## 3 Parameter-Optimal Iterative Learning Control with Quantized Information
: _There exists a tuning factor \(w_{k}\) (\(w_{k}>0\)) and an unique learning gain \(\mathbf{Y}_{k}\) of POGAILCQe in order for the output trajectory to converge uniformly to the desired trajectory, and the law (13) drives the system (3); satisfying \(\|\mathbf{e}_{k+1}\|\leq\rho\|\mathbf{e}_{k}\|\)_ with \(0<\rho<1\), the constant \(\rho\) is termed as the monotone convergence rate._  
Proof of Theorem 2.: Substituting this Equation (13) into the system (3), we obtain  
\[\mathbf{e}_{k+1}=\mathbf{e}_{k}-\mathbf{H}(\mathbf{u}_{k+1}-\mathbf{u}_{k})=\mathbf{e}_{k}- \mathbf{H}\mathbf{M}_{k}\mathbf{Y}_{k}. \tag{15}\]  
Thus,  
\[\|\mathbf{e}_{k+1}\|^{2}=\mathbf{e}_{k+1}^{\mathsf{T}}\mathbf{e}_{k+1}\]  
\[=(\mathbf{e}_{k}-\mathbf{H}\mathbf{M}_{k}\mathbf{Y}_{k})^{\mathsf{T}}(\mathbf{e}_{k}- \mathbf{H}\mathbf{M}_{k}\mathbf{Y}_{k}) \tag{16}\]  
\[=\|\mathbf{e}_{k}\|^{2}-\mathbf{e}_{k}^{\mathsf{T}}\mathbf{H}\mathbf{M}_{k}\mathbf{Y} _{k}-\mathbf{Y}_{k}^{\mathsf{T}}\mathbf{M}_{k}\mathbf{H}^{\mathsf{T}}\mathbf{e}_{k }+\mathbf{Y}_{k}^{\mathsf{T}}\mathbf{M}_{k}\mathbf{H}^{\mathsf{T}}\mathbf{H} \mathbf{M}_{k}\mathbf{Y}_{k}.\]  
Substituting equality (16) into (14), we have  
\[u_{k+1}(n)=u_{k}(n)+\gamma_{k}(n+1)(1+\Delta e_{k}(n+1))e_{k}(n+1). \tag{12}\]0Let \(\Delta J(\mathbf{Y}_{k})=\frac{\partial J(\mathbf{Y}_{k})}{\partial\mathbf{Y}_{k}}=0\), then  
\[u_{k+1}(n)=u_{k}(n)+\gamma_{k}(n+1)(1+\Delta e_{k}(n+1))e_{k}(n+1). \tag{12}\]1  
When the matrix \(\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{M}_{k}\) is nonnegative, the matrix \(\left(w_{k}\mathbf{I}+\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{ M}_{k}\right)\) is positive definite and thus nonsingular for \(w_{k}>0\), then the matrix \(\left(w_{k}\mathbf{I}+\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{ M}_{k}\right)\) is invertible. Therefore, there exists a unique solution \(\mathbf{Y}_{k}\) of Equation (17), and it is the parameter-optimal iteration time-varying learning-gain vector. The optimal solution is derived as  
\[u_{k+1}(n)=u_{k}(n)+\gamma_{k}(n+1)(1+\Delta e_{k}(n+1))e_{k}(n+1). \tag{12}\]2  
Be aware that the tracking error vector \(e_{k}\) is related to the matrix \(\mathbf{M}_{k}\). This indicates that the matrix \(\mathbf{M}_{k}\) is very likely to be singular when some components of \(e_{k}\) are zero. Equation (17) is indeterminate in the given situation, necessitating some changes for its solution. Under the particular circumstances, the solution of Equation (17) must be obtained when the matrix \(\mathbf{M}_{k}\) is nonsingular and singular, respectively. **Case 1. ** The matrix \(\mathbf{M}_{k}\) is nonsingular. Given the presumption \(\mathbf{C}\mathbf{B}\
eq 0\), which denotes that the Markov parameters matrix \(\mathbf{H}\) is nonsingular, it is certain that the matrix \(\mathbf{M}_{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{M}_{k}\) is positive definite. Substituting Equation (18) into (15), we obtain  
\[ \begin{split} e_{k+1}=e_{k}-\mathbf{H}\mathbf{M}_{k}\mathbf{Y}_{ k}\\ =e_{k}-\mathbf{H}\mathbf{M}_{k}\big{(}w_{k}\mathbf{I}+\mathbf{M} _{k}\mathbf{H}^{\mathrm{T}}\mathbf{H}\mathbf{M}_{k}\big{)}^{-1}\mathbf{M}_{k }\mathbf{H}^{\mathrm{T}}e_{k}\\ =\Big{(}\mathbf{I}-\Big{(}\mathbf{I}+w_{k}\mathbf{H}^{-\mathrm{ T}}\mathbf{M}_{k}^{-1}\mathbf{M}_{k}^{-1}\mathbf{H}^{-1}\Big{)}^{-1}\Big{)}e_{k}. \end{split}  \tag{19}\]  
Here, \(\mathbf{H}^{-\mathrm{T}}=\big{(}\mathbf{H}^{\mathrm{T}}\big{)}^{-1}\).