## 3 Internal Stability Analysis  
To reduce the computational burden of calculating the ETDMs of the vehicular platoon, we propose a Dixon resultant matrix-based method to determine the kernel and offspring hypersurfaces of the CTCR paradigm. In this section, it is shown that the dimension of our designed Dixon matrix is only half of that of the well-known Sylvester matrix. The closed-loop platoon dynamics described by (12) should be primarily stable for the delay-free case [29]. Therefore, before analyzing the internal stability for the delay case, we first analyze the internal stability for the delay-free case, and a prerequisite is given. For the delay-free case with \(\tau_{1}=\tau_{2}=0\), (14) becomes  
\[f_{i}(s)=s^{3}+\frac{1+k_{a}}{T}s^{2}+\frac{k_{\text{F}}+hk_{\text{F}}}{T}s+ \frac{k_{\text{F}}}{T} \tag{15}\]  
For a stable platoon, every \(f_{i}(s)=0\) in (15) should be Hurwitz stable. A lemma for the delay-free case is given as follows. **Lemma 1**. : _The closed-loop platoon dynamics (12) are internally stable for the delay-free case if the following inequality holds:_  
\[(1+k_{a})(k_{\text{p}}+hk_{\text{F}})-Tk_{\text{F}}>0 \tag{16}\]  
**Proof. ** The stability of \(f_{i}(s)=0\) in (15) is easy to examine based on the Routh-Hurwitz stability criterion, shown in  
\[ \begin{array}{ccc}s^{3}&1&\frac{k_{\text{p}}+hk_{\text{F}}}{T}\\ s^{2}&\frac{1+k_{\text{F}}}{T}&\frac{k_{\text{F}}}{T}\\ s^{1}&\frac{(1+k_{a})(k_{\text{p}}+hk_{\text{F}})-Tk_{\text{F}}}{T(1+k_{a})}\\ s^{0}&\frac{k_{\text{F}}}{T}\end{array}  \tag{17}\]  
By considering the definition that \(k_{r}>0\), \(k_{\text{p}}>0\), \(k_{a}>0\) in (10), the inequality in (16) is derived. **Remark 1**. : _Lemma 1 provides a necessary and sufficient internal stability condition of the closed-loop platoon dynamics described by (12) for the delay-free case, but it is just a necessary condition for the delay case. For \(\tau_{1}>0\) and \(\tau_{2}>0\), the internal stability of the closed-loop platoon dynamics described by (12) not only relies on the control gains, but also depends on the sensing delay \(\tau_{1}\) and communication delay \(\tau_{2}\)._  
As the stability analysis procedure is identical for each subsystem, the subscript \(i\) is omitted afterward for clarity. To establish the CTCR paradigm, we first replace the exponential terms in (14) by  
\[\mathrm{e}^{-\iota\cdot\tau_{k}\omega}=\cos(\tau_{k}\omega)-\iota\cdot\sin( \tau_{k}\omega),\ k=1,2 \tag{18}\]  
The cosine and sine functions are replaced by the half-angle tangent functions:  
\[\cos(\tau_{k}\omega)=\frac{1-z_{k}^{2}}{1+z_{k}^{2}},\ \sin(\tau_{k}\omega)= \frac{2z_{k}}{1+z_{k}^{2}},\ z_{k}=\tan\Bigl{(}\frac{\tau_{k}\omega}{2}\Bigr{)} \tag{19}\]  
where \(z_{k}\) are the auxiliary variables. Then, the infinite dimension polynomial (14) is transformed into a finite dimension polynomial in \(\omega\):  
\[f(\omega,z_{1},z_{2})=\sum_{m=0}^{3}b_{m}(z_{1},z_{2})\omega^{m}+\iota\cdot \sum_{m=0}^{3}c_{m}(z_{1},z_{2})\omega^{m} \tag{20}\]  
where \(b_{m}(z_{1},z_{2})\) and \(c_{m}(z_{1},z_{2})\) are the corresponding coefficients of \(\omega\). If there exists an imaginary solution for (14) at \(s=\iota\cdot\omega\), both the real and imaginary parts of (20) should be zero simultaneously:  
\[\psi_{1}(\omega)=\mathrm{Re}(f(\omega,z_{1},z_{2}))=\sum_{m=0}^{3}b_{m}(z_{1}, z_{2})\omega^{m}=0 \tag{21a}\] \[\psi_{2}(\omega)=\mathrm{Im}(f(\omega,z_{1},z_{2}))=\sum_{m=0}^{3}c_{m}(z_{1}, z_{2})\omega^{m}=0 \tag{21b}\]  
To reduce the computational complexity, we choose the Dixon resultant instead of the popular Sylvester resultant to solve \(z_{1}\), and \(z_{2}\). The necessary condition for (21a) and (21b) to have a common imaginary root, i.e., \(\iota\cdot\omega\), is described by the Dixon matrix \(D(\psi_{1}(\omega),\psi_{2}(\omega))\) in (2). By scanning the parameters \(\tau_{1}\omega,\ \tau_{2}\omega\) in \([0,\ 2\pi]\), the graphical description of \(\tau_{1}\omega,\ \tau_{2}\omega\) is derived for the case that (3) holds. Then, the values of \(z_{1},z_{2}\) are obtained according to (19).