## 3 The SRL-Based Control Strategy for CACC
### System Dynamics  
The primary control objective is to make the host vehicle follow the preceding vehicle at a desired distance \(d_{d}(t)\). The constant time headway spacing policy, which is the most commonly used, is adopted here [22; 23; 24]. The desired inter-vehicle distance is proportional to velocity:  
\[d_{d}(t)=d_{0}+hv_{h}(t) \tag{1}\]  
in which \(d_{0}\) is the desired safe distance at standstill, \(h\) is the time headway, and \(v_{h}(t)\) is the velocity of the host vehicle.  
The longitudinal dynamics model for the host vehicle must consider the powertrain, braking system, aerodynamic drag, longitudinal tire forces, and rolling resistances, etc. The following assumptions are made to obtain a suitable control-oriented model: (1) The tire longitudinal slip is negligible, and the lower level dynamics are lumped into a first order inertial system; (2) The vehicle body is rigid; (3) The influence of pitch and yaw motions is neglected. Then, the nonlinear longitudinal dynamics can be described as  
\[\left\{\begin{array}{c}\dot{s}_{h}(t)=v_{h}(t)\\ \dot{v}_{h}(t)=\frac{1}{m}\Big{(}\eta\frac{T(t)}{r}-\frac{C_{D}A}{21.15}v_{h}^ {2}(t)-mgf\Big{)}\\ \tau\dot{T}(t)+T(t)=T_{des}(t)\end{array}\right. \tag{2}\]  
in which \(s_{h}(t)\) is the position of the host vehicle; \(m\) is the vehicle mass; \(\eta\) is the efficiency of the driveline; \(T(t)\) and \(T_{des}(t)\) are the actual and desired driving/braking torque, respectively; \(r\) is the wheel radius; \(C_{D}\) is the aerodynamic coefficient; \(A\) is the frontal area; \(g\) is the acceleration of gravity; \(f\) is the rolling resistance coefficient; and \(\tau\) is the inertial delay of vehicle longitudinal dynamics.  
With the exact feedback linearization technique [41], the nonlinear model (2) is converted to  
\[T_{des}(t)=\frac{r}{\eta}\Big{(}\frac{C_{D}A}{21.15}v(t)(2\tau\dot{v}_{h}(t)+v _{h}(t))+mgf+mu(t)\Big{)} \tag{3}\]  
Figure 3: Schematic representation of the SRL-based CACC algorithm.  
in which \(u(t)\) is the control input after linearization, i.e., the desired acceleration, and we have  
\[u(t)=\tau\dot{a}_{h}(t)+a_{h}(t) \tag{4}\]  
in which \(a_{h}(t)=\dot{v}_{h}(t)\) denotes the acceleration of the host vehicle. The third-order state space model for the vehicle's longitudinal dynamics is derived as  
\[\left[\begin{array}{c}\dot{s}_{h}(t)\\ \dot{v}_{h}(t)\\ \dot{a}_{h}(t)\end{array}\right]=\left[\begin{array}{ccc}0&1&0\\ 0&0&1\\ 0&0&-1/\tau\end{array}\right]\left[\begin{array}{c}s_{h}(t)\\ v_{h}(t)\\ a_{h}(t)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1/\tau\end{array}\right]u(t) \tag{5}\]  
Using the Euler discretization approach, the state equation of continuous system (5) is discretized with a fixed sampling time \(\Delta t\) as follows:  
\[\left\{\begin{array}{c}s_{h}(t+1)=s_{h}(t)+v_{h}(t)\Delta t\\ v_{h}(t+1)=v_{h}(t)+a_{h}(t)\Delta t\\ a_{h}(t+1)=a_{h}(t)-\frac{\Delta t}{\tau}a_{h}(t)+\frac{\Delta t}{\tau}u(t) \end{array}\right. \tag{6}\]  
Considering the vehicle following system of CACC, we define the system state variables as \(x(t)=[e_{d}(t),\,v_{r}(t),\,a_{r}(t)]\). \(e_{d}(t)\), \(v_{r}(t)\), and \(a_{r}(t)\) are the inter-vehicle distance error, relative velocity, and relative acceleration, respectively, denoted as  
\[\left\{\begin{array}{c}e_{d}(t)=s_{p}(t)-s_{h}(t)-d_{d}(t)\\ v_{r}(t)=v_{p}(t)-v_{h}(t)\\ a_{r}(t)=a_{p}(t)-a_{h}(t)\end{array}\right. \tag{7}\]  
in which \(s_{p}(t)\), \(v_{p}(t)\), and \(a_{p}(t)\) are the position, velocity, and acceleration of the preceding vehicle, respectively.  
After taking the action \(u(t)\) in state \(x(t)\), the system will go to the next state \(x(t+1)\) according to the following transition equations:  
\[\left\{\begin{array}{c}e_{d}(t+1)=e_{d}(t)+v_{r}(t)\Delta t-ha_{h}(t)\Delta t \\ v_{r}(t+1)=v_{r}(t)+(a_{p}(t)-a_{h}(t))\Delta t\\ a_{r}(t+1)=a_{p}(t+1)-a_{h}(t)+\frac{\Delta t}{\tau}a_{h}(t)-\frac{\Delta t}{ \tau}u(t)\end{array}\right. \tag{8}\]