# Robust Guaranteed-Cost Preview Repetitive Control for Polytopic Uncertain Discrete-Time Systems +
## 1 Introduction  
In some control systems, it is hoped that the output can track the reference signal without steady-state error, even in the presence of uncertainty and/or exogenous disturbances. Preview control [1; 2] is one of the many ways to solve this kind of control problem. The basic idea of preview control is to use known future information of the reference or disturbance signals. It was found that preview control can enhance performance of the closed-loop system [3]. In recent years, the problem of preview control has received considerable attention and various control structures and algorithms have been proposed [4; 5; 6; 7]. As a typical regulator method, the linear quadratic regulator optimal control is a powerful technique for designing preview controllers [8; 9; 10]. In [8], a preview-based linear quadratic design method was investigated for reference-tracking motion control. In this way, the explicit implementation formulas for feedforward controllers were derived that can be applied to a range of rigid-body motion systems. In [9], a preview control using a feedforward-imperfect forecast measurement of a disturbance signal was proposed, in the context of discrete-time linear quadratic Gaussian control. In [10], the constrained optimal preview control of dual-stage actuators was proposed, and the problem was solved in terms of quadratic programming. In [11], using the reference and disturbance signal preview information, an optimal feed-forward preview controller was presented for single-input single-output systems. In order to estimate the unmeasurable state vector, a state observer-based optimal preview controller was developed in [12]. Other relevant work may referto [13]. However, as well known, the classical LQR-based optimal preview control cannot be directly applied to uncertain systems. For uncertain systems, it will become difficult to design preview controller. In [14], a robust tracking problem for a polytopic uncertain system subject to a previewable reference signal was considered. In [15], a linear matrix inequality (LMI)-based robust preview tracking problems for discrete systems with polytopic uncertainties was provided. In [16], the problem of multi-model adaptive preview control was proposed for discrete-time systems with unknown piecewise constant coefficients. In [17], the author overviewed the preview control research achievements and discussed the main existing problems. Very recently, for a class of polytopic uncertain discrete systems, by combining parameter-dependent Lyapunov function method with LMI techniques, a static output feedback preview tracking controller was presented in [18]. For a class of uncertain discrete-time systems, an LMI-based preview controller was proposed in [19]. The proposed method can be extended completely to preview control for time varying systems. Repetitive control (RC), which was first developed in the early 1980s [20], is a very useful strategy for tracking periodic reference signals [21; 22; 23]. Although the repetitive control method was originally proposed in continuous-time systems, it was realized by a digital computer in the actual applications. The direct design of a repetitive controller in the discrete time domain is of practical significance. In [24], a discrete-time variable-structure repetitive control was proposed. The presented control scheme ensures robust stability of the resultant quasi-sliding-mode dynamics to parameter variations and exogenous disturbances. The authors in [25] have investigated a positive realness problem, raised in discrete repetitive control. The sufficient condition of the repetitive control stability can be satisfied through constructing a discrete positive real transfer function. Based on two-dimensional system theory, the problem of designing discrete-time repetitive-control systems was studied in [26; 27]. By the same argument, the observer-based repetitive controller design method for discrete-time systems with uncertainties was presented in [28].