## 5 Results
### String Stability  
Platoon systems can show string stability issues. It represents a safety concern since a string stable platoon allows to avoid traffic collisions and traffic jams when a long string of vehicles travels along the same route. This evidence is investigated by different literature works, both experimentally and numerically. In particular, the methodology for the assessment of the platoon string stability is described in [11], which states that the sufficient condition to guarantee platoon string stability is:  
\[||H(j\omega)||_{\infty}<1\]  
where:  
\[H(j\omega)=\frac{E_{d,23}(j\omega)}{E_{d,12}(j\omega)}\]  
representing the frequency response function of two consecutive distance errors \(E_{d,23}\) and \(E_{d,12}\), which account for the propagation of perturbances along the platoon. Thus, \begin{table}
\begin{tabular}{l c c} \hline \hline
**Quantity** & **Symbol** & **Value** \\ \hline Equivalent vehicle mass & \(m_{toti}\) & 13,175 kg \\ \hline Wheel radius & \(r_{w,j}\) & 0.5715 m \\ \hline Electric motor maximum & \(p_{m,max}\) & 300 kW \\ power & \(T_{m,max}\) & 600 Nm \\ \hline Total transmission efficiency & \(\eta\) & 95\% \\ \hline Total transmission ratio & \(\tau\) & 19.74 \\ \hline Battery max capacity & \(Q_{batt}\) & 693 Ah \\ \hline Battery Nominal Voltage & \(V_{batt,n}\) & 500 V \\ \hline Initial SOC & \(SOC_{0}\) & 80\% \\ \hline Isolated vehicles drag & \(c_{x,0}\) & 0.57 \\ coefficient & \(\rho\) & 1.2 \\ \hline Vehicle frontal Area & \(A_{f,i}\) & 8.9 m\({}^{2}\) \\ \hline Road-type friction coefficient & \(\mu\) & 0.9 \\ \hline Rolling resistance coefficient & \(f_{0}\) & 0.0041 \\ \hline Rolling resistance coefficient & \(f_{2}\) & 0 \\ \hline Time headway & \(t_{h,des}\) & 1.4 s \\ \hline Nominal speed & \(v_{n}\) & 80 km/h \\ \hline Nominal inter-vehicle & \(d_{n}\) & 36 m \\ distance & \(d_{n}\) & 5\({}^{\circ}\) \\ \hline Nominal road slope & \(\alpha_{n}\) & \(5^{\circ}\) \\ \hline LQC: Q—matrix & Q & \(\text{diag}\Big{(}10^{2},10^{-5},10^{2},10^{-5},20,\,20\Big{)}\) \\ \hline LQC: R—matrix & R & \(10^{-5}\times\text{diag}(1,1)\) \\ \hline RL: reward function weight 1 & \(w_{1}\) & 1 \\ \hline RL: reward function weight 2 & \(w_{2}\) & 1 \\ \hline RL: Actor learning rate & \(lr_{a}\) & \(5\times 10^{-5}\) \\ \hline RL: Critic learning rate & \(lr_{c}\) & \(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Truck data and platooning controller parameters. the string stability is verified by testing the LQC and RL performance by exciting the platoon dynamics with a sine sweep disturbance applied by the lead vehicle velocity. The frequency range is set from \(10^{-4}\) to 5 Hz. The test is carried out at 40 km/h, and the amplitude of the lead vehicle speed disturbance is set to 1.5 km/h. The results in the time domain of the error distances (\(e_{d_{1,2}},e_{d_{2,3}}\)) are reported in Figure 12 for both controllers. The string stability condition is satisfied when the amplitude of \(e_{d_{2,3}}\) (blue line) is less than \(e_{d_{1,2}}\) (red line). The LQC is string-stable across the entire frequency range. On the other hand, the RL-based controller ensures a bandwidth of string stability of \(0-0.3\) Hz. In fact, as can be noted from the time history (top part of Figure 12), the blue line shows an amplitude of oscillation smaller than the red one until 2800 s (when the corresponding frequency is \(f\sim 0.3\) Hz). Nevertheless, this range is compliant with the typical longitudinal dynamics control of heavy-duty trucks, e.g., the frequency range investigated in [11], and was therefore considered satisfactory. Moreover, differently from the LQC, the RL signals are clearly affected by nonlinearities; in fact, the frequency content of the signals includes both fundamental harmonics due to the frequency of the excitation and super-harmonics (see spectrograms in Figure 13) because of control nonlinearity.