Clutches are widely used in various vehicle powertrains. The engagement process of a friction clutch has three phases, i.e., open, slipping, and sticking. Transitions between different phases introduce a discontinuity to the powertrain dynamics, which has been neglected in previous research. A model referenced adaptive controller (MRAC), based on Popov hyper-stability criterion, is designed to compensate the discontinuity. MRAC adjusts the frictional torque along with the errors of the state variables compared with those of a referenced model. The designed MRAC is applied to a clutch in a bus. Simulation and experimental results under fast and slow startup cases show that MRAC can simultaneously reduce vehicle jerk and frictional dissipation when compared with the conventional controller.