## 3 Design And Analysis of ILC Consensus Algorithm
With Assumptions 1 and 2, the following agents track the leader's trajectory as \(k\to\infty\), i.e., \(\lim_{k\to\infty}e_{k}(t)=0,t\in[0,T]\), if the following inequality holds,_  
\[\rho\left(I_{mn}-CB\Gamma(\left(L+S\right)\otimes I_{m})\right)<1. \tag{15}\]  
Proof. : Available from Assumption 2, we have \(x_{k+1}(0)-x_{k}(0)=0\). Take integral operation of dynamical system (9) with (14), and we get  
\[ \begin{array}{ll}&x_{k+1}(t)-x_{k}(t)\\ =&x_{k+1}(0)-x_{k}(0)+\int_{0}^{t}A\left(x_{k+1}(\tau)-x_{k}(\tau)\right)d \tau\\ &+\int_{0}^{t}B\left(u_{k+1}(\tau)-u_{k}(\tau)\right)d\tau\\ =&A\int_{0}^{t}\left(x_{k+1}(\tau)-x_{k}(\tau)\right)d\tau+B\Gamma(\left(L+S \right)\otimes I_{m})e_{k}(t)\\ &+BK\int_{0}^{t}\widetilde{sign}\left(\xi_{k}\left(\tau\right)\right)f^{\alpha }\left(\left(L+S\right)\otimes I_{m}\right)e_{k}\left(\tau\right)\right)d \tau.\end{array}  \tag{16}\]  
Next, we analyze the tracking error, and it follows from (3) that  
\[e_{i,k+1}\left(t\right) \tag{22}\] \[= y_{d}(t)-y_{i,k+1}(t)\] \[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]  
Recursively, following (16) yields  
\[e_{k+1}\left(t\right)-e_{k}\left(t\right)\] \[= C(x_{k}(t)-x_{k+1}(t))\] \[= CA\int_{0}^{t}\left(x_{k}(\tau)-x_{k+1}(\tau)\right)d\tau-CB\Gamma ((L+S)\otimes I_{m})e_{k}(t)\] \[-CBK\int_{0}^{t}\overline{sign}\left(\xi_{k}\left(\tau\right) \right)f^{\alpha}\left(\left(\left(L+S\right)\otimes I_{m}\right)e_{k}\left( \tau\right)\right)d\tau.\]  
Evidently, we get  
\[e_{k+1}\left(t\right)\] \[= (I_{mn}-CB\Gamma((L+S)\otimes I_{m}))e_{k}(t)+CA\int_{0}^{t} \left(x_{k}(\tau)-x_{k+1}(\tau)\right)d\tau\] \[-CBK\int_{0}^{t}\overline{sign}\left(\xi_{k}\left(\tau\right) \right)f^{\alpha}\left(\left(\left(L+S\right)\otimes I_{m}\right)e_{k}\left( \tau\right)\right)d\tau.\]  
To recall Lemma 2, we define the following operators; \(P:C_{r}[0,T]\to C_{r}[0,T]\) and \(Q_{k}:C_{r}[0,T]\to C_{r}[0,T]\),  
\[= y_{d}(t)-y_{i,k+1}(t)\]0  
and  
\[= y_{d}(t)-y_{i,k+1}(t)\]1  
It is obvious that \(P\) and \(Q_{k}\) satisfy the definitions in Lemma 2. Referring to Equations (25) and (26), Equation (24) becomes  
\[= y_{d}(t)-y_{i,k+1}(t)\]2 \[= y_{d}(t)-y_{i,k+1}(t)\]3 \[= y_{d}(t)-y_{i,k+1}(t)\]4 \[= y_{d}(t)-y_{i,k+1}(t)\]5 \[= y_{d}(t)-y_{i,k+1}(t)\]6Take the norm of both sides of Equation (26), and we substitute Equations (19) and (21) into it. Then, we get  
\[= y_{d}(t)-y_{i,k+1}(t)\]7 \[= y_{d}(t)-y_{i,k+1}(t)\]8 \[= y_{d}(t)-y_{i,k+1}(t)\]9 \[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]0 \[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]1 \[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]2 \[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]3  
where \(M=\max(1,\|CA\|\left(M_{1}+M_{2}T\right)+f_{0}\left\|CBK\right\|\|(L+S)\otimes I _{m}\|)\). Consequently, operator \(Q_{k}\) satisfies the condition of the operator defined in Lemma 2. Based on Lemma 2, we can conclude from (27) and (29) that  
\[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]4  
if \(\rho\left(I_{mn}-CB\Gamma(\left(L+S\right)\otimes I_{m}\right)\right)<1\). Hence, we complete the proof of Theorem 1. In the following, we consider two simple special cases for the heterogeneous multiagent systems (2) and ILC consensus algorithm (6). **Corollary 1**. : _Consider the heterogeneous multiagent systems (2) with algorithm (6), and suppose that \(C_{i}B_{i}\Gamma_{i}=\beta\times I_{m}\), where \(\beta\in R\) is a constant. With Assumptions 1 and 2, the following agents track the leader's trajectory with \(t\in[0,T]\) as \(k\rightarrow\infty\), if there exists a positive constant \(\beta>0\) satisfying that_  
\[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]5  
_where \(\lambda_{i},i=1,2,\cdots,n\) are eigenvalues of \(L+S\)._  
Proof. : When \(C_{i}B_{i}\Gamma_{i}=\beta\times I_{m}\), the convergence condition in Theorem 1 becomes  
\[= e_{i,k}\left(t\right)-(y_{i,k+1}(t)-y_{i,k}(t)).\]6  
Under Assumption 1, \(L+S\) a nonsingular matrix, and its eigenvalues have positive real parts [24].