# Autonomous Controller-Aware Scheduling of Intra-Platoon V2V Communications
## 3 System Model  
Let us assume a motorway scenario where a platoon of \(N\) vehicles (led by the so-called leader vehicle) drives in a coordinated way, with the longitudinal motion being controlled using the autonomous platoon controller, as shown in Figure 1. The system model can be then described in a discrete way, with the time step of \(\Delta t\), using the following formulas. \[\mathbf{s}(k+1)=\mathbf{As}(k)+\mathbf{u}(k)+\mathbf{z}(k), \tag{1}\]  
where:  
* \(\mathbf{s}(k)\) is the system state vector at time \(k\Delta t\), defined as: \[\mathbf{s}(k)=\big{[}x_{0}(k),v_{0}(k),a_{0}(k),x_{1}(k),v_{1}(k),a_{1}(k), \ldots,x_{N-1}(k),v_{N-1}(k),a_{N-1}(k)\big{]}^{T},\] (2) where \(x_{i}\), \(v_{i}\), and \(a_{i}\) stand for the longitudinal position, velocity, and acceleration of vehicle \(i\), respectively. * \(\mathbf{A}\) is the state transition matrix, representing the position, velocity. and acceleration changes according to the accelerated motion law: \[\mathbf{A}= \begin{bmatrix}1&\Delta t&\frac{\Delta t^{2}}{2}&0&0&0&\ldots&0&0 &0\\ 0&1&\Delta t&0&0&0&\ldots&0&0&0\\ 0&0&1&0&0&0&\ldots&0&0&0\\ 0&0&0&1&\Delta t&\frac{\Delta t^{2}}{2}&\ldots&0&0&0\\ 0&0&0&0&1&\Delta t&\ldots&0&0&0\\ 0&0&0&0&0&1&\ldots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ 0&0&0&0&0&0&\ldots&1&\Delta t&\frac{\Delta t^{2}}{2}\\ 0&0&0&0&0&0&\ldots&0&1&\Delta t\\ 0&0&0&0&0&0&\ldots&0&0&1\end{bmatrix} ,\] (3)* \(\mathbf{u}(k)\) is the controller actions vector (acceleration changes). * \(\mathbf{z}(k)=[z_{1}(k),z_{2}(k),\ldots,z_{3N}(k)]^{T}\) is the disturbances vector. In this work, we focus only on a case where the acceleration disturbance (acceleration change due to external reasons) may occur only to the leader vehicle, so: \(\forall_{i\
eq 3}\,z_{i}(k)=0\). A more general scenario with acceleration disturbances being applied also to other platoon vehicles is left for future study. We assume that all platoon vehicles employ a CACC platoon controller proposed in [6], where a leader-predecessor-follower model is considered. Therefore, each platoon vehicle, except for the leader, updates its acceleration based on the mobility information received from both the platoon leader and its predecessor. The controller actions \(\mathbf{u}(k)\) are then calculated according to the formula:  
\[u_{j}(k)= \begin{cases}\max\{\min\{\hat{a}_{i}(k),a_{\max}\},a_{\min}\}-a_{i}(k)& \text{if }j=3(i+1),i=0,\ldots,N-1\\ 0&\text{otherwise}\end{cases} , \tag{4}\]  
with \(a_{\min}\) and \(a_{\max}\) denoting the minimum and maximum acceleration constraints, respectively, and \(\hat{a}_{i}(k)\) representing the autonomous controller-calculated acceleration:  
\[\hat{a}_{i}(k)=\overbrace{\alpha_{1}(d_{d}-x_{i-1}(k)+x_{i}(k))}^ {\text{Location update}}-\overbrace{\alpha_{2}(v_{i-1}(k)-v_{i}(k))}^{\text{ Velocity update}}\] \[-\underbrace{\alpha_{3}(v_{0}(k)-v_{i}(k))}_{\text{Leader}- \text{related velocity update}}+\underbrace{\alpha_{4}a_{i-1}(k)+\alpha_{5}a_{0}(k)-a_{i}(k)} _{\text{Acceleration update}}, \tag{5}\]  
where \(\alpha_{l},l\in\{1,2,3,4,5\}\) are the controller coefficients (constants). Furthermore, we assume an ideal engine model, applying immediately the requested acceleration (the impacts of engine imperfections and their modeling are studied e.g., in [29]). The first term in (5), i.e., \(\alpha_{1}(d_{d}-x_{i-1}(k)+x_{i}(k))\) represents the impact of the inter-vehicle distance error. The second and third terms represent the influence of the velocity difference vs. the preceding and the leader vehicle, respectively. Similarly, the fourth and fifth terms indicate the impact of the preceding and leader vehicles' accelerations. Such a controller definition assumes that the updates are applied in a continuous manner. However, in a realistic platooning scenario, the information on the mobility of the platoon leader or the preceding car needs to be transmitted using wireless communication, and is updated only when a new intra-platoon message is received.