# Integrated Adaptive Cruise Control with Weight Coefficient Self-Tuning Strategy
## 4 Predictive Control Algorithm for IACC
### Final Expression of Predictive Control Algorithm  
In order to prevent the slack variables from increasing toward infinity, an additional quadratic term of slack variables is defined to penalize the violation of hard constraints so that it will be possible to make a balance between computing feasibility and the relaxation of constraints. The cost function in the finite predictive horizon, shown as Equation (24), is rewritten as  
\[J(\mathbf{y},\mathbf{u},\mathbf{\varepsilon})\,=J+\mathbf{\varepsilon}^{T}\mathbf{\rho}\mathbf{\varepsilon} \tag{28}\]  
where \(\mathbf{\varepsilon}=[\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon _{4},\varepsilon_{5}]^{T}\) is the slack variable vector, and \(\mathbf{\rho}=\text{diag}(\rho_{1},\rho_{2},\rho_{3},\rho_{4},\rho_{5})\) is the penalized matrix. Finally, the predictive optimization problem is to minimize the cost function Equation (28) subject to I/O constraints Equations (21), (22) and (25), which is eventually equivalent to solving Equation (29), as follows. \[\left\{\begin{array}{ll}\min_{u,\,\varepsilon,\,Q,\,R}J(\mathbf{y},\mathbf{u},\mathbf{ \varepsilon})\\ s.t.\,\,\,\textit{constraints}\,\,\eqref{eqFigures 8-10 show the longitudinal tracking response of speed, distance, acceleration and jerk, respectively. From Figures 8 and 9, one can see that the MPC-CW cannot provide a substantial braking force in time when the preceding vehicle decelerates emergently at 10 s, and accordingly the inter-vehicle distance becomes shorter than that of MPC-TW. As shown in Figure 10, the MPC-TW can take an immediate brake as a result of tunable weights and also avoid overshooting to some extent, accordingly improving the longitudinal ride comfort. Furthermore, during the steering maneuver, the tracking capability of MPC-CW (from 30 s or so in Figures 8 and 9) is somewhat sacrificed due to the extra brake pressure generated from the direct yaw moment. However, MPC-TW is able to weaken the impact of DYC applied to the longitudinal car-following performance by automatically tuning the weights of \(\Delta d\), \(\Delta v\) and \(a_{x,des}\) (see Figure 11), while maintaining a small lateral stability error range (see Figure 7). Moreover, from Figure 9, one can see that the introduction of a slack variable vector into the hard constraints (refer to Equation (23)) is able to satisfy the upper/lower boundary of inequality by increasing the corresponding slack variable when the boundary exceeds, shown as Equation (25), accordingly avoiding computing infeasibility. Figure 11 shows the adjustment factor of the weight coefficient based on the self-tuning strategy suggested as Equation (27). Under the emergent scenario (from 10 s to 20 s or so), the weight of the desired acceleration, as well as the speed error, decreases because the corresponding adjustment factor shifts to less than 1, which indicates that the penalty for \(a_{x,des}\) in the cost function becomes weakened, accordingly achieving a substantial braking force (see Figure 10) and also maintaining a better tracking capability (see Figures 8 and 9). Additionally, a small adhesion workload means that the tires are far from the adhesion limit and that the vehicle still has great capacity for acceleration/deceleration and lateral stability [12]. As shown in Figure 12, the peak values for MPC-TW and MPC-CW are about 59% and 52%, respectively, accordingly guaranteeing another hard constraint, shown as Equation (22). Dangerous situations, such as spin or drift, always happen when the adhesion workload tends to its peak [12]. Although the proposed model improves the longitudinal tracking capability and lateral stability, it therefore clearly reduces vehicle safety. That is to say, under the transient scenarios, the pursuit for the longitudinal tracking capability as well as lateral stability should be sought under the premise of sufficient road adhesion capacity. Additionally, this means that the designing of an upper/lower boundary of the adjustment factor, shown as Equation (27), is in relation to the adhesion workload.