## 2 Problem Formulation and Preliminaries
### Analysis in Iterative Domain  
ILC is a method to control systems operating in a repetitive mode over a finite interval with the requirement that the system accurately follows the desired output trajectory. The same actions are executed again and again. By using the previously executed information, the tracking accuracy is improved from repetition to repetition. It has long been recognized that ILC is a two-dimensional (2D) system in which information travels on two independent axes: the time axis and the iteration axis. [PERSON] and [PERSON] [21; 22] proposed ILC 2D analysis methods for discrete-time and continue-time systems. After Z-transformation of system dynamics, ILC is considered as a 2D system in frequency domain and iteration domain in this paper. The following analysis is based on the decomposition of 2D dynamics into two separate 1D problems. \(E_{k}(z)\) is considered as a 2D function \(E(k,z)\). Exchanging variable frequency and parameter iteration, the convergence of the system is considered as a 1D problem in the iteration domain. Thus, we exchange variable \(z\) and parameter \(k\). \(E_{k}(z)\) can be rewritten as \(E_{z}(k)\), that is, \(E_{z}(k)=[E_{1,z}(k),E_{2,z}(k),\cdots,E_{n,z}(k)]^{T}\), \(\xi_{k,ij}\) can be rewritten as \(\xi_{ij}(k)\). Then, (15) can be rewritten as  
\[E_{z}(k+1)-E_{z}(k)=-zH(z)[(\mathcal{L}(k)+\Phi)E_{z}(k)-\mathcal{A}(k)E_{z}(k -1)] \tag{17}\]  
where \(\mathcal{L}(k)=[l_{ij}(k)]\in\mathbb{R}^{n\times n}\) and \(\mathcal{A}(k)=[a_{ij}(k)]\in\mathbb{R}^{n\times n}\),  
\[l_{ij}(k)=\left\{\begin{array}{c}-a_{i,j}\xi_{ij}(k),\ j\kappa N_{i}\\ \sum\limits_{j\kappa N_{i}}a_{i,j},\ i=j\\ 0,\ others\end{array}\right., \tag{18}\] \[a_{ij}(k)=\left\{\begin{array}{c}a_{i,j}[1-\xi_{ij}(k)],\ j \kappa N_{i}\\ 0,\ others\end{array}\right..\]  
Take Z-transformation for (17)  
\[\mathbb{z}\mathbb{E}_{z}(\mathbb{z})-\mathbb{E}_{z}(\mathbb{z})=-zH(z)[( \mathcal{L}(\mathbb{z})+\Phi)\mathbb{E}_{z}(\mathbb{z})-\mathbb{z}^{-1} \mathcal{A}(\mathbb{z})\mathbb{E}_{z}(\mathbb{z})] \tag{19}\]  
where \(\mathcal{L}(\mathbb{z})=[l_{ij}(\mathbb{z})]\in\mathbb{R}^{n\times n}\) and \(\mathcal{A}(\mathbb{z})=[a_{ij}(\mathbb{z})]\in\mathbb{R}^{n\times n}\),  
\[l_{ij}(\mathbb{z})=\left\{\begin{array}{c}-a_{i,j}\xi_{ij}( \mathbb{z}),\ j\kappa N_{i}\\ \sum\limits_{j\kappa N_{i}}a_{i,j},\ i=j\\ 0,\ others\end{array}\right., \tag{20}\] \[a_{ij}(\mathbb{z})=\left\{\begin{array}{c}a_{i,j}[1-\xi_{ij}( \mathbb{z})],\ j\kappa N_{i}\\ 0,\ others\end{array}\right..\]  
(19) leads to  
\[\{(\mathbb{z}-1)I+zH(z)[\mathcal{L}^{\prime}(\mathbb{z})+\Phi]\}\mathbb{E}_{ z}(\mathbb{z})=0 \tag{21}\]  
where \(\mathcal{L}^{\prime}(\mathbb{z})=[l^{\prime}_{ij}(\mathbb{z})]\in\mathbb{R}^ {n\times n}\),  
\[l^{\prime}_{ij}(\mathbb{z})=\left\{\begin{array}{c}-a_{i,j}\mathbb{z}^{-1} [\mathbb{z}\xi_{ij}(\mathbb{z})+1-\xi_{ij}(\mathbb{z})],\ j\kappa N_{i}\\ \sum\limits_{j\kappa N_{i}}a_{i,j},\ i=j\\ 0,\ others\end{array}\right., \tag{22}\]From Equation (21), we can obtain characteristic equation of the system in iterative domain. In the next section, the convergence of the learning system is discussed based on the characteristic equation.