Article Modeling and Optimal Shift Control of a Planetary Two-Speed Transmission
Xinxin Zhao * and Jing Tang School of Mechanical Engineering, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, Beijing 100083, China; [email protected] * Correspondence: [email protected]
Received: 6 August 2019; Accepted: 6 September 2019; Published: 9 September 2019
Abstract: To improve the efficiency of electric vehicles (EVs), a planetary two-speed transmission is proposed, which consists of a brushless direct current (BLDC) motor, a turbo-worm reducer, two multi-disc wet brakes, and a Simpson planetary gearset. Based on the devised electronic actuator for shifting, the rotation direction of the BLDC shaft determines the gear ratio of the transmission. For acquiring smooth shift, the state-space equations with control variables of transmission are derived, and a three-stage algorithm is suggested. During the brake engagement process, the optimal control strategy has been developed using linear quadratic regulator control, considering the jerk and friction work of the brake. The simulation results show that the proposed optimal control strategy could reduce the slipping friction work of the brake and improve the shifting quality of EVs. The optimal control trajectory of the BLDC motor was conducted on the electronic shifting actuator bench test.
Keywords: electric vehicle; gear shifting; optimization; two-speed transmission; optimal quadratic control
1. Introduction Electric vehicles (EVs) are increasingly used due to internal combustion engine-related air pollution and global warming. Compared with traditional powertrain system, the driving motor of an EV is able to generate constant torque at low speed and constant power at high speed, which agrees with the requirements of EVs [1]. The cruising ranges of EVs are extended by increasing the energy-storage capacity or improving the efficiency of the transmission system. However, most EVs are equipped with a fixed ratio gearbox. For acquiring high speed and acceleration, the power and torque of driving motor need to be high enough. Reported studies have found that the gear ratio and shift schedule optimization strategies are related to the efficiency of the EVs equipped with transmissions [2–5]. In the paper, it is shown that the EV with a two-speed transmission has better performance than a fixed ratio gearbox, in terms of acceleration time, maximum speed, and energy economy [6]. Eberleh et al. found that the efficiency in common driving cycles could be increased when a motor coupled with a two-speed transmission is compared to a motor equipped with a fixed ratio gearbox [7]. Sorniotti and Subramanyan compared a conventional single-speed transmission layout, a two-speed layout, and a continuously variable transmission (CVT) layout 1. The two-speed transmission system showed significant advantages over the single-speed layout and CVT layout. Therefore, the EVs, coupled with a two-speed transmission, will become a development trend for enhanced powertrain efficiency. The layout and optimal shift control of the adoption of the two-speed transmission system for EVs have been performed. At present, there are several types of two-speed transmissions including manual transmissions (MTs) [8], automated manual transmissions (AMTs) [9–11], dual-clutch transmissions (DCTs) [12–15], and automatic transmissions (ATs) [16]. Sorniotti and Pilone et al. added a one-way
World Electric Vehicle Journal 2019, 10, 53; doi:10.3390/wevj10030053 www.mdpi.com/journal/wevj World Electric Vehicle Journal 2019, 10, 53 2 of 16 sprag clutch to the traditional MT, although it is difficult to shift seamlessly due to the MTs’ layout [17]. Two clutches are used in a DCT for removing the torque interruption during gear-shifting [18,19]. Ruan et al. [20] developed three case layouts of powertrain system for EVs, and they found that CVT can reduce energy consumption, and manufacturing cost. Actually, the stability of the steel strip restricts the development of the CVT in high power EVs. Shin et al. [16] presented a two-speed Simpson type planetary gearbox. The result of the vibration test, noise test, and transmission efficiency test was obtained, which meet the demand. As we all know, the planetary gear sets system with torque converter changes gear ratios smoothly [21]. The multiple plates clutch with the hydraulic system is often selected as a clutch for planetary transmission. However, the hydraulic system is not available in EVs. Considering the stability and accuracy of the controller, the motor-driven mechanism for gear-shifting is suitable for drive units for EVs, which permits controlling the pressure of the wet clutch with high accuracy. Therefore, it is necessary to develop an electronic actuator for planetary gear sets for a swift and smooth shift. For acquiring smooth shifting, researches have proposed optimization algorithms for different types of transmissions [22–25]. Alizadeh et al. [9] designed an observer to estimate the frictional torque transferred during gear-shifting, and the results demonstrated that the feedback control configuration is suitable to improve the shifting performance of the closed-loop system. The experimental study of the transient shift behavior of an EV coupled DCT was undertaken [12]. Using torque based control techniques, it is possible to achieve high-quality gear-shifting. The authors utilized the torque of the powertrain as an optimal shifting control target through deriving dynamic equations. Roozegar and Angeles used polynomial transition functions as the strategy of the angular velocities based on the calculus of variance [26]. The torques of the driving motor and the clutches during gear-shifting were calculated applying the polynomial transition functions based on the motor specifications considering acceleration and jerk. The optimal torque of the motor was proposed by a linear feed-forward control for inverse AMTs, and a mathematical model was developed for multispeed transmissions in EVs [27,28]. The results showed that the dynamic model of the planetary system could be utilized for optimal shifting control with the response of the transmission. Rahimi et al. designed an observer based on the back-stepping control following the optimal trajectory for minimizing shifting time [29,30]. The shifting controller was designed with the separate discrete-time PID, which improves shifting quality even in the condition of disturbances utilizing a genetic algorithm and trial-and-error [31]. This paper shows that the high fidelity mathematical model and adapted algorithm of a gear-shifting actuator could guarantee the performances of shifting. This paper presents the dynamic model of a two-speed transmission, which includes two brakes and an electronic shifting actuator and develops a three-stage (two sliding friction stages and gap stage) gear-shifting control algorithm. The linear-quadratic optimal control is applied to determine the optimal torque for shifting the motor during sliding friction stages. The speed trajectory of the BLDC motor for shifting was found during the shifting process. The remainder of this paper is arranged as follows. Section2 describes the topology and dynamic model of the proposed shifting actuator. Section3 presents the deducing process of the optimal controller for gear-shifting using the LQR (linear quadratic regulator) method. The simulation results in AMESim and discussions are presented in Section4. A conclusion is shown in Section5.
2. Statement of Planetary Two-Speed Transmission The two-speed planetary gear system with electronic actuators was established, as shown in Figure1. This transmission has a Simpson planetary gearset and two brakes which connect with the ring gears, and the detailed parameters are shown in the Table1. Based on this structure, the speed ratio was around 2, which achieved a smooth shift easily. When brake 1(B1) engaged, the ring gear of the left planetary system became static, and the planetary system became a differential gear train. When brake 2(B2) engaged, the ring gear of the right planetary system became static, and the gear ratio was the larger one. World Electric Vehicle Journal 2019, 10, 53 3 of 16 World Electric Vehicle Journal 2019, 10, 53 3 of 17
Figure 1. Schematic diagram of the planetary two-speed transmission. 1-driving motor; 2-sun gear Figure 1. Schematic diagram of the planetary two-speed transmission. 1-driving motor; 2-sun gear in in planetary system 1(S1); 3-carrier in planetary system 1 (C1); 4-brake 1(B1); 5-ring gear in planetary planetary system 1(S1); 3-carrier in planetary system 1 (C1); 4-brake 1(B1); 5-ring gear in planetary system 1 (R1); 6-worm shaft; 7-worm gear; 8-ring gear in planetary system 2 (R2); 9-carrier in planetary system 1 (R1); 6-worm shaft; 7-worm gear; 8-ring gear in planetary system 2 (R2); 9-carrier in system 2 (C2); 10-sun gear in planetary system 2(S2); 11-brake 2(B2). planetary system 2 (C2); 10-sun gear in planetary system 2(S2); 11-brake 2(B2). Table 1. Parameters of the planetary system. Table 1. Parameters of the planetary system. Component Name Module Number of Teeth Pitch Diameter/mm Mass/kg Inertia/kg.m2 2 Component Name Module Number of Teeth Pitch Diameter/mm Mass/kg Inertia4/kg.m P1 Sun gear s1 2 32 64 / 3.03 10− P1 Sun gear s1 2 32 64 / 3.03× × 410−4 P1 Planet gear p1 2 22 44 0.118 4.03 10− × −4 P1P1 Planet Ring gear gear r1 p1 2 76 22 152 44 / 0.118 0.01827 4.03 × 10 P1P1 Ring Carrier gear c1 r1 2 /// 76 152 / 5.538 0.0182710 6 × − P2 Sun gear s2 2 22 44 / 1.2 10 4 −6 P1 Carrier c1 2 / / / 5.538× − × 10 P2 planet gear p2 2 27 54 0.28 1.5 10 4 P2 Sun gear s2 2 22 44 / 1.2× ×− 10−4 P2 Ring gear r2 2 76 158 / 0.01515 −4 P2 planet gear p2 2 27 54 0.28 1.5 × 106 P2 Carrier c2 2 /// 7.207 10− P2 Ring gear r2 2 76 158 / 0.01515× P2 Carrier c2 2 / / / 7.207 × 10−6 2.1. Planetary Gear Sets Modeling PlanetaryAccording Gear Sets to theModeling Lagrange method, the dynamic model of planetary gear sets is built based on the structure of the two-speed transmission, which includes two sets of planetary gears shown According to the Lagrange method, the dynamic model of planetary gear sets is built based on in Figure2. Considering the generalized meshing equations, there are four constraint equations the structure of the two-speed transmission, which includes two sets of planetary gears shown in for the planetary systems, where α represents the angle of the S1 and S2, β represents the angle Figure 2. Considering the generalized1 meshing equations, there are four constraint1 equations for the of C1 and R2, γ1 represents the angle of pinions in the left planetary, θ1 represents the angle of R1, planetary systems, where α1 represents the angle of the S1 and S2, β1 represents the angle of C1 β2 represents the angle of C2 and γ2 represents the angle of pinions in the right planetary. We assume γ θ β thatand thereR2, is1 norepresents sliding betweenthe angle the of pinions sun gears in andthe left pinions planetary, and pinions 1 represents and ring the gears angle in theseof R1, two 2 planetaryrepresents system. the angle of C2 and γ2 represents the angle of pinions in the right planetary. We assume . . . that there is no sliding between theβ Rsuns1 gearsγ Rs1 an=dα pinions1Rs1 and pinions and ring gears in these two . 1 − 1 . , (1) planetary system. . β1(Rs1 + 2Rp1) + γ1Rp1 = θ1Rr1 . βγ RR.−= α. R R 11ssR 11= 11R s β2 s2 γ2 p2 α1 s2 . βγθ− ++=. . .(1) (2) β (R11(2)RR+sp2R ) 1 + γ 1111R R p= β RR r 2 s2 p2 2 p2 1 r,2
The system has two degrees of freedom (DOF) and (β2, α1) is chosen as the set of independent βγ −= α coordinates. Using the constraint equations, 22RRsp it 2 is 2easy 12 toR s find the relationship between the remaining (2) coordinates and the independent coordinates.βγβ++= 22(2)RRsp 2 2212 R p R r . . γ A (1 + A ) A (1 + A ) . 1 1 4 1β α 4 The system has two degrees of freedom (DOF) and− ( , ) is chosen . as the set of independent θ1 (1 + A2)(1 + A4) A4(1 + A2) A2 β coordinates. Using the constraint. = equations, it is easy− to find the relationship− . 2 between, the remaining (3) 1 + A A β1 4 4 α1 coordinates and the independent . coordinates.1 − 1 γ 2 A3 − A3 World Electric Vehicle Journal 2019, 10, 53 4 of 17
AA(1+−+ ) AA (1 ) γ 14 14 1 (1++−+−AAAAA )(1 ) (1 ) θ 24422β 1 +−2 = 1 AA44 β α (3) 1 1 11 γ − 2 AA World Electric Vehicle Journal 2019, 10, 53 33, 4 of 16 R R R R = s1 = s1 = p2 = s2 where A1 R A2 R A3 Rp2 A4 . where A = sR1 A = s1RA = RA = Rs2 .R 1 Rp1 p1 2 Rr1 r1 3 Rs2 s24 Rr2 r 2
FigureFigure 2. 2.The The Simpson Simpson planetary planetary gearset. gearset.
BasedBased on on the the kinetic kinetic energies energies equations equations in terms in terms of each of each gear, gear, the virtual the virtual work ofwork the systemof the system with thewith virtual the virtual displacements displacements of the independent of the independent coordinates coordinates is derived is derived as as δδδδδWT= +++ T T T δW = Ts1s2δssα121 +αβθβ 1Tc1r cr2 12δβ1 + 1Tr r1 1δθ 11 + cT 2c2δ 2β,2 , (4)(4) TT= TT= TT= wherewhere the the applied applied torque torque on on the the gears ge duears due to friction to friction elements elements is Ts1 sis2 =ss12Tin, T inc1,r 2 =cr12Tb2, bT 2r,1 =rb11Tb1, , =− TcTTcf=2 T .. The The equation equation of of dynamics dynamics and and constraint constraint for two-speedfor two-speed transmission transmission are written are written in matrix in matrix form. 2 − f form. Tin .. " # " T# D11 D12 α1 1 b1 A4 0 in Tb1 .. =α −− . (5) D DDD11 β 12 1 010bbA14b 1 Tb1 T 21 22 2 = 2 3 − b2 β − (5) DD21 22 2 01bb23 Tb2 T f Tf It can be seen that the D11, D12, D21, and D22 are constants from. Equation (5) and are almost always non-singular. Therefore, the input and output angular accelerations have a linear relation with It can be seen that the D11 , D12 , D21 , and D22 are constants from Equation (5) and are almost the friction torque of brakes and input torque. always non-singular. Therefore, the input and output angular accelerations have a linear relation 3.with Shifting the friction Actuator torque of brakes and input torque.
3. ShiftingThe gear-shifting Actuator actuator proposed in this paper consists of a BLDC motor, enveloping worm, worm gear with a screw surface, guide mechanism, and two brakes. Compared with hydraulic actuatorsThe for gear-shifting shifting, the actuator electronic proposed shifting in system this paper has theconsists following of a BLDC advantages: motor, simpleenveloping structure, worm, comfortableworm gear deployment, with a screw and surface, fast response. guide Thismechan electronicism, and actuator two brakes. with a wormCompared gear, whichwith hydraulic enables largeactuators reduction for shifting, ratios to the be electronic achieved, shifting presses syst theem plates has the of thefollowing brake foradvantages: changing simple the gear structure, ratio bycomfortable the BLDC motor. deployment, Due to the and self-locking fast response. characteristics This electronic of the worm actuator system, with the a BLDC worm motor gear, avoids which turningenables on large after reduction the brake isratios engaged. to be Theachieved, BLDC motorpresses drives the plates the worm of the shafts brake which for changing rotate following the gear theratio desired by the direction, BLDC andmotor. the Due range to ofthe the self-locking worm gear characteristics is around 120 degree.of the worm The worm system, gearing the BLDC and guidemotor mechanisms avoids turning with on helicoid after the implement brake is engage translationald. The BLDC motion motor instead drives of rotation, the worm and shafts it pushes which therotate plates following of the brakes the desired for shifting. direction, The structureand the ra ofnge the of electronic the worm shift gear system is around is shown 120 indegree. Figure 3The, butworm the pistonsgearing of and the brakesguide aremechanisms replaced by with the guidehelicoi mechanismd implement with translational a screw surface. motion instead of rotation, and it pushes the plates of the brakes for shifting. The structure of the electronic shift system is shown in Figure 3, but the pistons of the brakes are replaced by the guide mechanism with a screw surface. World Electric Vehicle Journal 2019, 10, 53 5 of 16 World Electric Vehicle Journal 2019, 10, 53 5 of 17
Figure 3. Structure diagram of an electronic actuator. Figure 3. Structure diagram of an electronic actuator. 3.1. The BLDC Motor Used for Shifting 3.1. The BLDC Motor Used for Shifting The BLDC motor drives the worm pairs through a coupler, which overcomes the moment of inertia and the momentThe BLDC of friction.motor drives The dynamic the worm equation pairs through of the BLDC a coupler, motor which displays overcomes as follows: the moment of inertia and the moment of friction. The dynamic equation of the BLDC motor displays as follows: . Jmωm =ωωTm =−Tm0 C −mωm, (6) JTTCmm − m− m0 mm, (6) J ω T wherewherem is J them is inertia the inertia of the of rotor, the ωrotor,m is them angular is the angular velocity, velocity,m is the T outputm is the torque output of torque the BLDC of the motor, Tm0 is the torque of the load, and Cm is the damping coefficient. BLDC motor, Tm0 is the torque of the load, and Cm is the damping coefficient. 3.2. Coupling Model 3.2. Coupling Model An elastic coupler connects the output shaft of the BLDC motor and worm shaft, and the torque An elastic coupler connects the output shaft of the BLDC motor and worm shaft, and the torque of the motor transfers to the shaft of the worm by overcoming the inertia moment of the coupler. of the motor transfers to the shaft of the worm by overcoming the inertia moment of the coupler. The The torque can be calculated as torque can be calculated as dωc Twsi = Tm0 Jc , (7) − dtdω TTJ=− c where T is the input torque of the worm shaft,wsiJ is m0 the c inertia of the coupler, and ω is the angular(7) ωsi c dt , c speed of the coupler. ω where Tωsi is the input torque of the worm shaft, J c is the inertia of the coupler, and c is the 3.3.angular Worm Gearing speed of with the Screw coupler. Surface Model The worm gear with screw surface not only converts the rotation to translation, but the translational motion3.3. Worm moves Gearing along with the screwScrew Surface surface Model to change the displacement accurately. Thus, the guide mechanismThe playsworm the gear role aswith a piston screw to surface change not the workingonly converts conditions the ofrotation the brakes. to translation, The equivalent but the dynamictranslational equation motion of worm moves gearing along system the screw can besurface presented to change as follows: the displacement accurately. Thus, the guide mechanism plays the role as a piston to change the working conditions of the brakes. The Tωso = T Jωsωωs, (8) equivalent dynamic equation of worm gearingwsi system− can be presented as follows: TTJ=−ω Tωgiωωωso= iT wsiωsoη, s s , (9)(8) T J where ωso is the output torque of the worm shaft, ωs =is theη inertia of the worm shaft, ωωs is the angular TiTωωgi so (9) speed of the worm shaft, Tωgi is the input torque of the, worm gear, i is the ratio of the worm ωωs gearing, i = , ωωg is the angular speed of the worm gear, and η is the transmission efficiency ωωg ω T Jω ω of thewhere worm ω system.so is the output torque of the worm shaft, s is the inertia of the worm shaft, s is the angular speed of the worm shaft, Tωgi is the input torque of the worm gear, i is the ratio of the ω i = ωs ω η worm gearing, ω , ω g is the angular speed of the worm gear, and is the transmission ωg efficiency of the worm system. World Electric Vehicle Journal 2019, 10, 53 6 of 16
Analyzing the dynamical performances of the worm gear, the equation of translation along the axis is presented as follows:
.. . MwgSwg = Fws cos γ Fn + Fsr cos α Cs wgSwg, (10) − − − where Mwg is the mass of the worm gear, Swg is the axial displacement of the worm gear, Fws is the support force by the worm, γ is the lead angle of the worm system, Fn is the normal force on plates of the brake, Fsr is the support force by the guide mechanism, α is the spiral angle between the worm gear and guide mechanism, and Cs wg is the axial damping coefficient. − The rotate motion of the worm gear is represented by following dynamical equation.
.. . Jwgθwg = Twg + T FsrRcp sin α Cwgθwg, (11) f − − .. where Jωg is the inertia of the worm gear, θωg is the angular acceleration of the worm gear, Tωg is the driving torque of the worm gear, T f is the friction torque between the worm gear and friction plates, Rcp is the equivalent radius of the friction plates, and Cωg is the damping coefficient of the rotation motion. Since the worm gear contacts with guide mechanism invariably, the geometric constraint between the rotation motion and translation motion is stated by the following equation, which is related to the screw pitch and lift range. The dynamical equation of worm gear could deduce by the mentioned Equations (10)–(12). θwg 8 Swg = 16 = θwg. (12) 2π π
3.4. Modeling of Multi-Disc Wet Brakes The guide mechanism driven by the worm gear presses the friction plates to shift gears. The force of friction plates is analyzed in the following:
.. . McpScp = Fcp CcpScp, (13) − where Mcp is the mass of the friction plates, Scp is the displacement of the plates, and Ccp is the damping coefficient. When the guide mechanism is not in contact with friction plates, the support force is zero. While the guide mechanism presses the plates, the static friction torque Tc f is shown as:
3 3 2 (R0 R1 ) T = µcNFcp − , (14) c f 3 (R 2 R 2) 0 − 1 where µc is the coefficient of the kinetic friction coefficient of static friction, N is the number of plates, and R0 and R1 are the radius of external diameter and inner diameter, respectively. The friction plates are copper-based, and the coefficient of kinetic friction is 0.06–0.08, and the maximum coefficient of static friction is 0.1–0.13. The model of the brake considers the change in the coefficient of friction in off-going states and on-coming states, but the heat loss during sliding mode and the friction resistance by oil stickiness are neglected. Since the normal pressure of plates is difficult to measure during shifting, the displacement of the guide mechanism could be used to judge the state of the brake.
4. Optimal Trajectories in the Gear-Shifting Process The proposed optimal controller for gear-shifting is elaborated in this section. Assuming fixed drag torque during shifting, the driving torque is decided by the angle of acceleration pedal. According to the indexes of shifting quality, the jerk and sliding friction work, the performance index function is designed by the state equations described in Section 2.1. The optimal quadratic control is applied to obtain the shifting controller. World Electric Vehicle Journal 2019, 10, 53 7 of 17
The proposed optimal controller for gear-shifting is elaborated in this section. Assuming fixed drag torque during shifting, the driving torque is decided by the angle of acceleration pedal. According to the indexes of shifting quality, the jerk and sliding friction work, the performance index function is designed by the state equations described in Section 2.1. The optimal quadratic control is applied to obtain the shifting controller.
4.1. Problem Statement of Gear-Shifting
Based on theWorld structure Electric of Vehicle an Journalelectronic2019, 10shifting, 53 actuator, the shifting process includes brake 7 of 16 off-going stage, gap stage, and brake on-coming stage. Since there is no pressure applied on the = = plates, the friction torques of B1 and B2 remain zero during the gap mode, T0b1 , T0b2 . 4.1. Problem Statement of Gear-Shifting However, friction heat is produced by the plates during the brake off-going stage and on-coming ≠ = stage. The friction worksBased of onbrakes the structureare T0b1 of anand electronic T0b2 when shifting B1 actuator,remains thein sliding shifting mode. process includes brake During B2 workingoff in-going the off-going stage, gap stage stage, and and on-coming brake on-coming stage, the stage. torques Since of there friction is no of pressure B1 and appliedB2 on the plates, the≠ friction torques of B1 and B2 remain zero during the gap mode, T = 0, T = 0. However, are T0b1 = and T0b2 , respectively. Therefore, the brake off-going stage and on-coming stageb1 take b2 a long time for smoothfriction shifting, heat is and produced the gap by stage the plates should during take thea short brake time off-going for avoiding stage and interruption. on-coming stage. The friction = The friction torqueworks of the of brakes brakes during are T bthe1 , off-going0 and Tb 2stage0 andwhen on-coming B1 remains stage in slidingare calculated mode. by During B2 working = Equation (14), whichin the corresponds off-going stageto the and torque on-coming of the BLDC stage, motor, the torques given of the friction parameters of B1 andof the B2 are Tb1 0 and brake and supportT b2force, 0 ,of respectively. the guide mechanism. Therefore, the To brakeimprove off-going the shifting stage andquality on-coming and reduce stage take a long time shifting time, the forperformance smooth shifting, indexes and include the gap the stage friction should work take of a shortthe brakes time for and avoiding jerk of interruption. EVs. The friction torque of the brakes during the off-going stage and on-coming stage are calculated by Equation (14), Building the quadratic optimal controller, the optimal trajectory of torque for brakes, T and T , which corresponds to the torque of the BLDC motor, given the parametersb1 of the brakeb2 and support force are obtained duringof shifting. the guide mechanism. To improve the shifting quality and reduce shifting time, the performance To improve theindexes shift quality, include the the optimal friction quadratic work of thecontrol brakes would and be jerk applied of EVs. for Buildingthe off-going the quadratic optimal stage and on-coming stage. During shifting, the torque of the driving motor changes in different controller, the optimal trajectory of torque for brakes, Tb1 and Tb2, are obtained during shifting. stages. Then the drivingTo improve motor theworks shift follow quality,ing the the optimal speed quadratic closed-loop control and would the two-speed be applied for the off-going n transmission beginsstage a down and on-coming shift when stage. the speed During increases shifting, to the the torque objective of the value driving 2 motor. The changesdriving in different stages. motor operates byThen direct the torque driving close-loop motor works control following after shifting. the speed The closed-loop control strategy and the of two-speedthe driving transmission begins motor is shown ina Figure down shift4a, and when Figure the speed 4b shows increases the tospeed the objectivecontrol strategy value n 2of. The the drivingBLDC motor operates by direct during shifting. torque close-loop control after shifting. The control strategy of the driving motor is shown in Figure4a,
and Figure4b showsWorld the Electric speed Vehicle control Journal strategy 2019, 10 of, 53 the BLDC motor during shifting. 8 of 17
(a) (b)
Figure 4. Ideal curve.Figure (a) States 4. Ideal of the curve. driving (a) States motor; of ( bthe) States driving of themotor; brushless (b) States DC of (BLDC) the brushless motor. DC (BLDC) motor.
Generally, the performanceGenerally, indexes the performance of brake consistindexes of of jerk brake and consist friction of work, jerk and which friction influence work, which influence the shifting qualitythe and shifting the work-life quality of theand brakes. the work-life The value of of thethe jerkbrakes. would The make value it uncomfortable of the jerk would make it for passengers, anduncomfortable it also has an for adverse passengers, effect on and the it powertrain also has an system.adverse Therefore,effect on the the powertrain jerk during system. Therefore, shifting should bethe ina jerk certain during range, shifting which should is given be asin a certain range, which is given as da rdTdT d T dT da rw dTO ==dTm wmdTb O =dTb +bb21 + , jaaa2 121 3 (15) j = = = a1 dt+iJa2 dt + a3 dt, dt dt (15) dt i0 JV dt dt 0 V dt dt
where where h i rw D11 r D21D rw D D 11 D21 a1 = w 11a2 = 21 2A4(1 + A3) + A4 rw DD11 21 i0 D22D11 aD21=D12 D11 i0 −D22D11 D21D12 D11 a= 2A() 1+ A + A −1 h− − i 2 43 4 rw D11 D21iDDDD-DA2A4 1 iDD-DD D a3 = 011222A2 11A4 + 212 12 + A2 + 4A2A4 1 + (1 + A3) . 022112112 11 i0 D22D11 D21D12 D11 · A1 A1 − rAADD 1 = w2411 21 ⋅ (). a3 2A24 A +2 + A 2 +4A 24 A 1+ 1+ A 3 iDD-DDD02211211211 A 1 A 1
To evaluate the jerk during shifting, g1 is the performance index, which is represented by t gjdt= 2 , where t is the time of shifting. In addition, the friction work has a relationship with the 1 0 work-life of the brakes, which is expressed as
tt12 WTwdtTwdt=+∣∣ ∣ ∣, (16) 00br11 b 2 r 2
where t1 is the time of B1 for off-going, t2 is the time of B2 for on-coming, and wr is the angular speed of the plates. Since there is a trade-off between the friction work and jerk during shifting, the expression of the comprehensive evaluation index can be represented as
1 t JW=+( α jdt2 ) , (17) 2 0 where α is a weighting coefficient, ( 01< 4.2. Optimal Trajectory Control for Brakes World Electric Vehicle Journal 2019, 10, 53 8 of 16 To evaluate the jerk during shifting, g1 is the performance index, which is represented R t g = j2dt t by 1 0 , where is the time of shifting. In addition, the friction work has a relationship with the work-life of the brakes, which is expressed as Z t1 Z t2 W = Tb1wr1 dt + Tb2wr2 dt, (16) 0 | | 0 | | where t1 is the time of B1 for off-going, t2 is the time of B2 for on-coming, and wr is the angular speed of the plates. Since there is a trade-off between the friction work and jerk during shifting, the expression of the comprehensive evaluation index can be represented as Z ! 1 t J = W + α j2dt , (17) 2 0 where α is a weighting coefficient, (0 < a < 1). When α is a large value, it is shown that the jerk takes a greater proportion during shifting, and it is easy to balance the proportion of friction work and jerk by tuning the coefficient. With regard to the weighting coefficient, α, we tune the coefficient to make the maximum values of the integral of the jerk during shifting close to the value of the friction work. 4.2. Optimal Trajectory Control for Brakes h i We set the state variables as X = x1 x2 x3 and control variable as U = [u], where the state x1 = wr1 variables are presented by x = w and control variable is given as u = dTb1 . Therefore, the state 2 m dt x3 = Tb1 . equation could be expressed by X = AX + BU + V, where A is the coefficient matrix of the state variables, B is the coefficient matrix of control variables and V is interference matrix which depends on the torque of the driving motor and the resistance torque of the vehicle. Because the matrix [ B AB A2B ] is a non-singular matrix, the system is completely controllable. 4.2.1. The B1 Off-Going Stage When B1 works in the off-going stage, the coefficient matrix of the state equation could be deduced by Equation (5), where (1+A +A )D +b D A A (1+A +A )(b D b D )+b A (b D b D ) 0 13 2 4 22 2 21 2 4 2 4 1 22− 2 12 2 2 1 21− 2 11 − 500 A2A4(D21D12 D22D11) D21D12 D22D11 13D − b D −b D A = 0 22 1 22− 2 11 , 500(D D D D ) D11D22 D12D21 21 12− 22 11 − 0 0 0 (1+A2+A4)D22+b2D21 (1+A2+A4)D12+b2D11 187.8 A A (D D D D ) A A (D D D D ) T f h iT 2 4 21 12− 22 11 − 2 4 12 21− 11 22 B = 0 0 1 , V = 258D22 D12 . D D− D D D D D D T f 12 21− 11 22 − 12 21− 11 22 0 Because the matrix [ B AB A2B ] is a non-singular matrix, the system is completely controllable. The cost function is selected as Z t ! Z t 2 1 2 1 J = W + αj dt = x1x3 + α(a3u) dt. (18) 2 0 2 0 R The linear-quadratic function can be expressed by J = (X(t)TQX(t) + U(t)TRU(t))dt, 0 0 0.5 h i 2 where the coefficient is Q = 0 0 0 and R = αa3 , and the designed Hamilton’s function is 0.5 0 0 R H = (X(t)TQX(t) + U(t)TRU(t) + λT(AX(t) + BU(t) + V))dt where λ(t) is the Lagrange multiplier. World Electric Vehicle Journal 2019, 10, 53 9 of 16 When the shifting ends, the torque of the brakes should be 0, which means the terminal constraint is x (tn) = 0. Applying the principle of the minimum, the undetermined matrix is represented 3 0 0 0 by P(tn) = 0 0 0 based on the state of the final value and terminal constraints. 0 0 0 The objective for optimal control is to find the trajectory of friction torque for the brakes while minimizing the cost function. Since there is an interference matrix, the related matrix should be substituted into the equation U(t) = R 1BTλ(t). The solved control variables matrix is the optimal − − trajectory of friction torque for B1 during the off-going stage, and the process of the solving process is developed by a quadratic problem with disturbance. World Electric Vehicle Journal 2019, 10, 53 10 of 17 4.2.2. The Gap Stage The initial state and terminal state of the angular displacement of the BLDC motor output shaft, The initial state and terminal state of= the angular= displacement= of the= BLDC motor output shaft, s()t should follow the equations st()0 0, s()tsff. , st()0 . 0, st()f 0. s(t) should follow the equations s(t0) = 0, s(t ) = s , s(t0) = 0, s(t ) = 0. The angular velocity of the motor tracksf thef optimal trajectory,f which falls in the field of the The angular velocity of the motor tracks the optimal trajectory, which falls in the field of the calculus calculus of variations. Researchers found that a quintic polynomial has good performances during of variations. Researchers found that a quintic polynomial has good performances during shifting. shifting. Therefore, the quintic polynomial, s=ax54321+++++ bx cx dx ex f , could be used as the Therefore, the quintic polynomial, s = ax5 + bx4 + cx3 + dx2 + ex1 + f , could be used as the optimal optimal trajectory during the gap mode for the angular velocity of the BLDC motor. According to the trajectory during the gap mode for the angular velocity of the BLDC motor. According to the constraints s()ttt=+6-151054 t 3 ofconstraints initial and terminalof initial states, and theterminal trajectory states, can bethe calculated trajectory by scan(t) =be6 tcalculated5 15t4 + 10 byt3 shown in Figure5, t t tt− − ( ) ≤≤τ =()τ − 0 ≤≤τ = = 0 =− whereshown 0 ins Figureτ 1, 5, 0 whereτ 1,0sT 1,,01 and T , tτf t0. , and Ttf t0 . ≤ ≤ ≤ ≤ − T 1 0.5 0 s -0.5 ds/dτ d2s/dτ2 d3s/dτ3 -1 0 0.2 0.4 0.6 0.8 1 τ Figure 5. Angular velocity in gap stage. Figure 5. Angular velocity in gap stage. 4.2.3. The B2 On-Coming Stage 4.2.3. The B2 On-Coming Stage The state equation of B2 is related to the dynamic function of planetary system P2, which is The state equation of B2 is related to the dynamic function of planetary system P2, which is the the same as the off-going stage. The state variables include wr2, wm, and wb2, and the control variable w w w same as the off-going stage. The state variables xinclude1 = wr2 r 2 , m , and b2 , and the control dTb2 dTb2 is , where the state variable is represented by x2 = wm and u== . The state equation is dt x12wr dt dT dTb2 = b2 variable is , where the state variable is representedx3 = Tb by = and u . The state 2 x2 wm represented bydt dt x = T A D +A b D +A D +(b D +A D )(1+1/A 32) b 13 4 22 4 − 3 12 4 22 3 11 4 21 4 0 500 (D D D D ) A (D D D D ) equation is represented− 21 12 by− 22 11 4 21 12− 22 11 h iT A = 13D22 b3D12+A4D22 , B = 0 0 1 0 ( ) D D D D 13500 DA+A21DD12 D22D11 −+++bD+AD()(11/) bD21 12 AD22 11 A 0 − 4 22 − 4 312 42221 311 −44 0500()DD 0- DDA () DD - D 0D T 21 12 22 114 21 12 22 11 = [ ] A D +A D12+(1+A4)D11 B 001 187.8 134D 22 4 + bD +ADT = (D D 22 D D ) A (D D D3122D )42f A 0 21 12− 22 11 4 12 21− 11 22 = 500()DD258-D DD22 D12 DD- DD V 21 12 22 11 21 T12f 22 11 D12D21 D11D22 D12D21 D11D22 00− 0 − − − 0 [ 2 ] , Because the matrix B AB A B is a non-singular matrix, the system is completely controllable. AccordingA+DD to theAAD state matrix++ of(1 thegear-shifting ) system, the optimal quadratic algorithm 187.8 4422 + 12 411T ()DD-- DDA () DD DD f 21 12 22 114 12 21 11 22 258DD = −−22 12 VTf DD- DD DD- DD 12 21 11 22 12 21 11 22 0 2 Because the matrix []BABAB is a non-singular matrix, the system is completely controllable. According to the state matrix of the gear-shifting system, the optimal quadratic algorithm is applied to solve the trajectory of angular velocity for the BLDC motor. Since there is limited shifting time and computational capabilities, the trajectory can be fitted in different conditions, which are saved into the controller. The optimal trajectory can be found rapidly for real-time control. WorldWorld Electric Electric Vehicle Vehicle Journal Journal2019 2019, 10, 10, 53, 53 1011 ofof 1617 World ForElectric improving Vehicle Journal the 2019 shift, 10, 53 quality, the optimal controller is designed based on quadratic11 of 17 is applied to solve the trajectory of angular velocity for the BLDC motor. Since there is limited shifting optimization in this section for the down shifting process. As mentioned, there are three stages of time and computational capabilities, the trajectory can be fitted in di erent conditions, which are saved shifting,For andimproving the linear-quadratic the shift quality, optimization the optimal is appliedcontroller for ff isthe designed B1 off-going based stageon quadratic and B2 intoon-comingoptimization the controller. stage. in this The section optimaloptimal for trajectory controlthe down rate can shifting of be the found brakesprocess. rapidly is Ascalculated for mentioned, real-time by control.thethere quadratic are three algorithm, stages of shifting,For improving and the thelinear-quadratic shift quality, the optimization optimal controller is applied is designed for the based B1on off-going quadratic stage optimization and B2 where VP() t is the feedback matrix and the gain of the controller. inon-coming this section stage. for The the downoptimal shifting control process. rate of the As brakes mentioned, is calculated there areby threethe quadratic stages of algorithm, shifting, andwhere the VP linear-quadratic() t is the feedback optimization matrix and is the applied gain of for theppp the controller. B1 off-going , stage and B2 on-coming =− =− 13 23 33 stage. The optimal control rate ofVP the() t brakes [ v123 v is v calculated ] [ by the quadratic ] algorithm, where VP(t) rrr, (19) is the feedback matrix and the gain of the controller. ppp WtVP()() t=−=− [ [w v ] v = v w ] =− [13 23 33 ] 123 (19) ( rrrp13 p23 p33 VP(t) = [v1 v2 v3] = [ r r r ] pij Wt()=−− [ w ] = w − , (19) where, is the elements of Wmatrix(t) = P [andw] = rw is the parameter of matrix R . Therefore, the control − variable p is given as =− + − + . Following the main steps in Figure 6, the where, iju is the elementsuvwvwvTw of matrix11rmb P and 2 31is the parameter of matrix . Therefore, the control where, Pij is the elements of matrix P and r is ther parameter of matrix R. Therefore,R the control variable uoptimalvariableis given trajectory as isu given= ofv 1stateaswr 1 + variable=−v2wm couldv +3Tb1 be+ w −obtained. Following + based. Following the on mainthe Riccati the steps main matrix. in Figuresteps 6in, theFigure optimal 6, the u − uvwvwvTw11−rmb 2 31 trajectory of state variable could be obtained based on the Riccati matrix. optimal trajectory of state variable could be obtained based on the Riccati matrix. Figure 6. Block diagram of an optimal controller for torque. According to the FigureoptimalFigure 6. 6. Block controlBlock diagramdiagram algorithm, of of an an optimal optimathe controllerl controller controller diagram for for torque. torque. of electronic actuators is shown in Figure 6. Because the shifting process lasts only for a short time, it is impossible to solve According to the optimal control algorithm, the controller diagram of electronic actuators is shown the optimalAccording trajectory to the optimalby optimal control quadratic algorithm, control the controllerfor the BLDC diagram motor of electronic in that process. actuators We is inshown Figure in6. Figure Because 6. the Because shifting the process shifting lasts process only forlasts a short only time,for a itshort is impossible time, it is to impossible solve the optimal to solve calculated the parameters VP() t and in different shifting conditions, then they were fitted trajectorythe optimal by optimal trajectory quadratic by optimal control forquadraticWt the() BLDC control motor infor that the process. BLDC Wemotor calculated in that the process. parameters We VPbycalculated( at) polynomialand W the(t) parametersin function. different shiftingThoseVP() t fit conditions,and coefficients Wt() then in are different they saved were in shifting fittedthe control by conditions, a polynomial unit, which then function. isthey used were Thoseto fitfitted the fit coecontrolfficients policy are savedduring in real-time the control control. unit, which is used to fit the control policy during real-time control. by a polynomial function. Those fit coefficients are saved in the control unit, which is used to fit the 5.5.control SimulationSimulation policy ResultsResultsduring real-time andand DiscussionsDiscussions control. The dynamic model of EVs included the power system module, two-speed transmission, 5. SimulationThe dynamic Results model and Discussionsof EVs included the power system module, two-speed transmission, vehiclevehicle system,system, andand driver-environmentdriver-environment module,module, whichwhich werewere built built in in AMESim. AMESim. TheThe simulationsimulation modelmodelThe of of EVs EVsdynamic isis shownshown model in in Figure Figure of EVs7 .7. included the power system module, two-speed transmission, vehicle system, and driver-environment module, which were built in AMESim. The simulation model of EVs is shown in Figure 7. FigureFigure 7. 7.Simulation Simulation model model in in AMESim. AMESim. Figure 7. Simulation model in AMESim. WorldWorld Electric Electric Vehicle Vehicle Journal Journal2019 2019, 10,, 10 53, 53 1112 of of 16 17 With NEDC (new European driving cycle) drive cycle, the input of the model was the desired With NEDC (new European driving cycle) drive cycle, the input of the model was the desired velocity of the vehicle, and the difference between the desired value and actual velocity exported the velocity of the vehicle, and the difference between the desired value and actual velocity exported signals of acceleration and brake to the drive-environment module. In addition, the vector control the signals of acceleration and brake to the drive-environment module. In addition, the vector control method was applied to the driving motor by the PI control for angular velocity and torque. The method was applied to the driving motor by the PI control for angular velocity and torque. The states states of the driving motor followed the desired torque and angular velocity by adjusting the current of the driving motor followed the desired torque and angular velocity by adjusting the current when when the desired torque and actual torque were compared. The shifting controller of the BLDC the desired torque and actual torque were compared. The shifting controller of the BLDC motor motor was similar to the driving motor controller, and the PI control was used to track the objective was similar to the driving motor controller, and the PI control was used to track the objective curve curve for gear-shifting. Actual values of the powertrain system parameters are given in the for gear-shifting. Actual values of the powertrain system parameters are given in the Nomenclature. Nomenclature. During the NEDC drive cycle, the simulation results showed that the velocity of the During the NEDC drive cycle, the simulation results showed that the velocity of the vehicle followed vehicle followed the desired value, which consisted of three times upshifts and downshifts. the desired value, which consisted of three times upshifts and downshifts. 5.1.5.1. Simulation Simulation Results Results Following Following NEDC NEDC Drive Drive Cycle Cycle Based on the simulation result with the NEDC drive cycle, we chose the first downshift, which Based on the simulation result with the NEDC drive cycle, we chose the first downshift, which is is shown in Figure 8 by an arrow. Two control policies were applied during downshift, and one is shown in Figure8 by an arrow. Two control policies were applied during downshift, and one is named named the original control, which used the BLDC motor speed fitted by the 3-4-5 polynomial the original control, which used the BLDC motor speed fitted by the 3-4-5 polynomial mentioned mentioned in Section 4.2 during shifting. The other is the optimal control, which used the BLDC in Section 4.2 during shifting. The other is the optimal control, which used the BLDC motor speed motor speed calculated by the optimal quadratic algorithm during the sliding stages and the shifting calculated by the optimal quadratic algorithm during the sliding stages and the shifting speed fitted speed fitted by the 3-4-5 polynomial during the gap stage. by the 3-4-5 polynomial during the gap stage. 50 desired speed actual speed 40 current gear 2 30 20 current gear vehicle speed/(km/h) 10 0 1 0 50 100 150 200 250 300 time/s Figure 8. Curves of gear number and angular velocity in the new European driving cycle (NEDC) Figure 8. Curves of gear number and angular velocity in the new European driving cycle (NEDC) driving cycle. driving cycle. The BLDC motor speed follows the curve shown in Figure4b, which is called the original control. The BLDC motor speed follows the curve shown in Figure 4b, which is called the original The angular velocity followed the desired velocity based on the optimal quadratic algorithm when control. The angular velocity followed the desired velocity based on the optimal quadratic algorithm the simulation results of the original control and optimal control were compared. The controller when the simulation results of the original control and optimal control were compared. The of the BLDC motor had a swift response for tracking drive cycle. During the first down-shifting, controller of the BLDC motor had a swift response for tracking drive cycle. During the first the maximum jerk with the optimal controller reduced by 35%. As can be seen from Figure8, the indexes down-shifting, the maximum jerk with the optimal controller reduced by 35%. As can be seen from of shifting quality have degraded gracefully, which led to acquiring a smooth and swift shift by tracking Figure 8, the indexes of shifting quality have degraded gracefully, which led to acquiring a smooth the desired trajectory. and swift shift by tracking the desired trajectory. 5.2. Verification for Optimal Control Trajectory 5.2. Verification for Optimal Control Trajectory According to the optimal angular velocity of the BLDC motor, the testbed of the actuator for shiftingAccording in our to lab the is shownoptimal in angular Figure9 .velocity The output of the shaft BLDC of the motor, BLDC the motor testbed connects of the theactuator worm for shaft,shifting and in the our brakes lab is system shown was in Figure neglected. 9. The In output the test, shaft the controllerof the BLDC of the motor BLDC connects motor outputsthe worm optimizedshaft, and angular the brakes velocity, system which was wasneglected. measured In the and test, shown the controller on a computer. of the ToBLDC drive motor the BLDCoutputs motor,optimized the NI-CRIO angular was velocity, adopted which as the was main measured controller. and LabVIEW shown on was a computer. applied for To communications drive the BLDC betweenmotor, thethe computerNI-CRIO was and adopted the NI-CRIO as the NI9401 main controller. boards for LabVIEW data acquisition. was applied The test for bench communications is shown inbetween the Figure the 10 computer. and the NI-CRIO NI9401 boards for data acquisition. The test bench is shown in the Figure 10. WorldWorld ElectricElectric VehicleVehicle JournalJournal 201920192019,,, 10 1010,,, 535353 131312 ofof 17 1617 ((a)) ((b)) Figure 9. PerformancesPerformances during during down-shifting. down-shifting. (aa)) (OptimalaOptimal) Optimal rotaryrotary rotary angleangle angle duduringring during down-shifting;down-shifting; down-shifting; ((b)) (JerkJerkb) Jerk withwith with differentdifferent different controcontro controllerllerller duringduring during down-shifting.down-shifting. down-shifting. Figure 10. TheThe response response test of the BLDC motor. The speed regulation of the BLDC motor was adjusted adjusted by by the the PWM PWM wave wave through through LabVIEW. LabVIEW. TheThe dutyduty ratioratio of of the the PWM PWM wave wave had had a lineara linear relationship relationship with with the angularthe angular velocity velocity after after testing testing with withdifferent different loads loads for the for worm the worm gear with gear a with screw a surfacescrewscrew surfacesurface and the andand worm thethe system. wormworm system.system. Tracking TrackingTracking the optimal thethe optimalangular velocityangular ofvelocity the BLDC of the motor, BLDC the motor, velocity the mode velocity was mode adopted was to adopted follow theto follow optimal the trajectory optimal trajectoryobtainedtrajectory withobtainedobtained the quadraticwithwith thethe quadraticquadratic algorithm. algorithm.algorithm. Figure 11 Figurea,b represent 11a,b represent the measured the measured result of result the actualof the actualrotary rotary angle andangle the and error the between error between the actual the angleactual and angleangle desired andand desireddesired angle. Dueangle.angle. to DueDue the inhibiting toto thethe inhibitinginhibiting device devicefor the actuator,for the actuator, the error the did error not existdid not at the exist end at of the the end test. of The the maximumtest. The maximum value of the value error of appearedthe error appearedat the end at of the the end gap of stage the whengap stage the polynomial when the po waslynomiallynomial applied, waswas and applied,applied, it began andand to it decreaseit beganbegan toto along decreasedecrease with alongthe optimal with the torque optimal control. torque Since control. the impact Since the often impact happened often inhappened the off-going in the stage off-going and on-coming stage and on-comingstage, the maximum stage, the maximum error of angular error of velocity angular failed veloci totyty influence failedfailed toto influenceinfluence the shifting thethe quality. shiftingshifting quality. Thequality. testbed TheThe verifiedtestbedtestbed verifiedverified the track thethe eff ecttracktrack with effecteffect optimal withwith quadraticoptimaloptimal quadquad control,raticratic and control,control, the response andand thethe response ofresponse BLDC motorofof BLDCBLDC achieved motormotor achievedthe real demands. the real demands. World Electric Vehicle Journal 2019, 10, 53 13 of 16 World Electric Vehicle Journal 2019, 10, 53 14 of 17 60 desired angle 50 actual angle 40 30 20 rotary angle/° 10 0 6.2 6.4 6.6 6.8 time/s (a) 1 0.5 0 -0.5 angle error/° -1 -1.5 6.2 6.4 6.6 6.8 time/s (b) FigureFigure 11.11.Test Test results results of of the the rotary rotary angle angle of the of BLDCthe BLDC motor. motor. (a) Rotary (a) Rotary angle; angle; (b) Error (b) ofError rotary of angle.rotary angle. 6. Conclusions 6. ConclusionsDue to the complex structure of a clutch compared to brakes, a planetary two-speed transmission was designedDue to forthe EVs,complex which structure was only coupledof a clutch with compared brakes instead to brakes, of clutches. a planetary The optimal two-speed control fortransmission the shifting was process designed of an electronic for EVs, actuator which withwas aonly screw coupled surface with was proposed.brakes instead By using of clutches. the Lagrange The method,optimal thecontrol dynamics for the model shifting of theprocess designed of an planetary electronic two-speed actuator with transmission a screw andsurface electronic was proposed. actuator modelBy using were developed.the Lagrange Then, method, a three-stage the optimaldynamics control model algorithm of the was designed developed planetary considering two-speed different stagestransmission for gear-shifting. and electronic The controllers actuator of model the BLDC were motor developed. were designed Then, usinga three-stage quadratic optimal optimal controlcontrol theoryalgorithm for off-goingwas developed and on-coming considering stages, different and variation stages for calculus gear-shifting. for the angular The controllers velocity wasof the applied BLDC inmotor the gap were stage. designed The optimal using torque quadratic trajectories optimal were control proposed theory by using for off-going the quadratic and method on-coming considering stages, theand external variation disturbances calculus for during the angular sliding velocity friction stages.was applied When in jerks the andgap frictionstage. The work optimal of the brakestorque weretrajectories compared, were it proposed was concluded by using that thethe optimalquadrati quadraticc method control considering for torque the improvedexternal disturbances the shifting qualityduring for sliding EVs. Basedfriction on stages. the test When bench ofjerks the and electronic friction actuator, work experimentalof the brakes results were demonstratedcompared, it thatwas theconcluded optimal that trajectory the optimal could bequadratic achieved control by the for designed torque controller improved of the the shifting BLDC motor. quality for EVs. Based Authoron the Contributions: test bench ofConceptualization, the electronic actuator, X.Z. and experimental J.T.; methodology, results X.Z.; demonstrated software, X.Z. and that J.T.; the validation, optimal X.Z.trajectory and J.T.; could formal be analysis, achieved X.Z.; by Writing—Original the designed controller draft preparation, of the BLDC X.Z.; motor. Writing—Review and editing, X.Z. and J.T.; funding acquisition, X.Z. Funding:Author Contributions:This work was Conceptualization, financially supported X.Z. by and the J.T.; Fundamental methodology, Research X.Z.; Fundssoftware, for X.Z. theCentral and J.T.; Universities validation, ofX.Z. China and (Grant J.T.; formal No. FRF-TP-18-036A1), analysis, X.Z.; writing—original the National Science draft preparation, Foundation forX.Z.; Young writin Scientistsg—review of Chinaand editing, (Grant X.Z. No. 51905031)and J.T.; funding and the acquisition, National key X.Z. research and development program (Grant No. 2018YFC0604402). The APC was funded by Grant No. FRF-TP-18-036A1. Funding: This work was financially supported by the Fundamental Research Funds for the Central Universities Conflicts of Interest: The authors declare no conflict of interest. of China (Grant No. FRF-TP-18-036A1), the National Science Foundation for Young Scientists of China (Grant No.51905031) and the National key research and development program (Grant No. 2018YFC0604402). The APC Nomenclature was funded by Grant No. FRF-TP-18-036A1. i final ratio 0Conflicts of Interest: The authors declare no conflict of interest. i2 1st gear ratio i 2 2nd gear ratio World Electric Vehicle Journal 2019, 10, 53 14 of 16 η1 efficiency of 1st gear η2 efficiency of 2nd gear η0 efficiency of reducer 2 J1 rotational inertia of input shaft (kg m ) 2 J2 rotational inertia of output shaft (kg m ) 2 JC inertial of brakes (kg m ) 2 Jm inertia of BLDC motor shaft (kg m ) TCAX maximum friction torque of brake (Nm) Cds friction coefficient of driving shaft Kds stiffness of driving shaft (Nm/◦) Cs damp coefficient of driving shaft (Nm/(r/min)) M mass of EV (kg) ρ air density (Kg/m3) A area of windward (m2) CD coefficient of air resistance F coefficient of rolling resistance R radius of wheel (m) α slope (◦) g acceleration of gravity (m/s2) 2 J0 inertia of electronic actuator (kg m ) 2 J1 inertial of powertrain system (kg m ) 2 J2 inertial of wheel (kg m ) ψfl flux of driving motor (Wb) Rs stator resistance (Ω) Ls inductance of driving motor (H) Lsd inductance of d axis (H) Lsq inductance of q axis (H) Imax maximum effective current (A) UDC direct voltage (V) P pole pairs R0 the radius of external diameter (m) R1 the radius of inner diameter (m) N the number of plates µs the coefficient of kinetic friction µd the maximum coefficient of static friction References 1. Sorniotti, A.; Subramanyan, S.; Turner, A.; Cavallino, C.; Viotto, F.; Bertolotto, S. Selection of the optimal gearbox layout for an electric vehicle. SAE Int. J. Engines 2011, 4, 1267–1280. [CrossRef] 2. Sorniotti, A.; Holdstock, T.; Pilone, G.L.; Viotto, F.; Bertolotto, S.; Everitt, M.; Barnes, R.J.; Stubbs, B.; Westby, M. Analysis and simulation of the gearshift methodology for a novel two-speed transmission system for electric powertrains with a central motor. Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 2012, 226, 915–929. [CrossRef] 3. Zhang, Z.; Zuo, C.; Hao, W.; Zuo, Y.; Zhao, X.; Zhang, M. Three-speed transmission system for purely electric vehicles. Int. J. Automot. Technol. 2013, 14, 773–778. [CrossRef] 4. Nicola, F.; Di Sorniotti, A.; Holdstock, T.; Viotto, F.; Bertolotto, S. Optimization of a multiple-speed transmission for downsizing the motor of a fully electric vehicle. SAE Int. J. Altern. Powertrains 2012, 1, 134–143. [CrossRef] 5. Wu, G.; Zhang, X.; Dong, Z. Impacts of two-speed gearbox on electric vehicle’s fuel economy and performance. SAE Tech. Pap. 2013. Available online: https://www.researchgate.net/publication/289058822_Impacts_of_ Two-Speed_Gearbox_on_Electric_Vehicle\T1\textquoterights_Fuel_Economy_and_Performance (accessed on 6 August 2019). 6. Gao, B.; Liang, Q.; Xiang, Y.; Guo, L.; Chen, H. Gear ratio optimization and shift control of 2-speed I-AMT in electric vehicle. Mech. Syst. Signal Process. 2015, 50, 615–631. [CrossRef] World Electric Vehicle Journal 2019, 10, 53 15 of 16 7. Eberleh, B.; Hartkopf, T. A high speed induction machine with two-speed transmission as drive for electric vehicles. In Proceedings of the Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM 2006), Taormina, Italy, 23–26 May 2006; pp. 249–254. 8. Ren, Q.; Crolla, D.A.; Morris, A. Effect of geared transmissions on electric vehicle drivetrains. In Proceedings of the ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, San Diego, CA, USA, 30 August–2 September 2009; pp. 959–966. 9. Alizadeh, H.V.; Mousavi, M.R.; Boulet, B. Synchromesh torque estimation in an electric vehicle’s clutchless automated manual transmission using unknown input observer. In Proceedings of the 2015 IEEE Vehicle Power and Propulsion Conference (VPPC), Montreal, QC, Canada, 17 December 2015; pp. 1–5. 10. Fang, S.; Song, J.; Song, H.; Tai, Y.; Li, F.; Nguyen, T.S. Design and control of a novel twospeed uninterrupted mechanical transmission for electric vehicles. Mech. Syst. Signal Process. 2016, 75, 473–493. [CrossRef] 11. Gao, B.; Xiang, Y.; Chen, H.; Liang, Q.; Guo, L. Optimal trajectory planning of motor torque and clutch slip speed for gear shift of a two-speed electric vehicle. Journal of Dynamic Systems. Meas. Control 2015, 137, 061016. [CrossRef] 12. Walker, P.; Zhu, B.; Zhang, N. Powertrain dynamics and control of a two speed dual clutch transmission for electric vehicles. Mech. Syst. Signal Process. 2017, 85, 1–15. [CrossRef] 13. Gu, Q.; Cheng, X.S. Study on double-clutch transmission gear shift control based on optimal control theory. In Proceedings of the 2011 IEEE 18th International Conference on Industrial Engineering and Engineering Management, Changchun, China, 10 October 2011; pp. 629–634. 14. Zhu, B.; Zhang, N.; Walker, P.; Zhan, W.; Zhou, X.; Ruan, J. Two-speed DCT electric powertrain shifting control and rig testing. Adv. Mech. Eng. 2013, 5, 323917. [CrossRef] 15. Heap, A.H.; Sah, J.J.F.; Kaminsky, L.A. Control of Engine Torque During Upshift and Downshift Torque Phase for a Hybrid Powertrain System. U.S. Patent 9,008,926, 14 April 2015. 16. Shin, J.W.; Kim, J.O.; Choi, J.Y.; Oh, S.H. Design of 2-speed transmission for electric commercial vehicle. Int. J. Automot. Technol. 2014, 15, 145–150. [CrossRef] 17. Sorniotti, A.; Pilone, G.L.; Viotto, F.; Bertolotto, S.; Everitt, M.; Barnes, R.; Morrish, I. A novel seamless 2- speed transmission system for electric vehicles: Principles and simulation results. SAE Int. J. Engines 2011, 4, 2671–2685. [CrossRef] 18. Cavina, N.; Corti, E.; Marcigliano, F.; Olivi, D.; Poggio, L. Control-Oriented Models for RealTime Simulation of Automotive Transmission Systems. Oil Gas Sci. Technol. Rev. díIFP Energies Nouv. 2015, 70, 67–90. [CrossRef] 19. Liang, Z. Reliability Analysis of Dual Clutch Transmission. Automob. Appl. Technol. 2016, 2016, 98–100. 20. Ruan, J.; Walker, P.D.; Wu, J.; Zhang, N.; Zhang, B. Development of continuously variable transmission and multi-speed dual-clutch transmission for pure electric vehicle. Adv. Mech. Eng. 2018, 10, 1687814018758223. [CrossRef] 21. Lazar, C.; Caruntu, C.F.; Balau, A.E. Modelling and predictive control of an electrohydraulic actuated wet clutch for automatic transmission. In Proceedings of the 2010 IEEE International Symposium on Industrial Electronics, Bari, Italy, 4–7 July 2010; pp. 256–261. 22. Heijden, A.V.D.; Serrarens, A.F.A.; Camlibel, M.K.; Nijmeijer, H. Hybrid optimal control of dry clutch engagement. Int. J. Control 2007, 80, 1717–1728. [CrossRef] 23. Dutta, A.; Ionescu, C.M.; De Keyser, R.; Wyns, B.; Stoev, J.; Pinte, G.; Symens, W. Robust and two-level (nonlinear) predictive control of switched dynamical systems with unknown references for optimal wet-clutch engagement. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 2014, 228, 233–244. [CrossRef] 24. Ngo, V.; Hofman, T.; Steinbuch, M.; Serrarens, A. Optimal control of the gearshift command for hybrid electric vehicles. IEEE Trans. Veh. Technol. 2012, 61, 3531–3543. [CrossRef] 25. Kum, D.; Peng, H.; Bucknor, N.K. Supervisory control of parallel hybrid electric vehicles for fuel and emission reduction. J. Dyn. Syst. Meas. Control 2011, 133, 061010. [CrossRef] 26. Roozegar, M.; Angeles, J. The optimal gearshifting for a multi-speed transmission system for electric vehicles. Mech. Mach. Theory 2017, 116, 1–13. [CrossRef] 27. Liang, Q.; Ren, L.; Zhao, H.; Gao, B.; Hong, C. Gearshifting Control of Battery Electric Vehicle with 2- Speed I-AMT Autimotive Engineering. In Proceedings of the 2013 China Automobile Engineering Conference, Beijing, China, 26 November 2013; pp. 1000–1003. World Electric Vehicle Journal 2019, 10, 53 16 of 16 28. Setiawan, Y.D.; Roozegar, M.; Zou, T.; Angeles, J. A Mathematical Model of Multispeed Transmissions in Electric Vehicles in the Presence of Gear-shifting. IEEE Trans. Veh. Technol. 2018, 67, 397–408. [CrossRef] 29. Mousavi, M.S.R.; Pakniyat, A.; Helwa, M.K.; Boulet, B. Observer-based backstepping controller design for gear shift control of a seamless clutchless two-speed transmission for electric vehicles. In Proceedings of the 2015 IEEE Vehicle Power and Propulsion Conference (VPPC), Montreal, QC, Canada, 17 December 2015; pp. 1–6. 30. Mousavi, M.S.R.; Boulet, B. Estimation of the state variables and unknown input of a two-speed electric vehicle driveline using fading-memory Kalman filter. IEEE Trans. Transp. Electrif. 2016, 2, 210–220. [CrossRef] 31. Roozegar, M.; Angeles, J. A two-phase control algorithm for gear-shifting in a novel multispeed transmission for electric vehicles. Mech. Syst. Signal Process. 2018, 104, 145–154. [CrossRef] © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).