# Resilient Consensus Control for Multi-Agent Systems: A Comparative Survey
## 3 Resilient Control for MASs
### Byzantine Attack
[82] proposes using the sliding window approach to transform the elastic consensus problem in the continuous domain into the discrete domain, making it possible to further improve the scheme through the event-triggered mechanism [84]. The main difficulty in applying event-triggered control to MSR is the processing of the difference between the current state value and the last passed state value. Ref. [84] regards the difference of the above state variables as non-attenuated noise and uses it as the basis for triggering. They give two control schemes to achieve elastic consensus of the system. However, the consensus still has bounded error. At each discrete time, the state value of the agent will be updated. Whether to broadcast it will be decided according to whether the triggered conditions are met: save the last broadcast value as an auxiliary state value; compare it with the updated state value; and trigger if the difference is greater than the threshold value. The author used the modified MSR algorithm and event-triggered mechanism to achieve consensus with bounded error, under the condition that the robustness of relevant topology is satisfied. Under the event-triggered mechanism, the system cannot achieve accurate consensus and must leave an upper bound of error \(c\) that grows exponentially with the number of normal agents. One can set the triggering parameter \(c_{0}\) by 0 to achieve error free consensus, but the number of triggers will increase. When \(t\rightarrow\infty\), the event-triggered mechanism will lose its effect due to attenuation. The final conclusion is almost the same as that in [82]. Consider reducing the conservatism of the conclusion by adjusting the update structure. Since the update structure reduces unnecessary state updates, under the same error \(c\), the number of system triggers will be less, but the convergence speed will be slower. In short, the accuracy of event-triggered mechanism and consensus cannot be reconciled well in the context of MSR algorithm. Compared with periodic sampling control, event triggering reduces the number of information exchanges. However, due to the existence of triggering parameters, there is always an error in the consensus state value. At the same time, for the threshold-triggered mechanism, it is also necessary to consider the balance between the convergence performance and the communication frequency caused by the threshold value. Ref. [85] proposes a sliding-mode control method, which only makes periodic judgment when the state is measurable, does not need continuous triggering condition detection and has better economic benefits, which also belongs to a feasible improvement direction. Ref. [86] focuses on the observation of the state rather than screening. They achieve the uniform ultimate boundedness of the system ouput. In consideration of malicious agent attacks and bounded sensor errors, it designs an observer so that the outputs of all followers in a heterogeneous system converge successfully to a dynamic convex hull composed of the outputs of complex leaders. For a heterogeneous system with leaders and attackers:  
\[\left\{ \begin{array}{c}\dot{x}_{i}=A_{i}x_{i}+B_{i}u_{i},\\ y_{i}=C_{i}x_{i}\end{array} \right.,\ i\in\mathcal{F}_{,}\left\{ \begin{array}{c} \dot{x}_{l}=Sx_{l},\\ y_{l}=Rx_{l}\end{array} ,\ l\in\mathcal{L},\ \left\{ \begin{array}{c}\dot{x}_{k}=f_{k}(x_{k}),\\ y_{k}=C_{k}x_{k}\end{array} \right.,\ k\in\bar{\mathcal{A}},\right. \tag{36}\]  
where \(\mathcal{F},\ \bar{\mathcal{L}},\ \bar{\mathcal{A}}\) are respectively a collection of followers, leaders and attackers. The controller is  
\[u_{i} = K_{i}\tilde{x}_{i}+H_{i}\tilde{z}_{i}-H_{i}\hat{d}_{i}, \tag{37}\] \[\dot{\tilde{z}}_{i} = F_{i}\tilde{z}_{i}+G_{i}\tilde{g}_{yi}, \tag{38}\]  
where \(\tilde{e}_{yi}=\sum_{j\in\mathcal{F}}a_{ij}(\tilde{y}_{j}-\tilde{y}_{i})+\sum _{l\in\mathcal{L}}a_{li}(y_{l}-\tilde{y}_{i})+\sum_{k\in\mathcal{L}}a_{ik}(y_{ k}-\tilde{y}_{i})\).