## 6 Conclusions  
Vehicle electrification has emerged as a definitive solution addressing air pollution issues in recent years; leading to a reduction in the reliance on non-renewable energy sources like gasoline that are commonly used in conventional vehicles. Including the various electrification options, hybrid electric vehicles (HEVs) have gained traction, particularly in the specialized domain of power distribution and transmission, they are often referred to as the HEV powertrain. HEVs utilize thermal and electrical energy sources to power their propulsion systems combining at least two distinct energy forms. In this paper, we aim to investigate the stability and behavior of the power generation set within the series equation of a hybrid electric vehicle. To assess the viability conditions, we employed various time-scale techniques. We specifically utilized a Positive Position Feedback (PPF) controller in the Series Hybrid Vehicle (SHR) scenario to mitigate vibrations within the main system. These investigations were carried out using the MATLAB application. Throughout our analysis, we showcased the efficacy of different PPF controller gains while examining various settings. Our findings have emphasized the superiority of numerical solutions over approximate ones in accurately capturing the system's behavior and stability characteristics. The analysis of hybrid electric vehicle powertrain stability findings are as follows:  
Our research focuses on using a positive position feedback controller to improve the stability of a Series Hybrid Vehicle powertrain, particularly in mitigating vibrations. Here is a breakdown of our key findings:  
Figure 17: Comparison between approximate solution (dashed line) and numerical solution (solid line) for the vibration amplitude of (**a**) the main system (**b**) the PPF controller. In order to design the PPF controller effectively, it is essential to adjust its natural frequency, denoted as \(\omega_{2}\), to match the frequencies of both the external force, represented by \(\omega\), and the natural frequency of the (HEVs), denoted as \(\omega_{1}\). This alignment ensures the optimal performance and stability of the controller in response to the external force and the inherent dynamics of the (HEVs) system. The positive position feedback (PPF) controller demonstrates notable effectiveness in mitigating high-amplitude vibrations within nonlinear systems. The SHR and IR case \(\omega\cong\frac{1}{2}\omega_{1}\) & \(\omega_{1}=\omega_{2}\) is one of the vibrating system's most severe resonance cases. The amplitude of the vibrating system has decreased by approximately 98% after utilizing the PPF feedback controller compared to its value without control. The effectiveness of the PPF feedback controller, denoted as \(\text{E}_{\text{a}}\), reaches approximately 90, showcasing its high efficacy in controlling the system's behavior. The response or behavior of the controlled system intensifies with the escalation of the external excitation force \(f\). The response of the main system has decreased with increasing the natural frequency \(\omega_{1}\). The curves are shifted to right with increasing the value of the PPF parameter \(\lambda_{1}\), which is advantageous in the performance of the PPF controller. For the PPF parameter, the amplitude of the controlled system decreases very slowly. The solutions obtained from the frequency response curves (FRC) align well with those derived from the Runge-Kutta 4 th order (RK-4) method. The closed loop response of relative displacement is obtained with PPF controller which comprises the peak-overshoot. The modified structure of the PPF controller is used for the control of relative displacement of suspension system. From the results, the PPF controller provided better closed loop performance in terms peak overshoot and settling time are minimized. Overall Impression:  
Our findings provide valuable insights into the stability characteristics of HEV powertrains and the effectiveness of using a PPF controller for vibration mitigation.