# Adaptive Cruise Control for Eco-Driving Based on Model Predictive Control Algorithm
## 2 Materials and Methods
### Framework of Model-Predictive Controller  
The fundamental working mechanism of the MPC can be generalized as follows: future system action is forecasted based on the predictive model and system current state at each sampling moment. The feedback scheme is used to optimize the forecasting error. At the same time, the inaccuracy of the prediction model is corrected in the light of output. Then, through solving an online optimization problem, the optimal control sequence can be obtained, and the first value of this sequence is applied to regulate the system. Next, the control horizon goes a step forward, so that the MPC working mechanism is repeated. Thus, the core characteristics of MPC can be summarized as the predictive model, receding horizon optimization, and feedback correction, which qualifies the MPC for excellent control effect and good robustness [34]. Furthermore, MPC can cope with multiple inputs and multiple outputs systems. Moreover, these inputs and outputs are allowed to have interactions with each other. It can also set constraints for input and output, which makes it a preferential choice when planning and making a detailed design for the upper-level strategy of the ACC system. Utilizing the ACC system longitudinal dynamics model as the prediction model, the predictive state can be calculated as [30]:  
\[\hat{\mathbf{X}}_{p}(t+p\big{|}t)=\overline{C_{0}}x(t)+\overline{C_{1}}\overline{c} (t+c_{h})+\overline{C_{2}}\overline{a_{i}}(t+p) \tag{21}\]  
\[\hat{\mathbf{Y}}_{p}(t+p\big{|}t)=\overline{C_{3}}x(t)+\overline{C_{5}}\overline{c }(t+c_{h})+\overline{C_{6}}\overline{a_{i}}(t+p)-\overline{C_{4}} \tag{22}\]  
attenuation function is applied as the reference variation trajectory of the optimized vector. The reference variation trajectory is defined as [32]:  
\[\mathbf{y}_{r}(t+i)=\rho^{i}\mathbf{y}(t) \tag{25}\]  
where \(\rho=e^{-\frac{T_{0}}{c}}\), \(T_{0}\) and \(c\) are sampling period and time constant of reference trajectory, respectively.  
The objective of eco-driving fuel economy improvement, driving safety and physical limitation of the vehicle are regarded as constraints of MPC [30]:  
\[s.t.\left\{\begin{array}{c}\Delta x(t)\geq x_{\min}\\ v_{\min}\leq v(t)\leq v_{\max}\\ a_{\min}\leq a(t)\leq a_{\max}\\ Jerk_{\min}\leq Jerk(t)\leq Jerk_{\max}\\ c_{\min}\leq c(t)\leq c_{\max}\end{array}\right. \tag{26}\]  
Accordingly, the multiple control objectives are synthesized based on the principle of MPC.  
The above optimization problem can be described as a quadratic programming (QP) question [35].  
In conclusion, at every sampling moment, the real-time vehicle inter-distance, speed difference, host vehicle speed, jerk, as well as acceleration are measured by the proposed ACC system. Using these measurements, the QP problem is solved to acquire the control sequence, and only the first value of the control sequence is adopted at every sampling moment to handle the whole ACC system. In such a way, the entire system operation course is iterated at the next sampling moment. The framework of the MPC-based ACC system is shown in Figure 3.