Jwω˙w = Tw − Ttrac (7)

where, Jw is the wheel inertia。 Ttrac the traction torque applied on the tire produced by the friction between the tire and road surface, i。e。 Ttrac = Ftracrw, and the driving torque Tw  the output shaft torque multiplied by the final drive  ratio i f , i。e。 Tw = i f Tf , and Tf the torque applied to the final differential。 Note that Tw is equal to the output shaft torque Ts under the assumption that the final differential is stiff。 In turn, overall vehicle is considered as a point mass mv。 Accordingly, vehicle longitudinal dynamics (8) with the rolling force Froll and the aerodynamic force Faero  is described  as

Mvv˙v = Ftrac − Froll − Faero − mvg sin θr (8)

III。 PROBLEM STATEMENT

A。 Shift Sequence

The gearshift operation particularly in the case of upshift can be pided into two phases in terms of transmitted torque shown in Fig。 2: one is the torque phase which is from the shift command to the lowest value of the output torque。 The other is the inertia phase where the speed is synchronized with some oscillation due to variation of the kinetic energy from newly connected gear set。 In this paper, two clutches are classified into the oncoming and the off-going clutch corresponding to their own functions。 The oncoming clutch and the off-going clutch only involve engagement and release operation, respectively。

In particular, the 1-2 upshift case is considered for the fea- sibility study of the proposed method。 In Fig。 3, the solid line denotes the engine speed, the upper dotted line ωc1a − ωc1c is

the speed of the oncoming clutch, and the lower dotted   line

ωc2a − ωc2c is the speed of the off-going clutch。 tc means the shift command point, tt  the end point of the torque phase,   ti

Engine, Clutch1,2 Speed

Time [s]Clutch Normal Force

Fig。 3。    Engine and clutch speed   characteristics

the end point of inertia phase, ts the syn with newly engaged clutch。 In t ∈ [tc, tt ], the torque is ramped up for engagement, w clutch torque is gradually reduced to zero。 oncoming clutch torque reaches the desired some slippage。 To minimize the shifting ti oncoming clutch torque should be large in t excessive torque change may lead to the driv

In t ∈ [ti, ts], the oncoming clutch slip may occur and the drive shaft may also oscillate。 After the synchronization point ts, the gear shifting operation is  completed。

B。 Open-loop Control

Since it is difficult to determine the magnitude of the clutch torque and the synchronization points of dual clutches, there exist open-loop gear shifting methods depending heav- ily on experimental calibrations。 In a  nominal  situation, gear shifting performance could be accepted without any undesirable effect。 However, there will be cases in which the unexpected behavior occurs associated with uncertainties, and so, the driver and passengers may feel   discomfort。

For comparison with the subsequent control strategy, the open-loop based control with ramp-type normal force tra- jectories as shown in Fig。 4 is simply designed as one of conventional ways。 After the oncoming clutch is engaged, some oscillations are observed in the clutch speed。 They may lead to the vehicle jerk that is undesirable。 Although such a profile can be further improved from a series of test  results, it has a limitation due to the lack of feedback information of both clutches in the controller。 If the feedback controller employs real-time information as well as prescribed torque trajectories, control performance will be  improved。