# Accelerated Iterative Learning Control of Speed Ripple Suppression for a Seeker Servo Motor
## 2 Description and Analysis of the Question
The controller block diagram is showing as Figure 1.  
**Lemma 1**. : (_Bellman-Goruvall inequality_) _[_26_]__: Let \(f_{1}(t)\), \(g_{1}(t)\), and \(h_{1}(t)\) be nonnegative continuous functions in interval \([0,T_{L}]\). Moreover, there is a nonnegative constant \(\eta\) that satisfies. Thus, \(f_{1}(t)\leq h_{1}(t)+\int_{0}^{t}\eta f_{1}(\tau)d\tau+\int_{0}^{t}g_{1}(t)d\tau\) \(f_{1}(t)\leq h_{1}(t)+\int_{0}^{t}\exp(\eta\cdot(t-\tau))[\eta h_{1}(\tau)+g_ {1}(\tau)]d\tau\)_  
**Theorem 1**. : _If the system described by Equations (1) and (3) satisfies the conditions_  
(I) \(\rho=\left\|\left(I+DKpe^{K_{E}}\right)^{-1}\right\|<1\) _and_  
(II) \(x_{k}(0)=x_{d}(0)\)_, \((k=0,1,2,\cdots)\)_  
_then_  
\[\left\|\delta_{[t_{k},\,T_{L}]}e_{k+1}(t)\right\|_{\lambda}\leq\rho\left\| \delta_{[t_{k},\,T_{L}]}e_{k}(t)\right\|_{\lambda} \tag{10}\]  
_when \(k\rightarrow\infty\), the output of system \(y_{k}(t)\) converges to the expected trajectory \(y_{d}(t)\) in \([0,T_{L}]\), i.e.,_  
\[\lim_{k\rightarrow\infty}y_{k}(t)=y_{d}(t)(t_{i}\in[0,T_{L}])\]  
Proof. : We presume that at the \((k+1)\)th iteration, the control time point stops at \(t=t_{i_{k+1}}\); Equation (6) indicates that \(y_{k+1}(t)=y_{k}(t)\) (\(t\in[0,t_{i_{k+1}}]\)). Therefore, when \(t\in[t_{i_{k+1}},T_{L}]\), the error can be expressed as  
\[\delta_{[t_{k},\,T_{L}]}e_{k+1}(t)=y_{d}(t)-y_{k+1}(t) \tag{11}\]  
When \(t\in[t_{i_{k+1}},T_{L}]\), from Equation (1) and condition (II), we obtain Equation (12):  
\[ \begin{array}{ll}e_{k+1}(t)&=y_{d}(t)-y_{k+1}(t)=(y_{d}(t)-y_{k}(t))-(y_{k+1 }(t)-y_{k}(t))\\ &=e_{k}(t)-C(x_{k+1}(t)-x_{k}(t))-D(u_{k+1}(t)-u_{k}(t))\\ &=e_{k}(t)-C\int_{0}^{t}f(x_{k+1}(\tau),\tau)-f(x_{k}(\tau),\tau))d\tau\\ &-C{B}\int_{0}^{t}(u_{k+1}(\tau)-u_{k}(\tau))d\tau-D(u_{k+1}(t)-u_{k}(t))\\ &=e_{k}(t)-C(\int_{0}^{t}f(x_{k+1}(\tau),\tau)-f(x_{k}(\tau),\tau))d\tau\\ &-C{B}\int_{0}^{t}Kpe^{-K_{E}(\frac{t_{k}-t}{l_{L}})}\delta_{[t_{k},\,T_{L}]} e_{k+1}(\tau)d\tau-DK_{Pe}e^{-K_{E}(\frac{t_{k}-t}{l_{L}})}\delta_{[t_{k},\,T_{L}]} e_{k+1}(t)\end{array}  \tag{12}\]  
Figure 1: The block diagram of the controller. We select a sufficiently large \(\lambda\) and let  
\[\left\|\delta_{[t_{k}\,,T_{L}]}e_{k+1}(t)\right\|_{{}_{\lambda}}\leq\rho\|\delta_ {[t_{k}\,,T_{L}]}e_{k}(t)\|_{{}_{\lambda}} \tag{20}\]  
After numbers of iterations, we obtain  
\[\left\|\delta_{[t_{k}\,,T_{L}]}e_{k+1}(t)\right\|_{{}_{\lambda}}\leq\rho^{k-1} \|\delta_{[t_{k}\,,T_{L}]}e_{1}(t)\|_{{}_{\lambda}} \tag{21}\]  
Therefore,  
\[\lim_{k\rightarrow\infty}\sup_{t_{L}\in[t_{k},T_{L}]}\left\|\delta_{[t_{k}\,, T_{L}]}e_{k}(t)\right\|_{{}_{\lambda}}=0 \tag{22}\]  
This result indicates that control law (3) adopts the appropriate scheme to shorten the learning interval and ensure monotonic convergence under the \(\lambda\)-norm. When \(\lambda\) is sufficiently large, as the number of iteration increases, we track the expected trajectory in a piecewise manner and conduct real-time correction of the learning gain. Moreover, the interval length which decreases and ultimately approaches zero. Finally, over the entire interval \([0,T_{L}]\), the system output tracks the expected trajectory. \(\square\)