## 3 Iterative Learning Control Based on the Open-Loop PD Regulation  
The open-loop PD control can be expressed as follows:  
\[u_{k+1}(t)=u_{k}(t)+\left(\varGamma\frac{\mathrm{d}}{\mathrm{d}t}+L\right)e_ {k}(t), \tag{13}\]  
where \(L\) and \(\varGamma\) are learning gain matrices, i.e., \(L=\mathrm{K_{P}}\times I_{E(6\times 6)}\), \(\varGamma=\mathrm{K_{d}}\times I_{E(6\times 6)}\),respectively; herein, \(\mathrm{K_{P}}\) and \(\mathrm{K_{d}}\) are the proportional and derivative parameters of the PD controller, respectively. \(e_{k}\) is the error in the \(k\)-th iteration process. The iterative learning control of the time-varying continuous system is based on the following theorem [25]. If the system described by Equations (12) and (13) meets the following conditions:1. \(\|I_{\rm E}-C(t)B(t)\Gamma(t)\|\leq\overline{p}<1\); (b) \(Y_{k}(0)=Y_{0}(k=1,2,3,\cdots),y_{0}(0)=y_{\rm d}(0)\), then, when \(k\to\infty\), there exist \(y_{k}(t)\to y_{\rm d}(t),\forall t\in[0,{\rm T}]\) the controller design and convergence analysis are as follows: From Equations (12) and (13), it can be seen that the system output is: \[y_{k+1}(0)={\rm C}Y_{k+1}(0)={\rm C}Y_{k}(0)=y_{k}(0),\] (14) then, \(e_{k}(0)=0(k=0,1,2,\cdots)\), that is, the initial conditions are met. The solution for differential equation \(\dot{Y}=AY+Bu\) is:  
\[ \begin{array}{rl}Y(t)&={\rm C}{\rm exp}\Big{(}\int_{0}^{t}A{\rm d}\tau\Big{)} +{\rm exp}\Big{(}\int_{0}^{t}A{\rm d}\tau\Big{)}\int_{0}^{t}B(\tau)u(\tau){\rm exp }\big{(}\int_{0}^{\tau}-Ad\delta\big{)}{\rm d}\tau\\ &\qquad\qquad={\rm C}{\rm exp}(At)+{\rm exp}(At)\int_{0}^{t}B(\tau)u(\tau){\rm exp }(-A\tau)d\tau\\ &\qquad\qquad\qquad={\rm C}{\rm exp}(At)+\int_{0}^{t}{\rm exp}(A(t-\tau)B(\tau )u(\tau){\rm d}\tau,\end{array}  \tag{15}\]  
let \(\[PERSON](t,\tau)={\rm exp}(A(t-\tau))\); then:  
\[Y_{k}(t)-Y_{k+1}(t)=\int_{0}^{t}\[PERSON](t,\tau)B(\tau)(u_{k}(\tau)-u_{k+1}(\tau) ){\rm d}\tau, \tag{16}\]  
Since \(e_{k}(t)=y_{\rm d}(t)-y_{k}(t),e_{k+1}(t)=y_{\rm d}(t)-y_{k+1}(t)\), then  
\[ \begin{array}{rl}e_{k+1}(t)-e_{k}(t)=y_{k}(t)-y_{k+1}(t)=C(t)(Y_{k}(t)-Y_{k+ 1}(t))=\\ \qquad\qquad\int_{0}^{t}C(t)\[PERSON](t,\tau)B(\tau)(u_{k}(\tau)-u_{k+1}(\tau)){ \rm d}\tau,\end{array}  \tag{17}\]  
the following formula exists:  
\[e_{k+1}(t)=e_{k}(t)-\int_{0}^{t}C(t)\[PERSON](t,\tau)B(\tau)(u_{k+1}(\tau)-u_{k}( \tau)){\rm d}\tau, \tag{18}\]  
Substitute Equation (13) into Equation (18); then, the error of the (\(i\)+1)-th output is:  
\[e_{k+1}(t)=e_{k}(t)-\int_{0}^{t}C(t)\Phi(t,\tau)B(\tau)\bigg{[}\Gamma(\tau)e_ {k}(\tau)+\dot{L}(\tau)e_{k}(\tau)\bigg{]}{\rm d}\tau, \tag{19}\]  
Let \(G(t,\tau)=C(t)B(\tau)\Gamma(\tau)\); then, according to the partial integration method, there exists:  
\[ \begin{array}{rl}\int_{0}^{t}C(t)B(t)\Gamma(t)\dot{e}_{k}(\tau){\rm d}\tau= \ G(t,\tau)e_{k}(\tau)|_{0}^{t}-\int_{0}^{t}\frac{\partial}{\partial\tau}G(t,\tau)e_{k}(\tau){\rm d}\tau\\ \qquad\qquad\qquad=(t)B(t)\Gamma(\tau)e_{k}(\tau)-\int_{0}^{t}\frac{\partial }{\partial\tau}G(t,\tau)e_{k}(\tau){\rm d}\tau,\end{array}  \tag{20}\]  
Substitute Equation (20) into Equation (19); then:  
\[ \begin{array}{rl}e_{k+1}(t)=[I_{\rm E}-C(t)B(t)\Gamma(t)]e_{k}(t)+\int_{0}^{ t}\frac{\partial}{\partial\tau}G(t,\tau)e_{k}(\tau){\rm d}\tau-\\ \qquad\qquad\qquad\int_{0}^{t}C(t)\Phi(t,\tau)B(\tau)L(\tau)e_{k}(\tau){\rm d} \tau,\end{array}  \tag{21}\]  
Perform the norm operation at both ends of Formula (21), and the following equation can be obtained:  
\[ \begin{array}{rl}\|e_{k+1}(t)\|\leq\|I_{\rm E}-C(t)B(t)\Gamma(t)\|\|e_{k}(t) \|+\int_{0}^{t}\Big{\|}\frac{\partial}{\partial\tau}G(t,\tau)\Big{\|}\|e_{k}( \tau)\|{\rm d}\tau\\ +\int_{0}^{t}\|C(t)\Phi(t,\tau)B(\tau)L(\tau)\|\|e_{k}(\tau)\|{\rm d}\tau\leq\|I _{\rm E}-C(t)B(t)\Gamma(t)\|\|e_{k}(t)\|\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\int_{0}^{t}b_{1} \|e_{k}(\tau)\|{\rm d}\tau,\end{array}