# Event-Triggered MFAILC Bipartite Formation Control for Multi-Agent Systems Under DoS Attacks
## 3 Design of the Controller Scheme
### Design of the MFAILLC Controller Based on the Event-Triggered Mechanism  
For the BFMASs, based on the CFDL data model (4) of agent \(p\) and the formation error \(\xi_{p}(k,i)\), the control update law is defined as follows [24]:  
\[u_{p}(k,i)=u_{p}(k,i-1)+\frac{\rho\Phi_{p}^{T}(k,i)}{\lambda_{s}+\big{\|}\Phi_ {p}(k,i)\big{\|}^{2}}\xi_{p}(k+1,i-1) \tag{14}\]  
According to Equations (8) and (14), combined with \(S_{p}<0\), this satisfies the event-triggered condition and the iteration sequence satisfying the event-triggered condition is recorded. Considering that \(\Phi_{p}(k,i)\) is difficult to obtain directly, its estimated value \(\hat{\Phi}_{p}(k,i)\) is used as a substitute. Incorporating the event-triggered mechanism, a further rewrite of the control input update law (14) is as follows:  
\[u_{p}(k,i_{s+1})=\begin{cases} u_{p}(k,i_{s})+\frac{\rho_{p}\hat{\Phi}_{p}^{T}(k,i_{s+1})}{ \lambda_{p}+\big{\|}\hat{\Phi}_{p}(k,i_{s+1})\big{\|}^{2}}\xi_{ap}(k+1,i_{s+1} ),&i=i_{s+1}\\ u_{p}(k,i_{s}),&i_{s}\leq i<i_{s+1}\end{cases} \tag{15}\]  
Here, \(\rho_{p}\) is the step factor and \(\lambda_{p}\) is the weight factor. The parameter \(\lambda_{p}\) has a large impact on the algorithm; the smaller the \(\lambda_{p}\), the faster the system responds while the amount of overshooting is larger. Conversely, the slower the response of the system, the smaller the amount of overshooting. For the \(\hat{\[PERSON]}_{p}(k,i)\) of Equation (15), an objective function is defined as below:  
\[\begin{split} J(\hat{\[PERSON]}_{p}(k,i))&=\big{\|}y_{p}( k+1,i-1)-y_{p}(k+1,i-2)-\hat{\[PERSON]}(k,i)\Delta u(k,i-1)\big{\|}^{2}\\ &\quad+\mu_{s}\big{\|}\hat{\[PERSON]}_{p}(k,i)-\hat{\[PERSON]}_{p}(k,i-1) \big{\|}^{2}\end{split} \tag{16}\]  
To proceed with the partial derivative of Equation (16) with respect to \(\hat{\[PERSON]}_{p}(k,i)\) and set \(\hat{\partial}(\big{(}\[PERSON]_{p}(k,i_{i})\big{)})/\partial(\[PERSON]_{p}(k,i))=0\), we can obtain the parameter estimation law as follows:  
\[\hat{\[PERSON]}_{p}(k,i)=\hat{\[PERSON]}_{p}(k,i-1)+\frac{\eta_{p}(\Delta y_{p}(k+1,i-1 )-\Phi_{p}(k,i-1)\Delta u_{p}(k,i-1))}{\mu_{p}+\big{\|}\Delta u_{p}(k,i-1) \big{\|}^{2}}\Delta u_{p}^{T}(k,i-1) \tag{17}\]  
In Equation (17), the weight factor \(\mu_{p}\) and the step factor \(\eta_{p}\) are used to update the PJM; the range of values of \(\eta_{p}\) and \(\mu_{p}\) is given in the proof section. The selection of \(\eta_{p}\) and \(\mu_{p}\) can be referred to \(\lambda_{p}\) and \(\rho_{p}\). To ensure the algorithm is general and applicable, the following two conditions need to be satisfied:  
\[\begin{cases}\hat{\[PERSON]}_{p,ll}(k,i)=\hat{\[PERSON]}_{p,ll}(k,1),if\text{sgn}(\hat{ \Phi}_{p,ll}(k,i))\
eq\text{sgn}(\hat{\Phi}_{p,ll}(k,1))\\ \hat{\Phi}_{p,lq}(k,i)=\hat{\Phi}_{p,lq}(k,1),if\text{sgn}(\hat{\Phi}_{p,lq}(k, i))\
eq\text{sgn}(\hat{\Phi}_{p,lq}(k,1))\end{cases} \tag{18}\]where \(\hat{\phi}_{p,l}(k,1)\) and \(\hat{\phi}_{p,lq}(k,1)\) are the initial values of \(\hat{\phi}_{p}(k,i)\) and \(q=1,2,\ldots,m\), \(l\
eq q\). Remark: Without the above reset condition there may be unstable control effects.