# Simulation Model and Method for Active Torsional Vibration Control of an HEV
## 1 Introduction
### Literature Review  
The literature shows that for low-frequency vibration and control, the 16-degree-of-freedom model based on previous work can be simplified to a three-mass model without substantially changing the low natural frequencies of the system [12]. [PERSON] [13] proposed a simple mathematical model to describe the dynamic response of the engine start-stop process, which only requires a small data set for parameterization.  
Models used to characterize the nonlinear characteristics of engine torque mainly include single-cylinder, mean value, and MAP models based on bench test data [14; 15; 16; 17; 18]. The single-cylinder model theoretically describes the real-time output torque of the engine clearly, but most of the model parameters have not been verified. The other two models give the average value of the engine's real-time output, which cannot reflect instantaneous fluctuation of the torque.  
Motor output torque fluctuations are mainly composed of electromagnetic torque and cogging torque. Compared with the engine, the fluctuation amplitude of the motor output torque is small, and it is often studied by analytical and finite element methods [19; 20; 21]. Its mean value can also be obtained from the controller area network (CAN) bus through the controller. Due to the delay between the controller and the actuator, there is a lag in the response of the motor or engine. This time delay affects the dynamic performance of the drivetrain, including comfort during mode switching, or the effect of active vibration control [11; 22].  
In the field of active torsional vibration control, the main method is to estimate the amplitude and phase of the target position torque fluctuation through the real-time status of the system. This can be done using the mode-switching signal, the real-time torsional angle, and the speed of each component using an active actuator to compensate the corresponding torque for reducing system torsional vibration [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. For example, the motor compensation torque can be determined by proportion integration (PI) control [23] or sliding mode control [24] to reduce system vibration by measuring the difference in rotational speed. [PERSON] et al. [25] employed fuzzy adaptive proportion integration differentiation (PID) control for engine speed and dynamic torque compensation control, which effectively suppressed the longitudinal degree of jerk during transition from motor-drive to engine-drive and engine-motor-drive. [PERSON] et al. [26] compared traditional vehicles and HEVs, showing that rapid motor response using PID algorithms can greatly reduce vibration during the shifting process. Aiming at the vibration problem caused by time-varying delay characteristics of CAN-bus data transmission, [PERSON] et al. [22] proposed a robust energy-peak controller to reduce the vibration of the integrated motor-transmission system. [PERSON] and [PERSON] et al. [27] established a predictive model of the power system based on component characteristics and ensured smoothness of power output and vehicle speed by controlling motor torque. [PERSON] et al. [28] designed a robust controller based on the mu-synthesis method to reduce vehicle jerk during HEV mode transitions, and a hardware-in-the-loop (HIL) simulation was performed to verify the proposed controller. [PERSON] et al. [29] designed an adaptive controller which used the fast response characteristics of the motor and the coordinated control of the engine and clutch to smooth the mode switching process. [PERSON] et al. [9] improved shift quality for the P3 HEV using the motor torque compensation method. [PERSON] and [PERSON] [30] used motor output torque composed of a linear combination of constant and time-varying components to offset the nonlinear response of the engine, thereby reducing vibration during engine restart.