800 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 4, APRIL 2009 A Passive Repetitive Controller for Discrete-Time Finite-Frequency Positive-Real Systems Ramon Costa-Castelló, Danwei Wang, and Robert Griñó Abstract—This work proposes and studies a new internal model for dis- crete-time passive or finite-frequency positive-real systems which can be used in repetitive control designs to track or to attenuate periodic signals. The main characteristic of the proposed internal model is its passivity. This property implies closed-loop stability when it is used with discrete-time pas- sive plants, as well as the broader class of discrete-time finite-frequency pos- itive real plants. This work discusses the internal model energy function and its frequency response. A design procedure for repetitive controllers based on the proposed internal model is also presented. Two numerical examples are included. Index Terms—Discrete-time control, discrete-time passivity, finite-fre- quency positive realness, repetitive control. I. INTRODUCTION Repetitive control is an established control design technique for sys- tems handling periodical signals. The most important component in a repetitive controller is the periodic signal internal model [4]. The main drawback of the conventional internal model is its high order that makes the stability analysis of the closed-loop system difficult. In the pio- neering work of Inoue [10], the stability of these systems is established by disecting the closed-loop system into three series-connected subsys- tems. The stability checking of the first two subsystems is straightfor- ward but, for the remaining third subsystem, the Small Gain Theorem needs to be used. Later, this approach was extended and modified to improve high frequency robustness [7] and conditions and proce- dures were established [19]. Lyapunov based analysis was also intro- duced in [14]. These works provide stability conditions for passive (i.e. Positive Real (PR) in linear systems) systems. These results have been extended to Almost Strictly Positive Real (ASPR) and Almost Strictly Negative Real (ASNR) systems [3]. Although a discrete-time formula- tion exists [1], most works have set out and developed repetitive control in continuous time. This work proposes a new structure for the internal model which is discrete-time passive (equivalently, it is Discrete-Time Positive Real). When this internal model is used as a repetitive controller with a dis- crete-time passive plant, the closed-loop system stability is guaranteed. Furthermore, feedback passivizable plants [16] can also benefit from this property. The passivity property of the proposed internal model implies a re- duced phase lag and this finds its application to finite frequency pos- itive-real (FFPR) and passive plants [11]. The FFPR property is less Manuscript received October 02, 2006; revised June 29, 2007. First pub- lished March 24, 2009; current version published April 08, 2009. This work was supported in part by the Comision Interministerial de Ciencia y Tecnología (CICYT) under Project DPI2007-62582. Recommended by Associate Editor M. Kothare. R. Costa-Castelló and R. Griñó are with the Institut d’Organització i Control de Sistemes Industrials (IOC), Universitat Politècnica de Catalunya (UPC), Barcelona 08028, Spain (e-mail: ramon.costa-castello@ieee.org; roberto.grino@upc.edu). D. Wang is with the School of Electrical and Electronics Engineering, Nanyang Technological University (NTU), Singapore 639798 (e-mail: ed- wwang@ntu.edu.sg). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2008.2009594 restrictive than passivity. In addition, being positive real in a certain frequency range is a necessary condition for good control performance in that frequency range [12]. The proposed internal model introduces configurable zeros to shape the open-loop frequency response and to provide an additional degree of freedom in control design. In view of time response, the proposed internal model will reduce the time delay of repetitive controllers due to its reduced phase lag. Furthermore, several internal models can be combined in parallel to obtain a multiperiodic repetitive control system. Besides this, the proposed internal model preserves all relevant proper- ties of repetitive controllers: trajectory tracking and disturbance rejec- tion capability, simple structure and low computational cost. This work analyses and characterises the proposed internal model and presents a design methodology for FFPR discrete-time plants which is illustrated with two numerical examples. II. PASSIVE INTERNAL MODEL A. Internal Model Structure The proposed internal model is described by the transfer function (1) where , , , and is a low-pass filter. For , the poles of (1) are 1 , , so they are uniformly distributed over a circumference of radius 2 . The frequencies associated to the poles are , so the poles are placed to cover all the harmonic frequencies of the fundamental one, . This pole placement is the same as the one obtained in the conventional internal model. The zeroes of (1) follow a similar placement [2]. Depending on the signs of and , we have the following cases: if , the poles and the zeroes are placed at the same fre- quencies, and if the poles and the zeroes are placed at shifted frequencies. In particular, in the later case, the frequencies as- sociated with the zeros are exactly the mean of the frequencies of the adjacent poles. However, in real world applications it is necessary to re- duce the controller gain in the high frequency band and for this reason, the internal model includes the low-pass filter . B. Energy Properties of the Passive Internal Model Let be the LTI discrete-time system (2) (3) where , ; or in input-output form 3 . Definition 1 (Discrete-Time Passivity (DTP), [13]): System (2), (3) is discrete-time passive with storage function ( -passive) if, and only if (4) Definition 2 ( -Dissipative [5]): DTP systems with (5) 1 equals 1 for and 1 for . 2 is necessary to assure the stability of the repetitive block. 3 being the th order identity matrix. 0018-9286/$25.00 © 2009 IEEE Authorized licensed use limited to: UNIVERSITAT POLITÈCNICA DE CATALUNYA. Downloaded on December 22, 2009 at 05:20 from IEEE Xplore. Restrictions apply.