## 3 Theory
### Event-Triggered Consistency Theory of Multi-Microwave Sources under DoS Attacks  
Typically, the energy of DoS attacks released by opponents is limited. Let \(N_{D}(0,t)\leq\
u+t/\tau_{D}\) represent the frequency of DoS attacks, and the \(\Delta(0,t)\leq\
u_{1}+t/T_{D}\) denotes the duration of DoS attacks, where \(\
u\geq 0\), \(\tau_{D}>0\) and \(\
u_{1}\geq 0\) are constants, and the \(T_{D}>1\). The network can be located at any point in the feedback loop. Inspired by references [26], the network is positioned between the triggering mechanism and the controller in this paper. As shown in Figure 2, when \(t\in\left[T_{n}^{on},T_{n}^{off}\right)\), the adversary initiates the DoS attacks to disrupt the communication between the controller network and the intelligent agent control system, if \(U_{i}(t)=0\)[26], it may decrease the system performance. To avoid this issue, ZOH generates a control signal \(U_{i}(i_{k-1}^{i})\) that denotes the most recently even-triggered moment as shown in Figure 3.  
On the other hand, when \(t\in\left[T_{n}^{off},T_{n+1}^{on}\right)\), if the event-triggered law \(f_{i}(t)\geq 0\) form Equation (8), \(P_{i}(t)\) is updated and sent to agent \(j\), using ZOH to save and send updated control signal \(U_{i}(i_{k}^{t})\) to actuator \(i\). When the value of \(f_{i}(t)<0\), \(P_{i}(t)\) remains the same until the \(f_{i}(t)\geq 0\). As shown in Figure 3, actuator \(i\) is controlled by control signal \(U_{i}\) to output digital control signal \(p\), which is then input into a comparator. When \(p\geq 1\), the High Voltage DC Magnetron Power Supply drives the magnetron to output at high-power state \(P_{2}\), for \(p<1\), it outputs at low-power state \(P_{1}\), enabling the collaborative switching of high and low-power magnetrons. The leader's state, denoted by \(P_{0}(t)\), is generally not influenced by \(f_{i}(t)\) and changes over time through updates. When \(f_{i}(t)<0\), the information exchange between microwave intelligent agents is reduced, achieving the uniform distribution of processor computing power. Figure 3: Flowchart of network-based agent control. Figure 2: Schematic of DoS attack signals. Therefore, the parameter \(d(t)\) describes the impact of DoS attacks, and ZOH maintains a closed-loop system. \[\bar{U}_{i}(t)=\left\{ \begin{array}{l}U_{i}\big{(}t_{k}^{i}\big{)},d(t)=0,others\\ U_{i}\big{(}t_{k-1}^{i}\big{)},d(t)=1,t\in[T_{n}^{on},T_{n}^{off})\end{array} \right. \tag{10}\]  
Then, Equation (6) can be rewritten as follows:  
\[\dot{\mathrm{P}}_{i}(t)=f(t,\mathrm{P}_{i}(t))+\bar{U}_{i}(t),t\in\Big{(}t_{k- 1}^{i},t_{k}^{i}\Big{)} \tag{11}\]  
When switching system state transitions, the minimum event-triggered time \(\tau_{\min}\)[27, 28] is introduced in an Event-triggered strategy to prevent Zeno behavior caused by DoS attacks. The definition of \(\tau_{\min}\) is as follows:  
\[\tau_{\min}\geq\frac{1}{\max\{c_{0},c_{1}\}}\ln\left(\frac{1}{n}\sqrt{\frac{ \chi(k_{1}-\frac{k_{2}}{\theta})}{(k_{1}+k_{2}\theta)}}+1\right) \tag{12}\]  
\[\left\{ \begin{array}{l}c_{0}\stackrel{{\Delta}}{{=}}S_{\max}( I_{N}\otimes BK)\\ c_{1}\stackrel{{\Delta}}{{=}}S_{\max}((I_{N}\otimes\mathrm{A})+(H \otimes BK))+S_{\max}(H\otimes BK)+S_{\max}(I_{N}\otimes\mathrm{A})\end{array} \right. \tag{13}\]  
\[\left\{ \begin{array}{l}k_{1}=S_{\min}(H^{3}\otimes(2\bar{p}BB^{T}\bar{p})-H^{ 2}\otimes\delta_{m1}I_{N})\\ k_{2}=\frac{1}{2}\{S_{\max}(H\otimes 2\delta_{m1}I_{N}-H^{2}\otimes(2\bar{p}BB^{T}\bar{p}))\} \end{array} \right. \tag{14}\]  
where \(0<\delta_{m1}<\delta_{\min},k_{1}>0,1>\chi>0\), and let \(\vartheta>k_{2}/\delta_{1}\), \(\beta=\sqrt{\chi(\delta_{1}-\frac{k_{2}}{\theta})/(k_{1}+k_{2}\theta)}\). \(S_{\min}()\) is the minimum singular value, and \(S_{\max}()\) is the maximum singular value of the independent matrix variable. Next, the following proves that the system does not exhibit Zeno behavior.