## 4 Numerical Example  
Example 1. Consider a SISO linear LDTI system in paper [34; 35] as follows  
\[\left\{ \begin{array}{l}\mathbf{x}_{k}(n+1)=\left[\begin{array}{cc}-0.8&-0.22\\ 1&0\end{array}\right]\mathbf{x}_{k}(n)+\left[\begin{array}{c}0.5\\ 1\end{array}\right]u_{k}(n),\\ y_{k}(n+1)=\left[\begin{array}{cc}1&0.5\end{array}\right]\mathbf{x}_{k}(n+1), \quad n\in N,\end{array} \right. \tag{72}\]  
where the operation time sampling is set as \(N=\{0,1,\cdots 100\}\) and the initial state as \([x_{1}(0),x_{2}(0)]^{\mathrm{T}}=\mathbf{0}\). The desired trajectory is chosen as \(y_{d}(n)=\sin(\frac{4n}{25})+sin(\frac{2n}{25}),\quad n\in[0,100]\). The parameters in the quantizer are given as \(s_{0}=2\), \(\eta_{e}=0.85\), then \(\sigma_{e}=0.08\). Setting the learning gain \(\Gamma\) as a concrete number, so numerical calculation yields that \(\rho=|1-\Gamma\mathbf{C}\mathbf{B}|+\sigma_{e}|\Gamma\mathbf{C}\mathbf{B}|=0.402<1\) when \(\Gamma=0.65\) and \(\rho=|1-\Gamma\mathbf{C}\mathbf{B}|+\sigma_{e}|\Gamma\mathbf{C}\mathbf{B}|=0.218<1\) when \(\Gamma=0.85\), satisfies the convergence condition of the ILCQe law (4) in Theorem 1, and the weighting factor of the POGAILCQe law is selected as \(w_{k}=0.01\cdot 0.5^{k}\) and \(w_{k}=0.05\cdot 0.5^{k}\) in Theorem 2. Figure 1 exhibits the tracking behavior comparison of the ILCQe law and the POGAILCQe law in the second iteration. The solid curve represents the desired trajectory, whereas the dashed and the dash-dotted curves stand for the output with \(\Gamma=0.65\) and \(\Gamma=0.85\), respectively. It is observed that the tracking behavior can be improved by fine-tuning the value of \(\Gamma\) in the ILCQe law. However, the tracking performance of the POGAILCQe law still performs better than the ILCQe law for the same number of iterations. Figure 1: Comparison of the outputs. The tracking error convergence curve is shown in Figure 2 with the tracking error of the POGAILCQe law plotted on the solid curve and the tracking error of the ILCQe law shown on the dotted curve, respectively. The tracking error of the POGAILCQe law with \(w_{k}=0.001\cdot 0.1^{k}\) produces faster convergence than the ILCQe law. Figure 3 displays comparable tracking errors, where the dash, dotted, and solid curves are tracking error tendencies of POGAILCQe with the tuning factor sequences \(w_{k}=0.01\cdot 0.5^{k}\) for the different quantized density being \(\eta_{e}=0.35\), \(\eta_{e}=0.65\) and \(\eta_{e}=0.85\), respectively. From Figures 1-3, it is observed that an optimal algorithm with an appropriate weighting factor can ensure a zero-error convergence in the ILC for discrete-time systems with error data quantization. We simulate the POGAILCQe method with the various tuning factors sequence using the same number of quantized densities to discuss the impact of the tuning factors on the convergence speed. In Figure 4, the parameters in the quantizer are given as \(s_{0}=2\), \(\eta_{e}=0.65\), then \(\sigma_{e}=0.21\). The dash tracking error for the lower tuning factor sequence \(w_{k}=0.01\cdot 0.5^{k}\) achieves faster convergence than that for the larger tuning factor sequence \(w_{k}=0.05\cdot 0.5^{k}\). Therefore, the simulation demonstrates that the convergence performance can be improved when we choose a lesser tuning factors sequence for the POGAILCQe law. This result is consistent with the theoretical analysis of Theorem 2. Figure 3: Tracking error tendency of POGAILCQe. Figure 2: Comparison of tracking error tendency. Example 2. A joint motion position control system of a robot manipulator is taken into consideration to demonstrate the validity and efficacy of the proposed POGAILCQe; the engineering example is derived from reference [42]. Figure 5 displays the structural diagram of the control system in the s-domain. In Figure 5, \(K_{P}\), \(K_{I}\) and \(K_{D}\) are the proportional, integral, and derivative gains for the conventional proportional-integral-derivative (PID) controller \(G_{c}(s)=K_{P}+\frac{K_{I}}{s}+K_{D}s\), respectively.