## 2 Preliminaries and Problem Formulation
### Control Objective  
For MJCPSs under DoS attacks, the control objective of this article is to develop the advanced resilient event-triggered rule and dual-mode security control protocol, which aims to ensure the safe operation of physical system in an unreliable network environment. Some useful definitions are given as follows.  
Figure 3: Typical DoS attack models.  
**Definition 1** ([38]).: _For any initial condition \((x(0),r(0))\), MJCPSs (1) can be said to be stochastically stable if they have a positive parameter \(\mathcal{M}(x(0),r(0))\) such that_  
\[\mathbb{E}\bigg{\{}\int_{0}^{\infty}x^{T}(t)x(t)\mathrm{dt}\mid(x(0),r(0)) \bigg{\}}\leq\mathcal{M}(x(0),r(0)).\]  
**Definition 2** ([38]).: _Given a positive parameter \(\gamma\in\mathbb{R}_{>0}\), MJCPSs (1) can be considered stochastically stable and to have an \(H_{\infty}\) disturbance attenuation level \(\gamma\) if this condition is met:_  
\[\mathbb{E}\bigg{\{}\int_{0}^{\infty}z^{T}(t)z(t)\mathrm{dt}\bigg{\}}\leq\gamma ^{2}\int_{0}^{\infty}\omega^{T}(t)\omega(t)\mathrm{dt}.\]  
**Definition 3** ([13]).: _The transmission sequence \(\{t_{k}\}_{k\in\mathbb{N}_{0}}\) is said to have a finite update rate if there are two positive scalars \(\Delta_{\min}\) and \(\Delta_{\max}\in\mathbb{R}_{>0}\) such that_  
\[\Delta_{\min}\leq t_{k+1}-t_{k}=\Delta_{k}\leq\Delta_{\max},\]  
_where \(\Delta_{\min}\) and \(\Delta_{\max}\) are the minimum and maximum update rates, respectively._  
**Remark 1**.: _Generally, DoS behaviors launched by malicious attackers have a certain concealment, which makes it difficult for defenders to predict attack intentions. Under such an unreliable communication network, the packets to be updated determined by ETM may be lost during the transmission, thereby reducing the security performance of the physical system. To defend against DoS attacks, several popular resilient ETC strategies have been proposed, which can adjust the triggering parameters online or in advance according to changes in the system state, see [34; 35; 39; 40; 41; 42] and the references therein. However, the invisibility of DoS attacks makes it difficult to adjust triggering parameters in real time, which leads to certain limitations of such methods. In contrast, setting a certain resilience margin for ETM to cope with the transmission failures caused by DoS attacks can break through the barriers of traditional control schemes. In addition, the triggering parameters do not need to be adjusted depending on whether DoS attacks occur. Hence, the method of designing a reasonable resilience margin for ETM in a unified framework may facilitate system analysis under DoS attacks._  
**Remark 2**.: _The control behavior in response to DoS attacks can be divided into two cases: one is to force the control signal to zero as mentioned in [12; 13], and the other is to maintain the historical control input by using a zero-order holder (ZOH), see [34; 38] and the references therein. Obviously, the former is an extremely conservative way to defend against DoS attacks because the controlled system may be in an open-loop unstable status for a long time. Conversely, the latter allows the physical system to obtain relatively satisfactory security performance in a degraded manner. That is, it can be concluded that the second way is more suitable as the core idea of the intrusion tolerance control scheme. On this basis, we can design a DoS-based dual-mode security controller by introducing a Dirac measure \(\alpha(t)\)[43], which is independent of the system mode \(r(t)\). The composite control scheme that includes two different control protocols is similar to hybrid control, but both protocols cannot take effect simultaneously. It should be noted that the event-triggered rules corresponding to the above control protocols are selected based on whether DoS attacks occur._