## 4 Structure of the First Diagram for Robot Control, the Iterative Learning Method
### The Closed-Loop System's Performance and Control Algorithm  
The following Algorithm 2 is established to implement the proposed model-free controller (16) where the \(\underline{\underline{\varrho}}\) and \(\underline{\underline{\mu}}\) values are obtained from expressions (26) and (13), respectively. In this control algorithm, the working time T of the robots is repeatedly tested by each While loop. Figure 2 shows the closed-loop system's output tracking performance with the output scenario \(\underline{\underline{\delta}}_{k}(n)\
eq\underline{0}\). Theorem 3: _The robot manipulators (1) output tracking error is driven by the proposed model-free control framework in Figure 1, which comprises the feedback linearization block via disturbance compensator (16), (23), and the ILC block (26) drives the robot manipulators (1) output tracking error \(|\underline{E}_{k}(\tau)|\) to \(T_{s}\) dependent neighborhood \(O\) of origin if \(d\) is limited and continuous. If a smaller \(t_{s}\) is selected, there will be a smaller \(O\)._  
Proof. : Because noise \(\underline{\underline{\Upsilon}}_{d}\) is continuous and bounded, the total noise \(\underline{\underline{\mu}}\) and component \(\underline{\underline{\delta}}\) are also continuous and bounded. The upper limit of \(O\) is \(\Delta\) and this limit can be reduced arbitrarily by reducing \(t_{s}\) according to Theorem 1. The validity of Theorem 2 was confirmed by:  
\[\underline{\dot{\omega}}=A\underline{\omega}+B\underline{\dot{\theta}}\text{ with } \underline{\omega}=vec(\underline{R},\underline{\dot{R}}) \tag{29}\]  
The subtraction of (29) from (17) yields:  
\[\underline{\dot{\sigma}}=A\underline{\sigma}-B\underline{\dot{\sigma}}\text{ where } \underline{\sigma}=vec(\underline{E},\underline{\dot{E}}) \tag{30}\]  
Using the Lyapaunov equation \(A^{T}P+PA=-Q\) with optional positive definite matrix \(Q\) we always have positive definite \(P\) because \(A\) is Hurwitz. From the positive definite function \(V(\underline{\sigma})=\underline{\sigma}^{T}P\underline{\sigma}\) we have the result:  
\[ \begin{array}{ll}\dot{V}&=(A\underline{\sigma}-B\underline{\dot{\sigma}})^{T }+\underline{\sigma}^{T}P(A\underline{\sigma}-B\underline{\dot{\sigma}})\\ &=\underline{\sigma}^{T}(A^{T}P+PA)\underline{\sigma}-2\underline{\sigma}^{T }P\underline{\dot{\sigma}}=-\underline{\sigma}^{T}(-A^{T}P-PA)\underline{ \sigma}-2\underline{\sigma}^{T}P\underline{\dot{\sigma}}\\ &=\leq-\lambda_{max}(-A^{T}P-PA)\left\|\underline{\sigma}\right\|^{2}+2\|PB \|\Delta\|=\|\underline{\varepsilon}\|\left[-\lambda_{max}(-A^{T}P-PA)\| \underline{\sigma}\right\|+2\|PB\|\Delta\right]\\ &&\text{with }\lambda_{max}(-A^{T}P-PA)\text{ is the maximum value of the }Q\text{ matrix.}\\ &&\text{Since }V(\underline{\varepsilon})<0\text{ as long as}\\ &&2\|PB\|\Delta<\lambda_{max}(-A^{T}P-PA)\|\underline{\varepsilon}\|\text{ or }\frac{2\|PB\|\Delta}{\lambda_{max}(-A^{T}P-PA)}<\|\underline{\varepsilon}\|\\ &&\text{The orbital error vector tends to the origin:}\\ &&O=\left\{\underline{\sigma}\in R^{2n}\bigg{|}\ \|\underline{\sigma}\|\leq\ \frac{2\|PB\|\Delta}{\lambda_{max}(-A^{T}P-PA)}\ \right\}\end{array}  \tag{31}\]  
This control algorithm contains all the necessary calculations and has the following structure:  
```
Choose two matrices \(A_{1},A_{2}\), given in (25) become Hurwitz. Determine \(\hat{A},\hat{B},\hat{C}\) given in (25) and \(\Phi\) given in (28). Choose \(0<t_{\text{s}}<<1\). Calculate \(S=T/t_{\text{s}}\). Determine \(D_{1}^{T}=1\)\(\Big{/}_{T_{\text{s}}}\); \(D_{0}^{T}=-D_{1}^{T}\) given in (23)  Choose learning \(\underline{\dot{\mu}}\) and tracking error \(\underline{E}_{0}\). Allocate \(\underline{\dot{\theta}}(\text{n})=\)\(\overline{\dot{\text{R}}}(\text{n})\),  with \(\text{n}=0,1,\ldots,\text{S}-1\) and \(\underline{Z}=\underline{0}\). Choose learning parameter \(K\) so that \(\Phi\) of (28) becomes Schur.