# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 5 An Overview of Attack-Resilient Control and Communication Techniques
### Resilient Control Approaches
#### 5.1.1 Event-Triggered Control  
In the literature, one of the most common control strategies used against denial-of-service attacks is the _event-triggered_ control. In the design and the analysis of event-triggered control schemes, the attack models presented in Sections 3 and 4.1 are utilized. In particular, for the continuous-time networked control problem discussed in Section 2.1.2, [PERSON] and [PERSON] [48] provided an event-triggered control mechanism that achieves asymptotic stabilization under any attack strategy that satisfies Assumptions 1 and 2. In their approach, a transmission of the state information is triggered when the error between the current state \(x(t)\) and the previously transmitted state exceeds a certain threshold. It is mentioned in [48] that the event-triggering approach requires continuous monitoring of the state. Specifically, there is a need to continuously monitor the state \(x(t)\) to check if the triggering condition is satisfied or not. To avoid continuous monitoring, self-triggering approach is helpful. In the self-triggering approach proposed in [48], the predicted value of the state is used for calculating the next transmission time \(t_{k+1}\). Notice that under DoS attacks, some of the transmissions may fail. The triggering approach in [48] takes into account the status of transmissions. If, for instance, the transmission attempt at time \(t_{k}\) fails, then the next transmission is attempted shortly afterwards. If, on the other hand, the transmission at time \(t_{k}\) is successful, the next transmission can be made after a longer duration. The parameters of the triggering schemes in [48] are designed to ensure that the overall control system is stable. Specifically, Lyapunov-function techniques are utilized for obtaining sufficient conditions concerning the event-triggering parameters that ensure global asymptotic stability under DoS attacks. An interesting resilient event-triggered control strategy is proposed in [49]. In one of the scenarios considered in that work, jamming attacks on a communication channel happen periodically where the attacker repeats cycles of jamming and sleeping. For such attacks, control and transmission strategy in [49] can identify the transition times between jamming and sleeping states of the attack; thus, it allows selection of times where the communication is guaranteed to be successful. As a result, number of transmission attempts is further reduced. In addition, event-triggered control of a discrete-time networked control system (see Section 2.1.1) is explored in our previous works [25; 38]. There, a Lyapunov-like function is utilized for determining the triggering time instants. Specifically, consider \(V\colon\mathbb{R}^{n}\to[0,\infty)\) given by \(V(x)\triangleq x^{\mathrm{T}}Px\), where \(P>0\). The network transmissions are attempted at times \(t_{0}=0\), and \(t_{i}\), \(i\in\mathbb{N}\), given by  
\[t_{i+1}\triangleq\min\Big{\{}t\in\{\tau_{i}+1,\tau_{i}+2,\ldots\}\colon t\geq t _{i}+\theta\;\;\text{or}\;\;V(Ax(t)+Bu(t_{i}))>\beta V(x(t_{i}))\Big{\}}, \tag{26}\]  
where \(\theta\in\mathbb{N}\) and \(\beta\in[0,1)\) are parameters of the event-triggered controller. The triggering condition \(V(Ax(t)+Bu(t_{i}))>\beta V(x(t_{i}))\) guarantees that \(V(x(t))\) stays within certain limits after a successful transmission at time \(t_{i}\). As indicated in Theorem 11, the scalar \(\beta\) can be chosen so that when the transmission attempt at time \(t_{i}\) is successful, then the closed-loop system shows stable behavior and the state goes inside a level set \(\{x\in\mathbb{R}^{n}\colon V(x)=\beta V(x(t_{i}))\}\) at the next time instant. This is illustrated in Figure 8 (left). Furthermore, the next transmission event is triggered only when the state is predicted to leave this level set. For the example case shown Figure 8 (right), a transmission event is triggered at time \(t_{i}+3\).