## 3 Parameter-Optimal Iterative Learning Control with Quantized Information
Calculating inner product to both sides of Equation (19) results in  
\[\|e_{k+1}\|^{2}=e_{k}^{\mathrm{T}}\bigg{(}\mathbf{I}-\Big{(}\mathbf{I}+w_{k} \mathbf{H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{k}^{-1}\mathbf{H}^{-  
Let  
\[\stackrel{{\sim}}{{\rho}}=\left(\frac{w_{k}\lambda_{\max}\left(\mathbf{ H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{k}^{-1}\mathbf{H}^{-1}\right)}{1+w_{k} \lambda_{\max}\left(\mathbf{H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{k} ^{-1}\mathbf{H}^{-1}\right)}\right)^{2}. \tag{24}\]  
Assuming \(\mathbf{CB}\
eq 0\) and the Markov parameters matrix \(\mathbf{H}\) is nonsingular, it is confirmed that the matrix \(\mathbf{H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{k}^{-1}\mathbf{H}^{-1}\) is positive definite. Therefore, \(\lambda_{\max}\left(\mathbf{H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{ k}^{-1}\mathbf{H}^{-1}\right)>0\). Then, Equation (24) indicates that \(0<\stackrel{{\sim}}{{\rho}}<1\). Therefore, expressions (20) deliver  
\[\|e_{k+1}\|^{2}\leq\stackrel{{\sim}}{{\rho}}\|e_{k}\|^{2}, \tag{25}\]  
then the tracking error monotone convergences to zero. **Case 2. ** The Matrix \(\mathbf{M}_{k}\) is singular. In this case, we cannot obtain \(\left(\mathbf{I}+w_{k}\mathbf{H}^{-\mathrm{T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{ k}^{-1}\mathbf{H}^{-1}\right)^{-1}\) from the expression \(\left(w_{k}\mathbf{I}+\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{M }_{k}\right)\)Y\({}_{k}=\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}e_{k}\) in (17). Then, the following row- and column-exchanging transformations can be used to evaluate the monotonicity of the tracking error energy in an indirect way. Suppose that \(\mathrm{Rank}(\mathbf{M}_{k})=r_{k}<N\) with \(e_{k}(i_{q})\
eq 0\) for \(q=1,\cdots,r_{k}\) and \(e_{k}(l_{m})=0\) for \(m=1,2,\cdots,N-r_{k}\). Here,  
\[i_{1}<i_{2}<\cdots<i_{r_{k}},\]  
\[l_{1}<l_{2}<\cdots<l_{N-r_{k}}.\]  
Denote  
\[\mathbf{R}_{k}=\left[\varepsilon_{i_{1}}\Big{|}\varepsilon_{i_{2}}\Big{|} \cdots\Big{|}\varepsilon_{i_{r_{k}}}\Big{|}\varepsilon_{i_{1}}\Big{|} \varepsilon_{i_{2}}\Big{|}\cdots\Big{|}\varepsilon_{i_{N-r_{k}}}\right]\]  
where \(\varepsilon_{i}\), for \(i=i_{1},i_{2},\cdots,i_{r_{k}},l_{1},\cdots,l_{N-r_{k}}\), in the unit vector, the \(i\)-th element is \(1\) and the others are \(0\). Obviously, the matrix \(\mathbf{R}_{k}\) is orthogonal, so that \(\mathbf{R}_{k}^{-1}=\mathbf{R}_{k}^{\mathrm{T}}\). We know if a matrix is multiplied right by the matrix \(\mathbf{R}_{k}\), then the \(i_{1}\)-th up to \(i_{r_{k}}\)-th and \(l_{1}\)-th up to \(l_{N-r_{k}}\)-th columns of the matrix may exchange to the first up to the \(N\)-th columns based on the structure of the matrix \(\mathbf{R}_{k}\). Similarly, the \(i_{1}\)-th up to \(i_{r_{k}}\)-th then \(l_{1}\)-th up to \(l_{N-r_{k}}\)-th rows of the matrix can be exchanged to the first up to the \(N\)-th rows by pre-multiplying \(\mathbf{R}_{k}^{\mathrm{T}}\). Thus, the matrices \(\mathbf{R}_{k}\) and \(\mathbf{R}_{k}^{\mathrm{T}}\) are called the column-exchanging and row-exchanging transformation matrices [42]. We have  
\[\mathbf{R}_{k}^{\mathrm{T}}\mathbf{H}\mathbf{R}_{k}=\left[ \begin{array}{ccccc} \mathbf{CB}&0&\cdots&0&*\cdots*\\ \mathbf{CA}^{i_{2}-i_{1}}\mathbf{B}&\mathbf{CB}&\cdots&0&*\cdots*\\ \vdots&\vdots&\ddots&\vdots&\vdots\cdots\\ \mathbf{CA}^{i_{r_{k}}-i_{1}}\mathbf{B}&\mathbf{CA}^{i_{r_{k}}-i_{2}}\mathbf{ B}&\cdots&\mathbf{CB}&*\cdots*\\ *&*&\cdots&*&\cdots*\\ \vdots&\vdots&\ddots&\vdots&\vdots\cdots\\ *&*&\cdots&*&*\cdots*\end{array} \right].\]  
In the top matrix, \"\(*\)\" are some elements of the Markov parameters matrix \(\mathbf{H}\). Let  
\[\mathbf{R}_{k}^{\mathrm{T}}\mathbf{H}\mathbf{R}_{k}=\left[ \begin{array}{cc} \stackrel{{-}}{{\mathbf{H}}}_{k}&\stackrel{{-}}{{ \mathbf{H}}}_{k}\\ \stackrel{{\sim}}{{\mathbf{H}}}_{k}&\stackrel{{-}}{{ \mathbf{H}}}_{k}\end{array} \right].