# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## Appendix A Proof of Theorem 1  
Based on (27) and (33), the following equation will be obtained:  
\[\widehat{x}(k+1)=[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}] \widehat{x}(k). \tag{11}\]  
Define a Lyapunov functional as  
\[V(\widehat{x}(k))=\widehat{x}(k)^{T}P(\theta)\widehat{x}(k), \tag{12}\]  
Figure 8: The reference signal and the output of the system with no preview.  
Figure 9: Tracking error with no preview.  
where \(P(\theta)\) is a positive definitive matrix. The difference of the Lyapunov function (A2) can be obtained as  
\[\begin{array}{c}\Delta V(\widehat{x}(k))=V(\widehat{x}(k+1))-V(\widehat{x}(k))\\ \qquad\qquad=\widehat{x}(k+1)^{T}P(\theta)\widehat{x}(k+1)-\widehat{x}(k)^{T}P (\theta)\widehat{x}(k)\\ \qquad\qquad=\widehat{x}(k)^{T}[\widehat{A}(\theta)+\widehat{B}(\theta)K( \alpha)C_{Z}]^{T}P(\theta)[\widehat{A}(\theta)+\\ \qquad\qquad\widehat{B}(\theta)K(\alpha)C_{Z}]\widehat{x}(k)-\widehat{x}(k)^{ T}P(\theta)\widehat{x}(k).\end{array} \tag{10}\]  
If the following condition  
\[\begin{array}{c}[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}]^{T}P (\theta)[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}]\\ -P(\theta)+Q_{\bar{x}}+C_{Z}^{T}K(\alpha)^{T}RK(\alpha)C_{Z}<0\end{array} \tag{11}\]  
holds, then it follows from (10) that one has  
\[\Delta V(\widehat{x}(k))<-\widehat{x}(k)^{T}(Q_{\bar{x}}+C_{Z}^{T}K(\alpha)^{ T}RK(\alpha)C_{Z})\widehat{x}(k)<0. \tag{12}\]  
Thus, according to [PERSON]'s stability theory, the closed-loop system (A1) is robust asymptotically stable. Furthermore, from (12), we have  
\[\widehat{x}(k)^{T}(Q_{\bar{x}}+C_{Z}^{T}K(\alpha)^{T}RK(\alpha)C_{Z})\widehat {x}(k)<-\Delta V(\widehat{x}(k)). \tag{13}\]  
Summing both sides of the above inequality from \(k=1\) to \(k=\infty\), it yields  
\[\bar{J}=\sum_{k=1}^{\infty}[\widehat{x}(k)^{T}Q_{\bar{x}}\widehat{x}(k)+ \Delta u(k)^{T}R\Delta u(k)]\leq\widehat{x}_{0}^{T}P(\theta)\widehat{x}_{0}. \tag{14}\]  
On the other hand, applying the [PERSON] complement lemma, condition (11) can be equivalent to  
\[\left(\begin{array}{ccc}-P(\theta)+Q_{\bar{x}}&*&*\\ \widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}&-P(\theta)^{-1}&*\\ R^{\frac{1}{2}}K(\alpha)C_{Z}&0&-I\end{array}\right)<0. \tag{15}\]  
Pre- and post-multiplying inequality (15) by \(\text{diag}\,\left\{G(\alpha)^{T},I,I\right\}\) and its transpose, respectively, one gets  
\[\left(\begin{array}{ccc}\Lambda_{11}&*&*\\ \left[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}\right]G(\alpha)&- P(\theta)^{-1}&*\\ R^{\frac{1}{2}}K(\alpha)C_{Z}G(\alpha)&0&-I\end{array}\right)<0, \tag{16}\]  
where \(\Lambda_{11}=-G(\alpha)^{T}P(\theta)G(\alpha)+G(\alpha)^{T}Q_{\bar{x}}G(\alpha)\).  
From the [PERSON] complement lemma, the above inequality (16) is equivalent to  
\[\begin{array}{c}G(\alpha)^{T}[\widehat{A}(\theta)+\widehat{B}(\theta)K( \alpha)C_{Z}]^{T}P(\theta)[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_ {Z}]G(\alpha)\\ -G(\alpha)^{T}P(\theta)G(\alpha)+G(\alpha)^{T}Q_{\bar{x}}G(\alpha)+G(\alpha)^{ T}C_{Z}^{T}K(\alpha)^{T}RK(\alpha)C_{Z}G(\alpha)<0.\end{array} \tag{17}\]  
It follows from Lemma 2 that  
\[-G(\alpha)^{T}P(\theta)G(\alpha)\leq-G(\alpha)^{T}-G(\alpha)+P(\theta)^{-1}. \tag{18}\]  
Therefore, if  
\[\begin{array}{c}G(\alpha)^{T}[\widehat{A}(\theta)+\widehat{B}(\theta)K( \alpha)C_{Z}]^{T}Pblue(\theta)[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha )C_{Z}]G(\alpha)\\ -G(\alpha)^{T}-G(\alpha)+P(\theta)^{-1}+G(\alpha)^{T}Q_{\bar{x}}G(\alpha)+G( \alpha)^{T}C_{Z}^{T}K(\alpha)^{T}RK(\alpha)C_{Z}G(\alpha)<0,\end{array} \tag{19}\]then inequality (A10) must hold. Using the Schur complement lemma again, inequality (A12) is equivalent to (34). The proof is completed.