# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 4 Design of the Guaranteed-cost Preview Repetitive Controller
\begin{array}{ccccc}-G_{j}-G_{j}^{T}+X_{i}&*&*&*&*\\ \widehat{A}_{i}G_{j}+\widehat{B}_{i}L_{j}Q_{\xi}&-X_{i}&*&*&*\\ R^{\frac{1}{2}}L_{j}Q_{\xi}&0&-I&*&*\\ C_{zi}G_{j}-UQ_{\xi}&\beta W^{T}L_{j}^{T}\widehat{B}_{i}^{T}&\beta W^{T}L_{j}^{T }R^{\frac{1}{2}}&\Psi_{44}&*\\ Q_{\xi}^{\frac{1}{2}}G_{j}&0&0&0&-I\end{array} \right), \tag{37}\]  
\(\Psi_{44}\) _is defined in Theorem 2, and \(Q_{\xi}\) and \(R\) are the weighting matrices in performance index (28), then the augmented system (27) is robustly asymptotically stable under the guaranteed-cost controller (32), and the gain matrix can be obtained by \(K_{i}=L_{i}U^{-1}\). Moreover, the corresponding closed-loop performance index satisfies \(\widehat{J}\leq\sum\limits_{i=1}^{s}\widehat{x}_{0}^{T}X_{i}^{-1}\widehat{x}_ {0}\)._  
Proof. : Setting \(X(\theta)=\sum\limits_{i=1}^{s}\theta_{i}X_{i}\), from (23), and taking into account the definition of \(K(\alpha)andG(\alpha)\), one gets  
\[\Pi(\theta,\alpha)=\sum\limits_{i=1}^{s}\sum\limits_{j=1}^{s}\theta_{i}x_{j} \Pi_{ij}. \tag{38}\]  
In fact, inequality (36) implies \(\Pi(\theta,\alpha)<0\). As a result, the condition of Theorem 2 holds. The proof is completed. **Remark 6**. : _Note that the controller is still generated, based on the traditional LQR approach. With different weighting matrices \(Q_{e}\) and \(R\), various gain matrices can be obtained, which indicates that the controller is non-optimal in a true sense. In recent studies, plenty of methods have been reported to solve this type of optimization problem, such as the adaptive particle swarm optimization method [42] and genetic algorithms [43]._  
Based on Theorem 3, the following conclusion can be obtained easily. **Corollary 1**. : _Given a scalar \(\beta\) and matrix \(W\), if there exist \(X>0\), \(U\), \(L\), and \(G\), with appropriate dimensions such that_  
\[\Psi_{i}<0\ (i\in 1,2,3,\ldots,s), \tag{39}\]  
_where_  
\[\Psi_{i}=\left( \begin{array}{ccccc}-G-G^{T}+X&*&*&*&*\\ \widehat{A}_{i}G+\widehat{B}_{i}LQ_{\xi}&-X&*&*&*\\ R^{\frac{1}{2}}LQ_{\xi}&0&-I&*&*\\ C_{zi}G-UQ_{\xi}&\beta W^{T}L^{T}\widehat{B}_{i}^{T}&\beta W^{T}L^{T}R^{\frac {1}{2}}&\Psi_{44}&*\\ Q_{\xi}^{\frac{1}{2}}G&0&0&0&-I\end{array} \right)<0,\]  
_then the augmented system (27) is robustly asymptotically stable under the guaranteed cost controller_  
\[\Delta u(k)=KZ(k)=LU^{-1}Z(k). \tag{40}\]  
_Moreover, the corresponding closed-loop performance index satisfies \(\widehat{J}\leq\widehat{x}_{0}^{T}X^{-1}\widehat{x}_{0}\)._  
**5. Numerical Example**  
Consider the discrete-time system (1) with \[A(\theta) =\left( \begin{array}{cc}0.60&0.10\\ 0.45&0.80\end{array} \right)\theta_{1}+\left( \begin{array}{cc}0.95&0\\ 0.10&0.96\end{array} \right)\theta_{2},\] \[B(\theta) =\left( \begin{array}{cc}0.35\\ 0.14\end{array} \right)\theta_{1}+\left( \begin{array}{cc}0.30\\ 0.34\end{array} \right)\theta_{2},\;\;E=\left( \begin{array}{cc}0.02\\ 0.05\end{array} \right),C=\left[ \begin{array}{cc}0.9&0.5\end{array} \right].\]  
Take the uncertain parameters \(\theta_{1}=0.20,\;\theta_{2}=0.80\). Set the initial value of \(x(k)\) as \(x(0)=[0\;\;0]^{\mathrm{T}}\). Assume that the reference signal is  
\[r(k)=\sin(\frac{2\pi}{10}k)+0.25\sin(\frac{4\pi}{10}k)+0.5\sin(\frac{6\pi}{10}k),\]  
and the disturbance signal is \(d(k)=\sin(\frac{2\pi}{10}k)\). As previously highlighted, one of the difficulties in an LQR design is the determination of the weighting matrices. In this study, we obtained the weighting matrices, after many attempts, and found that \(Q_{e}=10,\;R=0.01\) is a good alternative. The simulation results are first presented for the following three situations: Namely, (i) \(M_{r}\;=\;7,M_{d}=6\), (ii) \(M_{r}=5,\;M_{d}=4\) and (iii) \(M_{r}=2,\;M_{d}=1\). The adjustable parameters are given as \(\beta=0.40\) and \(\alpha_{1}=0.80,\;\alpha_{2}=0.20,\;W=0.3C_{Z}C_{Z}^{\mathrm{T}}\). By solving the LMIs (36) in Theorem 3, the feedback gain matrix \(K_{i}=L_{i}U^{-1}(i=1,2)\) can be derived, and then the feedback gain \(K=\alpha_{1}K_{1}+\alpha_{2}K_{2}\) is obtained.