## 2 Preliminaries and Problem Formulation  
### System Framework  
Consider MJCPSs defined on a complete probability space \((\Omega,\mathcal{F},\mathbb{P})\), whose dynamics can be captured by  
\[\begin{cases}\dot{x}(t)=A_{r(t)}x(t)+B_{r(t)}^{u}u(t)+B_{r(t)}^{\omega}\omega( t)\\ z(t)=C_{r(t)}x(t)+D_{r(t)}^{u}u(t)\end{cases} \tag{1}\]  
where \(x(t)\in\mathbb{R}^{n_{x}}\), \(u(t)\in\mathbb{R}^{n_{x}}\), and \(z(t)\in\mathbb{R}^{n_{z}}\) are the system state, control input, and regulated output variables, respectively. \(\omega(t)\in\mathbb{R}^{n_{\omega}}\) represents an exogenous disturbance belonging to \(\mathcal{L}_{2}[0,+\infty)\). \(A_{r(t)}\), \(B_{r(t)}^{u}\), \(B_{r(t)}^{\omega}\), \(C_{r(t)}\), and \(D_{r(t)}^{u}\) denote known matrices of compatible dimensions.  
Let \(\{r(t),t\geq 0\}\) represent a right-continuous Markov process taking values in a finite set \(\mathbb{S}\triangleq\{1,2,\ldots,S\}\), whose stationary mode transition rate matrix (TRM) \(\Pi\triangleq[\pi_{ij}]\in\mathbb{R}^{\mathbb{S}\times\mathbb{S}}\) can be given by  
\[\Pr\{r(t+\Delta t) =j|r(t)=i\} \tag{2}\] \[=\begin{cases}\pi_{ij}\Delta t+o(\Delta t),&i\
eq j\\ 1+\pi_{ii}\Delta t+o(\Delta t),&i=j\end{cases}\]  
where \(\Delta t>0\), \(\lim_{\Delta t\to 0}(o(\Delta t)/\Delta t)=0\), and TRs from \(i\) to \(j\) satisfy two conditions, that is, if \(i\
eq j\) then \(\pi_{ij}>0\); otherwise, \(\pi_{ii}=-\sum_{i\
eq j}\pi_{ij}\) for any \(i,j\in\mathbb{S}\).  
On the other hand, Figure 1 presents an advanced event-based control architecture for MJCPSs subject to energy-limited DoS attacks, where the virtual sensor system determined by a novel resilient event-triggered rule is developed to alleviate the heavy communication burden generated by using traditional sampled-data schemes, see refs. [29; 30; 31] and the references therein. For clarity, let \(e_{s}(t)=x(t_{k+1})-x(t_{k})\) represent the sampled error, then we review a general event-triggered scheme mentioned in [38] as follows:  
\[t_{k+1}=\inf\Bigl{\{}t\in\mathbb{R}_{>t_{k}}|\epsilon_{s}^{T}(t)\Theta_{a}e_{ s}(t)>\sigma x^{T}(t)\Theta_{b}x(t)\Bigr{\}}, \tag{3}\]  
where \(\sigma\in(0,1)\) is a given triggering parameter, \(\Theta_{a}\) and \(\Theta_{b}\) are the unknown weighting matrices to be designed, and \(\{t_{k}\}_{k\in\mathbb{N}_{b}}\) denotes the triggering sequence determined by (3).  
In this way, the control input applied to the control layer in the absence of DoS attacks can be described as  
\[u(t)=K_{r(t)}x(t_{k}),\ t\in[t_{k},t_{k+1}) \tag{4}\]  
where \(K_{r(t)}\) denotes the unknown control gain to be designed. Notice that the packets are transmitted on the sensor-controller (S-C) and controller-actuator (C-A) channels over the open and secure communication networks, respectively. This means that the S-C channel is vulnerable to remote intrusion by malicious attackers. From the perspective of system control performance, malicious intrusion on the S-C channel may cause information mismatch between the S-C and C-A channels. That is, the actuator may maintain historical control actions for a long time, which is enough to pose a serious threat to physical security. In order to ensure safe operation in an unreliable network environment, we will redesign an improved resilient event-triggered scheme and related security control protocol for MJCPs in the sequel.