The engine crank shaft and the clutch dynamics are given by

Jeω˙e = Te(ωe, α) − Td (1)

Jd ω˙d  = Td − Tc1 − Tc2 (2)

where, Je and Jd are the moment of inertia of the engine and the mass flywheel, respectively, Te the engine torque calculated from a steady-state map with respect to the engine speed ωe, and the throttle angle α, Tc1 the clutch 1 torque connected with the solid shaft, and Tc2 the clutch 2 torque connected with the hollow shaft。 Td represents a fly wheel compliance given by Td = kd (θe − θd )+ bd (ωe − ωd ) where kd and bd denote stiffness and viscous coefficients。 By assuming a Coulomb friction model, the clutch torque is represented as

Tck = μckRckFnksign(ωd − ωck), f or k = 1, 2 (3) where,  μc  is  the  friction  coefficient  of  the  clutch surface,

Rc the effective radius of the clutch disk, and Fn the clutch normal force。 The dynamics of the clutch is represented   as

This paper is the first manuscript of the invited session in 2011  American

Control Conference, San  Francisco。

The authors are with the Department of Mechanical Engineering,     KAIST,     Daejeon,     Korea    (jsk@kaist。ac。kr;

Fig。 1。    Vehicle driveline dynamics with a   DCT

are the equivalent moment of inertia    of

Fig。 2。    Schematic of the control concept for a    DCT

the solid and hollow shafts, which can be calculated   as

。 it2if 2 。2

Froll  = Kr Mvg cos θr, Faero =

is the vehicle mass, g the acceleration of  gravity,

。 it1if 1 。2

Jr

θr  road grade, Kr  the rolling stiffness coefficient, ρ  mass

density of air, Cd  coefficient of aerodynamic resistance,   AF

c2 = Jc2 + Jt2 +   i  i

(Jc1 + Jt1)。

t2  f 2

Here, it and i f denote the transmission gear ratio and the final differential ratio, and Jc and Jt stand for moment of  inertia of the clutch and transmission, respectively。 The drive shaft torque Ts  is modeled  as

the frontal area of vehicle, vx  the vehicle speed, and rw  is

the effective wheel radius。 This equation can be explained by force balance between the tractive force and the loads such as aerodynamic and rolling resistance。 Combining   equation

(7) and (8) yields

2   is  obtained by

where, the variable ks denotes the stiffness, bs the damping coefficient of the output shaft, θw the wheel angle, and ωw the wheel speed, respectively。 ωt  denotes the    transmission

adding the wheel inertias to the equivalent inertia of the ve- hicle mass。 The external load torque Tload in above equation is given by

output speed that is set to the corresponding clutch speed depending on the gear ratio (i。e。 ωt = ωc1 for odd number gearbox, and ωt = ωc2  for even gearbox)。

It is reasonable to assume that the tire is considered as

a rolling element without slip。 In addition, the distribution of traction force on the axles is also neglected。 These assumptions make the relationship between the wheel speed ωw and vehicle speed vv simple as vv = rwωw。 Subsequently, the wheel dynamics is