# Discrete-Time Design of Dual Internal Model-Based Repetitive Control Systems
## 1 Introduction  
Repetitive control (RC) is a learning control scheme utilizing an internal model principle by [PERSON] and [PERSON] [1] for accurate reference tracking or good disturbance rejection of the periodic signals. The inclusion of internal model inside the feedback loop establishes a generator of periodic control signal, enabling a null steady-state tracking error. The early applications of RC for reference tracking and disturbance rejection problems were listed by [PERSON] and [PERSON] [2], followed by [PERSON] et al. [3]. Recently, RC was developed and applied in many different applications such as wind-turbines [4], inverter compressor refrigeration fields [5], magnetically suspended rotor systems [6], functional electrical stimulation [7], nano-positioning systems [8], centrifugal compressors [9], and many others. In principle, RC design comprises two parts, i.e., a stabilizing controller and an internal model. In the RC-controlled system, the stabilizing controller, sometimes called the compensator/learning function, is required to stabilize the closed-loop model. In addition, the stabilizing controller also determines the convergence speed of the system error. On the other hand, the internal model represents the reference/disturbance model, which later behaves as a periodic signal generator. This internal model makes accurate reference tracking/good disturbance rejection possible. Some important works related to the internal model design can be found in [10, 11, 12, 13, 14, 15]. The first digital internal model obtained by discretizing the continuous-time delay was given in [10]. The internal model [10] has a finite-dimensional structure and provides a null steady-state tracking error for any periodic signal at a fundamental frequency and its harmonics up to Nyquist frequency components. The internal model [10] was successfully applied to the control of a single-axis electric servomechanism using a 16-bitmicroprocessor. In [11], a low-pass filter \(q(z)\) is added to discrete-time internal model [10] intended to improve the system robustness against high-frequency models. [PERSON] and [PERSON] [12] presented a low-order internal model targeting specific band-limited periodic references/disturbances. The discrete-time internal model specifically used to track/compensate odd-harmonic periodic references/disturbances was proposed in [13]. In [14], the discrete-time design of the internal model was developed to track or suppress periodic signals consisting of two or more fundamental frequencies. A discrete-time higher-order internal model for tracking/rejecting periodic signals with uncertain periods was introduced in [15]. In addition to some works on the internal models, several recent works on the development and application of RCs can be found in [16; 17; 18; 19]. A robust RC with an optimized-band-stop filter for achieving high accuracy in nanopositioning stages was proposed in [16]. In [17], the discrete-time RC with an active disturbance rejection control was implemented to realize high-precision permanent magnet synchronous motor drives. A design of higher-order RC with a phase lead stabilizing controller applied for a two-level grid-connected inverter was presented in [18]. [PERSON] et al. [19] developed the high-order selective harmonic RC scheme to improve the steady-state tracking accuracy of the pulse-width modulation converters. The works in [10; 11; 12; 13; 14; 15; 16; 17; 18; 19] aimed to develop and utilize the internal model-based RC for either periodic reference tracking or disturbance rejection. This also implies that the proposed RC designs are not intended to simultaneously track and reject periodic signals. In this work, we focus on the internal model design used for simultaneous reference tracking and disturbance rejection. More specifically, we develop a control strategy for tracking repetitive reference and rejection of uncertain periodic disturbance of a discrete-time linear system. This is motivated by the fact that tracking and rejection are common problems in control systems.