## 2 Materials and Methods
### Theory Basics
#### 2.2.2 Self-Triggered Model Predictive Control  
**Assumption 4**. : _Lipschitz continuous functions \(f_{a}\) and \(g_{a}\) are associated with Lipschitz constants \(L_{f,a}\) and \(L_{g,a}\), respectively. Additionally, there exists a \(K_{\infty}\) function \(y_{a}\) such that the condition \(L_{f,a}\geq y_{a}\parallel\chi_{a}-R_{L}\parallel\) holds true._  
**Lemma 1** ([32]). : _Assuming that Assumption 4 is satisfied, the Lipschitz constant \(L_{j,a}\) corresponding to the cost Function (10) satisfies the following conditions:_  
\[j_{a}^{*}(\chi_{a}(\tau_{a,\theta+1}),\mu_{a}(\tau_{a,\theta+1}))-j_{a}^{*}( \hat{\chi}_{a}(\tau_{a,\theta+1}),\hat{\mu}_{a}(\tau_{a,\theta+1}))<L_{j,a}E_{ a}(\sigma), \tag{12}\]  
_where \(L_{j,a}=\Sigma_{l=0}^{N_{p}}L_{f,a}\parallel e^{AIT}\parallel+L_{g,a} \parallel e^{A\text{N}_{p}T}\parallel,E_{a}(\sigma)=\frac{K\parallel \parallel}{\parallel A\parallel^{2}}(e^{\parallel A\parallel eT}-1)-\frac{K \parallel\parallel}{\parallel A\parallel}\sigma T\)._  
Let \(\tau_{a}\) denote the self-triggering instant, and \(\{\tau_{a},\theta\},\vartheta\in\{1,2,...,N\}\) represent the sequence of self-triggering instants, i.e., which means updating and optimizing at \(\tau_{\theta}\), and calculating the optimal control. Based on specific self-triggering conditions, the next self-triggering instant, \(\tau_{\theta+1}\), denoted as \(\tau_{\theta+1}=\tau_{\theta+1}+\sigma T,\sigma\in\{1,2,...,N_{p}\}\), is determined. During the time interval from \(\tau_{\theta}\) to \(\tau_{\theta+1}\), the controller ceases to update but continues to use the effective predicted control input obtained from the previous triggering instant to reduce communication overhead and computational burden. **Definition 1**. : _If Assumptions 3 and 4 are satisfied, there exist constants \(\alpha_{a}\in(0,1)\) and \(\sigma\in[1,2,...,N_{p}]\) such that_  
\[L_{j,a}E_{a}(\sigma)<\alpha_{a}\Delta_{1}. \tag{13}\]  
Hence, the subsequent triggering time \(t_{a,\theta+1}\) can be determined based on the following criteria:  
\[\tau_{a,\theta+1}=\tau_{a,\theta}+\min\{\sigma=inf\{L_{j,a}E_{a}(\sigma)- \alpha_{a}\Delta_{1}=0\},N_{p}\}, \tag{14}\]  
where \(L_{j,a}\) and \(E_{a}(\sigma)\) are defined in Equation (21). We construct the set of feasible control inputs, denoted as \(\hat{\mu}_{a}(l_{j};\tau_{a,\theta+1})\), at time \(\tau_{a,\theta+1}\). \[\hat{\mu}_{a}(l_{j};\tau_{a,\theta+1})=\left\{ \begin{array}{l}\mu_{a}^{*}(l_ {j};\tau_{a,\theta}),l_{\tau}\in[\tau_{a,\theta+1},\tau_{a,\theta}+N_{p}],\\ \hat{\mu}_{a}(l_{j};\tau_{a,\theta}+N_{p}),l_{\tau}\in[\tau_{a,\theta}+N_{p}, \tau_{a,\theta+1}+N_{p}].\end{array} \right. \tag{15}\]  
where \(\mu_{a}^{*}(l_{j};\tau_{a,\theta})\) represents the optimal control input obtained by solving the optimization problem at time \(\tau_{a,\theta}\), and \(\hat{\mu}_{a}(l_{j};\tau_{a,\theta}+N_{p})\) represents the feasible control input at the terminal time \(\tau_{a,\theta}+N_{p}\). These inputs are then applied to the system (8) to obtain the corresponding feasible state \(\hat{\chi}_{a}(l_{j};\tau_{a,\theta+1})\). Therefore, based on (8), we can construct the following cost function for the ST-MPC:  
\[J_{a}(\chi_{a}(\tau_{a,\theta+1}),\mu_{a}(\tau_{a,\theta+1}))\] \[= \sum_{l=0}^{N_{p}}\left[\parallel\chi_{a}(l;\tau_{a,\theta+1})-R_{a,l}(\tau_{a,\theta+1})\parallel_{Q_{a}}^{2}+\parallel\mu_{a}(l;\tau_{a,\theta+1 })\parallel_{R_{a}}^{2}\right] \tag{16a}\] \[+\parallel\chi_{a}(N_{p};\tau_{a,\theta+1})-R_{a,l}(\tau_{a,\theta+ 1})\parallel_{P_{a}}^{2},\] subject to \[\chi_{a}(l;\vartheta_{\tau}) =A\chi_{a}(l-1;\vartheta_{\tau})+B\mu_{a}(l-1;\vartheta_{\tau}),\] (16b) \[\mu_{a}(l;\vartheta_{\tau}) \in\mathcal{U}_{a},\] (16c) \[\chi_{a}(l;\vartheta_{\tau}) \in\mathcal{X}_{a},\] (16d) \[\chi_{a}(N_{p};\vartheta_{\tau}) \in\mathcal{X}_{a}^{f}\in\mathcal{X}_{a},\] (16e) \[R_{a,l}=\left\{ \begin{array}{ll}r_{0},&\quad\quad\quad\quad \text{layer}1,\\ \frac{\sum_{j\in\mathbb{K}_{a}}\chi_{j}}{\left|\mathbb{K}_{a}\right|},&\quad \quad\text{layer}2,3,\ldots,n,\end{array} \right.