## 3 Theory  
### Theory of Consistency by Multi-Microwave Sources Event-Triggered  
Generally, undirected graphs are used to represent information exchange between intelligent agents. In this article, \(G=(\mathcal{V},\,E,\,\mathcal{A})\) is the undirected graph, \(\mathcal{V}=\{0,\,1\ldots N\}\), \(E\subseteq\mathcal{V}\times\mathcal{V}\) denote the node set and the edge set, respectively. \(i,\,j\in\{0,\,1,\,2,\,\,\cdots,\,n\}\) denotes the agent label, and \(a_{ij}=a_{ji}\) denotes the weight of the edge, \(\mathcal{A}=[a_{ij}]\in E\), \(\mathcal{B}=diag\{b_{1},\,b_{2}\cdots b_{i}\}\) denote the adjacency matrices of the follower and the leader, respectively. When \(i\
eq j\), agent \(i\) and \(j\) are neighbors to each other if \(a_{ij}\
eq 0\). Let \(\mathcal{L}=\mathcal{D}-\mathcal{A}\), \(H=\mathcal{L}+\mathcal{B}\), \(\mathcal{D}=diag\{d_{1},\,d_{2},\,\,\cdots,\,d_{i}\},\,d_{i}=\deg_{in}( \mathcal{V}_{i})=\sum\limits_{j=1}^{i}a_{ij}.\) The problem is as follows [25].  
\[\dot{\mathrm{P}}_{i}(t)=f(t,\mathrm{P}_{i}(t))+U_{i}(t),t\in\left(t_{k-1}^{i}, t_{k}^{i}\right) \tag{6}\]  
\[\mathrm{Let}\,U_{i}(t)=\Gamma\times\delta_{i}p(t_{k}^{i})=\Gamma\sum\limits_{ j\in N_{i}}\mathrm{A}\big{(}\mathrm{P}_{i}(t_{k}^{i})-\mathrm{P}_{j}(t_{k}^{i}) \big{)}+B\big{(}\mathrm{P}_{i}(t_{k}^{i})-\mathrm{P}_{0}(t)\big{)}.\,\,f(t, \mathrm{P}_{i}(t))\]  
note the power state switching of microwave sources. Where \(\mathrm{A}\in R^{n\times n},B\in R^{n}\) and \(i\), \(U_{i}(t)\), \(\Gamma\) denote the number, control strategy, and controller gain of the microwave intelligent agent. A signal of \((0\pm\delta_{1})\sim(2\pm\delta_{2})\) V represents the output power status of the microwave source \(\mathrm{P}(t)\), and \(\delta_{1},\delta_{2}\) denote the signal margin. Let \(\delta_{\min}=Eig_{\min}(H=\mathcal{L}+\mathcal{B})\), \(\delta_{\min}\) is the minimum eigenvalue of \(\mathcal{H}\).  
\[\tilde{p}\mathrm{A}+\mathrm{A}^{T}\tilde{p}-2\delta_{\min}H_{1}BB^{T}\tilde{p} +\delta_{\min}I_{n}<0 \tag{7}\]  
If there is a positive definite matrix \(\tilde{p}>0\) in the Riccati inequality mentioned Equation (7). The gain \(\Gamma\) of the controller in Equation (6) is constructed as \(\Gamma=B^{T}\tilde{p}\).  
Next, the event-triggered strategy is defined as follows:  
\[t_{k}^{i}=\inf\{t\geq t_{k-1}^{i}+\tau_{\min}|f_{i}(t)\geq 0\} \tag{8}\]  
where \(f_{i}(t)=\|\mathrm{A}\|^{2}\times\|\delta e_{i}\mathrm{P}(t)\|^{2}\times e^{( 1-(1+2\|\mathrm{A}\|^{2})\frac{\gamma}{\delta})-\frac{\gamma}{2}}\|\delta_{i}p( t_{k}^{i})\|^{2}-\frac{\gamma}{\tilde{p}}\), \(\delta e_{i}\mathrm{P}(t)=\delta_{i}p(t_{k}^{i})-e_{i}p(t)\), let \(e_{i}p(t)=\sum\limits_{j\in N_{i}}\mathrm{A}\big{(}\mathrm{P}_{i}(t)-\mathrm{ P}_{j}(t)\big{)}+B\big{(}\mathrm{P}_{i}(t)-\mathrm{P}_{0}(t)\big{)}\). Where \(\beta>0,\eta>0\), \(t\) is the current time, \(t_{k-1}^{i}\) denotes the previous triggering time of agent \(i\), and \(\tau_{\min}>0\) denotes the lower bound of the triggering time, let \(\tau_{i}=t_{k}^{i}-t_{k-1}^{i}>\tau_{\min}\). For the next triggering moment \(t_{k}^{i}\), let \(t_{k}^{i}-t_{k-1}^{i}=\tau_{i}\). \(\tau_{i}\) is used to record the duration of \(t_{k-1}^{i}\) and reset to \(\tau_{\min}\) at the next triggering moment \(t_{k}^{i}\), let \(\tau_{i}-\tau_{\min}\geq 0\).  
When the microwave intelligent agent \(i\) triggers sampling at time \(k_{n}\), let \(t_{k_{n}}^{i}=i\); otherwise, \(t_{k_{n}}^{i}=0\). For all agents \(i\), the event-triggered times of the switching system were recorded as an event-triggered matrix \(\sigma_{k}\).  
\[\sigma_{k}=\left(\begin{array}{cccc}t_{k_{1}}^{1}&t_{k_{2}}^{1}&\cdots&t_{k_{ n}}^{1}\\ t_{k_{1}}^{2}&t_{k_{2}}^{2}&\cdots&t_{k_{n}}^{2}\\ \vdots&\vdots&\ldots&\vdots\\ t_{k_{1}}^{i}&t_{k_{2}}^{i}&\cdots&t_{k_{n}}^{i}\end{array}\right),k_{n}\in t \tag{9}\]