# Event-Triggered MFAILC Bipartite Formation Control for Multi-Agent Systems Under DoS Attacks
## 2 Preliminaries and Problem Formulation
### Problem Formulation  
For the repeatedly operated BFMASs, a multi-input multi-output (MIMO) agent \(p\) can be described as a nonlinear system as follows:  
\[y_{p}(k+1,i)=f_{p}(y_{p}(k,i),\ldots,y_{p}(k-n_{y},i),\quad u_{p}(k,i),u_{p}(k-n _{u},i)), \tag{1}\]  
here, \(k\) represents the sampling instant \(k\in[0,1,\ldots,T]\), \(p=1,2,\ldots,N\), where \(N\) is the number of agents. \(n_{y}\) and \(n_{u}\) are unknown positive integers, \(y_{p}(k,i)\in R^{n}\) and \(u_{p}(k,i)\in R^{n}\) are the system output and the control input of agent \(p\) in the \(i_{th}\) iteration and the sampling instant \(k\), respectively. \(f(\bullet)\) denotes an unknown nonlinear function. To obtain the equivalent linearized model of the MASs, the assumptions are as follows:  
**Assumption 1**. : \(f(\bullet)\) _has a continuous partial derivative with regard to \(u_{p}(k,i)\), i.e., \(\partial(f(\bullet))/\partial(u_{p}(k,i))\) exists and is continuous._  
**Assumption 2**. : _For all and, if, then the generalized Lipschitz condition holds for Equation (1) in the iterative domain, i.e.,_  
\[\left\|\Delta y_{p}(k+1,i)\right\|\leq b\left\|\Delta u_{p}(k,i)\right\| \tag{2}\]  
_here, \(\Delta y_{p}(k+1,i)=y_{p}(k+1,i)-y_{p}(k+1,i-1)\) and \(\Delta u_{p}(k,i)=u_{p}(k,i)-u_{p}(k,i-1)\). Both \(\left\|\Delta y_{p}(k+1,i)\right\|\) and \(\left\|\Delta u_{p}(k,i)\right\|\) are bounded, and \(b\) is a finite constant._  
**Assumption 3**. : _The communication topology \(\overline{\mathcal{G}}\) or \(\overline{\mathcal{G}}^{t}\) of the MASs is strongly connected, meaning that each agent has a communicable path to the virtual leader._  
**Remark 1**. : _For many practical control systems, Assumptions 1 and 2 are reasonable. Assumption 1 represents a typical constraint imposed on general nonlinear systems in control system design. Assumption 2 imposes an upper limit on the rate of change of the system output caused by variations in the control input along the iterative axis. These assumptions have also been discussed in the literature, such as [12; 13; 23]._  
**Theorem 1**. : _For the agent described in system (1) that satisfies Assumptions 1 and 2, when \(\left\|\Delta u_{p}(k,i)\right\|\
eq 0\), there must exist a time-varying pseudo-Jacobian matrix (PJM) \(\Phi_{p}(k,i)\) associated with the iterative axis, such that the agent can be equivalently represented by the following compact-form dynamic linearization (CFDL) data model:_  
\[\Delta y_{p}(k+1,i)=\Phi_{p}(k,i)\Delta u_{p}(k,i) \tag{3}\]  
Figure 1: Communication topology of the agents. _Here, \(\Phi_{p}(k,i)\) as below_  
\[\Phi_{p}(k,i)= \begin{bmatrix}\phi_{p,11}(k,i)&\cdots&\phi_{p,1m}(k,i)\\ \vdots&\vdots&\ddots\\ \phi_{p,m1}(k,i)&\cdots&\phi_{p,mm}(k,i)\end{bmatrix} \] \[\in\mathbb{R}^{m\times m}\]  
_Moreover, \(\left\|\Phi_{p}(k,i)\right\|\leq\overline{b}\), \(p\in[1,2,\ldots,N]\), \(\overline{b}\) denotes a positive constant._  
Consequently, the following linearized model can be used to rewrite the relationship between the nonlinear agent \(p\)'s system input and output:  
\[y_{p}(k+1,i)=y_{p}(k+1,i-1)+\Phi_{p}(k,i)\Delta u_{p}(k,i) \tag{4}\]  
For the agent \(p\) in Equation (1), the trajectory of the virtual leader is considered as the reference output curve, denoted by \(y_{d}(k)\), where \(k\in\{0,1,2,\ldots,T\}\) represents the leader's position at time \(k\). Considering that there exists a distance gap between the followers and the leader in the formation system, the tracking error of agent \(p\) in the bipartite formation system is defined as:  
\[e_{p}(k,i)=y_{d}(k)-\delta_{p}(y_{p}(k,i)-\tilde{y}_{p}(k)) \tag{5}\]  
In the above equation, \(\tilde{y}_{p}(k)\) is the relative distance between agent \(p\) and the virtual leader, \(y_{p}(k,i)\) is the output of the each agent, and \(\delta_{p}\) represents the node attribute of agent \(p\) in the communication topology graph, i.e., the membership indicator of agent \(p\) in the subset. The tracking error serves as a response to the agents' ability to track in accordance with their individual desired curves.