## 3 Stability Analysis  
### Time-Delay Independent Stability  
A time-delay dynamical system is said to be time-delay independent stable if it is asymptotically stable for any given \(\tau^{*}\geq 0\). According to the theory of time-delay differential equations, we obtain the following characteristic equation [36]:  
\[\det(s\mathbf{I}-\mathbf{A}-\mathbf{B}\mathrm{e}^{-s\tau^{*}})=0, \tag{9}\]  
where \(s\) is the characteristic root of the characteristic Equation (9); and \(\mathbf{I}\) is the identity matrix. The characteristic Equation (9) is expanded into a polynomial form, which can be expressed as  
\[D(s,\tau^{*})=P(s)+Q(s)\mathrm{e}^{-s\tau^{*}}, \tag{10}\]  
\(P(s)\) and \(Q(s)\) are real coefficient polynomials of the sixth and fourth order. These can be expressed as  
\[ \begin{array}{l}P(s)=s^{6}+p_{5}s^{5}+p_{4}s^{4}+p_{3}s^{3}+p_{2}s^{2}+p_{1} s+p_{0}\\ Q(s)=q_{4}s^{4}+q_{3}s^{3}+q_{2}s^{2}+q_{1}s+q_{0}\end{array}  \tag{11}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(\zeta_{1}\) & 0.6194 & \(g^{*}\) & \(-\)1.7647–1.7647 \\ \(\zeta_{2}\) & 0–0.0909 & \(k_{1}\) & 11.2941 \\ \(\beta\) & 42.5926 & \(k_{2}\) & 1.8783 \\ \(\gamma\) & 8.5185 & \(\tau\) & 0–7.0196 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensionless model parameters. where  
\[ \begin{array}{l}p_{5}=\zeta_{1}+\beta\zeta_{2}+\gamma\zeta_{1}+\gamma\zeta_{2}\\ p_{4}=1+\gamma+\beta k_{2}+\gamma k_{1}+\gamma k_{2}+\beta\zeta_{1}\zeta_{2}+ \gamma\zeta_{1}\zeta_{2}+\beta\gamma\zeta_{1}\zeta_{2}\\ p_{3}=\beta\zeta_{2}+\gamma\zeta_{2}+\beta\gamma\zeta_{2}+\beta\zeta_{1}k_{2}+ \gamma\zeta_{1}k_{1}+\gamma\zeta_{1}k_{2}+\beta\gamma\zeta_{1}k_{2}+\beta \gamma\zeta_{2}k_{1}\\ p_{2}=\beta k_{2}+\gamma k_{1}+\gamma k_{2}+\beta\gamma k_{2}+\beta\gamma k_{1}k_{ 2}+\beta\gamma\zeta_{1}\zeta_{2}k_{1}\\ p_{1}=\beta\gamma\zeta_{2}k_{1}+\beta\gamma\zeta_{1}k_{1}k_{2}\\ q_{4}=-\beta g^{*}\\ q_{3}=-(\beta\zeta_{1}g^{*}+\beta\gamma\zeta_{1}g^{*})\\ q_{2}=-(\beta\gamma g^{*}+\beta g^{*}+\beta\gamma g^{*}k_{1})\\ q_{1}=-\beta\zeta_{1}\gamma k_{1}g^{*}\\ q_{0}=-\beta\gamma k_{1}g^{*}\end{array} .\]  
According to the stability theory, the critical time-delay stability is the characteristic equation with pure imaginary roots \(s=i\omega_{0}\). For the time-delay independent stability of Equation (10), the following two conditions need to be satisfied simultaneously:  
**Condition 1**. : _When \(\tau^{*}=0\), the characteristic polynomial \(D(s,0)=P(s)+Q(s)\) is Routh-Hurwitz stable._  
**Condition 2**. : _When \(\tau^{*}>0\), the polynomial \(D(s,\tau^{*})=0\) has no real root \(\omega_{0}\)._  
For Condition 1, when \(\tau^{*}=0\), Equation (10) can be expressed as  
\[D(s,0)=P(s)+Q(s)=a_{6}s^{6}+a_{5}s^{5}+a_{4}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+ a_{0}, \tag{12}\]  
where \(a_{6}=1\), \(a_{5}=p_{5}\), \(a_{4}=p_{4}+q_{4}\), \(a_{3}=p_{3}+q_{3}\), \(a_{2}=p_{2}+q_{2}\), \(a_{1}=p_{1}+q_{1}\), and \(a_{0}=p_{0}+q_{0}\). According to the Routh-Hurwitz stability criterion, to ensure the asymptotic stability of the system, it is necessary to satisfy that the principal minor of each order of the determinant is greater than zero. The parameters of Table 3 are substituted into Equation (12), and the Routh-Hurwitz stability dividing line is judged to be obtained, as shown in line R\({}_{1}\) in Figure 3. Figure 3 is divided into two parts by lines R\({}_{1}\). The orange and green areas on the upper are the Routh-Hurwitz stability area, the blue area below does not meet the vehicle selection parameters. Figure 3: Time-delay independent stability: Time-delay independent stability region; Time-delay dependent stability region;  
For Condition 2, judge whether \(D(i\omega_{0},\tau^{*})=0\) has real roots. To facilitate the calculation, the real and imaginary parts of the polynomial can be separated by bringing [PERSON]'s formula \(\mathrm{e}^{-i\omega_{0}\tau^{*}}=\cos(\omega_{0}\tau^{*})-i\sin(\omega_{0}\tau^ {*})\) into Equation (10).