## 2 Problem Formulation and Preliminaries
### Problem Formulation
#### 2.2.2 Structure of ILC for Networked Multi-Agent Systems  
The following ILC law is used to achieve control objectives (7)  
\[u_{i,k+1}(t)=u_{i,k}(t)+\Gamma_{i}\!\left\{\sum_{j\in N_{i}}a_{i,j}[\delta_{i, k}(t+1)-\delta_{j,k}(t+1)]+\varphi_{i}\varepsilon_{i,k}(t+1)\right\} \tag{8}\]  
where \(\Gamma_{i}\in\mathbb{R}\) is the learning gain for \(i\)th agent. The formulation of control input of the ith agent requires the control input of the last iteration, error, and information of adjacent agents. Due to the limited bandwidth of communication channel, noise interference, and signal fading, when a multi-agent transmits information through the wireless network, random packet loss is introduced into the multi-agent. For the communication graph of multi-agent systems shown in Figure 1, a block diagram of ILC for multi-agent systems with loss of multiple communication packets is illustrated in Figure 2. When the \(i\)th agent and the \(j\)th agent exchange information with each other through the wireless network, the \(i\)th agent receives a signal from the \(j\)th agent, which is denoted as  
\[\dot{\delta}_{j,k}^{i}(t)=\xi_{k,ij}\delta_{j,k}(t) \tag{9}\]  
where \(\xi_{k,ij}\) obeys Bernoulli distribution and \(\delta_{j,k}(t)\) is the formation tracking error of the \(i\)th agent. When the \(j\)th agent sends a signal to the \(i\)th agent and there is no packet loss in the \(k\)th iteration, let \(\xi_{k,ij}=1\). Otherwise, \(\xi_{k,ij}=0\). Since the \(i\)th agent can obtain its own output \(y_{i,k}(t)\) and the desired output deviation \(d_{i}(t)\) without transmission network, there is no packet loss for the \(i\)th agent itself. That is, \(\delta_{i,k}(t+1)\) does not have packet loss for the \(i\)th agent. It is worth noting that, unlike a previous study [18], the scenario discussed in this paper is closer to the engineering practice, assuming that the packet loss rates of communication between agents are different. Figure 1: Communication graph of multi-agent system. Dropout Compensation ILC Method for Multi-Agent Systems with Loss of Multiple Communication Packets  
In this section, firstly, because the iterative data are changing in the iterative domain, the previous iterative data are used to compensate for the missing data in the iterative process. Secondly, considering that ILC is a two-dimensional system, the convergence of any frequency in the iterative dimension is analyzed by variable iteration and parameter iteration. We make the following assumptions:  
**Assumption 1**. : _The dynamics \(G_{i}\) for each agent is stable, that is, \(|A_{i}|<1\), \(i=1,\ 2,\cdots,\ n\). Without loss of generality, the system relative degree is one, that is, \(G_{i}\
eq 0\)._  
**Remark 1**. : _If an agent's dynamics \(G_{i}\) is not stable, we can use the local feedback of its output \(y_{i}\) to make it stable._  
**Assumption 2**. : _The initial reset condition satisfies all agents and the desired input for each iteration, that is, \(x_{i,k}(0)\) can be abbreviated to \(x_{i}(0)\) and \(C_{i}x_{i,k}(0)=y_{i}^{d}(0),\ k\in 1,\ 2,\ 3\cdots,\ i=1,\ 2,\cdots,\ n\)._  
**Remark 2**. : _Assumption 2 is common in ILC of multi-agent systems (see [13, 16, 19]). If the initial reset condition is not met, we can regard it as the problem of robustness to the initial shifts. In our future work, the ILC with initial-state learning for formation tracking control of multi-agent systems will be explored._  
**Assumption 3**. : _The graph \(\mathcal{G}\) of multi-agent system (1) is an undirected graph. In addition, its graph is connected, that is, there is a path between any two agents so that they can exchange information._  
**Lemma 1**. : [20]_: If an irreducible matrix \(M=(a_{i,j})\in\mathbb{C}^{n\times n},n\geq 1\) is weakly generalized diagonally dominant and at least one of the rows is strictly diagonally dominant, \(M\) is nonsingular._  
**Remark 3**. : _Assumption 3 implies that the \((\mathcal{L}+\Phi)\in\mathbb{C}^{n\times n}\).