# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 2 An Overview of Literature on Denial-of-Service in Control Systems
### Multi-Agent Consensus Problem  
The consensus problem for a multi-agent system under jamming attacks is considered in [71; 72; 73]. The multi-agent system in those works consists of \(n\) agents that possess scalar dynamics. In particular, the evolution of the \(i\)th agent's state is described by  
\[\dot{x}^{i}(t)=u^{i}(t),\quad t\geq 0, \tag{10}\]  
where \(x^{i}(t)\in\mathbb{R}\) is the state and \(u^{i}(t)\in\mathbb{R}\) is the local control input for agent \(i\). Inter-agent communication topology of the multi-agent system is characterized with an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where the nodes \(\mathcal{V}=\{1,\ldots,n\}\) represent the agents and the edges \(\mathcal{E}\subset\mathcal{V}\times\mathcal{V}\) represent the communication links between agents. The goal in [71; 72; 73] is to develop control and communication rules to guarantee practical consensus under jamming attacks. In practical consensus, agents states \([\quad x^{1}(t)\quad x^{2}(t)\quad\cdots\quad x^{n}(t)\quad]^{\mathrm{T}}\) converge to a vector \(x^{*}\in\mathbb{R}^{n}\) that belongs to a consensus set \(\mathcal{D}_{\varepsilon}\)  
\[\mathcal{D}_{\varepsilon}\triangleq\Big{\{}x\in\mathbb{R}^{n}:\Big{|}\sum_{ j\in\mathcal{N}^{1}}(x^{j}-x^{j})\Big{|}<\varepsilon,\;\;i\in\mathcal{V}\Big{\}}, \tag{11}\]  
where \(\varepsilon>0\) represents a predetermined required level of closeness between agents and \(\mathcal{N}^{i}\subset\mathcal{V}\) denotes the neighbors of agent \(i\), that is, the set of agents that share a communication link with agent \(i\). For this problem formulation, [PERSON] et al. [71] and [PERSON] et al. [72] considered the attack setup shown on the left-hand side of Figure 3. In this setup, the jamming attacker can target all communication links at once. Specifically when there is an attack, none of the agents can communicate with each other. In the control approach in [71; 72], each agent \(i\in\mathcal{V}\) tries to communicate with its neighbors at certain times denoted by \(t_{0}^{i}\), \(t_{1}^{i},\ldots\). Specifically, at time \(t_{k}^{i}\), agent \(i\) sends a packet to its neighbors to request their states. In the case where there is no jamming attack at that time, all neighbors receive this packet and they respond by sending back their states. Those states are then used by agent \(i\) for constructing a ternary control input (i.e., \(u^{i}(t)\in\{-1,0,1\}\)) for achieving consensus. For designing communication attempt times \(t_{0}^{i}\), \(t_{1}^{i},\ldots\), [PERSON] et al. [71] used a self-triggering approach, where communication is attempted by each agent \(i\) only when a triggering condition holds. For achieving consensus, a restriction on the average duration and the average frequency on the attacks is imposed. These duration and frequency restrictions are discussed in Section 3.1, and they follow the attack characterization proposed by [PERSON] and [PERSON] [48]. It is shown in [72] that the  
Figure 3: Multi-agent consensus in the presence of jamming attackers: (**Left**) single jamming attacker causes transmission failures on all inter-agent communication links; and (**Right**) multiple jamming attackers cause failures on different links). restriction on the attack frequency can be removed when the agents utilize _randomization_ in designing the communication attempt times. The utility of this randomization approach is further discussed in Section 5.2.1. Recently, [PERSON] et al. [73] explored the multi-agent consensus problem under the attack setup shown in Figure 3 (right). In this setup, there are multiple jamming attackers that can target individual communication links. We note that the consensus problem depicted in Figure 3 (left) is also explored for multi-agent systems with multi-dimensional dynamics in [74]. More recently, [PERSON] et al. [75] considered a game-theoretical formulation of multi-agent systems under jamming attacks that can target individual links.