## 1 Introduction  
Owing to the current climate change crisis, the development of automobiles is encountering several environmental issues. Particularly, it is challenging to satisfy customer demands for driving comfort with superior vehicle performance while ensuring low emissions and high energy efficiency. Moreover, the restrictions pertaining to CO2 emissions of automobiles have gradually become stricter worldwide. The use of hybrid electric vehicles (HEVs) represents a realistic and promising solution to these problems until the next generation of low emission applications, such as fuel cells, are developed because such vehicles utilize the existing infrastructure. HEVs generate mechanical power from an internal combustion engine (ICE) and electric motors. ICE converts the fuel combustion energy into the driving force. The reciprocating motion of the piston is converted to rotational motion using a crankshaft. However, owing to the periodic combustion and complex components of the drivetrain, ICEs generate torsional oscillations of multiple orders. Although HEVs have improved fuel efficiency and reduced emissions, the ICE is frequently switched on/off while driving. As these operations occur without driver input, the sudden operation may make certain passengers uncomfortable. This discomfort can be primarily attributed to the torsional oscillation of the engine torque. The sudden changes in the driving torque can be alleviated by reducing the oscillation transmitted to the power-train. Moreover, alternative techniques to reduce fuel consumption and emissions include downsizing and downspeeding. In the downsizing approach, the engine displacement and/or the number of cylinders is reduced, and the associated performance deterioration is compensated via turbocharging. Although energy loss can be reduced in this manner, the vibration may intensify owing to the high combustion pressure and low oscillation speed, which is undesirable.  
To achieve active vibration reduction of HEV powertrains by focusing on torque oscillation during engine start, this study proposes a motor control method using model predictive control (MPC). MPC is a control method that can determine the control input by solving the optimal control problem for a finite future at every control period. Because optimal control can be realized while satisfying the constraints, MPC is advantageous in various applications. However, it has been observed that real-time optimization cannot be realized unless the optimal control problem is solved numerically within the sampling period [1]. In the proposed approach, to facilitate the realization of real-time optimization, we determined the control input for the MPC to be applied in the future state in advance instead of conducting the optimization within the next control period. Specifically, we attempted to improve the control performance of MPC by using an echo state network (ESN) to learn the time trajectory of past disturbances. Additionally, by predicting future disturbances pertaining to the MPC, we aim to achieve active vibration control of the HEV powertrain.  
The remaining paper is organized as follows: Section 2 describes the research background and state-of-the-art solution techniques. Sections 3 and 4 describe the proposed control approach and simulation results for the case study, respectively. Section 5 discusses the results for the considered implementation, and Section 6 presents the concluding remarks.