## 2 Problem Formulation and the Convergence of the Quantized ILC Law  
Suppose that a repetitive single input-single output (SISO) linear time-invariant discrete system is described as follows  
\[\left\{ \begin{array}{l}x_{k}(n+1)=\textbf{A}x_{k}(n)+\textbf{B}u_{k}(n),\\ y_{k}(n+1)=\textbf{C}x_{k}(n+1),\\ x_{k}(0)=0,\;\;n\in S,\end{array} \right. \tag{1}\]  
where \(n\) is the discrete sampling time and \(S=\{0,1,2,\cdots,N-1\}\) stands for the collection of samples, and \(N\) represents the total number of samples, while the subscript \(k\) denotes the operation index or repetition, named as the iteration. \(x_{k}(n)\) and \(y_{k}(n)\) are the \(P\)-dimensional state vector and output for the \(k\)-th iteration; \(u_{k}(n)\) is the scalar input. **A**, **B** and **C** are matrices with the appropriate dimensions, respectively. In this paper, we will conduct the investigation for the case in which \(\textbf{C}\textbf{B}\
eq 0\), then the relative degree of the system (1) is united. For any input sequence \(\mathbf{u}_{k}=[u_{k}(0)|u_{k}(1)|\cdots|u_{k}(N-1)]^{\mathrm{T}}\), the output sequence of the system (1) is expressed as  
\[y_{k}(n+1)=\sum_{l=0}^{n}\mathbf{CA}^{n-l}\mathbf{B}u_{k}(l),\;n=0,1,\cdots,N-1. \tag{2}\]  
Let  
\[\mathbf{H}= \begin{bmatrix}\mathbf{CB}&\mathbf{0}&\mathbf{0}&\cdots&\mathbf{0 }\\ \mathbf{CAB}&\mathbf{CB}&\mathbf{0}&\cdots&\mathbf{0}\\ \mathbf{CA}^{2}\mathbf{B}&\mathbf{CAB}&\mathbf{CB}&\cdots&\mathbf{0}\\ \vdots&\vdots&\vdots&\vdots&\mathbf{0}\\ \mathbf{CA}^{N-1}\mathbf{B}&\mathbf{CA}^{N-2}\mathbf{B}&\mathbf{CA}^{N-3} \mathbf{B}&\cdots&\mathbf{CB}\end{bmatrix} ,\]  
and  
\[\mathbf{y}_{k}=[y_{k}(1)|y_{k}(2)|\cdots|y_{k}(N)]^{\mathrm{T}},\]  
\[\mathbf{u}_{k}=[u_{k}(0)|u_{k}(1)|\cdots|u_{k}(N-1)]^{\mathrm{T}}.\]  
Then, the system (1) or (2) becomes  
\[\mathbf{y}_{k}=\mathbf{H}\mathbf{u}_{k}. \tag{3}\]  
While the system (1) attempts to track a desired trajectory, an iterative learning control scheme is adapted. Suppose that \(y_{d}(n)\) is the desired trajectory and \(e_{k}(n)=y_{k}(n)-y_{d}(n)\) is the output error. The mechanism of the ILC is to compensate the control command \(u_{k}(n)\) by \(e_{k}(n)\) so that \(u_{k+1}(n)\) may drive the system to better the tracking performance. For the practical execution, the output signal will be quantized before transmission to the controller. To achieve this, we construct the updating law of the iterative learning control with quantized error (ILCQe) as follows  
\(u_{1}(n)\): given arbitrarily;  
\[u_{k+1}(n)=u_{k}(n)+\Gamma Q(e_{k}(n+1)),n\in D. \tag{4}\]  
Here, the parameter \(\Gamma\) is a derivative learning gain; it is a concrete number that was chosen using the \"trial and error method\" according to experience in the convergence conditions, and the \(Q(e_{k}(n+1))\) is a scalar function about a quantizer. In the parts that follow, we will discuss the quantizer. This study utilizes the logarithmic type of quantizer described in [40]; the set of the quantized level is represented by  
\[U=\left\{\pm s_{i}:s_{i}=\eta^{is}s_{0},\;i=0,\pm 1,\pm 2,\;\dots\right\}\cup \left\{0\right\},\;0<\eta<1,\;s_{0}>0.\]  
where \(s_{0}\) denotes the initial of quantized levels, and \(i\) corresponds to quantized series, \(s_{i}\) corresponds to the \(i\)-th segment of the quantizer to translate the quantized data in this segment to the appropriate quantized level. The signal parameter \(\eta\) is considered as a measure of quantized density in the logarithmic quantizer. The quantized level has a range; when the quantized density is lower, then the quantized level is also less, so it can be said that the quantizer is rough. The associated quantizer \(Q(\cdot)\) is defined as  
\[Q(v)=\left\{ \begin{array}{cl}s_{i}&if\;\frac{1}{1+\sigma}s_{i}<v<\frac{1}{1- \sigma}s_{i}\\ 0&if\;v=0\\ -Q(-v)&if\;v<0\end{array} \right., \tag{5}\]  
where \(\sigma=\frac{1-\eta}{1+\eta}\) is called sector bound. This sector bound method has been used to deal with the data quantization in the paper [41].