# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 3 Construction of Augmented Dynamic System  
In this section, we will construct an augmented dynamic system that includes previewed information, error vectors, and states of the system by using the \(L\)-order forward difference operator. Moreover, the augmented performance index will be also given. For all the system variables, define the \(L\)-order forward difference operator as  
\[\Delta\xi(k)=\xi(k)-\xi(k-L),\ \xi\in\{x,\ y,\ u,\ r,\ d\}. \tag{11}\]  
Taking the \(L\)-order difference operator on both sides of the state Equation (1) and error Equation (4), we derive  
\[\left\{ \begin{array}{c}e(k+1)=e(k+1-L)+\Delta r(k+1)-CA(\theta)\Delta x(k)\\ \quad\quad-CB(\theta)\Delta u(k)-CE\Delta d(k),\\ \Delta x(k+1)=A(\theta)\Delta x(k)+B(\theta)\Delta u(k)+E\Delta d(k),\\ \Delta y(k)=CAx(k).\end{array} \right. \tag{12}\]Defining  
\[\left\{ \begin{array}{l}x_{e}(k)=[e^{T}(k)\;e^{T}(k-1)\;\cdots\;e^{T}(k-L+1)]^{T},\\ \overline{x}(k)=[x_{e}^{T}(k)\;\;\Delta x^{T}(k)]^{T},\end{array} \right. \tag{13}\]  
and combining (12) and (13), we have  
\[\overline{x}(k+1)=\overline{A}(\theta)\overline{x}(k)+\overline{B}(\theta) \Delta u(k)+G_{r}\Delta r(k+1)+G_{d}\Delta d(k), \tag{14}\]  
where  
\[\overline{A}(\theta) =\left( \begin{array}{cccc}0&\cdots&0&I&-CA(\theta)\\ I&\cdots&0&0&0\\ \vdots&&\vdots&\vdots&\vdots\\ 0&\cdots&I&0&0\\ 0&\cdots&0&0&A(\theta)\end{array} \right),\] \[\overline{B}(\theta) =\left( \begin{array}{c}-CB(\theta)\\ 0\\ \vdots\\ 0\\ B(\theta)\end{array} \right),\;\;G_{r}=\left( \begin{array}{c}I\\ 0\\ \vdots\\ 0\\ 0\end{array} \right),\;\;G_{d}=\left( \begin{array}{c}-CE\\ 0\\ \vdots\\ 0\\ E\end{array} \right).\]  
From (2), the uncertain matrices \(\overline{A}(\theta)\) and \(\overline{B}(\theta)\) can be rewritten as  
\[\overline{A}(\theta) =\left( \begin{array}{cccc}0&\cdots&0&I&-C\sum\limits_{i=1}^{s} \theta_{i}A_{i}\\ I&\cdots&0&0&0\\ \vdots&&\vdots&\vdots&\vdots\\ 0&\cdots&I&0&0\\ 0&\cdots&0&0&\sum\limits_{i=1}^{s}\theta_{i}A_{i}\end{array} \right) \tag{15}\] \[=\sum\limits_{i=1}^{s}\theta_{i}\left( \begin{array}{cccc}0& \cdots&0&I&-CA_{i}\\ I&\cdots&0&0&0\\ \vdots&&\vdots&\vdots&\vdots\\ 0&\cdots&I&0&0\\ 0&\cdots&0&0&A_{i}\end{array} \right)=\sum\limits_{i=1}^{s}\theta_{i}\overline {A}_{i},\] \[\overline{B}(\theta) =\left( \begin{array}{c}-C\sum\limits_{i=1}^{s}\theta_{i}B_{i} \\ 0\\ \vdots\\ 0\\ \sum\limits_{i=1}^{s}\theta_{i}B_{i}\end{array} \right)=\sum\limits_{i=1}^{s} \theta_{i}\left( \begin{array}{c}-CB_{i}\\ 0\\ \vdots\\ 0\\ B_{i}\end{array} \right)=\sum\limits_{i=1}^{s}\theta_{i}\overline{B}_{i}. \tag{16}\]  
For the uncertain system (14), we specify the observation equation as  
\[e(k)=\overline{C}\overline{x}(k), \tag{17}\]  
where \(\overline{C}=[I_{q}\;\;0_{q\times[(L-1)q+n]}]\). Defining  
\[X_{r}(k)=\left( \begin{array}{c}\Delta r(k)\\ \Delta r(k+1)\\ \vdots\\ \Delta r(k+M_{r})\end{array} \right),\quad X_{d}(k)=\left( \begin{array}{c} \Delta r(k)\\ \Delta r(k+1)\\ \vdots\\ \Delta r(k+M_{d})\end{array} \right), \tag{18}\]  
it follows from assumptions A1 and A2 that \(X_{r}(k),\ X_{d}(k)\) satisfy  
\[X_{r}(k+1)=A_{r}X_{r}(k), \tag{19}\]  
\[X_{d}(k+1)=A_{d}X_{d}(k), \tag{20}\]  
where \(A_{r}\in\mathbb{R}^{q(M_{r}+1)\times q(M_{r}+1)},\ A_{d}\in\mathbb{R}^{p(M_{d} +1)\times p(M_{d}+1)}\), and  
\[A_{r}=\left( \begin{array}{cccc}0&I&0&\cdots&0\\ 0&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&\ddots&0\\ 0&\cdots&\cdots&0&I\\ 0&\cdots&\cdots&0&0\end{array} \right),\ A_{d}=\left( \begin{array}{cccc}0&I& 0&\cdots&0\\ 0&\ddots&\ddots&\ddots&\vdots\\ \vdots&&\ddots&\ddots&0\\ 0&\cdots&\cdots&0&I\\ 0&\cdots&\cdots&0&0\end{array} \right).\]  
Now, defining the augmented state vector  
\[\widehat{x}(k)=\left( \begin{array}{c}\overline{x}(k)\\ X_{r}(k)\\ X_{d}(k)\end{array} \right), \tag{21}\]  
from (14), (19) and (20), we obtain the augmented system  
\[\widehat{x}(k+1)=\widehat{A}(\theta)\widehat{x}(k)+\widehat{B}(\theta)\Delta u (k), \tag{22}\]  
where  
\[\widehat{A}(\theta)=\left(