# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## Appendix B Proof of Theorem 2  
Denote  
\[\widetilde{A}=\left(\begin{array}{c}\widehat{A}\\ 0\end{array}\right),\ \ \widetilde{B}=\left(\begin{array}{c}\widehat{B}\\ R^{\frac{1}{2}}\end{array}\right),\ \ \Phi(\theta)=\left(\begin{array}{cc}X(\theta)&0\\ 0&I\end{array}\right).\] (A13)  
Inequality (35) can be written as  
\[\left(\begin{array}{ccc}\Psi_{11}&*&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)L(\alpha)Q_{\xi}&-\Phi( \theta)&*\\ C_{Z}G(\alpha)-UQ_{\xi}&\beta W^{T}L(\alpha)^{T}\widetilde{B}(\theta)^{T}&\Psi_ {44}\end{array}\right)<0,\] (A14)  
or  
\[\left(\begin{array}{ccc}\left(\begin{array}{cc}\Psi_{11}&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)L(\alpha)Q_{\xi}&-\Phi( \theta)\end{array}\right)&*\\ \Xi_{31}&&\Psi_{44}\end{array}\right)<0,\] (A15)  
where  
\[\Xi_{31}=\beta W^{T}L(\alpha)^{T}\widetilde{B}(\theta)^{T}\left[\begin{array}[ ]{cc}0&I\end{array}\right]+UWW^{-1}U^{-1}(C_{Z}G(\alpha)-UQ_{\xi})\left[ \begin{array}{cc}I&0\end{array}\right].\]  
We further denote  
\[T=\left[\begin{array}{ccc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{\xi}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)L(\alpha)Q_{\xi}&-\Phi( \theta)\end{array}\right],\] \[N=UW,\ \ R=\left[\begin{array}{c}0\\ I\end{array}\right]\widetilde{B}(\theta)L(\alpha)W,\] \[S=W^{-1}U^{-1}(C_{Z}G(\alpha)-UQ_{\xi})\left[\begin{array}{cc} I&0\end{array}\right].\]  
By Lemma 3, inequality (A15) can guarantee that  
\[\left(\begin{array}{ccc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{\xi}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)L(\alpha)Q_{\xi}&-\Phi( \theta)\end{array}\right)\] (A16) \[+ sym\left(\left[\begin{array}{c}0\\ I\end{array}\right]\widetilde{B}(\theta)L(\alpha)WW^{-1}U^{-1}(C_{Z}G(\alpha)-UQ _{\xi})\left[\begin{array}{cc}I&0\end{array}\right]\right)\] \[= \left(\begin{array}{ccc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{\xi}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)L(\alpha)Q_{\xi}&-\Phi( \theta)\end{array}\right)\] \[+ sym\left(\left[\begin{array}{c}0\\ I\end{array}\right]\widetilde{B}(\theta)L(\alpha)U^{-1}(C_{Z}G(\alpha)-UQ_{ \xi})\left[\begin{array}{cc}I&0\end{array}\right]\right)\] \[= \left(\begin{array}{ccc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{\xi}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)&-\Phi(\theta)\end{array}\right)\] \[+ sym\left(\left[\begin{array}{c}0\\ I\end{array}\right]\widetilde{B}(\theta)L(\alpha)U^{-1}(C_{Z}G(\alpha)+UQ_{ \xi}-UQ_{\xi})\left[\begin{array}{cc}I&0\end{array}\right]\right)\] \[< 0.\]Setting \(K(\alpha)=L(\alpha)U^{-1}\), then inequality (A17) becomes  
\[\left(\begin{array}{cc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G(\alpha)^{T}Q_{x}G( \alpha)&*\\ \widetilde{A}(\theta)G(\alpha)&-\Phi(\theta)\end{array}\right)\] (A17) \[\quad+sym\left(\left[\begin{array}{cc}0\\ I\end{array}\right]\widetilde{B}(\theta)K(\alpha)(C_{Z}G(\alpha))\left[\begin{array} []{cc}I&0\end{array}\right]\right)\] \[=\left(\begin{array}{cc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{x}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)K(\alpha)C_{Z}G(\alpha)&- \Phi(\theta)\end{array}\right)<0.\]  
By calculation, it can be found that  
\[\left(\begin{array}{cc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{x}G(\alpha)&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)K(\alpha)C_{Z}G(\alpha) &-\Phi(\theta)\end{array}\right)\] (A18) \[=\left(\begin{array}{cc}-G(\alpha)-G(\alpha)^{T}+X(\theta)+G( \alpha)^{T}Q_{x}G(\alpha)&*&*\\ \widetilde{A}(\theta)G(\alpha)+\widetilde{B}(\theta)K(\alpha)C_{Z}G(\alpha) &-X(\theta)&*\\ R^{\frac{1}{2}}K(\alpha)C_{Z}G(\alpha)&0&-I\end{array}\right).\]  
Setting \(X(\theta)=P(\theta)^{-1}\), it follows from Theorem 1 that Theorem 2 holds.