## 3 Design And Analysis of ILC Consensus Algorithm  
Firstly, we define the tracking error of each agent as  
\[e_{i,k}\left(t\right)=y_{d}(t)-y_{i,k}(t), \tag{3}\]  
and  
\[e_{ij,k}\left(t\right)=y_{j,k}(t)-y_{i,k}(t),j\in N_{i}, \tag{4}\]  
where \(e_{i,k}\left(t\right)\) and \(e_{ij,k}\left(t\right)\) are the tracking errors with the leader and neighboring agents, respectively, of agent \(i\) at the \(k\)th iteration. Let \(\xi_{i,k}\left(t\right)\) denotes the information received or measured of the agent \(i\) at the \(k\)th iteration, and we get  
\[\xi_{i,k}\left(t\right)=\sum_{j\in N_{i}}a_{ij}e_{ij,k}(t)+s_{i}e_{i,k}\left(t \right), \tag{5}\]  
where \(a_{ij}\) is the adjacency elements associated with the edges of the digraph, and \(s_{i}\) is the coupling weight of the \(i\)th following agent to the leader. In terms of ILC algorithm, the tracking error continues to decrease with increasing iteration number, and the convergence rate will be slow when the tracking error is small. Hence, we are encouraged to appropriately amplifying the tracking error when the tracking error is small; the following agents can track the leader's trajectory more quickly. In this paper, we adopt the PD type ILC consensus algorithm and introduce a fractional-power proportional part as follows,  
\[u_{i,k+1}(t) \tag{6}\] \[= u_{i,k}(t)+\Gamma_{i}\xi_{i,k}\left(t\right)+K_{i}sign\left(\xi_{ i,k}\left(t\right)\right)\left|\xi_{i,k}\left(t\right)\right|^{\alpha},\]  
where \(\Gamma_{i}\in R^{r_{i}\times m}\), \(K_{i}\in R^{r_{i}\times m}\) are the learning gains, \(0<\alpha<1\), and the sign function \(sign\left(\xi_{i,k}\left(t\right)\right)\) is defined as  
\[sign\left(\xi_{i,k}\left(t\right)\right)=\left\{ \begin{array}{rl}&1\quad\xi_ {i,k}\left(t\right)>0,\\ &0\quad\xi_{i,k}\left(t\right)=0,\\ &-1\quad\xi_{i,k}\left(t\right)<0.\end{array} \right.\]  
Moreover, the concrete steps of the algorithm (6) are shown in Algorithm 1. ```
Parameters: agent i, iteration index k, \(a_{ij}\), \(s_{i}\), \(\Gamma_{i}\),\(K_{i}\), \(\alpha\) Input: the reference trajectory \(y_{d}(t)\) Output: the control input \(u_{i,k}(t)\)
1: initialize the iteration index k=1
2: while (\(e_{i,k}(t)\)! = \(0\), \(i\in V\), \(t\in[0,T]\))
3: initialize the initial state \(x_{i,k}(0)=0\)
4: calculate the output \(y_{i,k}(t)\) by the Equation (2),
5: calculate the tracking errors of each agent \(e_{i,k}(t)\) by the Equation (3),
6: calculate the tracking errors with the neighboring agents \(e_{ij,k}(t)\) by the Equation (4),
7: calculate \(\xi_{i,k}(t)\) by the Equation (5)
8: if k=1 \(u_{i,k}(t)=0\)
9: else \(u_{i,k}(t)=u_{i,k-1}(t)+\Gamma_{i}\xi_{i,k-1}\left(t\right)+K_{i}sign\left( \xi_{i,k-1}\left(t\right)\right)\left|\xi_{i,k-1}\left(t\right)\right|^{\alpha}\)
10: end if
11: set k=k+1
12: end while
13: print the control input \(u_{i,k}(t)\)
```  
**Algorithm 1** Iterative learning control (ILC) consensus algorithm based on fractional-power error signals  
**Remark 2**. : _[PERSON] et al. [23] used the sliding mode control to solve the finite-time consensus problem, where the sign function is used to force the state trajectories onto the sliding surface from any arbitrary initial location in the phase plane. However, the sign function in this paper is to extract the sign of the tracking errors, and then take fractional-power of the absolute value of the tracking errors. Fractional-power tracking errors can accelerate the convergence rate when the tracking errors are small._  
Then, it is obtained from (3) and (4) that  
\[e_{ij,k}(t)=e_{i,k}(t)-e_{j,k}(t), \tag{7}\]  
and then (5) turns to be  
\[\xi_{i,k}\left(t\right)=\sum_{j\in N_{i}}a_{ij}\left(e_{i,k}\left(t\right)-e_{ j,k}\left(t\right)\right)+s_{i}e_{i,k}\left(t\right). \tag{8}\]  
**Remark 3**.