## 3 Design And Analysis of ILC Consensus Algorithm
: _It should be pointed out that, in order to ensure the convergence of the multiagent systems, \(\alpha\) is in the range of 0 to 1 and cannot be too small._  
Let \(x_{k}(t)=[x_{1,k}^{T}(t),x_{2,k}^{T}(t),\cdots,x_{n,k}^{T}(t)]^{T}\), \(u_{k}(t)=[u_{1,k}^{T}(t),u_{2,k}^{T}(t),\cdots,u_{n,k}^{T}(t)]^{T}\), and the systems (2) are written in the compact-vector form as  
\[\left\{ \begin{array}{l}\dot{x}_{k}(t)=Ax_{k}(t)+Bu_{k}(t),\\ y_{k}(t)=Cx_{k}(t),\end{array} \right. \tag{9}\]  
where \(A=\text{diag}\{A_{1},A_{2},\cdots,A_{n}\}\), \(B=\text{diag}\{B_{1},B_{2},\cdots,B_{n}\}\) and \(C=\text{diag}\{C_{1},C_{2},\cdots,C_{n}\}\). Defining \(e_{k}(t)=[e_{1,k}^{T}(t),e_{2,k}^{T}(t),\cdots,e_{n,k}^{T}(t)]^{T}\) and \(\xi_{k}(t)=[\xi_{1,k}^{T}(t),\xi_{2,k}^{T}(t),\cdots,\xi_{n,k}^{T}(t)]^{T}\), and Equation (8) can be written as  
\[\xi_{k}\left(t\right)=\left(\left(L+S\right)\otimes I_{m}\right)e_{k}\left(t \right), \tag{10}\]  
where \(S=\text{diag}\{s_{1},s_{2},\cdots,s_{n}\}\), and \(L\) is the Laplacian matrix of \(G\). Next, we define the following operators \(\overline{sign}\left(\xi_{k}\left(t\right)\right):C_{r}[0,T]\to C_{r}[0,T]\) and \(f^{\alpha}\left(\xi_{k}\left(t\right)\right):C_{r}[0,T]\to C_{r}[0,T]\),  
\[\overline{sign}\left(\xi_{k}\left(t\right)\right) \tag{11}\] \[= \text{diag}\left\{sign\left(\xi_{1,k}\left(t\right)\right),\cdots,sign\left(\xi_{n,k}\left(t\right)\right)\right\},\]  
and  
\[f^{\alpha}\left(\xi_{k}\left(t\right)\right) \tag{12}\] \[= \Big{[}\big{|}\xi_{1,k}\left(t\right)\big{|}^{\alpha},\big{|}\xi_ {2,k}\left(t\right)\big{|}^{\alpha},\cdots,\big{|}\xi_{n,k}\left(t\right) \big{|}^{\alpha}\Big{]}^{T}.\]  
Consequently, the compact-vector version of algorithm (6) is  
\[u_{k+1}(t)\] \[= u_{k}(t)+\Gamma_{\xi_{k}}^{\delta}\left(t\right)+K\overline{ sign}\left(\xi_{k}\left(t\right)\right)f^{\alpha}\left(\xi_{k}\left(t \right)\right),\]  
where \(\Gamma=\text{diag}\{\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{n}\}\), and \(K=\text{diag}\{K_{1},K_{2},\cdots,K_{n}\}\). Substituting (10) into (13), we have  
\[ \begin{array}{ll}&u_{k+1}(t)\\ =&u_{k}(t)+\Gamma\left(\left(L+S\right)\otimes I_{m}\right)\dot{e}_{k}\left(t \right)\\ &+\widetilde{Kisign}\left(\xi_{k}\left(t\right)\right)f^{\alpha}\left(\left( \left(L+S\right)\otimes I_{m}\right)e_{k}\left(t\right)\right).\end{array}  \tag{14}\]  
To realize the consensus tracking, we need the following assumption on the initial states of agents. **Assumption 2**. : _The initial tracking error resetting condition is satisfied for all agents, i.e., \(\xi_{i,k}\left(0\right)=0\)._  
Now, we go to the main result, and some important lemmas are listed first. **Lemma 1**. : _[_25_]_ _Let \(x\left(t\right)\), \(c\left(t\right)\), and \(a\left(t\right)\) be a real valued continuous function on \([0,T]\), and \(a(t)\geq 0\), if_  
\[x\left(t\right)\leq c\left(t\right)+\int_{0}^{t}a\left(\tau\right)x\left(\tau \right)d\tau,t\in[0,T],\]  
_then,_  
\[x\left(t\right)\leq c\left(t\right)+\int_{0}^{t}a\left(\tau\right)c\left(t \right)e^{\int_{\tau}^{t}a(\sigma)d\sigma}d\tau,t\in[0,T].\]  
**Lemma 2**. : _[_25_]_ _Let \(\{b_{k}\}_{k\geq 0}(b_{k}\geq 0)\) be a constant sequence that converges to zero. Define an operator: \(Q_{k}:C_{r}[0,T]\to C_{r}[0,T]\) satisfying_  
\[\|Q_{k}(u)(t)\|\leq M(b_{k}+\int_{0}^{t}\|u(s)\|ds),\]  
_where \(M\geq 1\) is a constant, and the norm of \(r\) dimensional vector \(C_{r}[0,T]\) takes the maximum value. Suppose that \(P(t)\) is a continuous function matrix of \(r\times r\) dimension, and let \(P:C_{r}[0,T]\to C_{r}[0,T]\) satisfy_  
\[P(u)(t)=P(t)u(t).\]  
_Then, \(\lim_{n\to\infty}(P+Q_{n})\cdots(P+Q_{0})(u)(t)=0\) holds uniformly with \(t\), if \(\rho(P)<1\)._  
**Theorem 1**. : _Consider the heterogeneous multiagent systems (2) with the novel PD-type ILC consensus algorithm (6).