## 1 Introduction  
The plug-in hybrid electric vehicle (PHEV) is one kind of new energy vehicle that has attracted widespread attention from automobile manufacturers [1]. Compared with traditional internal combustion engine (ICE) vehicles, PHEVs have better system efficiency, fuel economy and emission performance because the driver can choose different working modes [2, 3]. During the mode-switching process of starting the engine, the vehicle driveline will often produce obvious shock and vibration. For a power-split PHEV, the engine ripple torque (ERT) will be directly transmitted to the powertrain system [4]. In the process of mode transition (MT), the transmitted ERT will cause obvious impacts and vibrations on the power transmission system, especially during engine starting [5, 6]. Such transients make driving less comfortable and reduce the service life of the transmission. Some control methods have been applied to the typical engine starting process to reduce vehicle shocks. [PERSON] [7] formulated dynamic and economic shift schedules in an MT process and coordinated the motor torque to compensate for the fluctuation torque. In [8], considering the inertial torque of the motor, a motor compensation coordinated control strategy was designed to reduce unwanted jerks. Taking the speed fluctuation of engine and the fuel consumption rate as the control objective, [PERSON] [9] designed a mode-switching control strategy based on a logical threshold. [PERSON] [10] proposed a control strategy of torque-dynamic coordination by means of a sliding mode control based on a disturbance compensation. These methods made the coordinated control strategy during MT process reduce vehicle jerks, but they ignored the influence of the ERT and elastic characteristics of the transmission system on the shocks of the powertrain. Based on the data from an engine start-up test, researchers in [11; 12] proposed a lookup table model of the ERT during engine start-up to estimate the engine pulsating torque and then used motors to compensate or offset the torque fluctuation of the powertrain. [PERSON] [13] and [PERSON] [14] set up a reduced-order engine thermodynamic model to estimate the ERT and used a belted starter/alternator to reduce the shocks and vibrations caused by ERT. [PERSON] [15] analyzed the MT process during the engine start-up and used engine torque estimation based on planetary dynamic equations to reduce vehicle jerks. In the implementation of such schemes, these methods helped estimate ERT to reduce vehicle jerks. [PERSON] [16] designed a coordinated control strategy to determine target engine torque, motor torque, clutch torque and the moment of fuel injection to suppress vehicle jerks and improve riding comfort during engine start. [PERSON] [17] divided the engine start into three phases, analyzed the influences of various parameters on the jerk in each phase, and then proposed a coordinated control strategy between the clutch pressure and motor torque and between the motor torque and engine torque. However, the influence of the elastic (stiffness-damping) characteristics of transmission on the powertrain vibrations was not taken into account. [PERSON] [18] used the approximate function of ERT and the proportional feedback control of the speed difference to reduce the fluctuations of the powertrain. [PERSON] [19] proposed an active damping-control strategy during the starting process to reduce the speed fluctuations of the transmission system. [PERSON] [20] used a staged torque estimation of the engine and an active damping-compensation control method based on drive shaft torque observations to reduce the jerks of the powertrain. [PERSON] [21] proposed a fuzzy control and a two degrees of freedom damping-control algorithm to reduce vehicle jerks. In these control methods, the ERT and the elastic characteristics of the transmission system were considered, but the influence of the limitations of battery-motor characteristics on vehicle jerks under different engine starting conditions was ignored.