# Accelerated Iterative Learning Control of Speed Ripple Suppression for a Seeker Servo Motor
## 2 Description and Analysis of the Question  
Consider a class of nonlinear systems:  
\[\left\{ \begin{array}{l}\dot{x}_{k}(t)=f(x_{k}(t),t)+Bu_{k}(t)\\ y_{k}(t)=Cx_{k}(t)+Du_{k}(t)\end{array} \right. \tag{1}\]  
where \(t\in[0,T_{L}]\); \(k\) is the number of iterations; \(x_{k}\in R^{n}\), \(u_{k}(t)\in R^{r}\), and \(y_{k}(t)\in R^{m}\) are the state variables, control variables, and output variables of the system, respectively; and \(B\), \(C\), and \(D\) are the matrices of the corresponding dimension. Control objective: Design a controller for the system and let the system output track the expected trajectory under this controller. For the convenience of proof, we make reasonable assumptions for the algorithm:  
**Assumption 1**. : _The unknown nonlinear function \(f(\cdot,\cdot)\) satisfies the Lipschitz condition, namely, there is a constant \(L>0\) that satisfies_  
\[\|f(x_{2}(t),t)-f(x_{1}(t),t)\|\leq L\|x_{2}(t)-x_{1}(t)\|\]  
**Assumption 2**. : _There is an ideal control \(u_{d}(t)\) that makes the state and output of the system be the expected values of \(x_{d}(t)\) and \(y_{d}(t)\), respectively._  
In the method proposed in this paper, we divide the interval \([0,T_{L}]\) into N equal sub-intervals, each with length \(h\). Let  
\[ \begin{array}{l}t_{0}=0,\ t_{N+1}=T_{L}\\ t_{i+1}=t_{i}+h,i=0,1,2,\cdots,N\end{array}  \tag{2}\]  
We presume that the controlled object (1) adopts the P-type learning law:  
\[ \begin{array}{l}u_{k+1}(t)=u_{k}(t)+\Delta u_{k+1}(t)\\ \Delta u_{k+1}(t)=\left\{\begin{array}{ll}0,&0\leq t\leq t_{i_{k}}\\ Kpe_{k+1}(t)e^{-K_{E}(\frac{t_{i_{k}}-t}{L})},&t_{i_{k}}<t\leq T_{L}\end{array}\right. \end{array}  \tag{3}\]  
where \(K_{P}>0\) is the proportional learning gain, \(K_{E}>0\) is the exponential correction factor, and \(t_{i_{k}}\in[0,T_{L}]\) is the cut-off time (note: in the \((k+1)\)th iteration, the error in interval \([0,t_{i_{k}}]\) satisfies the allowable range, and the error in the residual interval \([t_{i_{k+1}}\,T_{L}]\) does not satisfy the allowable range). Let \(\delta_{[t_{i_{k}}\,T_{L}]}\) be the indicative function, namely,  
\[\delta_{[t_{i_{k}}\,T_{L}]}(t)=\left\{ \begin{array}{ll}0,&t\
otin[t_{i_{k}}\,T_{L}]\\ 1,&t\in[t_{i_{k}}\,T_{L}]\end{array} \right. \tag{4}\]  
According to Equation (4), Equation (5) can be rewritten as  
\[u_{k+1}(t)=u_{k}(t)+K_{P}e^{-K_{E}(\frac{t_{i_{k}}-t}{L})}\delta_{[t_{i_{k}}\,T_{L}]}e_{k+1}(t) \tag{5}\]  
\[e_{k+1}(t)=\left\{ \begin{array}{ll}0,&0\leq t\leq t_{i_{k}}\\ y_{d}(t)-y_{k+1}(t),&t_{i_{k}}\leq t\leq T_{L}\end{array} \right. \tag{6}\]  
\[L_{i_{k}}=\left\{I\big{|}\|e_{k}[t_{i_{k}+l},t_{i_{k}+l_{1}}]\|_{\lambda} \
eq 0,0\leq l\leq N-i_{k}\right.\Big{\}} \tag{7}\]  
\[l_{i_{k}}=\left\{ \begin{array}{ll}\min L_{i_{k}}\,&L_{i_{k}}\
eq\phi\\ 0\,&L_{i_{k}}=\phi\end{array} \right. \tag{8}\]  
\[t_{i_{k+1}}=t_{i_{k}+l_{i_{k}}} \tag{9}\]  
For the aforementioned accelerated P-type iterative learning control, which is constructed with gain correction, the learning control law is described as follows: We presume that the initial operation input is \(u_{0}\), whose value is random in the entire operation interval \([0,T_{L}]\). In Equation (8), the introduction of \(l_{i_{k}}\) is aimed at obtaining the boundary point \(t_{i_{k}+l_{i_{k}}}\) that needs to be learned for the next iteration selection. The steps of this method are as following:  
Step 1: Calculate \(e_{k+1}(t)\) using Equation (6);  
Step 2: Calculate \(L_{i_{k}}\) using Equation (7);  
Step 3: Calculate \(l_{i_{k}}\) using Equation (8);  
Step 4: Calculate \(t_{i_{k+1}}\) using Equation (9);  
Step 5: Calculate \(u_{k+1}(t)\) using Equation (5);  
Step 6: Ending this iteration and starting the next. And repeat step 1-5. The structure of the control law (3) indicates that the interval that has been learned previously does not require continued learning, and we only need to learn the interval of the expected trajectory that is not completely tracked. As the number of iterations \(k\) increases, the required length of the learning interval gradually approaches zero.