# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 2 Problem Formulation and Preliminaries  
Consider the following uncertain SISO discrete-time system:  
\[\left\{ \begin{array}{l}x(k+1)=A(\theta)x(k)+B(\theta)u(k)+Ed(k),\\ y(k)=Cx(k),\end{array} \right. \tag{1}\]  
where \(x(k)\in\mathbb{R}^{n}\) is the state, \(u(k)\in\mathbb{R}^{m}\) is the control input, \(d(k)\in\mathbb{R}^{p}\) is the periodic disturbance, and \(y(k)\in\mathbb{R}^{q}\) is the output of the plant. \(E\) and \(C\) are constant matrices with appropriate dimensions. \(A(\theta)\) and \(B(\theta)\) are uncertain matrices with appropriate dimensions, and are given by  
\[\left[A(\theta)\quad B(\theta)\right]=\sum_{i=1}^{s}\theta_{i}\left[A_{i} \quad B_{i}\right], \tag{2}\]  
where \(A_{i}\) and \(B_{i}\), \((i=1,2,\ldots,s)\), are constant matrices with appropriate dimensions, and \(\theta=\left[\theta_{1},\ \theta_{2},\ \cdots,\ \theta_{s}\right]^{T}\) is the uncertain constant parameter vector satisfying  
\[\theta\in\Theta:=\left\{\theta\in\mathbb{R}^{s}|\ \theta_{i}\geq 0,(i=1,2, \ldots,s),\ \sum_{i=1}^{s}\theta_{i}=1\right\}. \tag{3}\]  
Let \(r(k)\in\mathbb{R}^{q}\) be a periodic reference signal to be tracked by \(y(k)\in\mathbb{R}^{q}\), and  
\[e(k)=r(k)-y(k) \tag{4}\]  
be the tracking error. Throughout this paper, two assumptions are made:  
**A1:**: The reference signal \(r(k)\) and disturbance signal \(d(k)\) are periodic, both of period \(L\). **A2:**: The preview lengths of the reference signal \(r(k)\) and disturbance signal \(d(k)\) are \(M_{r}\) and \(M_{d}\) (\(M_{r},\ M_{d}\) are less than \(L\)), respectively. That is, at each time \(k\), the \(M_{r}\) (\(M_{d}\)) future values, \(r(k+1)\), \(r(k+2)\), \(\ldots\), \(r(k+M_{r})\) (\(d(k+1)\), \(d(k+2)\), \(\ldots\), \(d(k+M_{d})\)), as well as the present and past values of the reference signal, are available. **Remark 1**. : _In fact, the disturbance signal and the reference signal are not necessarily the same period. In this case, the period of the repetitive controller can be set as the minimum common multiple of the period of reference signal and interference signal._  
**Remark 2**. : _Assumption A2 implies that the reference and the disturbance signals are previewable. Conventional discrete-time feedback control systems and the dynamic feedforward control scheme do not take full advantage of the known future values; that is, \(M_{r}=0\) and \(M_{d}=0\). Moreover, assumption A2 indicates that the previewable reference and the disturbance signals have significantly impacted performance in the control system only for a certain time period, during which the reference signal exceeds the preview length, and the impact is very small [2; 4; 5]._  
**Remark 3**. : _The future values of the disturbance signal are assumed to be measured in assumption A2. Admittedly, it is a very strong assumption. In most cases, the complete future disturbance signal cannot be measured directly. For some given control systems, the detailed mathematical modeling and estimation of future disturbance signal can refer to [36; 37]._  
The basic configuration of a discrete-time repetitive control system [38] is shown in Figure 1, where \(P(z)\) is the compensated plant, \(r(k)\) is a periodic reference input with period \(L\), and  
\[C_{R}(z)=\frac{1}{1-z^{-L}} \tag{5}\]  
is a repetitive controller. The output of the repetitive controller, \(v(k)\), is  
\[v(k)=\left\{ \begin{array}{ll}v(k-L)+e(k),&\quad k\geq L,\\ e(k),&\quad k<L.\end{array} \right. \tag{6}\]  
This paper concerns the repetitive-control system configuration in Figure 2, where \(G(z)\) is the controlled plant as described in System (1). \(K_{r}(j)\), \((j=0,\,1,\,2,\ldots,M_{r})\), and \(K_{d}(j)(j=0,\,1,\,2,\ldots,M_{d})\) are the gains of the reference signal and disturbance signal preview compensator, respectively. \(K_{e}(j)\), \((j=0,\,1,\,2,\ldots,L-1)\), and \(K_{y}\) are the gains of the repetitive controller and the output feedback controller, respectively.