Optimal Speed Profile for Minimum Vibration during Engine Start Using Pontryagin’s Minimum Principle Approach

An imperceptible engine start is critical to the acceptance of hybrid vehicles. This paper focusses on an optimal control problem that tries to reduce vibration during engine start. Efforts are made to obtain the optimal speed trajectory that could cause minimum vibration during engine start. In the first section, the target diesel powertrain is introduced. A four cylinder diesel engine is coaxially paralleled with an ISG motor. The ISG motor serves as the engine starter and engine flywheel. Its dynamic model is established using crank-link dynamics. Secondly, an index is brought out to evaluate the severity of vibration. The cylinder pressure variation is the main cause of engine torque ripple, which in turn results in engine speed fluctuation. The square of the angular acceleration is chosen as the index of vibration. The index shows a positive relation of cylinder pressure in terms of amplitude. Then, the author models this problem as a continuous-time optimal control problem with a fixed terminal time and a partially free terminal state, then solve it by the Pontryagin’s minimum principle. Finally, this paper presents an optimal engine start speed trajectory which is linearly increasing with time, and the corresponding control input should vary with engine torque ripple. The optimal control input of ISG torque consists of two parts. One is the constant value to counter the rotating inertial, and the other varies with engine torque to compensate for the torque ripple.