 35 over the traditional schedule with asatisfactory acceleration performance

KeywordsDynamic programming gear-shifting schedule recursive least-squares method terrain coefficient

Date received 3 November 2016 accepted 8 February 2017

Introduction

The transmission is one of the key components of thepowertrain system and plays an important role in theperformances of the vehicle such as the fuel economythe mobility and the driveability The convenience ofautomatic gear shift without driver intervention leadsto wide applications of automatic transmissions in pas-senger vehicles Besides the traditional hydraulic auto-matic transmission (AT) new automatic transmissionssuch as the dual-clutch transmission (DCT) the con-tinuously variable transmission (CVT) and the auto-mated manual transmission (AMT) are becomingincreasingly popular Instead of manual gear shift thetransmission controller selects the proper gear accord-ing to the gear-shifting schedule and shifts the gearautomatically to meet the driving demand It should benoted that the gear-shifting schedule affects the vehicleacceleration and the fuel economy significantly12

Most gear-shifting schedules are based on a fewparameters eg one-parameter and two-parametergear-shifting schedules3ndash7 For passenger cars with

automatic transmissions a two-parameter gear-shiftingschedule is widely used and the throttle position andthe vehicle (or engine) speed are commonly the selectedparameters When both the parameters satisfy theupshift (or downshift) conditions the transmission con-troller shifts the gear automatically In comparisonwith the one-parameter (eg the vehicle speed) sched-ule the two-parameter (eg the throttle position andthe vehicle speed) schedule improves the vehicle perfor-mance effectively

However some drivers focus more on the fuel econ-omy and others enjoy good driveability the gear-

1Mechanical Engineering Beijing Institute of Technology Beijing Peoplersquos

Republic of China2Mechanical Engineering Michigan State University East Lansing

Michigan USA

Corresponding author

Guoming G Zhu Mechanical Engineering and Electrical amp Computer

Engineering Michigan State University 1497 Engineering Research Court

Room E148 East Lansing MI 48824 USA

Email zhugegrmsuedu

shifting schedule can also be affected by the road condi-tions For example the upshift should occur at higherengine speed when driving uphill with a large roadgrade than when driving on a road with no grade andshould be limited when driving downhill because ofvehicle safety Unfortunately the two-parameter sched-ule cannot adapt to these different driving styles andenvironments well Therefore new scheduling para-meters are proposed such as the road grade8 the roadsurface9 the rate of change in the throttle10 the enginetorque11 and the vehicle acceleration12 The gear-shifting schedules based on these new scheduling para-meters can be classified into two groups driver-orientedschedules and terrain-oriented schedules

Driver-oriented schedules are associated with thedriver intention which is reflected in the throttle posi-tion its rate of change or the vehicle acceleration Theaverage acceleration and the mean square error ofacceleration were used to identify the driving style andan adaptive strategy was proposed on the basis of theidentified driving style to select the best gear-shiftingschedule13 Many vehicles are now equipped with a so-called lsquoeco-buttonrsquo for the driver to select the desiredgear-shifting schedule

Terrain-oriented schedules are optimized by usingthe road information such as the road surface and theroad grade The road grade was estimated on the basisof the vehicle dynamics and a modified gear-shiftingschedule based upon the fuzzy logic was proposed8

The gear-shifting schedule varies according to the roadgrade the road friction and turning or straight drivingand a hierarchical architecture of the intelligent gear-shifting scheme was developed to optimize the gear-shifting schedule9

The increased number of scheduling parametersshould improve the adaptability of the driving behaviorand the environment However the greater the numberof scheduling parameters are considered the morecomplicated the gear-shifting schedule becomes Theincreased complexity can also increase the computa-tional load for real-time control In this paper afterinvestigating the gear-shifting factors according to thedrivetrain dynamic equations the terrain coefficient isproposed as the third scheduling parameter and it isan aggregate variable which combines the road gradeand the resistance coefficient When this aggregate vari-able is combined with the other two widely used sche-duling parameters (eg the throttle position and thevehicle speed) it provides the three-parameter gear-shifting schedule which is proposed in this paper

Unfortunately the road grade and the resistancecoefficient included in the terrain coefficient cannot beobtained in real time by the on-board sensors directlyIntensive research has been carried out to estimate theroad conditions such as the road grade using sensor-based or model-based methods1415 For the sensor-based method additional sensors are used to estimatethe road condition parameters such as the road grade

These sensors include a level sensor a accelerometerand a Global Positioning System Real-time estimationof the road conditions is also developed on the basis ofthe vehicle dynamic model and real-time data such asthe engine torque the engine speed the vehicle speedand the gear ratio The sensor-based method is in gen-eral expensive and the model-based method is notmature yet In this paper a multi-step estimationmethod using the recursive least-squares (RLS) algo-rithm is proposed to estimate the terrain coefficient inreal time

After the scheduling parameters are selected themost important work is to design the upshift scheduleand the downshift schedule There are two basic rule-making methods for gear-shifting schedule design Onemethod uses the dynamic equation of the vehicle todesign the gear-shifting schedule directly A set of equa-tions corresponding to different acceleration pedalpositions are formulated to solve the gear-shifting con-ditions and then these gear-shifting conditions are usedto generate the gear-shifting schedule For example adesign tool called Cruise GSP7 was developed to pro-vide representative gear-shifting schedules for all avail-able gearboxes This method is usually applied to one-parameter and two-parameter gear-shifting schedulesThe number of equations increases significantly as thenumber of parameters becomes larger The othermethod is based on optimal control algorithms such asdynamic programming (DP)1116 genetic algorithm17

and fuzzy logic18 These methods are capable of obtain-ing a multi-parameter gear-shifting schedule but theyhave their own limitations For example the DPmethod is a simple and efficient method for calculatingthe optimal gear sequence for a given driving cycle withknown road conditions When the driving conditionsand the road conditions change the vehicle cannot pro-vide the expected performance with the shifting sched-ule generated offline In this paper the DP method isused to optimize the gear sequences for a set of drivingcycles and then the gear-shifting schedule is generatedfrom these optimized gear-shifting points using themoving least-squares (MLS) algorithm

In the rest of this paper a three-parameter gear-shifting schedule is designed to adapt to the drivingenvironment for improved fuel economy The effects ofthe road surface and the road grade on the gear-shiftingschedule are analyzed and a new terrain coefficient isdefined to reflect these effects The three parameters(namely the throttle position the vehicle speed and theterrain coefficient) are selected to generate the gear-shifting schedule and at the same time a multi-stepRLS algorithm is proposed for estimation of the terraincoefficient Next the DP method and the MLS methodare adopted to optimize the gear sequence and to gener-ate the gear-shifting schedule Finally simulations andvehicle tests were performed to validate the proposedgear-shifting schedule in order to improve the fueleconomy

522 Proc IMechE Part D J Automobile Engineering 232(4)

Powertrain modelling with an AMT

The gear-shifting schedule is designed for stepped auto-matic transmissions such as ATs AMTs and DCTs Itshould be noted that the gear-shifting control strategyfor CVTs is not the subject of this paper In this sectiona heavy-duty powertrain with a diesel engine and anine-speed AMT is shown in Figure 1 Gear C is onlyused for hill launch of the vehicle and therefore thegear-shifting schedule investigation is focused on theremaining eight gears The AMT is made up of a dryclutch an axis-fixed gearbox electrohydraulic actua-tors a transmission control unit (TCU) and feedbacksensors The gear shift is controlled by the TCU accord-ing to the gear-shifting schedule and implemented bythe solenoid valve actuators

In Figure 1 a is the throttle position Te is the enginetorque ne is the engine speed ug is the gear-shiftingcommand Tt is the driving torque nt is the speed of thewheels and Fr is the total driving resistance

A simplified vehicle dynamic equation can beexpressed as

dm _v=Teigi0ht

R fmg cos umg sin u 1

2CdrAv2 Fb

eth1THORN

where m is the vehicle mass d is the mass factor includ-ing the equivalence coefficient of the rotating mass ig is

the gear ratio of the AMT i0 is the final gear ratio r isthe radius of the wheels f is the rolling resistance coef-ficient u is the road grade and Fb is the braking forceThe numerical values of these parameters are given inTable 1

In order to analyze the effects of the road surfaceand the road grade on the fuel economy and the otherperformances of the vehicle a powertrain model whichis shown in Figure 2 is developed in SimulinkAccording to the powertrain topology (Figure 1) andthe dynamic equation (1) the powertrain dynamicmodel inside the dashed box in Figure 2 is implementedusing the SimDriveline toolbox in Simulink The enginepower is transmitted to the driving wheels of the vehicleby the clutch and the AMT The power is interruptedwhen the clutch is disengaged or the current gear is inneutral and the power recovers when the clutch andthe synchronizer are engaged19

To complete the powertrain and vehicle simulationsa driver model a hydraulic actuator model and apowertrain control model are required A simple drivermodel20 is constructed using two PI (proportional andintegral) controllers to follow the reference speed vref ofthe driving cycle and to generate the throttle position a

and the brake signals Tb The hydraulic actuator modelin the dot-dashed box in Figure 2 is implemented usingthe SimHydraulics toolbox in Simulink where theclutch and the gearbox actuators consist of hydrauliccylinders and solenoid valves Each control valvereceives control signals from the AMT controller andthe solenoid controls the hydraulic cylinder movementthen the cylinder displacement signals xc and xg aresent to the dry clutch and gearbox models The controlmodel is the most important part for this researchwhere the schematic diagram of control is shown in thedotted box in Figure 2 The main control tasks are todetermine the target gear the gear-shifting commandug the clutch control signal (current ic) and the gear-shifting control signal (voltage Ug) Different fromother powertrain models this model has a parameterestimation module which is used to estimate the terraincoefficient b for the gear-shifting schedule Detaileddefinition and estimation of b are discussed in the fol-lowing sections

Table 1 Specifications of the AMT vehicle

Signal Description Numerical value

m Mass 19900 kgR Radius of the wheels 059 mA Frontal area 65 m2

r Density of air 1225 kgm3

Cd Drag coefficient 08ig Gear ratio [793 600 443 343 231 175 129 100]d Mass factor [163 136 120 113 106 104 102 101]i0 Final gear ratio 463ht Transmission efficiency 08

Figure 1 Powertrain topology with an AMTAMT automated manual transmission

Miao et al 523

Gear-shifting scheduling parameters

Traditional two-parameter gear-shifting schedule

The early gear-shifting schedule is dependent on oneparameter eg the vehicle speed The gear selection isbased on the pre-selected upshift and downshift speedsFor example for a car driving in second gear if thecurrent vehicle speed is above the upshift speed thetransmission upshifts to third gear automatically or ifthe current vehicle speed is below the downshift speedthe transmission downshifts to first gear To improve thefuel economy and the driveability the two-parametergear-shifting schedule is adopted and nowadays used bymost automatic transmissions The throttle position a

and the vehicle speed v (or the engine speed ne) are thetwo popular scheduling parameters The gear-shiftinglogic of the two-parameter schedule is similar to that ofthe one-parameter schedule and it is required to satisfyboth gear-shifting conditions simultaneously

To optimize the two-parameter schedule for the dri-veability one important parameter is the vehicle accel-eration _v which can be calculated from the vehicledynamic equation (1) for different throttle positions witha fixed road grade u and a fixed resistance coefficient fIn this case the acceleration curves from first gear toeighth gear with 100 throttle is illustrated in Figure3(a) The best vehicle acceleration is achieved when thegear shift occurs at the intersection of the two neighbor-ing curves and the best gear-shifting points at partialthrottle position can also be obtained in a similar wayAs a result the two-parameter gear-shifting schedule isformed by connecting the gear-shifting points for

different throttle positions and is shown in Figure 3(b)for upshift conditions

It should be noted that the schedule shown in Figure3(a) is optimal only for road conditions with given u

and f However these two parameters vary in real timebecause of the variation in the road conditionsTherefore the optimal gear-shifting schedule should bea function of u and f Figure 3(a) shows the optimalgear-shifting points with 100 throttle for road condi-tions I (ie f = 001 and u = 0) and road conditionsII (f = 0015 and u = 2) respectively It indicates thatthe terrain parameters u and f do affect the gear-shifting schedule through the acceleration

The third scheduling parameter the terraincoefficient

From equation (1) it can be seen that the terrain para-meters u and f have similar effects on the vehicledynamics and hence a terrain coefficient b is definedto represent the effects of both u and f according to

b= f cos u+ sin u eth2THORN

Furthermore the terrain coefficient affects the fueleconomy of the vehicle for the two-parameter gear-shift-ing schedule This can be explained by using a simula-tion study for the Heavy Duty Urban DynamometerDriving Schedule (HDUDDS) between 291 s and 950 s(see Figure 6 later) Three HDUDDS-based drivingcycles with different terrain coefficients (0015 0025and 0035) are constructed and gear-shifting schedulesare designed for a terrain coefficient equal to 0025

Figure 2 Simulation model architecture of a heavy-duty vehicleAMT automated manual transmission


The fuel consumptions Q for the three driving cyclesare shown in Table 2 The desired driving torquechanges as the terrain coefficient varies (see equation(1)) leading to different fuel consumptions In order toobtain a fair comparison the relative fuel consumptiondifference DQ is calculated using

DQ=QQref

Qref3100 eth3THORN

where the reference fuel consumption Qref is determinedon the basis of the schedule for the known terrain coef-ficient (eg the terrain coefficient is 0015 for drivingcycle I) and the actual fuel consumption is obtainedusing the two-parameter schedule designed for a terraincoefficient of 0025 (driving cycle II)

From Table 2 it can be seen that as expected driv-ing cycle II has the best fuel economy since the actualterrain coefficient matches that used for designing thegear-shifting schedule The fuel consumptions for theother two driving cycles I and III increase by 459and 292 respectively in comparison with the refer-ence fuel consumption (for driving cycle II) This indi-cates that the fuel economy for the two-parameterschedule becomes worse as the terrain coefficient devi-ates from the designed value Therefore it is possible toimprove the fuel economy by utilizing the terrain coef-ficient for the gear-shifting schedule With the othertwo scheduling parameters v and a a three-parametergear-shifting schedule is proposed to optimize the fueleconomy of the vehicle

Terrain coefficient estimation

For the studied heavy-duty vehicle the AMT controllercommunicates with the other powertrain controllerssuch as the engine and brake controllers through thecontroller area network (CAN) communications with theSAE J1939_20120621 standard v and a are measuredand sampled by the AMT controller through the CANHowever the terrain coefficient b cannot be used directlysince u and f cannot be measured directly using the exist-ing sensors and it also cannot be calculated from equa-tion (1) because of the unknown vehicle mass m

Therefore a multi-step adaptive estimation schemeusing the RLS method is proposed to identify both mand b based on measurable signals such as _v Te v andthe gear In this method the buffered data sets are usedfor multi-step estimation to decrease the estimationerror caused by the measurement noise The RLSmethod is used to update the estimation based on thecurrent sampled data This estimation algorithm is ableto update only b or both m and b adaptively using anadaptive mass coefficient that is designed to detect thevariation in the mass from the current data

Equation (1) can be rearranged so that m and b areseparated into two terms according to

_v=Teigi0ht

dr CdrAv

2

2d Fb

d

1

m g

db eth4THORN

In comparison with the data sample rate and thevehicle dynamics m and b can be estimated for a slowerrate than the data sampling rate Therefore the batcheddata arrays are obtained by packing n sampled datapoints together In one batched data array m and b areconsidered to be constants and equation (1) can berewritten as

Y(k)=XT(k)F(k)+ e(k) eth5THORN

where e(k) is the error vector and

Y(k)= _v(k) _v(k 1) _v(k n+1)frac12 T

X(k)=X1(k)

X2(k)

=x1(k) x1(k 1) x1(k n+1)

x2(k) x2(k 1) x2(k n+1)

Figure 3 Traditional two-parameter gear-shifting schedule(a) shifting points for different road conditions (b) twoparameter gear-shifting schedule

Table 2 Comparison of the fuel consumptions for HDUDDs

Driving cycle b Q(l)

Qref

(l)DQ()

I 0015 4264 4077 + 459II 0025 4234 4234 0III 0035 4543 4414 + 292

Miao et al 525

x1x2

=

Teigi0ht

dr CdrAv

2

2d Fb

d

g

d

264

375

F = F1 F2frac12 T

=1

mb

T

For the studied heavy-duty vehicle the data sam-pling rate is 50 Hz The size of the batched data is setto n = 10 with the resulting update period of 02 s

The cost function J used for estimation is definedaccording to the least-squares method as

J=XNi=1

lNi Y(i) XT(i)F(i) 2 eth6THORN

where the forgetting factor l(0 l 1) is used to giveless weight to the old data and more weight to the newdata

The optimal estimation that minimizes the cost func-tion (6) can be obtained by setting the associated partialderivatives to zero according to

partJ

partF1=XNi=1

lNiX1(i) Y(i) XT1 (i)F1 XT

2 (i)F2

=0

partJ

partF2=XNi=1


2 (i)F2

=0

eth7THORN

Using the method described by Yang and Zhu20

equation (7) can be reformulated as

F1(k)+G1(k) XT2 (k)F2(k)

=F1(k 1)+G1(k) Y(k) XT1 (k)F1(k 1)

G2(k) XT

1 (k)F1(k)+F2(k)

=F2(k 1)+G2(k) Y(k) XT2 (k)F2(k 1)

eth8THORN

where

Gj(k)=Pj(k 1) l+Xj(k) XTj (k)Pj(k 1)

h i1Xj(k)

and

Pj(k)=Pj(k 1) Gj(k) XTj (k)Pj(k 1) j=1 2

It is easy to solve for F1(k) and F2(k) from equation(8) In practice the vehicle mass is a piecewise continu-ous function and usually remains unchanged in thedriving conditions An approach is developed whichassumes that the mass is unchanged after it convergesto the actual value and recursive estimation is utilized

only when the change is detected An adaptive masscoefficient s is used to detect the mass convergence andvariation from the current data and is defined as

s(k)= Y(k) XT1 (k)F1 k 1eth THORN XT

2 (k)F2(k)

F

eth9THORN

where if |s(k)| scon the estimation is determinedfrom

F1(k)=F1(k 1)

F2(k)=F2(k 1)+G2(k)Y(k) XT

1 (k)F1(k) XT2 (k)F2(k 1)

eth10THORN

(ie the mass is not updated) and if |s(k)| sest and|Ds(k)| sdet the mass estimation is found usingequation (8)

It should be noted that scon is the threshold for massconvergence sest is the threshold for variation and sdet

is the threshold for the rate of variation in s(k) For thisstudy they are set to 001 002 and 004 respectivelyThe proposed estimation algorithm is implemented inthe transmission control module for real-time applica-tion The implemented estimation algorithm is illu-strated in the form of the flow chart shown in Figure 4

Figure 5 shows the estimation results using threeRLS-based methods Trace I is the actual value andtraces II III and IV are estimated by the step-by-stepRLS method with one forgetting factor the step-by-step RLS method with two forgetting factors14 and theproposed multi-step RLS method respectively Fromtrace II in Figure 5(a) large fluctuations in both theestimated m and the estimated b can be observed as theactual value changes this indicates that the two esti-mated parameters are coupled However they convergeto the actual values eventually For trace III the RLSmethod with two forgetting factors14 tracks the actualvalues with a relatively good accuracy in comparisonwith trace II and they are almost decoupled (see curveIII in Figure 5(a)) However since the estimationdepends on the actual values of the mass and the ter-rain that need to be replaced by the previous estimatedvalues this leads to additional estimation errors Theproposed multi-step RLS estimation is able to decreasethe fluctuations in the estimation (see the open ellipsesin Figure 5(b)) According to the adaptive criteria pro-posed in equation (9) the estimation can be simplifiedto a single-parameter (b) method once the mass con-verges to the actual value and reverts to a dual-parameter (m and b) method when a change in themass is detected It can be seen that small fluctuationsoccur when m changes however the estimation con-verges to the actual values quickly In addition thesefluctuations can be further eliminated if b is assumedto be a constant when the mass changes (see near the150 s in Figure 5(a))


Three-parameter gear-shifting scheduledesign

After the three scheduling parameters v a and b areobtained the next step is to design the gear-shiftingschedule as a function of these three parameters Thetraditional design method for a gear-shifting scheduleis shown in Figure 3 Its design goal is either for goodacceleration or for fuel economy and it is suitable forthe one-parameter and two-parameter gear-shiftingschedules The DP-based method performs well formulti-parameter and multi-objective optimization usinga composite cost function In this section the DPmethod is used to optimize the gear sequences over aseries of driving cycles and the MLS method is pro-posed to generate the gear-shifting schedule based onthe optimal gear sequences

Gear sequences optimized by DP

According to the vehicle dynamic equation (1) thegear-shifting schedule is the only control variable forthe powertrain system (Figure 1) over a given drivingcycle Hence the improvement in the fuel economy fora given driving cycle can be achieved by an optimalgear sequence

For a given driving cycle v(k) with k = [0 N]the DP optimization problem is to find the optimal

gear-shifting commands U= ug(1) ug(2) u

g(N 1)

n oto minimize the fuel consumption cost function Jf givenby

Jf =XN1k=0

_mf(k) Dt+ gPshift(k) Dtshift

eth11THORN

where _mf = f neTeeth THORN is the rate of fuel consumptionobtained from the engine map PshiftDtshift is the energyloss during a gear shift Pshift is the instantaneous powerat clutch separation Dtshift is the gear-shifting intervaland g is the fuel equivalent coefficient For an accelera-tion process the driving power is interrupted duringgear shifts for AMT vehicles and hence the number ofgear shifts should be minimized to maintain a goodacceleration performance

Let the state variables be x(k) = ne(k) Te(k)ge(k) where ge(k) is the transmission gear at step kThe state transition equations and constraints are givenby equation (1) and

ge(k+1)= ge(k)+ ug(k) eth12THORN

ne(k)=30igethgeethkTHORNTHORNi0v(k)

pReth13THORN

Figure 5 Estimation results using three different methods(a) mass and terrain coefficient estimation (b) estimation errorof terrain coefficient

Figure 4 Flow chart of the RLS-based estimation algorithm forreal-time implementation

Miao et al 527

ug 2 f2 1 0 1g

ge 2 f1 2 3 4 5 6 7 8g

nemin4ne4nemax

04Te4Temax

neup51300

nedn41200

eth14THORN

It should be noted that to obtain a reasonable accel-eration performance an upshift is allowed for only onegear and a downshift can be for one or two gears foreach gear shift neup is the engine speed boundary forupshift to avoid frequent gear shifts and nedn is theengine speed boundary for downshift to avoid frequentgear shifts

In this study standard driving cycles such as theHDUDDS the Urban Dynamometer Driving Schedule(UDDS) the Federal Test Procedure (FTP) the NewYork City Cycle (NYCC) and the LA92 driving cycle(based on driving in Los Angeles in 1992) are chosento represent the driving speed profiles In addition areal vehicle driving cycle of 2400 s duration is used tooptimize the gear sequences It should be noted that thedriving cycle used for three-parameter schedule optimi-zation should include the throttle position the vehiclespeed and the terrain coefficient For this study theterrain coefficient is constructed in two steps First themaximum allowed terrain coefficient for the given driv-ing cycle is obtained using the 100 throttle positionand the gear-shifting schedule shown in Figure 3 andthen a series of driving cycles derived from the givendriving cycle are formed with different terrain coeffi-cients ranging from 0 to 015 with an incremental stepof 0005 If the terrain coefficient is larger than themaximum it is replaced by the maximum allowedvalue Figure 6 shows the optimal gear sequences H-IH-II and H-III for HDUDDS-based driving cycleswith terrain coefficients equal to 0015 0045 and0095 respectively

Three-parameter gear-shifting schedule design

The optimal gear-shifting points are represented bythree parameters they are the vehicle speed v the throt-tle position a and the terrain coefficient b The optimalupshift points from fourth gear to fifth gear are shownin Figure 7 It can be seen that the scattered data pointsare distributed quite evenly and the upshift surface canbe fitted using the MLS method as shown in Figure 7The upper boundary solid line shows the maximumallowed terrain coefficients for a gear shift from fourthgear to fifth gear

The MLS method is used to obtain a smooth gear-shifting surface The MLS cost function JM is based onthe ordinary least-squares method with an addedweighting function wi(xi x) which approximates thegiven distance for a moving point x (see the paper by

Schaefer et al22) where x = [x1 x2]T = [a 10b]T JM is

given by

JM =12

XNi=1

wi xi xeth THORN FT xieth THORN a(x) u xieth THORN 2 eth15THORN

where N is the total number of experimental points xiwi (xi x) is the weighting as a function of the distancebetween xi and x which is non-negative and decreasesmonotonically u(xi) is the actual value at point xiFT(xi) a(x) is the approximate value at point xi F is abasis function vector and a(x) is the coefficient vectorwhich needs to be determined The common choices forthe basis function set are linear and quadratic mono-mials For two variables a and b they are given by

F(x)= 1 x1 x2x212

x1x2x222

T

a= a1 a2 a6frac12 Teth16THORN

According to equation (15) the approximation islocal if wi decreases quickly to zero as the distance

Figure 6 Optimal gear-shifting sequences for HDUDDS-baseddriving cycles

Figure 7 Optimal upshift points and surface for a gear shiftfrom fourth gear to fifth gear


between x and xi increases and interpolation is achievedif wi has the maximum value when the distance is 0Thus for every point x the computation of the MLSmethod involves minimizing the error between theexperimental values and the approximate values Manyweighting functions have been proposed such as aspline function a Gaussian function and an exponen-tial function In this study the spline function used is

wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN

where d = |xindashx|r and r = k max(|xindashx|) It should benoted that k affects the characteristics of the interpo-lated gear-shifting surface and a value of k between 06and 07 is recommended23

Using the MLS method the three-parameter gear-shifting schedule is obtained and shown in Figure 8 Itcan be compiled as a lookup table for software imple-mentation Figure 8 shows the schedule with terraincoefficients range from 0 to 015 which cover most roadconditions It should be noted that the two-parametergear-shifting schedule can be regard as a special case ofthe three-parameter gear-shifting schedule with a fixedterrain coefficient (see the bold curves at b = 0 inFigure 8) In practical applications if the terrain coeffi-cient is greater than 015 the gear-shifting schedule

remains the same as that at 015 Also the gear-shiftingschedule for a negative terrain coefficient is differentfrom that shown in Figure 8 A negative terrain coeffi-cient means downhill driving and upshift is not allowedat that time Therefore the schedule is just kept in thecurrent gear to maintain a safe speed with the help ofthe engine brake

Simulations and experimental validation

This section studies the improvement in the fuel econ-omy for a vehicle with an AMT using the proposedthree-parameter gear-shifting schedule First the simu-lation study is carried out for the UDDS and US06standard driving cycles for the feasible schedule andfeasible improvement in the fuel economy Second twovehicle tests are conducted to verify the fuel economyand acceleration performance of the vehicle for the pro-posed three-parameter gear-shifting schedule

Validation in the simulation environment

The AMT simulation model is shown in Figure 2 Inthis model the gear-shifting schedule is modeled by alookup table and it can be used for the two-parameterschedule by setting b = 0 and for the three-parameterschedule by using the estimated b The simulationswere carried out for the UDDS driving cycle and theUS06 driving cycle with the terrain coefficients set to0015 The simulation results are shown in Figure 9 andFigure 10 and the fuel consumptions are listed inTable 3

Figure 9 and Figure 10 show the reference speed andthe actual speed of the UDDS driving cycle and theUS06 driving cycle respectively together with the asso-ciated gear sequences obtained using the two-parametergear-shifting schedule S-I and the three-parametergear-shifting schedule S-II and the optimal DP gear-shifting sequence S-DP which is used as the global opti-mal reference From Figure 9 and Figure 10 it can beseen that gear sequence S-II is closer to the optimalsequence S-DP than is gear sequence S-I It should benoted that there are some small deviations between thetracking speed and the driving cycle due to the power-train output power (see Figure 9 and Figure 10) Thefuel consumption and the total number of gear shiftsfor the three gear-shifting sequences plotted in Figure 9and Figure 10 are shown in Table 3 and Table 4respectively

In Table 3 QI QII and QDP are the fuel consump-tions of the sequences S-I S-II and S-DP respectivelyThe fuel consumption QDP is optimized by the DP algo-rithm offline and is the optimal fuel consumption forthe given UDDS and US06 driving cycles Both DQI

and DQII are calculated using equation (3) and repre-sent the percentage increments of QI and QII respec-tively in comparison with the optimal reference QDP Itcan be seen that the fuel consumption QII for the three-parameter gear-shifting schedule is very close to the

Figure 8 Three-parameter gear-shifting schedule (a) gearupshift schedule (b) gear downshift schedule

Miao et al 529

optimal fuel consumption QDP In comparison with thetwo-parameter gear-shifting schedule the proposedschedule decreases the fuel consumption by 333 and272 for the UDDS driving cycle and the US06 driv-ing cycle respectively

Table 4 shows the number of gear upshifts for thethree sequences It can be observed that the number ofgear upshifts in the low-gear region (first gear to fourthgear) for all three sequences are very similar and thenumber of gear upshifts in the high-gear region (fifth

gear to eighth gear) of sequence S-II is greater than thatof sequence S-I but smaller than that of sequence S-DPGear shifts in the high-gear region have less influenceon the vehicle comfort and the power performance thando those in the low-gear region because of the relativelysmall change in the vehicle acceleration according toequation (1) This indicates that the proposed scheduleis able to improve the fuel economy without degradingthe comfort and the power performance

Experimental results

Experiments on the gear-shifting schedules were per-formed on a heavy-duty vehicle equipped with anAMT on a test track To make a fair comparisonexperiments using the two-parameter schedule andexperiments using the three-parameter schedule shouldbe in the same driving conditions in terms of the roadsurface the road grade the vehicle speed the vehicleparameters the driving behavior etc Different fromthe simulations or vehicle dynamometer tests the vehi-cle is controlled by the driver for the experiments car-ried out on the test track and it is very difficult tomaintain a reproducible vehicle speed for differentshifting schedules To make the test reproducible refer-ence speed signs similar to the speed limit signs onhighways were placed on the test track for the driver tofollow the reference speeds For the fuel economyexperiment one rough road (nearly 2500 m) was usedas the test route and the test track consists of two typesof road surface with several road grades and differentcurvatures Reference speed signs were placed at thestart and the end locations with a grade and curvaturechange and others were placed uniformly at 50 mintervals as shown by the open circles in Figure 11The reference speed was obtained from the actual vehi-cle data collected from the target route Two groups ofvehicle experiments were carried out by the same driverfor the two-parameter gear-shifting schedule and thethree-parameter gear-shifting schedule for the sameroute and the same speed reference respectively Sometest results are shown in Figure 11 and Figure 12

In addition according to the acceleration perfor-mance requirement of the powertrain system the accel-eration time from 0 kmh to 32 kmh (T0ndash32) and theacceleration time from 0 kmh to 60 kmh (T0ndash60) onthe cement road should be less than 12 s and 26 srespectively Therefore another experiment wasdesigned to verify the acceleration performance In thisexperiment the vehicle was launched at the higheststarting gear (third gear) with a fully pressed accelera-tion pedal and continued to travel with a fully pressedacceleration pedal until the vehicle speed reaches 65kmh Then the vehicle was stopped and the test wasrepeated five times The experimental results are shownin Figure 13

Figure 11 shows the speed profiles obtained fromthe vehicle tests for the two-parameter schedule and thethree-parameter schedule The simulated speed profile

Figure 10 Simulation results for the US06 driving cycle

Figure 9 Simulation results for the UDDS driving cycle

Table 3 Comparison of the fuel consumptions

Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006

UDDS Urban Dynamometer Driving Schedule US06 Supplemental

Federal Test Procedure


in both graphs uses the same target speed profile Onthis test route 49 reference speed signs were placedand the distance between the signs ranges from 34 m to69 m From Figure 11 it can be seen that for both vehi-cle tests the vehicle tracks the target speed profile wellwith the help of the speed reference signs Figure 12shows the gear sequences of the two-parameter sched-ule and the three-parameter schedule from the vehicletests and the simulations It can be seen that the gearsof the proposed three-parameter gear-shifting scheduleare higher than those of the two-parameter gear-shift-ing schedule According to the vehicle dynamics equa-tion (1) the higher gear corresponds to a lower enginespeed and a higher engine torque As a result theengine is operated in the high-efficiency domain whenthe vehicle speed is low (v 40 kmh for this vehicle)and it is the opposite if the vehicle speed is high (v

80 kmh for this vehicle) Furthermore the three-parameter gear-shifting schedule has a smaller numberof gear shifts than does the two-parameter gear-shiftingschedule Figure 13 also shows that the proposed sched-ule operates the vehicle in higher gears than the two-parameter gear-shifting schedule does in general thiscan reduce the acceleration performance (Table 5) Itshould be noted that the observed vehicle deceleratesbecause of the AMT power interruption during gearshifts

Table 5 compares the improvements in the fuel econ-omy and the acceleration duration of the two vehicleexperiments Qtest is the fuel consumption in the vehicletest on the rough road Qsim is the simulated fuel con-sumption based on the reference speed profile T0ndash32 isthe average acceleration durations of the vehicle from 0kmh to 32 kmh and T0ndash60 is the average accelerationdurations of the vehicle from 0 kmh to 60 kmhrespectively It can be seen that both acceleration dura-tions of the two-parameter gear-shifting schedule areshorter than those of the proposed schedule but theyall meet the design requirements for the accelerationperformance In particular the fuel consumptions ofthe proposed schedule are decreased by 353 and357 respectively in comparison with those for thetwo-parameter gear-shifting schedule in both the vehi-cle experiments and the simulations This confirms thatthe proposed three-parameter gear-shifting scheduleimproves the fuel economy with a satisfactory accelera-tion performance

Conclusions

In this paper two gear-shifting schedules for vehiclesequipped with stepped automatic transmissions werestudied They are the two-parameter gear-shiftingschedule and the newly proposed three-parameter gear-

Figure 11 Vehicle speed with the two-parameter scheduleand the three-parameter scheduleSim simulations

Figure 12 Comparison of the gear sequences for the vehicletests and the simulations

Table 4 Comparison of the numbers of gear shifts

Driving cycle Gear shift Sequence S-I Sequence S-II Sequence S-DP

UDDS First gear to fourth gear 72 72 72Fifth gear to eight gear 24 44 47

US06 First gear to fourth gear 32 31 31Fifth gear to eight gear 10 15 18

UDDS Urban Dynamometer Driving Schedule US06 Supplemental Federal Test Procedure

Miao et al 531

shifting schedule The terrain coefficient is defined andestimated in real time by the proposed multi-step recur-sive least-squares algorithm and it is used as an addi-tional scheduling parameter to improve the fueleconomy of the vehicle The estimation algorithm forthe proposed terrain coefficient also has potentialapplications to longitudinal control cruise control andanti-lock brake control of the vehicle The proposedschedule was validated using simulation studies for theUDDS driving cycle and the US06 driving cycle andusing vehicle experiments for the fuel economy and theacceleration performance The simulations and theexperimental results show that the proposed scheduleimproves the fuel economy by about 35 with a satis-factory acceleration performance Currently the three-parameter gear-shifting schedule map is optimized off-line on the basis of the given driving cycles and futurework will focus on online optimization of the three-parameter gear-shifting schedule

Acknowledgements

The authors would like to thank Yinong Zhao Lei Jinand Shijin Dong from the Beijing Institute ofTechnology for their help with performing the experi-ments The authors would also like to thank the ChinaScholarship Council for supporting ChengshengMiaorsquos visit to Michigan State University as a visitingscholar

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This work was partially supportedby the National Natural Science Foundation of China(grant number E050201)

References

1 Kim D Peng H Bai S et al Control of integrated power-

train with electronic throttle and automatic transmissionIEEE Trans Control Systems Technol 2007 15(3) 474ndash482

2 Eckert JJ Santiciolli FM Costa ES et al Vehicle gearshifting co-simulation to optimize performance and fuelconsumption in the Brazilian standard urban drivingcycle Blucher Engng Proc 2014 1(2) 615ndash631

3 Liu Y Qin D Jiang H et al Shift schedule optimizationfor dual clutch transmissions In IEEE vehicle power and

propulsion conference Dearborn Michigan USA 7ndash10September 2009 paper 0940038 pp 1071ndash1078 NewYork IEEE

4 Vagg C Brace CJ Wijetunge R et al Development of anew method to assess fuel saving using gear shift indica-tors Proc IMechE Part D J Automobile Engineering

2012 226(12) 1630ndash16395 Wang Y Zhu G and Zhang F Powertrain control para-

meter optimization using HIL simulations of a heavy-

duty vehicle Int J Powertrains 2013 2(1) 1ndash256 Kulkarni M Shim T and Zhang Y Shift dynamics and

control of dual-clutch transmissions Mechanism Mach

Theory 2007 42(2) 168ndash1827 Le Guen D Weck T Balihe A et al Definition of gear-

shift pattern Innovative optimization procedures usingsystem simulation SAE paper 2011-01-0395 2011

8 Wu G and Zhang D Shifting rule modification strategyof automatic transmission based on driverndashvehiclendashroad

environment Chin J Mech Engng Engl Edn 2010 23(3)346ndash352

9 Hui J and Anlin G A new vehicle intelligent shift systemAdv Mech Engng 2013 5 1ndash9

10 Mashadi B Kazemkhani A and Lakeh RB An auto-matic gear-shifting strategy for manual transmissionsProc IMechE Part I J Systems and Control Engineering

2007 221(5) 757ndash76811 Ngo VD Hofman T Steinbuch M et al Gear shift map

design methodology for automotive transmissions Proc

IMechE Part D J Automobile Engineering 2014 228(1)50ndash72

12 Xiang Y Guo L Gao B et al A study on gear shiftingschedule for 2-speed electric vehicle using dynamic pro-gramming In 2013 25th Chinese control and decision con-

ference Guiyang Peoplersquos Republic of China 25ndash27 May2013 paper 13647203 pp 3805ndash3809 New York IEEE

13 Wu S Zhu E Li Q et al Study on intelligent shift control

strategy of automobile based on genetic-fuzzy algorithmIn 3rd international conference on innovative computing

Figure 13 Experimental results for acceleration

Table 5 Comparison of the fuel consumptions and theaccelerations

Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)

Two-parameter 2633 2631 928 2314Three-parameter 2540 2537 1136 2578Comparison ndash353 ndash357 + 212 + 264


information and control Dalian Peoplersquos Republic ofChina 18ndash20 June 2008 paper 10184837 pp 402ndash402New York IEEE

14 Vahidi A Stefanopoulou A and Peng H Recursive leastsquares with forgetting for online estimation of vehiclemass and road grade theory and experiments Veh Sys-

tem Dynamics 2005 43(1) 31ndash5515 Mahyuddin MN Na J Herrmann G et al Adaptive

observer-based parameter estimation with application toroad gradient and vehicle mass estimation IEEE Trans

Ind Electron 2014 61(6) 2851ndash286316 Skugor B Deur J and Ivanovic V Dynamic

programming-based design of shift scheduling map tak-ing into account clutch energy losses during shift transi-ents SAE paper 2016-01-1116 2016

17 Fofana A Haas O Ersanilli V et al Multi-objectivegenetic algorithm for an automatic transmission gear

shift map IFAC-Papers Online 2016 49(3) 123ndash12818 Kong H Study on AMT fuzzy shifting strategy and reali-

zation In IEEE international conference on automation

and logistics Qingdao Peoplersquos Republic of China 2008pp 1870ndash1875 New York IEEE

19 Song XY Sun ZX Yang XJ et al Modelling controland hardware-in-the-loop simulation of an automatedmanual transmission Proc IMechE Part D J Automobile

Engineering 2010 224(2) 143ndash16020 Yang J and Zhu GG Adaptive recursive prediction of the

desired torque of a hybrid powertrain IEEE Trans Veh

Technol 2015 64(8) 3402ndash341321 SAE J1939_201206 Serial control and communications

heavy duty vehicle network ndash top level document Warren-dale SAE International 2012

22 Schaefer S McPhail T and Warren J Image deformationusing moving least squares ACM Trans Graphics 200625(3) 533ndash540

23 Nogueira X Cueto-Felgueroso L Colominas I et al Anew shock-capturing technique based on moving least

squares for higher-order numerical schemes on unstruc-tured grids Comput Meth Appl Mech Engng 2010199(37) 2544ndash2558

Miao et al 533

shifting schedule can also be affected by the road condi-tions For example the upshift should occur at higherengine speed when driving uphill with a large roadgrade than when driving on a road with no grade andshould be limited when driving downhill because ofvehicle safety Unfortunately the two-parameter sched-ule cannot adapt to these different driving styles andenvironments well Therefore new scheduling para-meters are proposed such as the road grade8 the roadsurface9 the rate of change in the throttle10 the enginetorque11 and the vehicle acceleration12 The gear-shifting schedules based on these new scheduling para-meters can be classified into two groups driver-orientedschedules and terrain-oriented schedules

Driver-oriented schedules are associated with thedriver intention which is reflected in the throttle posi-tion its rate of change or the vehicle acceleration Theaverage acceleration and the mean square error ofacceleration were used to identify the driving style andan adaptive strategy was proposed on the basis of theidentified driving style to select the best gear-shiftingschedule13 Many vehicles are now equipped with a so-called lsquoeco-buttonrsquo for the driver to select the desiredgear-shifting schedule

Terrain-oriented schedules are optimized by usingthe road information such as the road surface and theroad grade The road grade was estimated on the basisof the vehicle dynamics and a modified gear-shiftingschedule based upon the fuzzy logic was proposed8

The gear-shifting schedule varies according to the roadgrade the road friction and turning or straight drivingand a hierarchical architecture of the intelligent gear-shifting scheme was developed to optimize the gear-shifting schedule9

The increased number of scheduling parametersshould improve the adaptability of the driving behaviorand the environment However the greater the numberof scheduling parameters are considered the morecomplicated the gear-shifting schedule becomes Theincreased complexity can also increase the computa-tional load for real-time control In this paper afterinvestigating the gear-shifting factors according to thedrivetrain dynamic equations the terrain coefficient isproposed as the third scheduling parameter and it isan aggregate variable which combines the road gradeand the resistance coefficient When this aggregate vari-able is combined with the other two widely used sche-duling parameters (eg the throttle position and thevehicle speed) it provides the three-parameter gear-shifting schedule which is proposed in this paper

Unfortunately the road grade and the resistancecoefficient included in the terrain coefficient cannot beobtained in real time by the on-board sensors directlyIntensive research has been carried out to estimate theroad conditions such as the road grade using sensor-based or model-based methods1415 For the sensor-based method additional sensors are used to estimatethe road condition parameters such as the road grade

These sensors include a level sensor a accelerometerand a Global Positioning System Real-time estimationof the road conditions is also developed on the basis ofthe vehicle dynamic model and real-time data such asthe engine torque the engine speed the vehicle speedand the gear ratio The sensor-based method is in gen-eral expensive and the model-based method is notmature yet In this paper a multi-step estimationmethod using the recursive least-squares (RLS) algo-rithm is proposed to estimate the terrain coefficient inreal time

After the scheduling parameters are selected themost important work is to design the upshift scheduleand the downshift schedule There are two basic rule-making methods for gear-shifting schedule design Onemethod uses the dynamic equation of the vehicle todesign the gear-shifting schedule directly A set of equa-tions corresponding to different acceleration pedalpositions are formulated to solve the gear-shifting con-ditions and then these gear-shifting conditions are usedto generate the gear-shifting schedule For example adesign tool called Cruise GSP7 was developed to pro-vide representative gear-shifting schedules for all avail-able gearboxes This method is usually applied to one-parameter and two-parameter gear-shifting schedulesThe number of equations increases significantly as thenumber of parameters becomes larger The othermethod is based on optimal control algorithms such asdynamic programming (DP)1116 genetic algorithm17

and fuzzy logic18 These methods are capable of obtain-ing a multi-parameter gear-shifting schedule but theyhave their own limitations For example the DPmethod is a simple and efficient method for calculatingthe optimal gear sequence for a given driving cycle withknown road conditions When the driving conditionsand the road conditions change the vehicle cannot pro-vide the expected performance with the shifting sched-ule generated offline In this paper the DP method isused to optimize the gear sequences for a set of drivingcycles and then the gear-shifting schedule is generatedfrom these optimized gear-shifting points using themoving least-squares (MLS) algorithm

In the rest of this paper a three-parameter gear-shifting schedule is designed to adapt to the drivingenvironment for improved fuel economy The effects ofthe road surface and the road grade on the gear-shiftingschedule are analyzed and a new terrain coefficient isdefined to reflect these effects The three parameters(namely the throttle position the vehicle speed and theterrain coefficient) are selected to generate the gear-shifting schedule and at the same time a multi-stepRLS algorithm is proposed for estimation of the terraincoefficient Next the DP method and the MLS methodare adopted to optimize the gear sequence and to gener-ate the gear-shifting schedule Finally simulations andvehicle tests were performed to validate the proposedgear-shifting schedule in order to improve the fueleconomy






dm _v=Teigi0ht


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Miao et al 523
















DQ=QQref

Qref3100 eth3THORN







_v=Teigi0ht

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1

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db eth4THORN




Y(k)= _v(k) _v(k 1) _v(k n+1)frac12 T

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X2(k)

=x1(k) x1(k 1) x1(k n+1)

x2(k) x2(k 1) x2(k n+1)




Qref

(l)DQ()

I 0015 4264 4077 + 459II 0025 4234 4234 0III 0035 4543 4414 + 292

Miao et al 525

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=1

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G2(k) XT

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eth8THORN

where


h i1Xj(k)

and





2 (k)F2(k)

F

eth9THORN


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eth10THORN












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eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533





dm _v=Teigi0ht


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Y(k)= _v(k) _v(k 1) _v(k n+1)frac12 T

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x2(k) x2(k 1) x2(k n+1)




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I 0015 4264 4077 + 459II 0025 4234 4234 0III 0035 4543 4414 + 292

Miao et al 525

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F(x)= 1 x1 x2x212

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T







wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533
















DQ=QQref

Qref3100 eth3THORN







_v=Teigi0ht

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1

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Y(k)= _v(k) _v(k 1) _v(k n+1)frac12 T

X(k)=X1(k)

X2(k)

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eth8THORN

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T







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eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533


DQ=QQref

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d

1

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db eth4THORN




Y(k)= _v(k) _v(k 1) _v(k n+1)frac12 T

X(k)=X1(k)

X2(k)

=x1(k) x1(k 1) x1(k n+1)

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Miao et al 525

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d

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=F1(k 1)+G1(k) Y(k) XT1 (k)F1(k 1)

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eth8THORN

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and





2 (k)F2(k)

F

eth9THORN


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1 (k)F1(k) XT2 (k)F2(k 1)

eth10THORN












g(N 1)


Jf =XN1k=0


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pReth13THORN



Miao et al 527

ug 2 f2 1 0 1g

ge 2 f1 2 3 4 5 6 7 8g

nemin4ne4nemax

04Te4Temax

neup51300

nedn41200

eth14THORN







given by

JM =12

XNi=1



F(x)= 1 x1 x2x212

x1x2x222

T







wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533

x1x2

=

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2

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d

g

d

264

375

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J=XNi=1




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2 (k)F2(k)

F

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1 (k)F1(k) XT2 (k)F2(k 1)

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g(N 1)


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ug 2 f2 1 0 1g

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given by

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XNi=1



F(x)= 1 x1 x2x212

x1x2x222

T







wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533







g(N 1)


Jf =XN1k=0


eth11THORN





pReth13THORN



Miao et al 527

ug 2 f2 1 0 1g

ge 2 f1 2 3 4 5 6 7 8g

nemin4ne4nemax

04Te4Temax

neup51300

nedn41200

eth14THORN







given by

JM =12

XNi=1



F(x)= 1 x1 x2x212

x1x2x222

T







wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533

ug 2 f2 1 0 1g

ge 2 f1 2 3 4 5 6 7 8g

nemin4ne4nemax

04Te4Temax

neup51300

nedn41200

eth14THORN







given by

JM =12

XNi=1



F(x)= 1 x1 x2x212

x1x2x222

T







wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533


wi(d)=1 3d2 +2d3 04d 10 d51

eth17THORN












Miao et al 529











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533











Driving cycle Q I

(l)Q II

(l)Q DP

(l)DQ I

()DQII

()

UDDS 5116 4939 4936 + 365 + 006US06 3515 3419 3417 + 287 + 006







Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533




Conclusions









Miao et al 531


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533


Acknowledgements




Funding


References


























Schedule Qtest

(l)Qsim

(l)T0ndash32

(s)T0ndash60

(s)





















Miao et al 533



















Miao et al 533

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Three-parameter transmission gear-shifting schedule for ... · Traditionally the transmission gear-shifting schedule is based upon the throttle position and the vehicle (or engine)