Research ArticleOptimal Clutch Pressure Control in Shifting Process of AutomaticTransmission for Heavy-Duty Mining Trucks
Heng Zhang 1 Xinxin Zhao1 and Jianning Sun12
1School of Mechanical Engineering University of Science and Technology Beijing Beijing 100083 China2China Intelligent and Connected Vehicles (Beijing) Research Institute Co Ltd Beijing 100176 China
Correspondence should be addressed to Heng Zhang zhanghengustbhotmailcom
Received 15 May 2020 Revised 6 August 2020 Accepted 18 September 2020 Published 12 October 2020
Academic Editor Michele Guida
Copyright copy 2020 Heng Zhang et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
)e optimal control of automatic transmission plays an important role in the shifting smoothness and fuel economy of heavy-dutymining trucks In this paper a dynamic model of the powertrain system is built to study the clutch pressure control during theshifting process A linear-quadratic optimal regulator is used to achieve the optimum control pressure of clutches where shiftingjerk and clutch friction loss are chosen to a form quadratic performance index function Besides a detailed solution of the linear-quadratic problem with the disturbance matrix in the state equations is provided)is paper also carries out a software simulationand verification of the normal condition (no load without slope) and the extreme condition (full load with maximum slope)Compared with the preset reference trajectory control the simulation results show that the proposed optimal clutch pressurecontrol can effectively reduce jerk and friction loss during the shifting process and has good robustness to differentoperating conditions
1 Introduction
Transmission is the key component of a vehicle thattransmits different torque ratios at different speeds [1] Inorder to make heavy-duty mining trucks face the problemsof bad working conditions poor fuel economy and highlabor intensity of drivers AT is widely used in this field withthe characteristics of strong transmission capability smoothshifting process and high reliability [2 3] In addition underthe condition of increasingly mature lockup clutch tech-nology the efficiency of AT has also been significantlyimproved [4]
In order to avoid power interruption shift overlappingand overheating of the clutch plate caused by incoordina-tion between on-coming and off-going clutches some re-searchers have studied the clutch-to-clutch shifting processand divided it into four parts (preshift phase torque phaseinertia phase and postshift phase) [5ndash7] Meng et al [8]analyzed the ATrsquos electrohydraulic control system andproposed a two-degree-of-freedom PID control method tooptimize the duty ratio of proportional solenoid valves for
the speed difference of clutch discs following a preset tra-jectory Song and Sun [9] focused on the nonlinear dynamiccharacteristics of the wet clutch and designed a sliding modecontroller to control the on-coming clutch pressure to en-sure a smooth clutch-to-clutch shifting in the inertia phaseTo track and control the relative speed of the clutcheliminate the uncertainty of system parameters and improvethe robustness with respect to changing driving conditionsrobust controllers postfeedback controllers and distur-bance compensators were designed by Sanada et al [10]Zhao and Li [11] used the subspace identification method todescribe the prediction equation of the AT powertrainsystem which solved the difficulty of modeling an accurateclutch-to-clutch shifting process and then applied the modelprediction controller to improve the shift quality in hard-ware-in-the-loop tests Considering power comfortabilityand robustness during the clutch engagement Wurm andBestle [12] adopted a multiobjective genetic algorithm toachieve their balance and finally obtain Pareto optimality Insummary since the torque signal is hard to be measured by asensor in the torque phase current studies mostly track a
HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8618759 9 pageshttpsdoiorg10115520208618759
preset reference trajectory of the on-coming clutchrsquos relativespeed for a smooth shift process in the inertial phaseHowever under a wide variation of working conditions forheavy-duty mining trucks the preset reference trajectorycannot be determined as the optimal solution of shifting jerkand clutch friction loss which are the most important shiftperformance indicators
)e linear-quadratic optimal regulator (LQR) has beenused in the field of vehicle powertrain system control suc-cessfully To reduce the torsional vibration of electric vehicletransmission systems the method of an infinite-time LQRwas proposed by Lin et al [13] to adjust the output torque ofthe drive motor dynamically Gao et al [14] used an offlinefinite-time LQR to track the clutch relative speed during thegear shift of an electric vehicle equipped with AMT For thetorque phase and the inertia phase of the DCT upshiftprocess Li and Gorges [15] put forward control methodsbased on LQR and integral LQR respectively to reduce thejerk of shifting to make the vehicle shift more smoothlyHowever friction loss of clutch as another importantperformance index is not considered
)erefore this paper proposes a linear-quadratic opti-mization-based clutch pressure trajectory for the shiftingprocess of heavy-duty mining trucks equipped with AT toreduce the jerk of shifting and friction loss of clutch si-multaneously In Section 2 the dynamic modeling ofpowertrain systems is carried out and the ATshifting processis analyzed followed by the design of the LQR controller forthe trajectory of clutch pressure in the inertial phase (Section3) )en Section 4 presents the optimized results for twotypical working conditions of heavy-duty mining trucks anda comparison with the preset reference trajectory Con-cluding remarks are finally given in Section 5
2 Modeling for Powertrain Systems of Heavy-Duty Mining Trucks
Equipped with automatic transmission the powertrainsystem of heavy-duty mining trucks can be divided intoinput power module (composed of a diesel engine and atorque converter) wet clutch pressure control moduleplanetary gear set module and output power module (driveaxle and vehicle longitudinal dynamic model) [16ndash18] asshown in Figure 1
21 e Transmission Input Power Module )e coworkingoutput characteristics of the diesel engine and the torqueconverter (TC) determine the input power of the trans-mission )e output torque of the diesel engine TE is relatedto its rotational speed ωE and throttle opening θ and acts onthe converter pump as follows
TE ωE θ( 1113857 TP + JP _ωP (1)
where TP is the pump torque JP is the equivalent inertia ofthe turbine and ωP is the pump speed
)e dimensionless characteristics of the torque converter(including the pump torque coefficient λP TC torque ratio
KTC speed ratio iTC and efficiency ηTC) are used to describetheir dynamic equation [19] as follows
TP λPρTCgD5TCω
2P
TT KTCTP
ωT iTCωP
PT TTωT ηTCPP KTCiTCTPωP
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(2)
where ρTC is the fluid density of the torque converter DTC isthe effective diameter of the torque converter TT is theturbine torque ωT is the turbine speed and PT and PP arethe power of the turbine and pump
22 Wet Clutch Pressure Control Module Using Stribeckfriction models (shown in Figure 2) the clutch frictionworking mode switches among ldquoin motion ①rdquo ldquocapturedand accelerating②rdquo and ldquocaptured and static③rdquo [20 21] asshown in Figure 3
(1) When the relative speed of the clutch |ω| is greaterthan a threshold ωtol the clutch is the ldquoin motionrdquomode and the friction torque Tf is the same as thedynamic torque Td which is given as follows
Tf Td sign(Δω)f(Δω)SRNp
sign(Δω)f(Δω)kdp(3)
where f(Δω) 006 + 004e(minus 036Δω) is the frictionfactor for the copper-based surface and S R N andp are equivalent area equivalent radius number offriction pairs and clutch pressure respectivelyConstants can be summarized as a pressure pro-portional coefficient kd SRN
(2) When the clutch relative speed |ω| is less than thethreshold ωtol and the required torque Ttp isgreater than the dynamic torque Td the clutch is inthe ldquocaptured and acceleratingrdquo mode and thefriction torque Tf is the same as the dynamic torqueTd
(3) When the clutch relative speed |ω| is less than thethreshold ωtol and the required torque Ttp is lessthan the dynamic torque Td the clutch is in theldquocaptured and staticrdquo mode and the friction torqueTf is the same as the required torque Ttp
23 PlanetaryGear SetModule In this study 1st gear to 2ndgear upshift is chosen as the example of an AT shiftingprocess for heavy-duty mining trucks In the first gear theclutch CS is engaged to connect the sun gear S1 to the carrierC1 of the first planetary gear set P1 at the same speed )etorque is input from the turbine and output from the ring R1of P1 When the transmission control unit (TCU) issues anupshift command CS is disengaged by releasing pressureand the on-coming brake BS is engaged gradually It finallyfixes S1 to the transmission housing to stop its movementand enters the second gear state With brake BL keepingengaged during the first and second gear the ring R3 of the
2 Mathematical Problems in Engineering
3rd planetary gear set P3 is fixed on the transmissionhousing while the torque is transferred from the carrier C3to the transmission output shaft At all time the planetaryrow P3 can be regarded as a reducer with the speed ratio ip3
By analyzing the transmission input shaft (torqueconverter turbine shaft) the sun shaft and ring shaft of P1and the transmission output shaft the dynamic equations ofthe gearbox can be written as follows
TT + TCS minus TC1 JT _ωT
TS1 minus TCS minus TBS JS1 _ωBS
TR1 kp1TS1
TR1 minus TS3 JR1 _ωR1
TO ip3TS3
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where TCS and TBS are the torques of CS and BS JS1 JR1 andJT are the inertias of the sun and ring of P1 and thetransmission input shaft TS1 TC1 and TR1 are the torque ofsun carrier and ring of P1 respectively ωBS is the speed ofBS kp1 is the gear ratio ofR1 to S1TS3 is the torque of S3 andTO is the torque of the transmission output shaft
24 e Transmission Output Power Module )e torqueoutput TO by the AT is finally transmitted to the wheelsthrough the main reducer (speed ratio iFD) For modelingconvenience the longitudinal vehicle model is simplifiedwithout considering the vehiclersquos pitch yaw and othermotion directions resulting in
TO Jw _ωw + TV
iFD (5)
where ωw is the wheel speed rw is the wheel radius and Jw isthe equivalent inertia of wheels When the heavy-dutymining truck is under the nonbraking condition the lon-gitudinal resistance torque TV can be summarized fromrolling air and climbing resistance as follows
TV rw Mgfroll +CDSV
2115v2V + Mg sin α1113874 1113875 (6)
where rw is the radius of the wheel M is the full load massfroll is the rolling resistance coefficient CD is the air re-sistance coefficient SV is the windward area vV is the vehiclespeed and α is the road slope respectively
Out
Clutch
Planetary gearIn
R1
S1
P1
R3
S3
CS
BS BL
C1 P3 C3
Engine
Torque converter
Main reducer
TT
JT
JS1
TC1TCS
TCS TS1 TBS
TR1
JR1
TS3
TE TP
JP
Wheel
TViFDTO
JwTO
Figure 1 Simplified dynamic model of the powertrain consisting of engine torque converter two planetary gear sets (P1 and P3) clutches(CS BS and BL) and vehicle
Tf
Static
Dynamic
Coulomb friction Viscous friction
Static frictionStribeck friction
ω
Figure 2 Stribeck friction model
In motion
Captured and accelerating
Captured and static
Ttp gt Td
Ttp gt Td
Ttp le Td
Ttp le TdTf = Ttp
Tf = Td (pΔω)
Tf = sign (Ttp)|Td|
|Δω| gt Δωtol |Δω| gt Δωtol
Figure 3 Clutch friction working modes and switching accordingto different transmitted torques and relative speeds of clutches
Mathematical Problems in Engineering 3
3 LQR Controller Design
)e whole shifting process of the heavy-duty mining truckshift can be divided into four stages which are rapiddraining oil (preshift phase) torque phase inertia phase andrapid boosting oil (postshift phase) as shown in Figure 4)epreshift phase (t1 sim t2) and postshift phase (t4 sim t5) aredesigned for eliminating the gap between the clutch platesand hoisting transmission capacity respectively In thetorque phase the 1st gear speed ratio is still maintained andtorque is transferred from CS to BS At the end of the torquephase if TCS(t3)gt 0 this forward torque will cause shiftshock otherwise if TCS(t3)lt 0 this negative torque willcause power loss )erefore in order to make TCS equal tozero BS should be able to undertake the transmission ofoutput torque independently
TBS t3( 1113857 1
kp1TT (7)
Open-loop control is adopted in preshift postshift andtorque phases because there is no change in the clutch speed[22ndash24] However torque and speed change drastically lastlonger and will produce greater shift jerk and clutch frictionloss in the inertial phase which is why the method based onthe LQR is used for this key phase in this paper
31 Establishment of State-Space Model )e turbine speedωT clutch speed ωBS and clutch pressure pBS of on-comingbrake BS are selected as the state variables of the shiftingprocess model
x ωT ωBS pBS1113858 1113859T (8)
)e change rate of pB is selected as the control variablebecause it is related to the shift jerk and friction loss of theclutch
u dpBS
dt (9)
From the gearbox dynamic equation the clutch pressurecontrol equation and transmission power input and outputequations (1)ndash(6) the powertrain dynamic model can bederived as follows
_ωT D22KTC
D24ωT +
D12D23
D24pBS
+D12D14 minus D13D22( 1113857TL minus D22TT
D24
_ωBS D12KTC
D24ωT +
minus D11D23
D24pBS
+D11D14 + D13D12( 1113857TL + D12TT
D24
_pBS dpBS
dt
(10)
where D11 Jw(1+ kp1)2k2p1i2p3 + JT D12 Jw1+ kp1k2
p1i2p3
D13 minus 1+ kp1k2p1ip3 D14 minus 1kp1ip3 D21 minus D12 D22
minus (Jw1k2p1i
2p3 + JS1) D23 f(ΔωBS)kd and D24 minus D2
12 minus
D11D22)e equation of the state-space model in the inertia
phase can be written as follows
_x Ax + Bu + Γ (11)
where A
D22KTCD24 0 D12D23D24D12KTCD24 0 minus D11D23D24
0 0 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ B
001
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ and
Γ
(D12D14 minus D13D22)TL minus D22TTD24(D11D14 + D13D12)TL + D12TTD24
0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
32ProblemStateand theQuadraticCostFunctionDefinition)e shifting process is not completed instantaneously and ashifting shock is inevitable To achieve a smooth shift with alow dynamic load of the system the vehicle jerk which is thederivative of acceleration wrt time should be undercontrol During the 1-2 upshift process it can be derivedfrom equations (4) and (5) as follows
j da
dt rw
d _ωw
dt
rwiFDdTO
Jwdt
rwiFD
Jw
kp1ip3d f ΔωBS( 1113857kdpBS( 1113857
dt c1u
(12)
where c1 rwiFDJwkp1ip3f(ΔωBS)kdAlthough increasing the time of the shift process can
reduce the vehicle jerk this method is not advisable becauselong-term slipping of the clutch generates friction heat andwear and damages the clutch plates finally In reference toequation (3) clutch friction loss for BS is modeled as follows
Preshift
1stgear
Torquephase
t0 t1 t2 t3 t4 t5 t6
2ndgear
Postshift
Inertiaphase
0
0
TO
TT
ndashTdBS
ndashTBS
TCS
ωT
ωBS
ωCS
pCS
pBS
Figure 4 Shifting process with four phases
4 Mathematical Problems in Engineering
W 1113946t
0TfBSωBS
11138681113868111386811138681113868
11138681113868111386811138681113868dt 1113946t4
t3
f ωBS( 1113857kdpBSωBS1113868111386811138681113868
where c2 f(ΔωBS)kd and t3 and t4 represent the start andterminal time of the inertia phase respectively
As can be seen from equations (12) and (13) the jerk forthe shift in the inertial phase is directly proportional to thecontrol variable whereas clutch friction loss for BS isproportional to the product of its speed ωBS x2 andpressure pBS x3 )erefore the finite-time linear-qua-dratic optimization performance index for the shifting in-ertia phase is shown as follows
J 12
1113946t4
t3
pBSωBS + ru2
1113872 1113873dt 12
1113946t4
t3
x2x3 + ru2
1113872 1113873dt
12
1113946t4
t3
xTQx + ru
21113872 1113873dt
(14)
where Q
0 0 00 0 050 05 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
)e first term in the integral function represents theclutch friction loss and the second represents the jerk Byadjusting the comprehensive weight coefficient r (weightcoefficient multiplied by a normalized coefficient) the LQRcontroller can achieve a proper balance between them
33 e Solution of Clutch Pressure Trajectory Based on LQRAccording to equations (11) and (14) we introduce theHamiltonian function [25] as follows
H 12
xTQx + ru
21113872 1113873 + λ(t)
T(Ax + Bu + Γ) (15)
It is assumed that the shifting proportional solenoidvalve of automatic transmission used in this study has arapid response [26 27] and there is no limit to the controlvariable u According to the maximum principle we obtain
zH
zu ru(t) + B
Tλ(t) 0 (16)
resulting in the optimal control trajectory
ulowast(t) minus
BTλ(t)
r (17)
)e normalized equations are as follows
_x zH
zλ Ax + Bu + Γ (18)
_λ(t) minuszH
zx minus Qx minus A
Tλ(t) (19)
By substituting equation (17) into (18) we obtain
_x(t) Ax(t) minusBB
Tλ(t)
r+ Γ (20)
At the end of the inertial phase clutch BS speed is zero sothat terminal constraint function
where μ is the undetermined Lagrange multiplier Due to theinterference matrix Γ the solution of the conventional LQRwhich is letting λ(t) P(t)x(t) cannot be used Here weset
λ(t) P(t)x(t) + M(t)μ + h(t) (23)
gT K(t)x(t) + L(t)μ + η(t) (24)
According to equations (22) (23) and (24) we obtain
P t4( 1113857
0 0 0
0 0 0
0 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M t4( 1113857
0
1
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ h t4( 1113857
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
K t4( 1113857 0 1 01113858 1113859 L t4( 1113857 0 η t4( 1113857 0
(25)
Substitution of equation (23) into (19) yields_λ(t) minus Q minus A
TP(t)1113960 1113961x(t) minus A
TM(t)μ minus A
Th(t) (26)
Substituting equations (20) and (23) into the derivativeof equation (23) we obtain affirmatively
_λ(t) _P(t)x(t) + P(t) _x(t) + _M(t)μ + _h(t)
_P(t) + P(t)A minusP(t)BBT
P(t)
r1113890 1113891x(t)
+ _M(t) minusP(t)BB
TM(t)
r1113890 1113891μ
minusP(t)BBT
rh(t) + P(t)Γ + _h(t)
(27)
Comparing equations (26) and (27) wrt x(t) μ andrest we obtain
_P(t) minus P(t)A minus ATP(t) +
P(t)BBTP(t)
rminus Q (28)
_M(t) P(t)BBT
rminus A
T1113890 1113891M(t) (29)
_h(t) P(t)BBT
rminus A
T1113890 1113891h(t) minus P(t)Γ (30)
When the terminal time t4 is finite these equations arenonlinear and time varying )erefore the differential
Mathematical Problems in Engineering 5
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
preset reference trajectory of the on-coming clutchrsquos relativespeed for a smooth shift process in the inertial phaseHowever under a wide variation of working conditions forheavy-duty mining trucks the preset reference trajectorycannot be determined as the optimal solution of shifting jerkand clutch friction loss which are the most important shiftperformance indicators
)e linear-quadratic optimal regulator (LQR) has beenused in the field of vehicle powertrain system control suc-cessfully To reduce the torsional vibration of electric vehicletransmission systems the method of an infinite-time LQRwas proposed by Lin et al [13] to adjust the output torque ofthe drive motor dynamically Gao et al [14] used an offlinefinite-time LQR to track the clutch relative speed during thegear shift of an electric vehicle equipped with AMT For thetorque phase and the inertia phase of the DCT upshiftprocess Li and Gorges [15] put forward control methodsbased on LQR and integral LQR respectively to reduce thejerk of shifting to make the vehicle shift more smoothlyHowever friction loss of clutch as another importantperformance index is not considered
)erefore this paper proposes a linear-quadratic opti-mization-based clutch pressure trajectory for the shiftingprocess of heavy-duty mining trucks equipped with AT toreduce the jerk of shifting and friction loss of clutch si-multaneously In Section 2 the dynamic modeling ofpowertrain systems is carried out and the ATshifting processis analyzed followed by the design of the LQR controller forthe trajectory of clutch pressure in the inertial phase (Section3) )en Section 4 presents the optimized results for twotypical working conditions of heavy-duty mining trucks anda comparison with the preset reference trajectory Con-cluding remarks are finally given in Section 5
2 Modeling for Powertrain Systems of Heavy-Duty Mining Trucks
Equipped with automatic transmission the powertrainsystem of heavy-duty mining trucks can be divided intoinput power module (composed of a diesel engine and atorque converter) wet clutch pressure control moduleplanetary gear set module and output power module (driveaxle and vehicle longitudinal dynamic model) [16ndash18] asshown in Figure 1
21 e Transmission Input Power Module )e coworkingoutput characteristics of the diesel engine and the torqueconverter (TC) determine the input power of the trans-mission )e output torque of the diesel engine TE is relatedto its rotational speed ωE and throttle opening θ and acts onthe converter pump as follows
TE ωE θ( 1113857 TP + JP _ωP (1)
where TP is the pump torque JP is the equivalent inertia ofthe turbine and ωP is the pump speed
)e dimensionless characteristics of the torque converter(including the pump torque coefficient λP TC torque ratio
KTC speed ratio iTC and efficiency ηTC) are used to describetheir dynamic equation [19] as follows
TP λPρTCgD5TCω
2P
TT KTCTP
ωT iTCωP
PT TTωT ηTCPP KTCiTCTPωP
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
(2)
where ρTC is the fluid density of the torque converter DTC isthe effective diameter of the torque converter TT is theturbine torque ωT is the turbine speed and PT and PP arethe power of the turbine and pump
22 Wet Clutch Pressure Control Module Using Stribeckfriction models (shown in Figure 2) the clutch frictionworking mode switches among ldquoin motion ①rdquo ldquocapturedand accelerating②rdquo and ldquocaptured and static③rdquo [20 21] asshown in Figure 3
(1) When the relative speed of the clutch |ω| is greaterthan a threshold ωtol the clutch is the ldquoin motionrdquomode and the friction torque Tf is the same as thedynamic torque Td which is given as follows
Tf Td sign(Δω)f(Δω)SRNp
sign(Δω)f(Δω)kdp(3)
where f(Δω) 006 + 004e(minus 036Δω) is the frictionfactor for the copper-based surface and S R N andp are equivalent area equivalent radius number offriction pairs and clutch pressure respectivelyConstants can be summarized as a pressure pro-portional coefficient kd SRN
(2) When the clutch relative speed |ω| is less than thethreshold ωtol and the required torque Ttp isgreater than the dynamic torque Td the clutch is inthe ldquocaptured and acceleratingrdquo mode and thefriction torque Tf is the same as the dynamic torqueTd
(3) When the clutch relative speed |ω| is less than thethreshold ωtol and the required torque Ttp is lessthan the dynamic torque Td the clutch is in theldquocaptured and staticrdquo mode and the friction torqueTf is the same as the required torque Ttp
23 PlanetaryGear SetModule In this study 1st gear to 2ndgear upshift is chosen as the example of an AT shiftingprocess for heavy-duty mining trucks In the first gear theclutch CS is engaged to connect the sun gear S1 to the carrierC1 of the first planetary gear set P1 at the same speed )etorque is input from the turbine and output from the ring R1of P1 When the transmission control unit (TCU) issues anupshift command CS is disengaged by releasing pressureand the on-coming brake BS is engaged gradually It finallyfixes S1 to the transmission housing to stop its movementand enters the second gear state With brake BL keepingengaged during the first and second gear the ring R3 of the
2 Mathematical Problems in Engineering
3rd planetary gear set P3 is fixed on the transmissionhousing while the torque is transferred from the carrier C3to the transmission output shaft At all time the planetaryrow P3 can be regarded as a reducer with the speed ratio ip3
By analyzing the transmission input shaft (torqueconverter turbine shaft) the sun shaft and ring shaft of P1and the transmission output shaft the dynamic equations ofthe gearbox can be written as follows
TT + TCS minus TC1 JT _ωT
TS1 minus TCS minus TBS JS1 _ωBS
TR1 kp1TS1
TR1 minus TS3 JR1 _ωR1
TO ip3TS3
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where TCS and TBS are the torques of CS and BS JS1 JR1 andJT are the inertias of the sun and ring of P1 and thetransmission input shaft TS1 TC1 and TR1 are the torque ofsun carrier and ring of P1 respectively ωBS is the speed ofBS kp1 is the gear ratio ofR1 to S1TS3 is the torque of S3 andTO is the torque of the transmission output shaft
24 e Transmission Output Power Module )e torqueoutput TO by the AT is finally transmitted to the wheelsthrough the main reducer (speed ratio iFD) For modelingconvenience the longitudinal vehicle model is simplifiedwithout considering the vehiclersquos pitch yaw and othermotion directions resulting in
TO Jw _ωw + TV
iFD (5)
where ωw is the wheel speed rw is the wheel radius and Jw isthe equivalent inertia of wheels When the heavy-dutymining truck is under the nonbraking condition the lon-gitudinal resistance torque TV can be summarized fromrolling air and climbing resistance as follows
TV rw Mgfroll +CDSV
2115v2V + Mg sin α1113874 1113875 (6)
where rw is the radius of the wheel M is the full load massfroll is the rolling resistance coefficient CD is the air re-sistance coefficient SV is the windward area vV is the vehiclespeed and α is the road slope respectively
Out
Clutch
Planetary gearIn
R1
S1
P1
R3
S3
CS
BS BL
C1 P3 C3
Engine
Torque converter
Main reducer
TT
JT
JS1
TC1TCS
TCS TS1 TBS
TR1
JR1
TS3
TE TP
JP
Wheel
TViFDTO
JwTO
Figure 1 Simplified dynamic model of the powertrain consisting of engine torque converter two planetary gear sets (P1 and P3) clutches(CS BS and BL) and vehicle
Tf
Static
Dynamic
Coulomb friction Viscous friction
Static frictionStribeck friction
ω
Figure 2 Stribeck friction model
In motion
Captured and accelerating
Captured and static
Ttp gt Td
Ttp gt Td
Ttp le Td
Ttp le TdTf = Ttp
Tf = Td (pΔω)
Tf = sign (Ttp)|Td|
|Δω| gt Δωtol |Δω| gt Δωtol
Figure 3 Clutch friction working modes and switching accordingto different transmitted torques and relative speeds of clutches
Mathematical Problems in Engineering 3
3 LQR Controller Design
)e whole shifting process of the heavy-duty mining truckshift can be divided into four stages which are rapiddraining oil (preshift phase) torque phase inertia phase andrapid boosting oil (postshift phase) as shown in Figure 4)epreshift phase (t1 sim t2) and postshift phase (t4 sim t5) aredesigned for eliminating the gap between the clutch platesand hoisting transmission capacity respectively In thetorque phase the 1st gear speed ratio is still maintained andtorque is transferred from CS to BS At the end of the torquephase if TCS(t3)gt 0 this forward torque will cause shiftshock otherwise if TCS(t3)lt 0 this negative torque willcause power loss )erefore in order to make TCS equal tozero BS should be able to undertake the transmission ofoutput torque independently
TBS t3( 1113857 1
kp1TT (7)
Open-loop control is adopted in preshift postshift andtorque phases because there is no change in the clutch speed[22ndash24] However torque and speed change drastically lastlonger and will produce greater shift jerk and clutch frictionloss in the inertial phase which is why the method based onthe LQR is used for this key phase in this paper
31 Establishment of State-Space Model )e turbine speedωT clutch speed ωBS and clutch pressure pBS of on-comingbrake BS are selected as the state variables of the shiftingprocess model
x ωT ωBS pBS1113858 1113859T (8)
)e change rate of pB is selected as the control variablebecause it is related to the shift jerk and friction loss of theclutch
u dpBS
dt (9)
From the gearbox dynamic equation the clutch pressurecontrol equation and transmission power input and outputequations (1)ndash(6) the powertrain dynamic model can bederived as follows
_ωT D22KTC
D24ωT +
D12D23
D24pBS
+D12D14 minus D13D22( 1113857TL minus D22TT
D24
_ωBS D12KTC
D24ωT +
minus D11D23
D24pBS
+D11D14 + D13D12( 1113857TL + D12TT
D24
_pBS dpBS
dt
(10)
where D11 Jw(1+ kp1)2k2p1i2p3 + JT D12 Jw1+ kp1k2
p1i2p3
D13 minus 1+ kp1k2p1ip3 D14 minus 1kp1ip3 D21 minus D12 D22
minus (Jw1k2p1i
2p3 + JS1) D23 f(ΔωBS)kd and D24 minus D2
12 minus
D11D22)e equation of the state-space model in the inertia
phase can be written as follows
_x Ax + Bu + Γ (11)
where A
D22KTCD24 0 D12D23D24D12KTCD24 0 minus D11D23D24
0 0 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ B
001
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ and
Γ
(D12D14 minus D13D22)TL minus D22TTD24(D11D14 + D13D12)TL + D12TTD24
0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
32ProblemStateand theQuadraticCostFunctionDefinition)e shifting process is not completed instantaneously and ashifting shock is inevitable To achieve a smooth shift with alow dynamic load of the system the vehicle jerk which is thederivative of acceleration wrt time should be undercontrol During the 1-2 upshift process it can be derivedfrom equations (4) and (5) as follows
j da
dt rw
d _ωw
dt
rwiFDdTO
Jwdt
rwiFD
Jw
kp1ip3d f ΔωBS( 1113857kdpBS( 1113857
dt c1u
(12)
where c1 rwiFDJwkp1ip3f(ΔωBS)kdAlthough increasing the time of the shift process can
reduce the vehicle jerk this method is not advisable becauselong-term slipping of the clutch generates friction heat andwear and damages the clutch plates finally In reference toequation (3) clutch friction loss for BS is modeled as follows
Preshift
1stgear
Torquephase
t0 t1 t2 t3 t4 t5 t6
2ndgear
Postshift
Inertiaphase
0
0
TO
TT
ndashTdBS
ndashTBS
TCS
ωT
ωBS
ωCS
pCS
pBS
Figure 4 Shifting process with four phases
4 Mathematical Problems in Engineering
W 1113946t
0TfBSωBS
11138681113868111386811138681113868
11138681113868111386811138681113868dt 1113946t4
t3
f ωBS( 1113857kdpBSωBS1113868111386811138681113868
where c2 f(ΔωBS)kd and t3 and t4 represent the start andterminal time of the inertia phase respectively
As can be seen from equations (12) and (13) the jerk forthe shift in the inertial phase is directly proportional to thecontrol variable whereas clutch friction loss for BS isproportional to the product of its speed ωBS x2 andpressure pBS x3 )erefore the finite-time linear-qua-dratic optimization performance index for the shifting in-ertia phase is shown as follows
J 12
1113946t4
t3
pBSωBS + ru2
1113872 1113873dt 12
1113946t4
t3
x2x3 + ru2
1113872 1113873dt
12
1113946t4
t3
xTQx + ru
21113872 1113873dt
(14)
where Q
0 0 00 0 050 05 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
)e first term in the integral function represents theclutch friction loss and the second represents the jerk Byadjusting the comprehensive weight coefficient r (weightcoefficient multiplied by a normalized coefficient) the LQRcontroller can achieve a proper balance between them
33 e Solution of Clutch Pressure Trajectory Based on LQRAccording to equations (11) and (14) we introduce theHamiltonian function [25] as follows
H 12
xTQx + ru
21113872 1113873 + λ(t)
T(Ax + Bu + Γ) (15)
It is assumed that the shifting proportional solenoidvalve of automatic transmission used in this study has arapid response [26 27] and there is no limit to the controlvariable u According to the maximum principle we obtain
zH
zu ru(t) + B
Tλ(t) 0 (16)
resulting in the optimal control trajectory
ulowast(t) minus
BTλ(t)
r (17)
)e normalized equations are as follows
_x zH
zλ Ax + Bu + Γ (18)
_λ(t) minuszH
zx minus Qx minus A
Tλ(t) (19)
By substituting equation (17) into (18) we obtain
_x(t) Ax(t) minusBB
Tλ(t)
r+ Γ (20)
At the end of the inertial phase clutch BS speed is zero sothat terminal constraint function
where μ is the undetermined Lagrange multiplier Due to theinterference matrix Γ the solution of the conventional LQRwhich is letting λ(t) P(t)x(t) cannot be used Here weset
λ(t) P(t)x(t) + M(t)μ + h(t) (23)
gT K(t)x(t) + L(t)μ + η(t) (24)
According to equations (22) (23) and (24) we obtain
P t4( 1113857
0 0 0
0 0 0
0 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M t4( 1113857
0
1
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ h t4( 1113857
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
K t4( 1113857 0 1 01113858 1113859 L t4( 1113857 0 η t4( 1113857 0
(25)
Substitution of equation (23) into (19) yields_λ(t) minus Q minus A
TP(t)1113960 1113961x(t) minus A
TM(t)μ minus A
Th(t) (26)
Substituting equations (20) and (23) into the derivativeof equation (23) we obtain affirmatively
_λ(t) _P(t)x(t) + P(t) _x(t) + _M(t)μ + _h(t)
_P(t) + P(t)A minusP(t)BBT
P(t)
r1113890 1113891x(t)
+ _M(t) minusP(t)BB
TM(t)
r1113890 1113891μ
minusP(t)BBT
rh(t) + P(t)Γ + _h(t)
(27)
Comparing equations (26) and (27) wrt x(t) μ andrest we obtain
_P(t) minus P(t)A minus ATP(t) +
P(t)BBTP(t)
rminus Q (28)
_M(t) P(t)BBT
rminus A
T1113890 1113891M(t) (29)
_h(t) P(t)BBT
rminus A
T1113890 1113891h(t) minus P(t)Γ (30)
When the terminal time t4 is finite these equations arenonlinear and time varying )erefore the differential
Mathematical Problems in Engineering 5
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
3rd planetary gear set P3 is fixed on the transmissionhousing while the torque is transferred from the carrier C3to the transmission output shaft At all time the planetaryrow P3 can be regarded as a reducer with the speed ratio ip3
By analyzing the transmission input shaft (torqueconverter turbine shaft) the sun shaft and ring shaft of P1and the transmission output shaft the dynamic equations ofthe gearbox can be written as follows
TT + TCS minus TC1 JT _ωT
TS1 minus TCS minus TBS JS1 _ωBS
TR1 kp1TS1
TR1 minus TS3 JR1 _ωR1
TO ip3TS3
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎩
(4)
where TCS and TBS are the torques of CS and BS JS1 JR1 andJT are the inertias of the sun and ring of P1 and thetransmission input shaft TS1 TC1 and TR1 are the torque ofsun carrier and ring of P1 respectively ωBS is the speed ofBS kp1 is the gear ratio ofR1 to S1TS3 is the torque of S3 andTO is the torque of the transmission output shaft
24 e Transmission Output Power Module )e torqueoutput TO by the AT is finally transmitted to the wheelsthrough the main reducer (speed ratio iFD) For modelingconvenience the longitudinal vehicle model is simplifiedwithout considering the vehiclersquos pitch yaw and othermotion directions resulting in
TO Jw _ωw + TV
iFD (5)
where ωw is the wheel speed rw is the wheel radius and Jw isthe equivalent inertia of wheels When the heavy-dutymining truck is under the nonbraking condition the lon-gitudinal resistance torque TV can be summarized fromrolling air and climbing resistance as follows
TV rw Mgfroll +CDSV
2115v2V + Mg sin α1113874 1113875 (6)
where rw is the radius of the wheel M is the full load massfroll is the rolling resistance coefficient CD is the air re-sistance coefficient SV is the windward area vV is the vehiclespeed and α is the road slope respectively
Out
Clutch
Planetary gearIn
R1
S1
P1
R3
S3
CS
BS BL
C1 P3 C3
Engine
Torque converter
Main reducer
TT
JT
JS1
TC1TCS
TCS TS1 TBS
TR1
JR1
TS3
TE TP
JP
Wheel
TViFDTO
JwTO
Figure 1 Simplified dynamic model of the powertrain consisting of engine torque converter two planetary gear sets (P1 and P3) clutches(CS BS and BL) and vehicle
Tf
Static
Dynamic
Coulomb friction Viscous friction
Static frictionStribeck friction
ω
Figure 2 Stribeck friction model
In motion
Captured and accelerating
Captured and static
Ttp gt Td
Ttp gt Td
Ttp le Td
Ttp le TdTf = Ttp
Tf = Td (pΔω)
Tf = sign (Ttp)|Td|
|Δω| gt Δωtol |Δω| gt Δωtol
Figure 3 Clutch friction working modes and switching accordingto different transmitted torques and relative speeds of clutches
Mathematical Problems in Engineering 3
3 LQR Controller Design
)e whole shifting process of the heavy-duty mining truckshift can be divided into four stages which are rapiddraining oil (preshift phase) torque phase inertia phase andrapid boosting oil (postshift phase) as shown in Figure 4)epreshift phase (t1 sim t2) and postshift phase (t4 sim t5) aredesigned for eliminating the gap between the clutch platesand hoisting transmission capacity respectively In thetorque phase the 1st gear speed ratio is still maintained andtorque is transferred from CS to BS At the end of the torquephase if TCS(t3)gt 0 this forward torque will cause shiftshock otherwise if TCS(t3)lt 0 this negative torque willcause power loss )erefore in order to make TCS equal tozero BS should be able to undertake the transmission ofoutput torque independently
TBS t3( 1113857 1
kp1TT (7)
Open-loop control is adopted in preshift postshift andtorque phases because there is no change in the clutch speed[22ndash24] However torque and speed change drastically lastlonger and will produce greater shift jerk and clutch frictionloss in the inertial phase which is why the method based onthe LQR is used for this key phase in this paper
31 Establishment of State-Space Model )e turbine speedωT clutch speed ωBS and clutch pressure pBS of on-comingbrake BS are selected as the state variables of the shiftingprocess model
x ωT ωBS pBS1113858 1113859T (8)
)e change rate of pB is selected as the control variablebecause it is related to the shift jerk and friction loss of theclutch
u dpBS
dt (9)
From the gearbox dynamic equation the clutch pressurecontrol equation and transmission power input and outputequations (1)ndash(6) the powertrain dynamic model can bederived as follows
_ωT D22KTC
D24ωT +
D12D23
D24pBS
+D12D14 minus D13D22( 1113857TL minus D22TT
D24
_ωBS D12KTC
D24ωT +
minus D11D23
D24pBS
+D11D14 + D13D12( 1113857TL + D12TT
D24
_pBS dpBS
dt
(10)
where D11 Jw(1+ kp1)2k2p1i2p3 + JT D12 Jw1+ kp1k2
p1i2p3
D13 minus 1+ kp1k2p1ip3 D14 minus 1kp1ip3 D21 minus D12 D22
minus (Jw1k2p1i
2p3 + JS1) D23 f(ΔωBS)kd and D24 minus D2
12 minus
D11D22)e equation of the state-space model in the inertia
phase can be written as follows
_x Ax + Bu + Γ (11)
where A
D22KTCD24 0 D12D23D24D12KTCD24 0 minus D11D23D24
0 0 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ B
001
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ and
Γ
(D12D14 minus D13D22)TL minus D22TTD24(D11D14 + D13D12)TL + D12TTD24
0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
32ProblemStateand theQuadraticCostFunctionDefinition)e shifting process is not completed instantaneously and ashifting shock is inevitable To achieve a smooth shift with alow dynamic load of the system the vehicle jerk which is thederivative of acceleration wrt time should be undercontrol During the 1-2 upshift process it can be derivedfrom equations (4) and (5) as follows
j da
dt rw
d _ωw
dt
rwiFDdTO
Jwdt
rwiFD
Jw
kp1ip3d f ΔωBS( 1113857kdpBS( 1113857
dt c1u
(12)
where c1 rwiFDJwkp1ip3f(ΔωBS)kdAlthough increasing the time of the shift process can
reduce the vehicle jerk this method is not advisable becauselong-term slipping of the clutch generates friction heat andwear and damages the clutch plates finally In reference toequation (3) clutch friction loss for BS is modeled as follows
Preshift
1stgear
Torquephase
t0 t1 t2 t3 t4 t5 t6
2ndgear
Postshift
Inertiaphase
0
0
TO
TT
ndashTdBS
ndashTBS
TCS
ωT
ωBS
ωCS
pCS
pBS
Figure 4 Shifting process with four phases
4 Mathematical Problems in Engineering
W 1113946t
0TfBSωBS
11138681113868111386811138681113868
11138681113868111386811138681113868dt 1113946t4
t3
f ωBS( 1113857kdpBSωBS1113868111386811138681113868
where c2 f(ΔωBS)kd and t3 and t4 represent the start andterminal time of the inertia phase respectively
As can be seen from equations (12) and (13) the jerk forthe shift in the inertial phase is directly proportional to thecontrol variable whereas clutch friction loss for BS isproportional to the product of its speed ωBS x2 andpressure pBS x3 )erefore the finite-time linear-qua-dratic optimization performance index for the shifting in-ertia phase is shown as follows
J 12
1113946t4
t3
pBSωBS + ru2
1113872 1113873dt 12
1113946t4
t3
x2x3 + ru2
1113872 1113873dt
12
1113946t4
t3
xTQx + ru
21113872 1113873dt
(14)
where Q
0 0 00 0 050 05 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
)e first term in the integral function represents theclutch friction loss and the second represents the jerk Byadjusting the comprehensive weight coefficient r (weightcoefficient multiplied by a normalized coefficient) the LQRcontroller can achieve a proper balance between them
33 e Solution of Clutch Pressure Trajectory Based on LQRAccording to equations (11) and (14) we introduce theHamiltonian function [25] as follows
H 12
xTQx + ru
21113872 1113873 + λ(t)
T(Ax + Bu + Γ) (15)
It is assumed that the shifting proportional solenoidvalve of automatic transmission used in this study has arapid response [26 27] and there is no limit to the controlvariable u According to the maximum principle we obtain
zH
zu ru(t) + B
Tλ(t) 0 (16)
resulting in the optimal control trajectory
ulowast(t) minus
BTλ(t)
r (17)
)e normalized equations are as follows
_x zH
zλ Ax + Bu + Γ (18)
_λ(t) minuszH
zx minus Qx minus A
Tλ(t) (19)
By substituting equation (17) into (18) we obtain
_x(t) Ax(t) minusBB
Tλ(t)
r+ Γ (20)
At the end of the inertial phase clutch BS speed is zero sothat terminal constraint function
where μ is the undetermined Lagrange multiplier Due to theinterference matrix Γ the solution of the conventional LQRwhich is letting λ(t) P(t)x(t) cannot be used Here weset
λ(t) P(t)x(t) + M(t)μ + h(t) (23)
gT K(t)x(t) + L(t)μ + η(t) (24)
According to equations (22) (23) and (24) we obtain
P t4( 1113857
0 0 0
0 0 0
0 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M t4( 1113857
0
1
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ h t4( 1113857
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
K t4( 1113857 0 1 01113858 1113859 L t4( 1113857 0 η t4( 1113857 0
(25)
Substitution of equation (23) into (19) yields_λ(t) minus Q minus A
TP(t)1113960 1113961x(t) minus A
TM(t)μ minus A
Th(t) (26)
Substituting equations (20) and (23) into the derivativeof equation (23) we obtain affirmatively
_λ(t) _P(t)x(t) + P(t) _x(t) + _M(t)μ + _h(t)
_P(t) + P(t)A minusP(t)BBT
P(t)
r1113890 1113891x(t)
+ _M(t) minusP(t)BB
TM(t)
r1113890 1113891μ
minusP(t)BBT
rh(t) + P(t)Γ + _h(t)
(27)
Comparing equations (26) and (27) wrt x(t) μ andrest we obtain
_P(t) minus P(t)A minus ATP(t) +
P(t)BBTP(t)
rminus Q (28)
_M(t) P(t)BBT
rminus A
T1113890 1113891M(t) (29)
_h(t) P(t)BBT
rminus A
T1113890 1113891h(t) minus P(t)Γ (30)
When the terminal time t4 is finite these equations arenonlinear and time varying )erefore the differential
Mathematical Problems in Engineering 5
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
3 LQR Controller Design
)e whole shifting process of the heavy-duty mining truckshift can be divided into four stages which are rapiddraining oil (preshift phase) torque phase inertia phase andrapid boosting oil (postshift phase) as shown in Figure 4)epreshift phase (t1 sim t2) and postshift phase (t4 sim t5) aredesigned for eliminating the gap between the clutch platesand hoisting transmission capacity respectively In thetorque phase the 1st gear speed ratio is still maintained andtorque is transferred from CS to BS At the end of the torquephase if TCS(t3)gt 0 this forward torque will cause shiftshock otherwise if TCS(t3)lt 0 this negative torque willcause power loss )erefore in order to make TCS equal tozero BS should be able to undertake the transmission ofoutput torque independently
TBS t3( 1113857 1
kp1TT (7)
Open-loop control is adopted in preshift postshift andtorque phases because there is no change in the clutch speed[22ndash24] However torque and speed change drastically lastlonger and will produce greater shift jerk and clutch frictionloss in the inertial phase which is why the method based onthe LQR is used for this key phase in this paper
31 Establishment of State-Space Model )e turbine speedωT clutch speed ωBS and clutch pressure pBS of on-comingbrake BS are selected as the state variables of the shiftingprocess model
x ωT ωBS pBS1113858 1113859T (8)
)e change rate of pB is selected as the control variablebecause it is related to the shift jerk and friction loss of theclutch
u dpBS
dt (9)
From the gearbox dynamic equation the clutch pressurecontrol equation and transmission power input and outputequations (1)ndash(6) the powertrain dynamic model can bederived as follows
_ωT D22KTC
D24ωT +
D12D23
D24pBS
+D12D14 minus D13D22( 1113857TL minus D22TT
D24
_ωBS D12KTC
D24ωT +
minus D11D23
D24pBS
+D11D14 + D13D12( 1113857TL + D12TT
D24
_pBS dpBS
dt
(10)
where D11 Jw(1+ kp1)2k2p1i2p3 + JT D12 Jw1+ kp1k2
p1i2p3
D13 minus 1+ kp1k2p1ip3 D14 minus 1kp1ip3 D21 minus D12 D22
minus (Jw1k2p1i
2p3 + JS1) D23 f(ΔωBS)kd and D24 minus D2
12 minus
D11D22)e equation of the state-space model in the inertia
phase can be written as follows
_x Ax + Bu + Γ (11)
where A
D22KTCD24 0 D12D23D24D12KTCD24 0 minus D11D23D24
0 0 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ B
001
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ and
Γ
(D12D14 minus D13D22)TL minus D22TTD24(D11D14 + D13D12)TL + D12TTD24
0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
32ProblemStateand theQuadraticCostFunctionDefinition)e shifting process is not completed instantaneously and ashifting shock is inevitable To achieve a smooth shift with alow dynamic load of the system the vehicle jerk which is thederivative of acceleration wrt time should be undercontrol During the 1-2 upshift process it can be derivedfrom equations (4) and (5) as follows
j da
dt rw
d _ωw
dt
rwiFDdTO
Jwdt
rwiFD
Jw
kp1ip3d f ΔωBS( 1113857kdpBS( 1113857
dt c1u
(12)
where c1 rwiFDJwkp1ip3f(ΔωBS)kdAlthough increasing the time of the shift process can
reduce the vehicle jerk this method is not advisable becauselong-term slipping of the clutch generates friction heat andwear and damages the clutch plates finally In reference toequation (3) clutch friction loss for BS is modeled as follows
Preshift
1stgear
Torquephase
t0 t1 t2 t3 t4 t5 t6
2ndgear
Postshift
Inertiaphase
0
0
TO
TT
ndashTdBS
ndashTBS
TCS
ωT
ωBS
ωCS
pCS
pBS
Figure 4 Shifting process with four phases
4 Mathematical Problems in Engineering
W 1113946t
0TfBSωBS
11138681113868111386811138681113868
11138681113868111386811138681113868dt 1113946t4
t3
f ωBS( 1113857kdpBSωBS1113868111386811138681113868
where c2 f(ΔωBS)kd and t3 and t4 represent the start andterminal time of the inertia phase respectively
As can be seen from equations (12) and (13) the jerk forthe shift in the inertial phase is directly proportional to thecontrol variable whereas clutch friction loss for BS isproportional to the product of its speed ωBS x2 andpressure pBS x3 )erefore the finite-time linear-qua-dratic optimization performance index for the shifting in-ertia phase is shown as follows
J 12
1113946t4
t3
pBSωBS + ru2
1113872 1113873dt 12
1113946t4
t3
x2x3 + ru2
1113872 1113873dt
12
1113946t4
t3
xTQx + ru
21113872 1113873dt
(14)
where Q
0 0 00 0 050 05 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
)e first term in the integral function represents theclutch friction loss and the second represents the jerk Byadjusting the comprehensive weight coefficient r (weightcoefficient multiplied by a normalized coefficient) the LQRcontroller can achieve a proper balance between them
33 e Solution of Clutch Pressure Trajectory Based on LQRAccording to equations (11) and (14) we introduce theHamiltonian function [25] as follows
H 12
xTQx + ru
21113872 1113873 + λ(t)
T(Ax + Bu + Γ) (15)
It is assumed that the shifting proportional solenoidvalve of automatic transmission used in this study has arapid response [26 27] and there is no limit to the controlvariable u According to the maximum principle we obtain
zH
zu ru(t) + B
Tλ(t) 0 (16)
resulting in the optimal control trajectory
ulowast(t) minus
BTλ(t)
r (17)
)e normalized equations are as follows
_x zH
zλ Ax + Bu + Γ (18)
_λ(t) minuszH
zx minus Qx minus A
Tλ(t) (19)
By substituting equation (17) into (18) we obtain
_x(t) Ax(t) minusBB
Tλ(t)
r+ Γ (20)
At the end of the inertial phase clutch BS speed is zero sothat terminal constraint function
where μ is the undetermined Lagrange multiplier Due to theinterference matrix Γ the solution of the conventional LQRwhich is letting λ(t) P(t)x(t) cannot be used Here weset
λ(t) P(t)x(t) + M(t)μ + h(t) (23)
gT K(t)x(t) + L(t)μ + η(t) (24)
According to equations (22) (23) and (24) we obtain
P t4( 1113857
0 0 0
0 0 0
0 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M t4( 1113857
0
1
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ h t4( 1113857
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
K t4( 1113857 0 1 01113858 1113859 L t4( 1113857 0 η t4( 1113857 0
(25)
Substitution of equation (23) into (19) yields_λ(t) minus Q minus A
TP(t)1113960 1113961x(t) minus A
TM(t)μ minus A
Th(t) (26)
Substituting equations (20) and (23) into the derivativeof equation (23) we obtain affirmatively
_λ(t) _P(t)x(t) + P(t) _x(t) + _M(t)μ + _h(t)
_P(t) + P(t)A minusP(t)BBT
P(t)
r1113890 1113891x(t)
+ _M(t) minusP(t)BB
TM(t)
r1113890 1113891μ
minusP(t)BBT
rh(t) + P(t)Γ + _h(t)
(27)
Comparing equations (26) and (27) wrt x(t) μ andrest we obtain
_P(t) minus P(t)A minus ATP(t) +
P(t)BBTP(t)
rminus Q (28)
_M(t) P(t)BBT
rminus A
T1113890 1113891M(t) (29)
_h(t) P(t)BBT
rminus A
T1113890 1113891h(t) minus P(t)Γ (30)
When the terminal time t4 is finite these equations arenonlinear and time varying )erefore the differential
Mathematical Problems in Engineering 5
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
W 1113946t
0TfBSωBS
11138681113868111386811138681113868
11138681113868111386811138681113868dt 1113946t4
t3
f ωBS( 1113857kdpBSωBS1113868111386811138681113868
where c2 f(ΔωBS)kd and t3 and t4 represent the start andterminal time of the inertia phase respectively
As can be seen from equations (12) and (13) the jerk forthe shift in the inertial phase is directly proportional to thecontrol variable whereas clutch friction loss for BS isproportional to the product of its speed ωBS x2 andpressure pBS x3 )erefore the finite-time linear-qua-dratic optimization performance index for the shifting in-ertia phase is shown as follows
J 12
1113946t4
t3
pBSωBS + ru2
1113872 1113873dt 12
1113946t4
t3
x2x3 + ru2
1113872 1113873dt
12
1113946t4
t3
xTQx + ru
21113872 1113873dt
(14)
where Q
0 0 00 0 050 05 0
⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦
)e first term in the integral function represents theclutch friction loss and the second represents the jerk Byadjusting the comprehensive weight coefficient r (weightcoefficient multiplied by a normalized coefficient) the LQRcontroller can achieve a proper balance between them
33 e Solution of Clutch Pressure Trajectory Based on LQRAccording to equations (11) and (14) we introduce theHamiltonian function [25] as follows
H 12
xTQx + ru
21113872 1113873 + λ(t)
T(Ax + Bu + Γ) (15)
It is assumed that the shifting proportional solenoidvalve of automatic transmission used in this study has arapid response [26 27] and there is no limit to the controlvariable u According to the maximum principle we obtain
zH
zu ru(t) + B
Tλ(t) 0 (16)
resulting in the optimal control trajectory
ulowast(t) minus
BTλ(t)
r (17)
)e normalized equations are as follows
_x zH
zλ Ax + Bu + Γ (18)
_λ(t) minuszH
zx minus Qx minus A
Tλ(t) (19)
By substituting equation (17) into (18) we obtain
_x(t) Ax(t) minusBB
Tλ(t)
r+ Γ (20)
At the end of the inertial phase clutch BS speed is zero sothat terminal constraint function
where μ is the undetermined Lagrange multiplier Due to theinterference matrix Γ the solution of the conventional LQRwhich is letting λ(t) P(t)x(t) cannot be used Here weset
λ(t) P(t)x(t) + M(t)μ + h(t) (23)
gT K(t)x(t) + L(t)μ + η(t) (24)
According to equations (22) (23) and (24) we obtain
P t4( 1113857
0 0 0
0 0 0
0 0 0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ M t4( 1113857
0
1
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ h t4( 1113857
0
0
0
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
K t4( 1113857 0 1 01113858 1113859 L t4( 1113857 0 η t4( 1113857 0
(25)
Substitution of equation (23) into (19) yields_λ(t) minus Q minus A
TP(t)1113960 1113961x(t) minus A
TM(t)μ minus A
Th(t) (26)
Substituting equations (20) and (23) into the derivativeof equation (23) we obtain affirmatively
_λ(t) _P(t)x(t) + P(t) _x(t) + _M(t)μ + _h(t)
_P(t) + P(t)A minusP(t)BBT
P(t)
r1113890 1113891x(t)
+ _M(t) minusP(t)BB
TM(t)
r1113890 1113891μ
minusP(t)BBT
rh(t) + P(t)Γ + _h(t)
(27)
Comparing equations (26) and (27) wrt x(t) μ andrest we obtain
_P(t) minus P(t)A minus ATP(t) +
P(t)BBTP(t)
rminus Q (28)
_M(t) P(t)BBT
rminus A
T1113890 1113891M(t) (29)
_h(t) P(t)BBT
rminus A
T1113890 1113891h(t) minus P(t)Γ (30)
When the terminal time t4 is finite these equations arenonlinear and time varying )erefore the differential
Mathematical Problems in Engineering 5
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
equations are replaced by the difference equations and thevalue at each step can be calculated singly in the reversedirection with (minus Δt) as the time interval from equation (25)
Substituting equations (20) and (23) into the time derivativeof equation (24) yields the following
_gT _K(t)x(t) + K(t) _x(t) + _L(t)μ + _η(t)
_K(t) + K(t)A minusK(t)BBT
P(t)
r1113890 1113891x(t) _L(t) minus
K(t)BBTM(t)
r1113890 1113891μ
minusK(t)BBT
h(t)
r+ K(t)Γ + _η(t) 0
(31)
Equation (31) holds for any x(t) and μ if
_K(t) K(t)BBT
P(t)
rminus A1113890 1113891 (32)
_L(t) K(t)BBT
M(t)
r (33)
_η(t) K(t)BBT
h(t)
rminus Γ1113890 1113891 (34)
Comparing equation (29) with (32) and M(t4) withK(t4) in equation (25) shows
K(t) MT(t) (35)
)en we substitute the initial value of the system inequation (24)
μ gT t3( 1113857 minus K(0)x(0) minus η(0)
L(0) (36)
Finally the optimal control trajectory in equation (17) is
ulowast(t) minus
BTλ(t)
r minus
BT
r[P(t)x(t) + M(t)μ + h(t)]
minusB
TP(t)
rx(t) minus
BT
r[M(t)μ + h(t)]
V(t)x(t) + W(t)
(37)
where control parameters V(t) minus BTP(t)rx(t) andW(t) minus BTr[M(t)μ + h(t)]
4 Simulation Results and Discussion
)e LQR controller model and the model of the powertrainsystem for the heavy-duty mining truck were built inMatlabSimulink with the major simulation parametersshown in Table 1 No load without road slope under 50throttle opening and full load with α 6∘ road slope under100 throttle opening are chosen as the normal conditionand the extreme condition respectively to test the presetreference trajectory [8] and optimal trajectory under vari-ations of comprehensive weight coefficients r
41 Normal Working Condition )e no-load mass of theheavy-duty mining truck in this paper is 30 tons Afterstarting on a flat road with a 50 throttle opening thesimulation results are shown in Figure 5
At 10 s where the pressure of CS drops to zero andenters the inertia phase the BS pressure is controlled by theLQR optimal trajectory described above )e entire shifttime is about 09 s as shown in Figure 5(a) Although thespeed of the turbine shaft and BS begins to decrease unlikethe torque phase they are no longer the same in the inertialphase because of the clutch CS slipping When the inertialphase ends the speed of BS drops to zero and the turbinespeed begins to rise again Compared with the preset ref-erence trajectory the optimized speed of BS changes moreslowly (as shown in Figure 5(b)) Figure 5(c) shows thetrends of BS CS and turbine torque in the shifting processWith the rise in BS torque the torque of CS graduallydecreases to zero at the end of the torque phase )e reasonfor the sudden BS torque change at about 15 s (inertia phaseends) is that the clutch friction working mode of BS switchesfrom the ldquocaptured and acceleratingrdquo mode to the ldquocapturedand staticrdquo mode which means the sliding friction becomesstatic friction
)e result of shifting jerk and clutch friction loss for theLQR controller is compared with that of the preset trajectorycontroller in Figure 5(d) For r 1lowast 10minus 6 the peak value ofthe optimized jerk jmax 133ms3 is slightly smaller thanthe peak value jmax 137ms3 before optimization whichsatisfies the design requirement of jle plusmn 5ms3 At the endof the inertial phase the loss of clutch friction is W 247 kJafter optimization which is about 263 less thanW 335 kJ for the preset reference controller By adjustingthe weight coefficients to r 35lowast 10minus 6 shift jerk drops tojmax 105ms3 while clutch friction loss increases toW 313 kJ which shows the proposed controller can op-timize both objectives
42 Extreme Working Condition )e extreme workingconditions of the heavy-duty mining truck studied inthis paper are 72 tons at full load and 100 throttleopening on α 6∘ road slope )is simulation is forchecking the robustness of the LQR controller as shownin Figure 6
6 Mathematical Problems in Engineering
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
Due to the small acceleration of the vehicle under thiscondition the required shifting speed is reached later startingfrom 13 s and ending at 21 s Compared with normal con-ditions the speed and torque of CS BS and other transmissioncomponents are at higher values)e peak speed of BS exceeds1700 rmin and the torque reaches 1000Nm as shown inFigures 6(a)ndash6(c) In Figure 6(d) it can be seen that after the
LQR controller optimizes the clutch pressure the peak value ofthe jerk is jmax 197ms3 and the final value of clutch frictionloss is W 423 kJ at the weight coefficient r 285lowast 10minus 6which are less than jmax 233ms3 and W 522 kJ beforeoptimization respectively Under different working conditionsthe comparison results of the LQR controller with the referencetrajectory controller are shown in Table 2
ReferenceJerkFriction lossProposed r = 1lowast10ndash6
02 04 06 08 16 180 121 214Time (s)
ndash2
ndash1
0
1
2
0
1
2
3
4
Fric
tion
loss
(Nmiddotm
)
(d)
Figure 5 Simulation result with mass m 30t throttle opening θ 50 and road slope α 0∘
Mathematical Problems in Engineering 7
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
5 Conclusions
)is paper proposes a clutch pressure control method bylinear-quadratic optimization for the inertia phase of an ATshifting process for a heavy-duty mining truck )e pow-ertrain systemmodel and the LQR controller model are builtin MatlabSimulink )e results show that the LQR opti-mized clutch pressure control trajectory can reduce the jerkand the clutch friction loss for both normal and extremeworking conditions Compared with the preset referencetrajectory their maximum value can drop by 234 for jerkand by 263 for clutch energy loss under the normalworking condition and jmax 197ms3 and W 423 kJ atextreme conditions which indicates that the optimizationmethod results in effective and robust control
Data Availability
)e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
)e authors declare no potential conflicts of interest withrespect to research authorship andor publication of thisarticle
Acknowledgments
)is work was supported by the National Natural ScienceFoundation of China (grant no 51905031) the Fundamental
08 1 12 14 16 18 2 22 24 26 28Time (s)
CSBS
0
02
04
06
08Cl
utch
pre
ssur
e (M
Pa)
(a)
08 1 12 14 16 18 2 22 24 26 28Time (s)
0
500
1000
1500
2000
Spee
d (r
min
)
Turbine shaftBS_propsedBS_reference
(b)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash1000ndash500
0500
1000150020002500
Torq
ue (N
middotm)
BSCSTurbine shaft
(c)
08 1 12 14 16 18 2 22 24 26 28Time (s)
ndash3
ndash2
ndash1
0
1
2
3
Jerk
(ms
3 )
0
1
2
3
4
5
6
Fric
tion
loss
(Nmiddotm
)
Jerk_proposedJerk_reference
Friction loss_proposedFriction loss_reference
times104
(d)
Figure 6 Simulation result with mass m 72t throttle opening θ 100 road slope α 6∘ and comprehensive weight coefficientr 285lowast 10minus 6
Table 2 Comparison results of the LQR controller with the reference trajectory controller under different conditions
Condition Controller jmax (ms3) Jerk improve W (kJ) Clutch energy loss improve
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
Mathematical Problems in Engineering 9
Research Funds for the Central University of China (grantno FRF-TP-18-036A1) and the National Key Research andDevelopment Program of China (grant no2018YFC0604402)
References
[1] P Dong Y Liu P Tenberge and X Xu ldquoDesign and analysisof a novel multi-speed automatic transmission with fourdegrees-of-freedomrdquo Mechanism and Machine eoryvol 108 pp 83ndash96 2017
[2] Y Zhang and W Ma ldquoShift control system of heavy-dutyvehicle automatic transmissionrdquo Journal of Networks vol 8no 12 2013
[3] F Meng H Zhang D Cao and H Chen ldquoSystem modelingand pressure control of a clutch actuator for heavy-dutyautomatic transmission systemsrdquo IEEE Transactions on Ve-hicular Technology vol 65 no 7 pp 4865ndash4874 2016
[4] X Zhao W Zhang Y Feng and Y Yang ldquoOptimizing gearshifting strategy for off-road vehicle with dynamic pro-grammingrdquoMathematical Problems in Engineering vol 2014Article ID 642949 9 pages 2014
[5] Y Cheng P Dong S Yang and X Xu ldquoVirtual clutchcontroller for clutch-to-clutch shifts in planetary-type Au-tomatic transmissionrdquo Mathematical Problems in Engineer-ing vol 2015 Article ID 213162 16 pages 2015
[6] V Ranogajec J Deur V Ivanovic and H E Tseng ldquoMulti-objective parameter optimization of control profiles for au-tomatic transmission double-transition shiftsrdquo Control En-gineering Practice vol 93 Article ID 104183 2019
[7] J-Y Oh J-Y Park J-W Cho J-G Kim J-H Kim andG-H Lee ldquoInfluence of a clutch control current profile toimprove shift quality for a wheel loader automatic trans-missionrdquo International Journal of Precision Engineering andManufacturing vol 18 no 2 pp 211ndash219 2017
[8] F Meng G Tao T Zhang Y Hu and P Geng ldquoOptimalshifting control strategy in inertia phase of an automatictransmission for automotive applicationsrdquo Mechanical Sys-tems and Signal Processing vol 60-61 pp 742ndash752 2015
[9] X Song and Z Sun ldquoPressure-based clutch control for au-tomotive transmissions using a sliding-mode controllerrdquoIEEEASME Transactions on Mechatronics vol 17 no 3pp 534ndash546 2012
[10] K Sanada B Gao N Kado H Takamatsu and K ToriyaldquoDesign of a robust controller for shift control of an automatictransmissionrdquo Proceedings of the Institution of MechanicalEngineers Part D Journal of Automobile Engineering vol 226no 12 pp 1577ndash1584 2012
[11] X Zhao and Z Li ldquoData-driven predictive control applied togear shifting for heavy-duty vehiclesrdquo Energies vol 11 no 8p 2139 2018
[12] A Wurm and D Bestle ldquoRobust design optimization forimproving automotive shift qualityrdquo Optimization and En-gineering vol 17 no 2 pp 421ndash436 2016
[13] C Lin S Sun P Walker and N Zhang ldquoOff-line optimi-zation based active control of torsional oscillation for electricvehicle drivetrainrdquo Applied Sciences vol 7 no 12 p 12612017
[14] B Gao Q Liang Y Guo and H Chen ldquoGear ratio opti-mization and shift control of 2-speed I-AMT in electric ve-hiclerdquo Mechanical Systems and Signal Processing vol 50-51pp 615ndash631 2015
[15] G Li and D Gorges ldquoOptimal control of the gear shiftingprocess for shift smoothness in dual-clutch transmissionsrdquo
Mechanical Systems and Signal Processing vol 103 pp 23ndash382018
[16] H Hwang and S Choi ldquoDynamic driveline torque estimationduring whole gear shift for an automatic transmissionrdquoMechanism and Machine eory vol 130 pp 363ndash381 2018
[17] Y Lei K Liu Y Zhang et al ldquoAdaptive gearshift strategybased on generalized load recognition for automatic trans-mission vehiclesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 614989 12 pages 2015
[18] V Ranogajec V Ivanovic J Deur and H E Tseng ldquoOpti-mization-based assessment of automatic transmission dou-ble-transition shift controlsrdquo Control Engineering Practicevol 76 pp 155ndash166 2018
[19] H Jin-Oh and K Lee ldquoNonlinear robust control of torqueconverter clutch slip system for passenger vehicles usingadvanced torque estimation algorithmsrdquo Vehicle SystemDynamics vol 37 no 3 pp 175ndash192 2002
[20] H Jian W Wei H Li and Q Yan ldquoOptimization of apressure control valve for high power automatic transmissionconsidering stabilityrdquo Mechanical Systems and Signal Pro-cessing vol 101 pp 182ndash196 2018
[21] J Kim and S B Choi ldquoDesign and modeling of a clutchactuator system with self-energizing mechanismrdquo IEEEASME Transactions on Mechatronics vol 16 no 5 pp 953ndash966 2011
[22] L Y Fu and X Z Li ldquoResearch on integrated shift controlstrategy for automatic transmissionrdquo Applied Mechanics andMaterials vol 835 pp 687ndash692 2016
[23] L Li X Wang X Qi X Li D Cao and Z Zhu ldquoAutomaticclutch control based on estimation of resistance torque forAMTrdquo IEEEASME Transactions on Mechatronics vol 21no 6 pp 2682ndash2693 2016
[24] S Li C Wu and Z Sun ldquoDesign and implementation ofclutch control for automotive transmissions using terminal-sliding-mode control and uncertainty observerrdquo IEEETransactions on Vehicular Technology vol 65 no 4pp 1890ndash1898 2016
[25] T Bubnicki Modern Control eory Springer Berlin Ger-many 2005
[26] T Ouyang G Huang S Li J Chen and N Chen ldquoDynamicmodelling and optimal design of a clutch actuator for heavy-duty automatic transmission considering flow forcerdquo Mech-anism and Machine eory vol 145 Article ID 103716 2020
[27] T Ouyang S Li G Huang F Zhou and N Chen ldquoMath-ematical modeling and performance prediction of a clutchactuator for heavy-duty automatic transmission vehiclesrdquoMechanism and Machine eory vol 136 pp 190ndash205 2019
