## 4 Iterative Learning Control with Adaptive Sliding Mode Control (ILC-ASMC)
### PD-Type Iterative Learning Controller Design  
ILC is suitable for controlled objects with periodic motion and does not rely on accurate mathematical models. It has a significant control effect on systems with strong coupling, model uncertainty, and high-precision tracking requirements. Reference [30] uses P-type iterative learning to deal with repeated disturbance during the operation of CNC machine tools. For this purpose, this paper proposes an improved exponential convergence PD-type iterative learning control. It is suitable for FTS systems and has faster response capabilities and real-time adjustment mechanisms. The improved iterative learning control is combined with the ASMC method to obtain a composite control that combines the advantages of the two control methods, gradually eliminating the non-periodic and periodic disturbances of the system, and obtaining better control effects. The system control structure is shown in Figure 3.  
When using ILC, the state equation at the \(k\)-th iteration can be written as follows according to Equation (4). \[\left\{ \begin{array}{l}\dot{x}_{k}(t)=Ax_{k}(t)+Bu_{k}(t)+f(t,x_{k}(t))\\ y_{k}(t)=Cx_{k}(t)\end{array} \right. \tag{20}\]  
where \(t\in[0,T]\), \(k\) is the number of iterations \((k=0,1,2,3\cdots)\), \(x_{k}(t)\in R^{n}\) is the system state vector, \(u_{k}(t)\in R^{m}\) is the system control vector, \(y_{k}(t)\in R^{q}\) is the system output vector, \(f(t,x_{k}(t))\) is the total disturbance of the system, \(A= \begin{bmatrix}0&1\\ -a_{1}&-a_{2}\end{bmatrix} \), \(B= \begin{bmatrix}0\\ b\end{bmatrix} \), and \(C= \begin{bmatrix}1&0\end{bmatrix} \) are the matrices of corresponding dimensions. Define the error of the \(k\)-th iteration trajectory tracking as  
\[e_{k}(t)=y_{d}(t)-y_{k}(t) \tag{21}\]  
where \(y_{d}(t)\) represents the expected trajectory, and \(y_{k}(t)\) represents the output of the \(k\)-th iteration of the system. Control signal \(u_{k}(t)(k=0,1,\cdots)\) consists of two parts during any iteration: SMC \(u_{smc,k}(t)\) and ILC \(u_{ILC,k+1}(t)\). \[u_{k}(t)=u_{smc,k}(t)+\frac{1}{b}(u_{ILC,k+1}(t)) \tag{22}\]  
Generally, the iterative learning controller is designed as  
\[u_{ILC,k+1}(t)=Q(u_{k}(t))+L(e_{k}(t)) \tag{23}\]  
Figure 3: Adaptive sliding mode control with iterative learning control. where \(L\) is the learning gain, and \(Q\) is the control input filter. \(k\) is the iteration number and \(u_{k}(t)\) denotes the \(k\)-th control input. Different types of ILC controllers can be obtained by modifying the matrices \(L\) and \(Q\). Taking \(Q\) as the identity matrix, we get  
\[u_{ILC,k+1}(t)=u_{k}(t)+L(e_{k}(t)) \tag{24}\]  
where the current period control input is a function of the error and control input of the previous period. Take the closed-loop PD-type ILC law  
\[u_{ILC,k+1}(t)=u_{k}(t)+Kpe_{k}(t)+K_{D}\dot{e}_{k}(t) \tag{25}\]  
where parameters \(K_{P}\) and \(K_{D}\) are the learning gains. PD-type ILCs are very similar to traditional PD controllers, but instead of handling system response, PD-type ILCs handle errors while maintaining a linear effect on the past input signal. Most learning laws prove that the sufficient condition for learning convergence is the number of iterations \(k\rightarrow\infty\). It is necessary to consider making the iterative learning process converge to the expected value faster. Therefore, the ILC based on exponential gain is proposed by adding to the gain of the control law. The correction coefficient speeds up the convergence speed of ILC. Take the closed-loop PD-type ILC law based on exponential gain  
\[u_{ILC,k+1}(t)=u_{k}(t)+K_{P}e^{\lambda_{0}t}e_{k}(t)+K_{D}e^{\lambda_{0}t}e_{ k}(t) \tag{26}\]  
where \(\lambda_{0}>0\) is the gain correction coefficient.