# Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors
## 5 Conclusions  
In this paper, a comprehensive study of TV systems for an HEV based on different control methodologies was addressed. Analysis and comparison of the performances obtained were conducted using PID, LQR, and different SMC strategies for the TV applied to an FWD HEV with two electric motors integrated in the rear wheels. All the considered control architectures proved to be suitable for achieving the DYC objectives corresponding to increasing lateral acceleration and improving vehicle responsiveness. The extension of the steering linear region is conditioned by the saturation of the Ems, which generates the torque bias.  
The chattering issue was encountered when plain FOSM was applied and it was successfully reduced by the proposed techniques, especially when the continuous FOSM algorithm was used. The best controllers with respect to the proposed PF were continuous FOSM, LQR, and suboptimal SOSM. Interestingly, the SMC with properly tuned control gains achieved performances very similar to those obtained by LQR, which theoretically should provide important advantages, being a full-state controller. The application of the SMC strategies with a continuous approximation of the sign proved to be particularly effective in terms of reducing the chattering as well as complexity of the controller, proving to be a valid option for the DYC.  
Future work will be devoted to the development of TV systems for HEVs based on innovative and compliant SMC strategies, such as Simplex-based algorithms [25]. Attention will be paid to the inclusion of strategies based on MPC or machine learning, as well as to the experimental and DiL validation [30] of the obtained results.  
Conceptualization, H.d.C.P., M.C. and E.P.; methodology, H.d.C.P., M.C. and E.P.; software, H.d.C.P.; validation, H.d.C.P., M.C. and E.P.; formal analysis, H.d.C.P., M.C. and E.P.; investigation, H.d.C.P., M.C. and E.P.; resources, M.C. and E.P.; data curation, H.d.C.P.; writing--original draft preparation, H.d.C.P.; writing--review and editing, H.d.C.P., M.C. and E.P.; visualization, H.d.C.P.; supervision, M.C. and E.P. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Controller** & **Step 50\({}^{\circ}\)** & **Step 80\({}^{\circ}\)** & **Ramp Steering** \\ \hline OFF & 3.740 & 7.258 & 58.543 \\ PID & 1.000 & 1.469 & 35.981 \\ FOSM lowpass & 0.926 & 1.577 & 35.445 \\ FOSM continuous & 0.641 & 1.009 & 35.011 \\ LQR & 0.608 & 1.104 & 35.042 \\ SOSM twisting & 0.728 & 2.084 & 34.776 \\ SOSM suboptimal & 0.525 & 1.491 & 34.802 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Numerical comparison of the control systems in the three conditions studied using the proposed PF.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Acknowledgments:** The authors want to acknowledge the support of the VI-Grade Italy team during the execution of the work and extend thanks for the availability of the simulation software used in its development.  
**Conflicts of Interest:** The authors declare no conflict of interest.