# Resilient Consensus Control for Multi-Agent Systems: A Comparative Survey
## 3 Resilient Control for MASs
### Spoofing Attack
The selection of coupling strength \(c\) and pulse interval parameter \(h_{2}\) are related to the global information of communication topology, so the control protocol is not completely distributed. In addition, the system dynamics contains random variables about deception attack as \(\beta_{i}\). Thus, it can only realize the target of bounded synchronous tracking in the mean square sense. Compared with the case of no attack, \(\hat{\beta}\) increases the upper bound of the synchronization error and narrows the pulse spacing, which also puts forward corresponding requirements for \(c\). That is why \(\tilde{\beta}\) is always restricted. Figure 3: The deception attack in cyber-physical system. According to the secific location of the deception attack, the figure shows three different deception attacks, where \(\xi_{i}\) represents the deception attack on the controller itself or controller–actuator channel (tampering with the control signal \(u_{i}\)), \(q_{i}\) represents the deception attack on the sensor–controller channel (tampering with the output signal \(y_{i}\)), and \(d_{i}\) represents the deception attack on the transmission channel of the neighbor agents (tampering with the state value \(x_{j}\) with \(j\in\mathcal{N}_{i}\)). Compared with [67], ref. [68] studies the overlay spoofing attack on the sensor-controller channel, as shown in Figure 3. The attack causes the normal state value to be superimposed with the error value \(q_{i}(t)\):  
\[u_{i}(t)=\sum_{k=1}^{\infty}[c\sum_{j=1}^{N}[-l_{ij}x_{j}(t)+\beta_{ij}q_{i}(t) ]-cd_{i}(t)[x_{i}(t)-x_{l}(t)]]\delta(t-t_{k}). \tag{28}\]  
Ref. [68] also achieves mean square bounded synchronization, but only the size of the upper bound of the error is affected by \(\tilde{\beta}\). Regardless of the attack mode, the synchronization performance is mainly affected by the coupling strength, degree matrix, attack probability and pulse spacing. In particular, the first two terms play a decisive role in reducing synchronization error when the probability of each side being attacked is the same. Most of the above research fail to effectively eliminate the impact of spoofing attacks on MASs, resulting in the boundedness of their final consensus. As for that, ref. [69] proposes a distributed filter with adaptive compensator as follows:  
\[\left\{ \begin{array}{l}\dot{\psi}_{i}(t)=E\psi_{i}(t)+\dot{\varepsilon}_{i}( t)\\ \dot{\varepsilon}_{i}(t)=-\dot{\varepsilon}_{i}(t)+\varepsilon_{q,i}^{a}(t)+ \rho_{a,i}(t)\\ \rho_{a,ij}(t)=-\frac{\dot{\varepsilon}_{i}^{T}\dot{\varepsilon}_{i}(t)\dot{ \varepsilon}_{a,ij}^{2}(t)}{\left\|\dot{\varepsilon}_{i}^{T}\dot{\varepsilon} _{i}(t)\right\|^{2}\dot{\varepsilon}_{i}(t)\right\|^{2}\dot{\varepsilon}_{i}( t)+\dot{\sigma}_{ij}(t)}\end{array} \right. \tag{29}\]  
where \(\psi_{i}(t)\) and \(\dot{\varepsilon}_{i}(t)\) are the states of the observer and filter, respectively. \(\varepsilon_{q,i}^{a}(t)\) is the relative observation, and \(\rho_{a,i}(t)=\sum_{j\in\mathcal{N}_{i}}a_{ij}\rho_{a,ij}(t)\) is the compensation term. \(\dot{\delta}_{a,ij}(t)=-\lambda\sigma_{ij}(t)\dot{\delta}_{a,ij}(t)+\lambda \left\|F_{i}^{T}\dot{\varepsilon}_{i}(t)\right\|\), \(\sigma_{ij}(t)\) is a bounded signal. \(\dot{\delta}_{a,ij}(t)\) is the estimate of the spoofing attack signal. Under the function of the filter, the system can compensate the offset of consensus error variables caused by spoofing attacks and achieve accurate consensus result without error. The distributed controller only uses the local information of the agent itself and does not need to use the random characteristic and upper bound of the attack signal, which is more conducive to distributed applications. However, the design complexity of the controller is related to the dimension of the system state, which makes it extremely difficult to design its control parameters for complex systems. The assumption that the attack is bounded and obeys Bernoulli distribution made in [67; 68] is very conservative.