# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 4 Design of the Guaranteed-cost Preview Repetitive Controller  
Note that the proposed preview repetitive controller (7) can be written as  
\[\Delta u(k) = \sum_{j=0}^{L-1}K_{e}(j)e(k-j)+K_{y}\Delta y(k) \tag{29}\] \[+ \sum_{j=0}^{M_{r}}K_{r}(j)\Delta r(k+i)+\sum_{j=0}^{M_{d}}K_{d}(j )\Delta d(k+j).\]  
Set  
\[K_{e}(j)=\sum_{i=1}^{s}\alpha_{i}K_{ei}(j),\;\;K_{y}=\sum_{i=1}^ {s}\alpha_{i}K_{yi},\] \[K_{r}(j)=\sum_{i=1}^{s}\alpha_{i}K_{ri}(j),\;\;K_{d}(j)=\sum_{i=1 }^{s}\alpha_{i}K_{di}(j), \tag{30}\]  
where \(\alpha_{i}\) (\(i=1,2,\ldots,s\)) are adjustable variables and \(\alpha_{i}\geq 0\), \(\sum\limits_{i=1}^{s}\alpha_{i}=1\). Furthermore, denoting  
\[K_{i} = [K_{ei}(0),\;K_{ei}(1),\;\;\cdots,\;K_{ei}(L-1),\;K_{yi}, \tag{31}\] \[K_{ri}(0),\;K_{ri}(1),\;\;\cdots,\;K_{ri}(M_{r}),\] \[K_{di}(0),\;K_{di}(1),\;\;\cdots,\;K_{di}(M_{d})],\]  
it follows from (29)-(31) that we have  
\[\Delta u(k)=(\sum_{i=0}^{s}\alpha_{i}K_{i})Z(k). \tag{32}\]  
If we denote \(\alpha=(\alpha_{1}\;\alpha_{2}\;\ldots\alpha_{s})^{T}\in R^{s}\) and \(K(\alpha)=\sum\limits_{i=1}^{s}\alpha_{i}K_{i}\), then  
\[\Delta u(k)=K(\alpha)Z(k). \tag{33}\]Therefore, the guaranteed-cost preview repetitive control problem is transformed into the guaranteed-cost static output feedback control problem for the augmented dynamic system (27). In the following, we will give some sufficient conditions for the existence of a guaranteed-cost static output feedback controller (33) for the augmented dynamic system (27). **Theorem 1**. : _For the augmented dynamic system (27) with performance index (28), if there exists a positive defined parameter-dependent matrix \(P(\theta)\) and an invertible matrix \(G(\alpha)=\sum\limits_{i=1}^{s}\alpha_{i}G_{i}\) such that_  
\[\left( \begin{array}{ccc}\Pi_{11}&*&*\\ \left[\widehat{A}(\theta)+\widehat{B}(\theta)K(\alpha)C_{Z}\right]G(\alpha)& -P(\theta)^{-1}&*\\ R^{\frac{1}{2}}K(\alpha)C_{Z}G(\alpha)&0&-I\end{array} \right)<0, \tag{34}\]  
_where_  
\[\Pi_{11}=-G(\alpha)^{T}-G(\alpha)+P(\theta)^{-1}+G(\alpha)^{T}Q_{\xi}G(\alpha),\]  
\(Q_{\xi}\) _and \(R\) are the weighting matrices in performance index (28), then the static output feedback controller (33) is a guaranteed-cost controller and the performance index satisfies \(\widehat{J}\leq\widehat{x}_{0}^{T}P(\theta)\widehat{x}_{0}\), where \(\widehat{x}_{0}\) is the initial state of augmented error system (27)._  
Proof. : The proof is given in the Appendix A. The following result is equivalent to Theorem 1. **Theorem 2**. : _For the dynamic system (27) with performance index (28), if there exists a positive defined parameter-dependent matrix \(X(\theta)\); invertible matrices \(W\), \(U\), and \(G(\alpha)\); and a scalar \(\beta\), such that_  
\[\Pi(\theta,\alpha)=\left( \begin{array}{cccc}\Psi_{11}&*&*&*\\ \Psi_{21}&-X(\theta)&*&*\\ R^{\frac{1}{2}}L(\alpha)Q_{\xi}&0&-I&*\\ C_{z}G(\alpha)-UQ_{\xi}&\Psi_{42}&\beta W^{T}L(\alpha)^{T}R^{\frac{1}{2}}&\Psi_ {44}\end{array} \right)<0, \tag{35}\]  
_where_  
\[\Psi_{11} = -G(\alpha)-G(\alpha)^{T}+X(\theta)+G(\alpha)^{T}Q_{\xi}G(\alpha),\] \[\Psi_{21} = \widehat{A}(\theta)G(\alpha)+\widehat{B}(\theta)L(\alpha)Q_{\xi},\] \[\Psi_{42} = \beta W^{T}L(\alpha)^{T}\widehat{B}(\theta)^{T},\] \[\Psi_{44} = -\beta UW-\beta U^{T}W^{T},\]  
_and \(Q_{\xi}\) and \(R\) are the weighting matrices in performance index (28), then the conclusion of Theorem 1 holds._  
Proof. : The proof is given in the Appendix B. It should be pointed out that the condition (35) in Theorem 2 is not a convex problem. The following theorem can translate it into an LMI problem. **Theorem 3**. : _Given a scalar \(\beta\) and matrix \(W\), if there exists \(X_{i}>0\), \(U\), \(L_{i}\), and \(G_{i}\), \((i=1,2,3\ldots,s)\), with appropriate dimensions such that_  
\[\Pi_{ij}<0\ (i,j\in 1,2,\cdots,s), \tag{36}\]_where_  
\[\Pi_{ij}=\left(