# A New Driver Model Based on Driver Response
## 3 Stability Analysis  
In this section, the stability of the ID and proposed model is examined. A homogeneous infinite road with identical drivers and vehicles is considered so the alignment to forward conditions is the same, i.e., the acceleration during alignment is uniform, and vehicles maintain the same distance headway \(D\) and corresponding equilibrium velocity \(v_{e}(D)\). Small changes in distance headway are denoted by \(y\) and the corresponding small changes in velocity by \(u(t)\). The distance headway during a change in velocity is then  
\[s=D+y, \tag{14}\]  
and the corresponding velocity is  
\[v=v_{e}(D)+u. \tag{15}\]  
Temporal changes in the distance headway produce changes in velocity during alignment [35] which can be characterized as  
\[y(t)=\frac{dy}{dt}=u_{L}-u_{F}, \tag{16}\]  
where the subscripts \(L\) and \(F\) denote the leading and following vehicles, respectively. \(u(t)\) is a function of the change in distance headway [41] and for a smooth flow is given by  
\[u(t)=\frac{du}{dt}=f_{s}y_{F}+(f_{v}+f_{\Delta v})u_{F}-f_{\Delta v}u_{L}, \tag{17}\]  
with \(f_{s}=\frac{\partial f}{\partial s}\), \(f_{v}=\frac{\partial f}{\partial v}\), and \(f_{\Delta v}=\frac{\partial f}{\partial x}\). For small changes in distance headway and velocity  
\[y(t)=\partial e^{\lambda t+ik}, \tag{18}\]  
\[u(t)=\partial e^{\lambda t+ik}, \tag{19}\]  
Equations (18) and (19) can be expressed as  
\[ \begin{pmatrix}y(t)\\ u(t)\end{pmatrix} = \begin{pmatrix}\dot{y}\\ \dot{u}\end{pmatrix} e^{\lambda t+ik}, \tag{20}\]  
where \(\alpha=\sigma+i\omega\) is the growth rate of change during alignment with \(i=\sqrt{-1}\). The real part \(\sigma\) is the change in amplitude, the imaginary part \(\omega=2\pi/T\) is the corresponding frequency, and \(k\) is the phase shift, i.e., the delay between traffic waves that a driver experiences [41]. \(\hat{y}\) and \(\hat{u}\) are the amplitudes of the changes in distance headway and velocity, respectively. Substituting (20) in (16) and (17) gives  
\[y(t)=u-ue^{ik}, \tag{21}\]  
\[u(t)=f_{y}ye^{ik}+(f_{\hat{v}}+f_{\Delta v})ue^{ik}-f_{\Delta v}u. \tag{22}\]  
A model is stable if the eigenvalues of the Jacobian matrix \(J\) have negative real parts. The eigenvalues can be obtained from  
\[|J-\lambda I|=0, \tag{23}\]  
where \(I\) is the identity matrix and \(J\) is the Jacobian matrix given by  
\[J= \begin{pmatrix}j_{11}&j_{12}\\ j_{21}&j_{22}\end{pmatrix} ,\]  
where \(j_{11}\) and \(j_{21}\) are the differentials of (21) and (22) with respect to \(y\) and \(j_{12}\) and \(j_{22}\) are the differentials of (21) and (22) with respect to \(u\) which gives  
\[J=e^{ik} \begin{pmatrix}0&e^{-ik}-1\\ f_{s}&(f_{\hat{v}}+f_{\Delta v})-f_{\Delta v}e^{-ik}\end{pmatrix}  \tag{24}\]  
Substituting this in (23) results in  
\[\left|e^{ik} \begin{pmatrix}0&e^{-ik}-1\\ f_{s}&(f_{\hat{v}}+f_{\Delta v})-f_{\Delta v}e^{-ik}\end{pmatrix} - \begin{pmatrix} \lambda&0\\ 0&\lambda\end{pmatrix} \right|=0\]  
\[ \begin{pmatrix}\lambda&1-e^{-ik}\\ -f_{s}&\lambda-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\end{pmatrix} =0, \tag{25}\]  
therefore, the eigenvalues are the solutions of  
\[\lambda^{2}+\left(-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\right)\lambda+f_{s} \Big{(}1-e^{-ik}\Big{)}=0. \tag{26}\]  
Let \(A(k)=-f_{v}-f_{\Delta v}+f_{\Delta v}e^{-ik}\) and \(B(k)=f_{s}\Big{(}1-e^{-ik}\Big{)}\) which gives  
\[\lambda^{2}+A(k)\lambda+B(k)=0. \tag{27}\]  
The eigenvalues are then  
\[\lambda_{1,2}=-\frac{A(k)}{2}\Bigg{(}1\pm\sqrt{1-\frac{4B(k)}{A^{2}(k)}}\Bigg{)}, \tag{28}\]  
A group of two or more vehicles maintaining a distance headway is called a string [42]. A traffic model is string stable if the real parts of the eigenvalues (28) are negative [41]. Conversely, a model is unstable if changes in the string increase such as stop and go traffic [16]. With a stable string, changes in flow do not grow. A stable string is necessary for efficient and effective traffic control, and indicates that congestion can be controlled in a flow that maintains the desired distance headway [42].