# Game Theory-Based Interactive Control for Human-Machine Cooperative Driving
## 4 Vehicle Interactive Control Strategy
### Interactive Control Based on DMPC and Non-Cooperative Game
\[\mathbf{x}^{V_{i}}(k+1)=\mathbf{A}\mathbf{x}^{V_{i}}(k)+\mathbf{B} \mathbf{u}^{V_{i}}(k)\] (33c) \[\mathbf{z}^{V_{i}}(k)=\mathbf{C}\mathbf{x}^{V_{i}}(k)\] (33d) \[-u^{V_{lim}^{i}}_{lim}\leq\mathbf{u}^{V_{i}}(k)\leq u^{V_{i}}_{lim}\] (33e) \[-\Delta u^{V_{i}}_{lim}\leq\Delta\mathbf{u}^{V_{i}}_{lim}\leq \Delta u^{V_{i}}_{lim}\] (33f) \[\mathbf{x}^{e}(k+1)=\mathbf{A}\mathbf{x}^{e}(k)+\lambda\mathbf{B }\mathbf{u}^{h}(k)+(1-\lambda)\mathbf{B}\mathbf{u}^{e}(k)\] (33g) \[\mathbf{z}^{e}(k)=\mathbf{C}\mathbf{x}^{e}(k)\] (33h) \[-u^{e}_{lim}\leq\mathbf{u}^{e}(k)\leq u^{e}_{lim}\] (33i) \[-\Delta u^{e}_{lim}\leq\Delta\mathbf{u}^{e}(k)\leq\Delta u^{e}_{ lim} \tag{33j}\]  
where Equation (33a) and (33b) are the cost functions of the SVs and ego vehicle, respectively. \(\mathbf{Q}^{V_{i}}\), \(\mathbf{P}^{V_{i}}\), \(\mathbf{R}^{V_{i}}\) is the weight matrix in the cost function of vehicle \(V_{i}\). And \(\mathbf{Q}^{e}\), \(\mathbf{P}^{e}\), \(\mathbf{R}^{e}\) is the weight matrix in the cost function of the ego vehicle. \(u^{V_{i}}_{lim}\) and \(\Delta u^{V_{i}}_{lim}\) contain the constraints of vehicle \(V_{i}\), wheel angle amplitude \(\delta^{V_{i}}_{lim}\), acceleration amplitude \(u^{V_{i}}_{lim}\), and their change rate constraint \(\Delta\delta^{V_{i}}_{lim}\), \(\Delta u^{V_{i}}_{lim}\). The structure of ego vehicles \(u^{e}_{lim}\) and \(\Delta u^{e}_{lim}\) is the same as that of vehicle \(V_{i}\). The detailed values of their parameters can be viewed in \[\min_{\mathbf{u}^{e}}\Pi^{e} =\sum_{j=1}^{N_{p}}\
orm{\mathbf{z}^{e}(k+j)-\mathbf{z}_{ref}^{e} (k+j)}_{\mathbf{Q}^{e}}^{2}+\sum_{j=0}^{N_{c}-1}\
orm{\mathbf{u}^{e}(k+j)}_{ \mathbf{R}^{e}}^{2}+\sum_{j=1}^{N_{p}}\
orm{I^{e}(k+j)}_{\mathbf{P}^{e}}^{2}\]0  
It can be observed from Equation (33) that each player optimizes their objective function under relevant constraints, and each player's control output not only affects their future state but also imposes an impact on other players. This is a coupled optimization problem, and the optimal solution can be searched based on the idea of Nash equilibrium. Its core definition is that the Nash equilibrium solution is obtained when no player can benefit more by unilaterally changing their strategy. In general, Nash equilibrium is complicated to calculate and cannot be found in arbitrary problems. The iterative optimal response method is a popular method for approximating Nash equilibrium, and related work can be found in [42; 43]. The detailed solution process is shown in Algorithm 2. Its core idea is that players solve their optimal response solution in each iteration while fixing other players' strategies and then update their own strategy in this iteration and pass it to the remaining players to solve the optimal response solution. Until all players obtain the optimal response, we proceed to the next iteration. As the number of iterations increases, the player's strategy gradually approaches convergence. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameters** & **Value** & **Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(m\) & 1200 kg & \(N_{p}\) & 20 & \(\Delta a^{e}_{lim}\) & 0.3 m/s\({}^{2}\) \\ \(l_{z}\) & 1300 kg & \(N_{c}\) & 10 & \(a^{V_{i}}_{lim}\) & 8 m/s\({}^{2}\) \\ \(l_{a}\) & 1.050 m & \(\delta^{e}_{lim}\) & 10 deg & \(\Delta a^{V_{i}}_{lim}\) & 0.8 m/s\({}^{2}\) \\ \(l_{b}\) & 1.90 m & \(\Delta\delta^{e}_{lim}\) & 1 deg & \(\delta^{e}_{p}\) & 0.6 \\ \(C_{f}\) & 5600 N/rad & \(\delta^{V_{i}}_{lim}\) & 15 deg & \(\omega_{n}\) & 6\(\pi\) rad/s \\ \(C_{r}\) & 5600 N/rad & \(\Delta\delta^{e}_{lim}\) & 1.25 deg & \(\tau_{d1}\) & 0.08 s \\ \(T_{s}\) & 0.01 s & \(a^{e}_{lim}\) & 3 m/s\({}^{2}\) & \(\tau_{d2}\) & 0.15 s \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters for vehicle and driver characteristics.