# Torque Allocation of Hybrid Electric Trucks for Drivability and Transient Emissions Reduction
## 4 Results
### Reference Tracking Performance  
The main purpose of the torque allocation control is to satisfy drivers' requests in terms of vehicle longitudinal acceleration. The driver's intention, expressed in terms of a reference longitudinal acceleration \(a_{x,ref}=\ddot{\theta}_{Ref}R_{w}\), is converted into a total torque request \(T_{e}\), which is obtained by the sum of the FF and FB contributions. The simulation results obtained during a tip-in manoeuvre--i.e., an instantaneous wheel torque request--are shown in Figures 8 and 9, where the influence of the two controller contributions is highlighted. Figure 7: Control effort distribution between the ICE and EM, according to the dynamic limits of the engine. Figure 8: Vehicle state tracking performance during a tip-in manoeuvre with only the FF torque contribution (**a**) and with FF + FB integration (**b**): longitudinal acceleration (top left), front wheels, rear wheels, and primary shaft speeds (top right), rear axle deformation (down left) and torque \(T_{i}\). Figure 9: Control torque request (top), jerk (centre) and weighted states errors (down) during a tip-in manoeuvre with only the FF torque contribution (**a**) and with FF + FB integration (**b**). The control, with and without the activation of the FB contribution, provided a satisfactory reference tracking performance for the speed states. However, the presence of the FB contribution improved the transient vehicle response, with an evident oscillation reduction for the whole set of states. This also implies a considerable improvement in vehicle acceleration performance and in the consequent amplitude of the corresponding jerk (i.e., derivative of the vehicle acceleration), with beneficial implications for vehicle comfort. The advantages provided by the FB integration were paid off with a peak torque request of about 20% higher than that of controller with only the FF term activated. The FB intervention also represented a fundamental contribution to the rejection of external disturbances that may affect controller tracking performance. Indeed, the FF formulation was based on the simplified linear model, and did not include motion resistance increments due to sudden road slopes. The influence of step disturbances on the road slope is analysed in Figure 10.  
The application of the torque requested by only the FF term would compromise the desired performance of the torque controller if external disturbances or uncertain/unmodelled dynamics affected the vehicle dynamics. Indeed, the FF formulation did not include the influence of the road slope, since this is a quantity that is difficult to measure or estimate--thus not representing a suitable input for feeding into a control logic. For this reason, the intervention of the FB term is essential to compensate for the higher vehicle motion resistance by applying an additional torque contribution that is able to bring the longitudinal acceleration back to the reference value; this was automatically achieved  
Figure 10: Control torque request (**a**), road slope (**b**), longitudinal acceleration (**c**) and weighted states errors (**d**) during a tip-in manoeuvre with the FF + FB controller mode. with an acceptable worsening of state errors and without any information related to the road profile. The effect of the FB contribution was, however, influenced by the design of the LQR cost function and the selection of the weights \(q_{i}\) and \(r\). The optimal control theory [35; 36] suggests that the ratio \(q_{i}/r\) is responsible for the trade-off between reference tracking performance and energy consumption. A sensitivity analysis of the weight \(r\) on the torque controller performance is shown in Figure 11.  
For low \(r\) values, the FB contribution aims at improving the reference tracking performance by reducing state errors and the amplitude of the vehicle acceleration and jerk oscillations. This advantage is obtained with a transient peak power request, i.e., the product \(T_{e}\ \dot{\theta}_{e}\) of the engine and the electric motor.