For mathematical simplicity, most dynamical models of traffic systems are examined on ring roads where the last vehicle in a platoon affects the first vehicle as if it was in front of it. Moreover, control systems for autonomous vehicles, such as those proposed by Daniel Work et al., have been exclusively tested physically on ring roads. It is assumed that straight line roads are essentially ring roads with infinite radius, but there is no rigorous mathematical justification for the claim that there is no qualitative different between traffic systems on ring roads and line roads in the current literature on traffic analysis. Hence we have reason to doubt that controllers designed with ring roads in mind may perform similarly in real world traffic conditions.
For this reason, our research effort going forward will be to design and analyze controllers on line roads. We begin with a discussion of linear stability of dynamical models of traffic flow, and attempt to answer the question of whether the stability of the system changes when it is subject to different boundary conditions. The analysis below is modifed version of the one conducted by Wang. et al [1].
As always, we consider a system of \(N\) vehicles, labeled with indices \(i=1, 2, \dots, N\) arranged in order of increasing index, so vehicle \(i - 1\) is in front of vehicle \(i\). The dynamics of the system satisfy
\[ \dot{x}_i = v_i \]
\[ \dot{v}_i = f(x_{i - 1} - x_i, v_{i - 1} - v_i, v_i) \]
Where \(f\) is some function governing the acceleration of the vehicle satisfying the following property: For any \(h^* > 0\), there exists a unique \(V(h^*)\) such that \(f(h^*, 0, V(h^*)) = 0\). We call the corresponding function \(V\) the optimal velocity, though \(f\) need not give the Optimal Velocity Model. For simplicity we perform the following change of coordinates
\[ \Delta x_i(t) = x_i(t) - (ih^* + V(h^*)t) \qquad \Delta v_i(t) = v_i(t) - v^*\]
so that
\[ \Delta \dot{x}_i(t) = \Delta v_i(t)\]
\[ \Delta \dot{v}_i(t) = f(\Delta x_{i - 1}(t) - \Delta x_i(t), \Delta v_{i - 1}(t) - \Delta v_i(t), \Delta v_i(t) + V(h^*))\]
Now we compute the linearization of the model. For \(i=2, \dots, N-1\) we have
\[ \dot{X}_i=\begin{bmatrix} \Delta \dot{x}_i(t) \\ \Delta \dot{v}_i(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -\frac{\partial f}{\partial h} & -\frac{\partial f}{\partial v}-\frac{\partial f}{\partial \dot{h}} \end{bmatrix}\begin{bmatrix} \Delta \dot{x}_i(t) \\ \Delta \dot{v}_i(t) \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ \frac{\partial f}{\partial h} & \frac{\partial f}{\partial \dot{h}} \end{bmatrix} \begin{bmatrix} \Delta \dot{x}_{i-1}(t) \\ \Delta \dot{v}_{i-1}(t) \end{bmatrix} = MX_i + NX_{i-1}\]
where
\[ M = \begin{bmatrix} 0 & 1 \\ -\frac{\partial f}{\partial h} & -\frac{\partial f}{\partial v}-\frac{\partial f}{\partial \dot{h}} \end{bmatrix} \qquad N = \begin{bmatrix} 0 & 0 \\ \frac{\partial f}{\partial h} & \frac{\partial f}{\partial \dot{h}} \end{bmatrix}\]
and the expressions for \(i=1\) and \(i=N\) are determined by the boundary conditions. The full state vector is \(X = [X_1^T, X_2^T, \dots, X_N^T]^T\). For a ring road, this gives us the autonomous linear dynamical system \(\dot{X} = RX\) and \(\dot{X} = LX\) for a line road where the lead vehicle moves with constant velocity:
\[ R = \begin{bmatrix} M & 0 & 0 & \cdots & 0 & N \\ N & M & 0 & \cdots & 0 & 0 \\ 0 & N & M & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & N & M \end{bmatrix} \qquad L = \begin{bmatrix} C & 0 & 0 & \cdots & 0 & 0 \\ N & M & 0 & \cdots & 0 & 0 \\ 0 & N & M & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & N & M \end{bmatrix} \qquad C = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\]
References
- Wang, L., Horn, B. K. P. and Strang, G. (2017), Eigenvalue and Eigenvector Analysis of Stability for a Line of Traffic.