## 5 Simulation Example
From this, the average transmission period is calculated as 0.7463 s. Note that the proposed resilient event-triggered rule (8) is equivalent to the static event-triggered rule when DoS attacks do not occur. Once malicious attackers inject DoS attacks into the communication network, Figure 9 provides schematic diagrams of DoS attacks and system modes, where \(\alpha(t)=0\) and \(\alpha(t)=1\) represent the active and dormant DoS intervals, respectively. Then, based on the different values of \(\mathcal{J}_{d}\), DoS attacks can be divided into two types, namely low-intensity and high-intensity DoS attacks. Assume that the upper bound of performance loss caused by low-intensity DoS attacks is \(\mathcal{J}_{d}=1.0\times 10^{-4}\). In this case, Figure 10 shows the state response under the proposed dual-mode event-triggered security controller (10). Figure 11 presents the event-triggered instants determined by the resilient event-triggered condition (8) under low-intensity DoS attacks, where the number of the successful transmissions is 21. From this, the average transmission period is calculated as 2.3810 s. Compared with the situation without DoS attacks, the system state can only converge to a bounded range when the time approaches infinity, indicating that DoS attacks caused a serious negative impact on the system performance. Although the event triggering interval does not exceed the maximum inter-execution interval \(\Delta_{\max}\) calculated in Theorem 2, the controller remains in an unreleased state for a long time. On the other hand, assume that the upper bound of performance loss caused by low-intensity DoS attacks is \(\mathcal{J}_{d}=0.1\). In this case, Figure 12 shows the state response under the proposed dual-mode event-triggered security controller (10). Compared with the above two situations, it can be seen that the system state cannot converge to zero when the time tends to infinity. This means that high-intensity DoS attacks can cause irreversible damage to the control performance of the system. Figure 13 shows the triggering instant and transmission sequence under high-intensity DoS attacks, where the number of the successful transmissions is 13 and the average transmission period is calculated as 3.8462 s. From this, it can be seen that once the event-triggering interval exceeds the maximum allowable inter-execution interval \(\Delta_{\max}\), the system will completely lose control performance due to the controller not being updated for a long time. This means that the proposed event dual-mode event-triggered security controller (10) can ensure the safe and stable operation of the system under a specific DoS attack intensity. Figure 4: PWM-driven boost converter. Figure 5: State responses without DoS attacks. Figure 6: Evolution of the regulated output without DoS attacks. Figure 7: Evolution of the actual state error without DoS attacks. Figure 8: Update instants without DoS attacks. Figure 10: State responses under DoS attacks with \(\mathcal{J}_{d}=10^{-4}\). Figure 9: Top: (**a**) DoS signal. Bottom: (**b**) Switching signal. Figure 11: Update instants under DoS attacks with \(\mathcal{J}_{d}=10^{-4}\). Figure 12: State responses under DoS attacks with \(\mathcal{J}_{d}=0.1\). Figure 13: Update instants under DoS attacks with \(\mathcal{J}_{d}=0.1\). On the other hand, in order to further demonstrate the effectiveness of the proposed control strategy, we consider a DC motor-driven inverted pendulum systems, which is modeled in [45]. The system parameters are as follows:  
\[A_{1} =\left[ \begin{array}{ccc}-0.1397&-0.0256&0\\ 0.5121&-0.0373&0\\ -20&-4&-1\\ \end{array} \right],\;B_{1}=\left[ \begin{array}{c}0\\ 0\\ 1\\ \end{array} \right]\] \[A_{2} =\left[ \begin{array}{ccc}-0.0113&-0.0037&0\\ 0.1470&-0.0181&0\\ -20&-4&-5\\ \end{array} \right],\;B_{2}=\left[ \begin{array}{c}0\\ 0\\ 1\\ \end{array} \right]\]  
and the other parameters remain unchanged. Then, the security controller gains and weighting matrices can be calculated as  
\[K_{1}^{a} =\left[