# Optimal Clutch Pressure Control in Shifting Process of Automatic Transmission for Heavy-Duty Mining Trucks

Authors: Heng Zhang; Xinxin Zhao; Jianning Sun
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The optimal control of automatic transmission plays an important role in the shifting smoothness and fuel economy of heavy-duty mining trucks. In this paper, a dynamic model of the powertrain system is built to study the clutch pressure control during the shifting process. A linear-quadratic optimal regulator is used to achieve the optimum control pressure of clutches, where shifting jerk 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. This paper also carries out a software simulation and 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 pressure control can effectively reduce jerk and friction loss during the shifting process and has good robustness to different operating conditions.


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## Body

## 1. Introduction

Transmission is the key component of a vehicle that transmits different torque ratios at different speeds [1]. In order to make heavy-duty mining trucks face the problems of bad working conditions, poor fuel economy, and high labor intensity of drivers, AT is widely used in this field with the characteristics of strong transmission capability, smooth shifting process, and high reliability [2, 3]. In addition, under the condition of increasingly mature lockup clutch technology, the efficiency of AT has also been significantly improved [4].In order to avoid power interruption, shift overlapping, and overheating of the clutch plate, caused by incoordination between on-coming and off-going clutches, some researchers have studied the clutch-to-clutch shifting process and divided it into four parts (preshift phase, torque phase, inertia phase, and postshift phase) [5–7]. Meng et al. [8] analyzed the AT’s electrohydraulic control system and proposed a two-degree-of-freedom PID control method to optimize the duty ratio of proportional solenoid valves for the speed difference of clutch discs following a preset trajectory. Song and Sun [9] focused on the nonlinear dynamic characteristics of the wet clutch and designed a sliding mode controller to control the on-coming clutch pressure to ensure a smooth clutch-to-clutch shifting in the inertia phase. To track and control the relative speed of the clutch, eliminate the uncertainty of system parameters, and improve the robustness with respect to changing driving conditions, robust controllers, postfeedback controllers, and disturbance compensators were designed by Sanada et al. [10]. Zhao and Li [11] used the subspace identification method to describe the prediction equation of the AT powertrain system, which solved the difficulty of modeling an accurate clutch-to-clutch shifting process and then applied the model prediction controller to improve the shift quality in hardware-in-the-loop tests. Considering power, comfortability, and robustness during the clutch engagement, Wurm and Bestle [12] adopted a multiobjective genetic algorithm to achieve their balance and finally obtain Pareto optimality. In summary, since the torque signal is hard to be measured by a sensor in the torque phase, current studies mostly track a preset reference trajectory of the on-coming clutch’s relative speed for a smooth shift process in the inertial phase. However, under a wide variation of working conditions for heavy-duty mining trucks, the preset reference trajectory cannot be determined as the optimal solution of shifting jerk and clutch friction loss, which are the most important shift performance indicators.The linear-quadratic optimal regulator (LQR) has been used in the field of vehicle powertrain system control successfully. To reduce the torsional vibration of electric vehicle transmission systems, the method of an infinite-time LQR was proposed by Lin et al. [13] to adjust the output torque of the drive motor dynamically. Gao et al. [14] used an offline finite-time LQR to track the clutch relative speed during the gear shift of an electric vehicle equipped with AMT. For the torque phase and the inertia phase of the DCT upshift process, Li and Görges [15] put forward control methods based on LQR and integral LQR, respectively, to reduce the jerk of shifting to make the vehicle shift more smoothly. However, friction loss of clutch, as another important performance index, is not considered.Therefore, this paper proposes a linear-quadratic optimization-based clutch pressure trajectory for the shifting process of heavy-duty mining trucks equipped with AT to reduce the jerk of shifting and friction loss of clutch simultaneously. In Section2, the dynamic modeling of powertrain systems is carried out and the AT shifting process is analyzed, followed by the design of the LQR controller for the trajectory of clutch pressure in the inertial phase (Section 3). Then, Section 4 presents the optimized results for two typical working conditions of heavy-duty mining trucks and a comparison with the preset reference trajectory. Concluding remarks are finally given in Section 5.


## 2. Modeling for Powertrain Systems of Heavy-Duty Mining Trucks

Equipped with automatic transmission, the powertrain system of heavy-duty mining trucks can be divided into input power module (composed of a diesel engine and a torque converter), wet clutch pressure control module, planetary gear set module, and output power module (drive axle and vehicle longitudinal dynamic model) [16–18], as shown in Figure 1.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.


### 2.1. The Transmission Input Power Module

The coworking output characteristics of the diesel engine and the torque converter (TC) determine the input power of the transmission. The output torque of the diesel engineTE is related to its rotational speed ωE and throttle opening θ and acts on the converter pump as follows:(1)TEωE,θ=TP+JPω˙P,where TP is the pump torque, JP is the equivalent inertia of the turbine, and ωP is the pump speed.The dimensionless characteristics of the torque converter (including the pump torque coefficientλP , TC torque ratio KTC, speed ratio iTC, and efficiency ηTC) are used to describe their dynamic equation [19] as follows:(2)TP=λPρTCgDTC5ωP2,TT=KTCTP,ωT=iTCωP,PT=TTωT=ηTCPP=KTCiTCTPωP,where ρTC is the fluid density of the torque converter, DTC is the effective diameter of the torque converter, TT is the turbine torque, ωT is the turbine speed, and PT and PP are the power of the turbine and pump.


### 2.2. Wet Clutch Pressure Control Module

Using Stribeck friction models (shown in Figure2), the clutch friction working mode switches among “in motion ①”, “captured and accelerating ②,” and “captured and static ③” [20, 21], as shown in Figure 3.(1)
When the relative speed of the clutch△ω is greater than a threshold △ωtol, the clutch is the “in motion” mode, and the friction torque Tf is the same as the dynamic torque Td which is given as follows:(3)Tf=Td=signΔωfΔωSRNp=signΔωfΔωkdp,wherefΔω=0.06+0.04e−0.36Δω is the friction factor for the copper-based surface and S, R, N, and p are equivalent area, equivalent radius, number of friction pairs, and clutch pressure, respectively. Constants can be summarized as a pressure proportional coefficient kd=SRN.(2)
When the clutch relative speed△ω is less than the threshold △ωtol and the required torque Ttp is greater than the dynamic torque Td, the clutch is in the “captured and accelerating” mode, and the friction torque Tf is the same as the dynamic torque Td.(3)
When the clutch relative speed△ω is less than the threshold △ωtol and the required torque Ttp is less than the dynamic torque Td, the clutch is in the “captured and static” mode, and the friction torque Tf is the same as the required torque Ttp.Figure 2
Stribeck friction model.Figure 3
Clutch friction working modes and switching according to different transmitted torques and relative speeds of clutches.


### 2.3. Planetary Gear Set Module

In this study, 1st gear to 2nd gear upshift is chosen as the example of an AT shifting process for heavy-duty mining trucks. In the first gear, the clutchCS is engaged to connect the sun gear S1 to the carrier C1 of the first planetary gear set P1 at the same speed. The torque is input from the turbine and output from the ring R1 of P1. When the transmission control unit (TCU) issues an upshift command, CS is disengaged by releasing pressure and the on-coming brake BS is engaged gradually. It finally fixes S1 to the transmission housing to stop its movement and enters the second gear state. With brake BL keeping engaged during the first and second gear, the ring R3 of the 3rd planetary gear set P3 is fixed on the transmission housing, while the torque is transferred from the carrier C3 to the transmission output shaft. At all time, the planetary row P3 can be regarded as a reducer with the speed ratio ip3.By analyzing the transmission input shaft (torque converter turbine shaft), the sun shaft and ring shaft ofP1, and the transmission output shaft, the dynamic equations of the gearbox can be written as follows:(4)TT+TCS−TC1=JTω˙T,TS1−TCS−TBS=JS1ω˙BS,TR1=kp1TS1,TR1−TS3=JR1ω˙R1,TO=ip3TS3,where TCS and TBS are the torques of CS and BS; JS1, JR1, and JT are the inertias of the sun and ring of P1 and the transmission input shaft; TS1, TC1, and TR1 are the torque of sun, carrier, and ring of P1, respectively; ωBS is the speed of BS; kp1 is the gear ratio of R1 to S1; TS3 is the torque of S3; and TO is the torque of the transmission output shaft.


### 2.4. The Transmission Output Power Module

The torque outputTO by the AT is finally transmitted to the wheels through the main reducer (speed ratio iFD). For modeling convenience, the longitudinal vehicle model is simplified without considering the vehicle's pitch, yaw, and other motion directions resulting in(5)TO=Jwω˙w+TViFD,where ωw is the wheel speed, rw is the wheel radius, and Jw is the equivalent inertia of wheels. When the heavy-duty mining truck is under the nonbraking condition, the longitudinal resistance torque TV can be summarized from rolling, air, and climbing resistance as follows:(6)TV=rwMgfroll+CDSV21.15vV2+Mgsinα,where rw is the radius of the wheel, M is the full load mass, froll is the rolling resistance coefficient, CD is the air resistance coefficient, SV is the windward area, vV is the vehicle speed, and α is the road slope, respectively.


## 2.1. The Transmission Input Power Module

The coworking output characteristics of the diesel engine and the torque converter (TC) determine the input power of the transmission. The output torque of the diesel engineTE is related to its rotational speed ωE and throttle opening θ and acts on the converter pump as follows:(1)TEωE,θ=TP+JPω˙P,where TP is the pump torque, JP is the equivalent inertia of the turbine, and ωP is the pump speed.The dimensionless characteristics of the torque converter (including the pump torque coefficientλP , TC torque ratio KTC, speed ratio iTC, and efficiency ηTC) are used to describe their dynamic equation [19] as follows:(2)TP=λPρTCgDTC5ωP2,TT=KTCTP,ωT=iTCωP,PT=TTωT=ηTCPP=KTCiTCTPωP,where ρTC is the fluid density of the torque converter, DTC is the effective diameter of the torque converter, TT is the turbine torque, ωT is the turbine speed, and PT and PP are the power of the turbine and pump.


## 2.2. Wet Clutch Pressure Control Module

Using Stribeck friction models (shown in Figure2), the clutch friction working mode switches among “in motion ①”, “captured and accelerating ②,” and “captured and static ③” [20, 21], as shown in Figure 3.(1)
When the relative speed of the clutch△ω is greater than a threshold △ωtol, the clutch is the “in motion” mode, and the friction torque Tf is the same as the dynamic torque Td which is given as follows:(3)Tf=Td=signΔωfΔωSRNp=signΔωfΔωkdp,wherefΔω=0.06+0.04e−0.36Δω is the friction factor for the copper-based surface and S, R, N, and p are equivalent area, equivalent radius, number of friction pairs, and clutch pressure, respectively. Constants can be summarized as a pressure proportional coefficient kd=SRN.(2)
When the clutch relative speed△ω is less than the threshold △ωtol and the required torque Ttp is greater than the dynamic torque Td, the clutch is in the “captured and accelerating” mode, and the friction torque Tf is the same as the dynamic torque Td.(3)
When the clutch relative speed△ω is less than the threshold △ωtol and the required torque Ttp is less than the dynamic torque Td, the clutch is in the “captured and static” mode, and the friction torque Tf is the same as the required torque Ttp.Figure 2
Stribeck friction model.Figure 3
Clutch friction working modes and switching according to different transmitted torques and relative speeds of clutches.


## 2.3. Planetary Gear Set Module

In this study, 1st gear to 2nd gear upshift is chosen as the example of an AT shifting process for heavy-duty mining trucks. In the first gear, the clutchCS is engaged to connect the sun gear S1 to the carrier C1 of the first planetary gear set P1 at the same speed. The torque is input from the turbine and output from the ring R1 of P1. When the transmission control unit (TCU) issues an upshift command, CS is disengaged by releasing pressure and the on-coming brake BS is engaged gradually. It finally fixes S1 to the transmission housing to stop its movement and enters the second gear state. With brake BL keeping engaged during the first and second gear, the ring R3 of the 3rd planetary gear set P3 is fixed on the transmission housing, while the torque is transferred from the carrier C3 to the transmission output shaft. At all time, the planetary row P3 can be regarded as a reducer with the speed ratio ip3.By analyzing the transmission input shaft (torque converter turbine shaft), the sun shaft and ring shaft ofP1, and the transmission output shaft, the dynamic equations of the gearbox can be written as follows:(4)TT+TCS−TC1=JTω˙T,TS1−TCS−TBS=JS1ω˙BS,TR1=kp1TS1,TR1−TS3=JR1ω˙R1,TO=ip3TS3,where TCS and TBS are the torques of CS and BS; JS1, JR1, and JT are the inertias of the sun and ring of P1 and the transmission input shaft; TS1, TC1, and TR1 are the torque of sun, carrier, and ring of P1, respectively; ωBS is the speed of BS; kp1 is the gear ratio of R1 to S1; TS3 is the torque of S3; and TO is the torque of the transmission output shaft.


## 2.4. The Transmission Output Power Module

The torque outputTO by the AT is finally transmitted to the wheels through the main reducer (speed ratio iFD). For modeling convenience, the longitudinal vehicle model is simplified without considering the vehicle's pitch, yaw, and other motion directions resulting in(5)TO=Jwω˙w+TViFD,where ωw is the wheel speed, rw is the wheel radius, and Jw is the equivalent inertia of wheels. When the heavy-duty mining truck is under the nonbraking condition, the longitudinal resistance torque TV can be summarized from rolling, air, and climbing resistance as follows:(6)TV=rwMgfroll+CDSV21.15vV2+Mgsinα,where rw is the radius of the wheel, M is the full load mass, froll is the rolling resistance coefficient, CD is the air resistance coefficient, SV is the windward area, vV is the vehicle speed, and α is the road slope, respectively.


## 3. LQR Controller Design

The whole shifting process of the heavy-duty mining truck shift can be divided into four stages, which are rapid draining oil (preshift phase), torque phase, inertia phase, and rapid boosting oil (postshift phase) as shown in Figure4. The preshift phase (t1∼t2) and postshift phase (t4∼t5) are designed for eliminating the gap between the clutch plates and hoisting transmission capacity, respectively. In the torque phase, the 1st gear speed ratio is still maintained and torque is transferred from CS to BS. At the end of the torque phase, if TCSt3>0, this forward torque will cause shift shock; otherwise, if TCSt3<0, this negative torque will cause power loss. Therefore, in order to make TCS equal to zero, BS should be able to undertake the transmission of output torque independently:(7)TBSt3=1kp1TT.Figure 4
Shifting process with four phases.Open-loop control is adopted in preshift, postshift, and torque phases because there is no change in the clutch speed [22–24]. However, torque and speed change drastically, last longer, and will produce greater shift jerk and clutch friction loss in the inertial phase, which is why the method based on the LQR is used for this key phase in this paper.


### 3.1. Establishment of State-Space Model

The turbine speedωT, clutch speed ωBS, and clutch pressure pBS of on-coming brake BS are selected as the state variables of the shifting process model:(8)x=ωTωBSpBST.The change rate ofpB is selected as the control variable because it is related to the shift jerk and friction loss of the clutch:(9)u=dpBSdt.From the gearbox dynamic equation, the clutch pressure control equation, and transmission power input and output equations (1)–(6), the powertrain dynamic model can be derived as follows:(10)ω˙T=D22KTCD24ωT+D12D23D24pBS+D12D14−D13D22TL−D22TTD24,ω˙BS=D12KTCD24ωT+−D11D23D24pBS+D11D14+D13D12TL+D12TTD24,p˙BS=dpBSdt,where D11=Jw1+kp12/kp12ip32+JT, D12=Jw1+kp1/kp12ip32, D13=−1+kp1/kp12ip3, D14=−1/kp1ip3, D21=−D12, D22=−Jw1/kp12ip32+JS1, D23=fΔωBSkd, and D24=−D122−D11D22.The equation of the state-space model in the inertia phase can be written as follows:(11)x˙=Ax+Bu+Γ,where A=D22KTC/D240D12D23/D24D12KTC/D240−D11D23/D24000, B=001, and Γ=D12D14−D13D22TL−D22TT/D24D11D14+D13D12TL+D12TT/D240.


### 3.2. Problem State and the Quadratic Cost Function Definition

The shifting process is not completed instantaneously, and a shifting shock is inevitable. To achieve a smooth shift with a low dynamic load of the system, the vehicle jerk, which is the derivative of acceleration w.r.t. time, should be under control. During the 1-2 upshift process, it can be derived from equations (4) and (5) as follows:(12)j=dadt=rwdω˙wdt=rwiFDdTOJwdt=rwiFDJwkp1ip3dfΔωBSkdpBSdt=c1u,where c1=rwiFD/Jwkp1ip3fΔωBSkd.Although increasing the time of the shift process can reduce the vehicle jerk, this method is not advisable because long-term slipping of the clutch generates friction heat and wear and damages the clutch plates finally. In reference to equation (3), clutch friction loss for BS is modeled as follows:(13)W=∫0tTf,BSωBSdt=∫t3t4fωBSkdpBSωBSdt=c2∫t3t4pBSωBSdt,where c2=fΔωBSkd and t3 and t4 represent the start and terminal time of the inertia phase, respectively.As can be seen from equations (12) and (13), the jerk for the shift in the inertial phase is directly proportional to the control variable, whereas clutch friction loss for BS is proportional to the product of its speed ωBS=x2 and pressure pBS=x3. Therefore, the finite-time linear-quadratic optimization performance index for the shifting inertia phase is shown as follows:(14)J=12∫t3t4pBSωBS+ru2dt=12∫t3t4x2x3+ru2dt=12∫t3t4xTQx+ru2dt,where Q=000000.500.50.The first term in the integral function represents the clutch friction loss, and the second represents the jerk. By adjusting the comprehensive weight coefficientr (weight coefficient multiplied by a normalized coefficient), the LQR controller can achieve a proper balance between them.


### 3.3. The Solution of Clutch Pressure Trajectory Based on LQR

According to equations (11) and (14), we introduce the Hamiltonian function [25] as follows:(15)H=12xTQx+ru2+λtTAx+Bu+Γ.It is assumed that the shifting proportional solenoid valve of automatic transmission used in this study has a rapid response [26, 27], and there is no limit to the control variable u. According to the maximum principle, we obtain(16)∂H∂u=rut+BTλt=0,resulting in the optimal control trajectory(17)u∗t=−BTλtr.The normalized equations are as follows:(18)x˙=∂H∂λ=Ax+Bu+Γ,(19)λ˙t=−∂H∂x=−Qx−ATλt.By substituting equation (17) into (18), we obtain(20)x˙t=Axt−BBTλtr+Γ.At the end of the inertial phase, clutchBS speed is zero so that terminal constraint function(21)gTxt4,t4=x2t4=0.Here, we obtain(22)λt4=λ1λ2λ3=μ∂gTxt4,t4∂xt4=0μ0,where μ is the undetermined Lagrange multiplier. Due to the interference matrix Γ, the solution of the conventional LQR, which is letting λt=Ptxt, cannot be used. Here, we set(23)λt=Ptxt+Mtμ+ht,(24)gT=Ktxt+Ltμ+ηt.According to equations (22), (23), and (24), we obtain(25)Pt4=000000000,Mt4=010,ht4=000,Kt4=010,Lt4=0,ηt4=0.Substitution of equation (23) into (19) yields(26)λ˙t=−Q−ATPtxt−ATMtμ−ATht.Substituting equations (20) and (23) into the derivative of equation (23), we obtain affirmatively(27)λ˙t=P˙txt+Ptx˙t+M˙tμ+h˙t=P˙t+PtA−PtBBTPtrxt+M˙t−PtBBTMtrμ−PtBBTrht+PtΓ+h˙t.Comparing equations (26) and (27) w.r.t xt, μ, and rest, we obtain(28)P˙t=−PtA−ATPt+PtBBTPtr−Q,(29)M˙t=PtBBTr−ATMt(30)h˙t=PtBBTr−ATht−PtΓWhen the terminal timet4 is finite, these equations are nonlinear and time varying. Therefore, the differential equations are replaced by the difference equations, and the value at each step can be calculated singly in the reverse direction with −Δt as the time interval from equation (25). Substituting equations (20) and (23) into the time derivative of equation (24) yields the following:(31)g˙T=K˙txt+Ktx˙t+L˙tμ+η˙t=K˙t+KtA−KtBBTPtrxtL˙t−KtBBTMtrμ−KtBBThtr+KtΓ+η˙t=0.Equation (31) holds for any xt and μ, if(32)K˙t=KtBBTPtr−A,(33)L˙t=KtBBTMtr,(34)η˙t=KtBBThtr−Γ.Comparing equation (29) with (32) and Mt4 with Kt4 in equation (25) shows(35)Kt=MTt.Then, we substitute the initial value of the system in equation (24):(36)μ=gTt3−K0x0−η0L0.Finally, the optimal control trajectory in equation (17) is(37)u∗t=−BTλtr=−BTrPtxt+Mtμ+ht=−BTPtrxt−BTrMtμ+ht=Vtxt+Wt,where control parameters Vt=−BTPt/rxt and Wt=−BT/rMtμ+ht.


## 3.1. Establishment of State-Space Model

The turbine speedωT, clutch speed ωBS, and clutch pressure pBS of on-coming brake BS are selected as the state variables of the shifting process model:(8)x=ωTωBSpBST.The change rate ofpB is selected as the control variable because it is related to the shift jerk and friction loss of the clutch:(9)u=dpBSdt.From the gearbox dynamic equation, the clutch pressure control equation, and transmission power input and output equations (1)–(6), the powertrain dynamic model can be derived as follows:(10)ω˙T=D22KTCD24ωT+D12D23D24pBS+D12D14−D13D22TL−D22TTD24,ω˙BS=D12KTCD24ωT+−D11D23D24pBS+D11D14+D13D12TL+D12TTD24,p˙BS=dpBSdt,where D11=Jw1+kp12/kp12ip32+JT, D12=Jw1+kp1/kp12ip32, D13=−1+kp1/kp12ip3, D14=−1/kp1ip3, D21=−D12, D22=−Jw1/kp12ip32+JS1, D23=fΔωBSkd, and D24=−D122−D11D22.The equation of the state-space model in the inertia phase can be written as follows:(11)x˙=Ax+Bu+Γ,where A=D22KTC/D240D12D23/D24D12KTC/D240−D11D23/D24000, B=001, and Γ=D12D14−D13D22TL−D22TT/D24D11D14+D13D12TL+D12TT/D240.


## 3.2. Problem State and the Quadratic Cost Function Definition

The shifting process is not completed instantaneously, and a shifting shock is inevitable. To achieve a smooth shift with a low dynamic load of the system, the vehicle jerk, which is the derivative of acceleration w.r.t. time, should be under control. During the 1-2 upshift process, it can be derived from equations (4) and (5) as follows:(12)j=dadt=rwdω˙wdt=rwiFDdTOJwdt=rwiFDJwkp1ip3dfΔωBSkdpBSdt=c1u,where c1=rwiFD/Jwkp1ip3fΔωBSkd.Although increasing the time of the shift process can reduce the vehicle jerk, this method is not advisable because long-term slipping of the clutch generates friction heat and wear and damages the clutch plates finally. In reference to equation (3), clutch friction loss for BS is modeled as follows:(13)W=∫0tTf,BSωBSdt=∫t3t4fωBSkdpBSωBSdt=c2∫t3t4pBSωBSdt,where c2=fΔωBSkd and t3 and t4 represent the start and terminal time of the inertia phase, respectively.As can be seen from equations (12) and (13), the jerk for the shift in the inertial phase is directly proportional to the control variable, whereas clutch friction loss for BS is proportional to the product of its speed ωBS=x2 and pressure pBS=x3. Therefore, the finite-time linear-quadratic optimization performance index for the shifting inertia phase is shown as follows:(14)J=12∫t3t4pBSωBS+ru2dt=12∫t3t4x2x3+ru2dt=12∫t3t4xTQx+ru2dt,where Q=000000.500.50.The first term in the integral function represents the clutch friction loss, and the second represents the jerk. By adjusting the comprehensive weight coefficientr (weight coefficient multiplied by a normalized coefficient), the LQR controller can achieve a proper balance between them.


## 3.3. The Solution of Clutch Pressure Trajectory Based on LQR

According to equations (11) and (14), we introduce the Hamiltonian function [25] as follows:(15)H=12xTQx+ru2+λtTAx+Bu+Γ.It is assumed that the shifting proportional solenoid valve of automatic transmission used in this study has a rapid response [26, 27], and there is no limit to the control variable u. According to the maximum principle, we obtain(16)∂H∂u=rut+BTλt=0,resulting in the optimal control trajectory(17)u∗t=−BTλtr.The normalized equations are as follows:(18)x˙=∂H∂λ=Ax+Bu+Γ,(19)λ˙t=−∂H∂x=−Qx−ATλt.By substituting equation (17) into (18), we obtain(20)x˙t=Axt−BBTλtr+Γ.At the end of the inertial phase, clutchBS speed is zero so that terminal constraint function(21)gTxt4,t4=x2t4=0.Here, we obtain(22)λt4=λ1λ2λ3=μ∂gTxt4,t4∂xt4=0μ0,where μ is the undetermined Lagrange multiplier. Due to the interference matrix Γ, the solution of the conventional LQR, which is letting λt=Ptxt, cannot be used. Here, we set(23)λt=Ptxt+Mtμ+ht,(24)gT=Ktxt+Ltμ+ηt.According to equations (22), (23), and (24), we obtain(25)Pt4=000000000,Mt4=010,ht4=000,Kt4=010,Lt4=0,ηt4=0.Substitution of equation (23) into (19) yields(26)λ˙t=−Q−ATPtxt−ATMtμ−ATht.Substituting equations (20) and (23) into the derivative of equation (23), we obtain affirmatively(27)λ˙t=P˙txt+Ptx˙t+M˙tμ+h˙t=P˙t+PtA−PtBBTPtrxt+M˙t−PtBBTMtrμ−PtBBTrht+PtΓ+h˙t.Comparing equations (26) and (27) w.r.t xt, μ, and rest, we obtain(28)P˙t=−PtA−ATPt+PtBBTPtr−Q,(29)M˙t=PtBBTr−ATMt(30)h˙t=PtBBTr−ATht−PtΓWhen the terminal timet4 is finite, these equations are nonlinear and time varying. Therefore, the differential equations are replaced by the difference equations, and the value at each step can be calculated singly in the reverse direction with −Δt as the time interval from equation (25). Substituting equations (20) and (23) into the time derivative of equation (24) yields the following:(31)g˙T=K˙txt+Ktx˙t+L˙tμ+η˙t=K˙t+KtA−KtBBTPtrxtL˙t−KtBBTMtrμ−KtBBThtr+KtΓ+η˙t=0.Equation (31) holds for any xt and μ, if(32)K˙t=KtBBTPtr−A,(33)L˙t=KtBBTMtr,(34)η˙t=KtBBThtr−Γ.Comparing equation (29) with (32) and Mt4 with Kt4 in equation (25) shows(35)Kt=MTt.Then, we substitute the initial value of the system in equation (24):(36)μ=gTt3−K0x0−η0L0.Finally, the optimal control trajectory in equation (17) is(37)u∗t=−BTλtr=−BTrPtxt+Mtμ+ht=−BTPtrxt−BTrMtμ+ht=Vtxt+Wt,where control parameters Vt=−BTPt/rxt and Wt=−BT/rMtμ+ht.


## 4. Simulation Results and Discussion

The LQR controller model and the model of the powertrain system for the heavy-duty mining truck were built in Matlab/Simulink with the major simulation parameters shown in Table1. No load without road slope under 50% throttle opening and full load with α=6∘ road slope under 100% throttle opening are chosen as the normal condition and the extreme condition, respectively, to test the preset reference trajectory [8] and optimal trajectory under variations of comprehensive weight coefficients r.Table 1
Simulation parameters.
SymbolValueUnitJw417kg⋅m2JS10.31kg⋅m2JR10.68kg⋅m2JP2kg⋅m2kp12—ip32.67—kd0.04752—iFD22.4—rw0.97mJT2kg⋅m2M72000kgfroll0.03—CD0.8—SV15m3


### 4.1. Normal Working Condition

The no-load mass of the heavy-duty mining truck in this paper is 30 tons. After starting on a flat road with a 50% throttle opening, the simulation results are shown in Figure5.Figure 5
Simulation result with massm=30t, throttle opening θ=50%, and road slope α=0∘.
(a)(b)(c)(d)At 1.0 s, where the pressure ofCS drops to zero and enters the inertia phase, the BS pressure is controlled by the LQR optimal trajectory described above. The entire shift time is about 0.9 s as shown in Figure 5(a). Although the speed of the turbine shaft and BS begins to decrease, unlike the torque phase, they are no longer the same in the inertial phase because of the clutch CS slipping. When the inertial phase ends, the speed of BS drops to zero, and the turbine speed begins to rise again. Compared with the preset reference trajectory, the optimized speed of BS changes more slowly (as shown in Figure 5(b)). Figure 5(c) shows the trends of BS, CS, and turbine torque in the shifting process. With the rise in BS torque, the torque of CS gradually decreases to zero at the end of the torque phase. The reason for the sudden BS torque change at about 1.5 s (inertia phase ends) is that the clutch friction working mode of BS switches from the “captured and accelerating” mode to the “captured and static” mode, which means the sliding friction becomes static friction.The result of shifting jerk and clutch friction loss for the LQR controller is compared with that of the preset trajectory controller in Figure5(d). For r=1∗10−6, the peak value of the optimized jerk jmax=1.33m/s3 is slightly smaller than the peak value jmax=1.37m/s3 before optimization, which satisfies the design requirement of j≤±5m/s3. At the end of the inertial phase, the loss of clutch friction is W=24.7kJ after optimization, which is about 26.3% less than W=33.5kJ for the preset reference controller. By adjusting the weight coefficients to r=3.5∗10−6, shift jerk drops to jmax=1.05m/s3, while clutch friction loss increases to W=31.3kJ, which shows the proposed controller can optimize both objectives.


### 4.2. Extreme Working Condition

The extreme working conditions of the heavy-duty mining truck studied in this paper are 72 tons at full load and 100% throttle opening onα=6∘ road slope. This simulation is for checking the robustness of the LQR controller as shown in Figure 6.Figure 6
Simulation result with massm=72t, throttle opening θ=100%, road slope α=6∘, and comprehensive weight coefficient r=2.85∗10−6.
(a)(b)(c)(d)Due to the small acceleration of the vehicle under this condition, the required shifting speed is reached later, starting from 1.3 s and ending at 2.1 s. Compared with normal conditions, the speed and torque ofCS, BS, and other transmission components are at higher values. The peak speed of BS exceeds 1700r/min and the torque reaches 1000Nm as shown in Figures 6(a)–6(c). In Figure 6(d), it can be seen that after the LQR controller optimizes the clutch pressure, the peak value of the jerk is jmax=1.97m/s3 and the final value of clutch friction loss is W=42.3kJ at the weight coefficient r=2.85∗10−6, which are less than jmax=2.33m/s3 and W=52.2kJ before optimization, respectively. Under different working conditions, the comparison results of the LQR controller with the reference trajectory controller are shown in Table 2.Table 2
Comparison results of the LQR controller with the reference trajectory controller under different conditions.
ConditionControllerjmax (m/s3)Jerk improveW (kJ)Clutch energy loss improveNormalPreset reference1.37—33.5—LQR (r=1∗10−6)1.332.9%24.726.3%LQR (r=3.5∗10−6)1.0523.4%31.36.7%ExtremePreset reference2.33—52.2—LQR (r=2.85∗10−6)1.9715.5%42.319.0%


## 4.1. Normal Working Condition

The no-load mass of the heavy-duty mining truck in this paper is 30 tons. After starting on a flat road with a 50% throttle opening, the simulation results are shown in Figure5.Figure 5
Simulation result with massm=30t, throttle opening θ=50%, and road slope α=0∘.
(a)(b)(c)(d)At 1.0 s, where the pressure ofCS drops to zero and enters the inertia phase, the BS pressure is controlled by the LQR optimal trajectory described above. The entire shift time is about 0.9 s as shown in Figure 5(a). Although the speed of the turbine shaft and BS begins to decrease, unlike the torque phase, they are no longer the same in the inertial phase because of the clutch CS slipping. When the inertial phase ends, the speed of BS drops to zero, and the turbine speed begins to rise again. Compared with the preset reference trajectory, the optimized speed of BS changes more slowly (as shown in Figure 5(b)). Figure 5(c) shows the trends of BS, CS, and turbine torque in the shifting process. With the rise in BS torque, the torque of CS gradually decreases to zero at the end of the torque phase. The reason for the sudden BS torque change at about 1.5 s (inertia phase ends) is that the clutch friction working mode of BS switches from the “captured and accelerating” mode to the “captured and static” mode, which means the sliding friction becomes static friction.The result of shifting jerk and clutch friction loss for the LQR controller is compared with that of the preset trajectory controller in Figure5(d). For r=1∗10−6, the peak value of the optimized jerk jmax=1.33m/s3 is slightly smaller than the peak value jmax=1.37m/s3 before optimization, which satisfies the design requirement of j≤±5m/s3. At the end of the inertial phase, the loss of clutch friction is W=24.7kJ after optimization, which is about 26.3% less than W=33.5kJ for the preset reference controller. By adjusting the weight coefficients to r=3.5∗10−6, shift jerk drops to jmax=1.05m/s3, while clutch friction loss increases to W=31.3kJ, which shows the proposed controller can optimize both objectives.


## 4.2. Extreme Working Condition

The extreme working conditions of the heavy-duty mining truck studied in this paper are 72 tons at full load and 100% throttle opening onα=6∘ road slope. This simulation is for checking the robustness of the LQR controller as shown in Figure 6.Figure 6
Simulation result with massm=72t, throttle opening θ=100%, road slope α=6∘, and comprehensive weight coefficient r=2.85∗10−6.
(a)(b)(c)(d)Due to the small acceleration of the vehicle under this condition, the required shifting speed is reached later, starting from 1.3 s and ending at 2.1 s. Compared with normal conditions, the speed and torque ofCS, BS, and other transmission components are at higher values. The peak speed of BS exceeds 1700r/min and the torque reaches 1000Nm as shown in Figures 6(a)–6(c). In Figure 6(d), it can be seen that after the LQR controller optimizes the clutch pressure, the peak value of the jerk is jmax=1.97m/s3 and the final value of clutch friction loss is W=42.3kJ at the weight coefficient r=2.85∗10−6, which are less than jmax=2.33m/s3 and W=52.2kJ before optimization, respectively. Under different working conditions, the comparison results of the LQR controller with the reference trajectory controller are shown in Table 2.Table 2
Comparison results of the LQR controller with the reference trajectory controller under different conditions.
ConditionControllerjmax (m/s3)Jerk improveW (kJ)Clutch energy loss improveNormalPreset reference1.37—33.5—LQR (r=1∗10−6)1.332.9%24.726.3%LQR (r=3.5∗10−6)1.0523.4%31.36.7%ExtremePreset reference2.33—52.2—LQR (r=2.85∗10−6)1.9715.5%42.319.0%


## 5. Conclusions

This paper proposes a clutch pressure control method by linear-quadratic optimization for the inertia phase of an AT shifting process for a heavy-duty mining truck. The powertrain system model and the LQR controller model are built in Matlab/Simulink. The results show that the LQR optimized clutch pressure control trajectory can reduce the jerk and the clutch friction loss for both normal and extreme working conditions. Compared with the preset reference trajectory, their maximum value can drop by 23.4% for jerk and by 26.3% for clutch energy loss under the normal working condition andjmax=1.97m/s3 and W=42.3kJ at extreme conditions, which indicates that the optimization method results in effective and robust control.



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*Source: 8618759-2020-10-12.xml*