## 4 Iterative Learning Control with Adaptive Sliding Mode Control (ILC-ASMC)
### Convergence Analysis  
In this section, in order to update the control law of SMC, the convergence analysis is studied. The control effect in Equation (22) is achieved using the control in Equations (7) and (26). Substitute the combined control Equation (22) into Equation (14).  
\[s\sigma_{k}-\sigma_{k}(0)=w_{k}-u_{ILC,k} \tag{28}\]  
Among them, \(s\) represents the Laplacian operator. At the same time, \(w_{k}=-\hat{\eta}sat(\sigma_{k}/\Delta)\)\(-k_{n}\sigma_{k}-f\) is a bounded disturbance term. Assuming that all signals are available in the frequency domain, Equation (28) in the \(k+1\)-th iteration is as follows  
\[s\sigma_{k+1}=\sigma_{k+1}(0)-u_{ILC,k+1}+w_{k} \tag{29}\]  
By substituting the exponential convergence iterative learning control update law in Equation (26) into Equation (28), we get the following equation  
\[s\sigma_{k+1}=\sigma_{k+1}(0)-u_{ILC,k+1}+w_{k+1}-Kpe^{\lambda_{0}t}e_{k}(t)- K_{D}e^{\lambda_{0}t}\dot{e}_{k}(t) \tag{30}\]Equation (28) can be organized as  
\[u_{IL,k}(t)=\sigma_{k}(0)-s\sigma_{k}+w_{k} \tag{31}\]  
By substituting Equation (31) into Equation (30), we obtain the following equation  
\[\sigma_{k+1}=\frac{1}{s}\Big{(}\sigma_{k+1}(0)-\sigma_{k}(0)-s\sigma_{k}+w_{k}+ w_{k+1}-K_{P}e^{\lambda_{0}t}e_{k}(t)-K_{D}e^{\lambda_{0}t}e_{k}(t)\Big{)} \tag{32}\]  
The initial conditions for each iteration are \(e_{k}(0)=\dot{e}_{k}(0)=0\), when \(k=0,1,2,\ldots\), \(\sigma_{k}(0)=0\). Therefore, \(\sigma_{k+1}(0)=\sigma_{k}(0)=0\). Equation (32) is simplified to  
\[\sigma_{k+1}=\sigma_{k}-\frac{K_{P}e^{\lambda_{0}t}+sK_{D}e^{\lambda_{0}t}}{s }e_{k}+\frac{w_{k+1}-w_{k}}{s} \tag{33}\]  
Regard the sliding surface as  
\[\sigma_{k}=\mathcal{L}\bigg{(}\lambda_{1}e_{k}+\dot{e}_{k}+\lambda_{2}\int e _{k}d\tau\bigg{)}=(\lambda_{1}+s+\frac{\lambda_{2}}{s})e_{k} \tag{34}\]  
The tracking error can be represented as  
\[e_{k+1}=\rho e_{k}+\frac{\Delta_{k}}{s(s+\lambda_{1}+\frac{\lambda_{2}}{s})} \tag{35}\]  
where \(\rho\) and \(\Delta_{k}\) are defined as  
\[\rho=\Bigg{(}1-\frac{K_{P}e^{\lambda_{0}t}+sK_{D}e^{\lambda_{0}t}}{s(s+ \lambda_{1}+\frac{\lambda_{2}}{s})}\Bigg{)} \tag{36}\]  
\[\Delta_{k}=w_{k+1}-w_{k}\]  
It can be deduced by a recursive algorithm  
\[\begin{array}{l}e_{2}=\frac{\Delta_{1}}{s(s+\lambda_{1}+\frac{\lambda_{2}}{ s})}+pe_{1}\\ e_{3}=\frac{\Delta_{2}}{s(s+\lambda_{1}+\frac{\lambda_{2}}{s})}+p\frac{\Delta_{ 1}}{s(s+\lambda_{1}+\frac{\lambda_{2}}{s})}+p^{2}e_{1}\\ \vdots\\ e_{k+1}=\sum\limits_{i=1}^{k}\left(p^{k-i}\frac{\Delta_{i}}{s(s+\lambda_{1}+ \frac{\lambda_{2}}{s})}\right)+p^{k}e_{1}\end{array} \tag{37}\]  
In a nonlinear uncertain iterative system, when \(\left|1-\frac{K_{P}e^{\lambda_{0}t}+sK_{D}e^{\lambda_{0}t}}{s(s+\lambda_{1}+ \frac{\lambda_{2}}{s})}\right|<1\), and the disturbance remains unchanged throughout iterations, from Equations (35) and (37), it can be seen that the time-varying disturbance \(\Delta_{k}=w_{k+1}-w_{k}\) becomes zero. Where \(\frac{1}{s(s+\lambda_{1}+\frac{\lambda_{2}}{s})}\) acts as a low-pass filter, significantly reducing the high-frequency chattering generated by sliding mode control. When the coefficient \(K_{P}\), \(\lambda_{0}\), \(\lambda_{1}\), \(\lambda_{2}\) satisfy \(|p|<1\). The proposed exponential convergence iterative learning control law based on adaptive sliding mode control can guarantee that after \(k\)-th iterations, the output error asymptotically converges to a neighborhood close to zero, satisfying the following equation  
\[\lim_{k\rightarrow\infty}e_{k}=0 \tag{38}\]