# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 2 Problem Formulation and Preliminaries
The objective is to design a guaranteed-cost preview control law, with the form  
\[u(k)=\sum_{j=0}^{L-1}K_{e}(j)v(k-j)+K_{y}y(k)+\sum_{j=0}^{M_{r}}K_{r}(j)r(k+j) +\sum_{j=0}^{M_{d}}K_{d}(j)d(k+j). \tag{7}\]  
**Remark 4**. : _The preview repetitive controller (7) consists of four parts: The first part is repetitive control, the second term represents output feedback, the third and the fourth represent the preview action based on the future values of the previewable reference signal and the disturbance signal. Besides, considering that the system states are not measurable, the static output feedback is used instead of taking the state feedback._  
**Remark 5**. : _Based on a state feedback controller with integral action and a dynamic feedforward controller, a new ILC structure was proposed in [31]. Similarly, the future values of the reference trajectory \(r(k)\) and the integral  
Figure 1: Basic discrete-time repetitive-control system. Figure 2: Configuration of repetitive-control system with preview compensation. of the learning action are contained in the preview repetitive controller (7), which can be viewed as a feedforward controller and help the system reduce static errors._  
For an uncertain discrete-time system (1), we introduce a quadratic cost function  
\[J=\sum_{k=0}^{\infty}\left[e(k)^{T}Q_{e}e(k)+\Delta u(k)^{T}R\Delta u(k)\right], \tag{8}\]  
where \(\Delta u(k)=u(k)-u(k-L)\), \(Q_{e}\), and \(R\) are the given symmetric, positive definite weighting matrices. A guaranteed-cost preview controller is defined as follows. **Definition 1**. : _For the uncertain discrete-time system (1) and the cost function (8), if there exists a preview repetitive-control law (7) that produces a control input \(u^{*}(k)\), a positive constant \(J^{*}\) such that \(\lim\limits_{k\to\infty}e(k)=0\), and, for all admissible uncertainties, the closed-loop system is robustly stable and the value of the cost function (8) satisfies \(J\leq J^{*}\), then \(J^{*}\) is said to be a guaranteed cost and \(u^{*}(k)\) is said to be a guaranteed-cost preview repetitive-controller for the cost function (8)._  
To obtain our main results, the following lemmas will be used. **Lemma 1** (Schur complement [39]). : _For a real matrix \(\Sigma=\Sigma^{T}\), the following assertions are equivalent:_  
\(i\)_. \(\Sigma:=\left[ \begin{array}{cc}\Sigma_{11}&\Sigma_{12}\\ \star&\Sigma_{22}\end{array} \right]>0\). \(ii\)_. \(\Sigma_{11}>0\), and \(\Sigma_{22}-\Sigma_{12}^{T}\Sigma_{11}^{-1}\Sigma_{12}>0\). \(iii\)_. \(\Sigma_{22}>0\), and \(\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{12}^{T}>0\)._  
**Lemma 2** ([40]). : _For the matrices \(L\in\mathbb{R}^{n\times n}\), \(H\in\mathbb{R}^{n\times n}\), and \(H>0\), one has_  
\[-L^{T}H^{-1}L\leq-L-L^{T}+H. \tag{9}\]  
**Lemma 3** ([41]). : _For appropriately dimensioned matrices \(T\), \(S\), \(R\), \(N\), and scalar \(\beta\), \(T+S^{T}R^{T}+RS<0\) if the following condition holds:_  
\[\left[ \begin{array}{cc}T&(\beta R^{T}+NS^{T})^{T}\\ \beta R^{T}+NS^{T}&-\beta N-\beta N^{T}\end{array} \right]<0. \tag{10}\]