A NEW OPTIMALITY BASED REPETITIVE CONTROL ALGORITHM FOR DISCRETE-TIME SYSTEMS J. Hätönen *,† , D. H. Owens † , R. Ylinen * ∗ Systems Engineering Laboratory, University of Oulu, P.O.BOX 4300, FIN-90014 University of Oulu, Finland † Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK Keywords: Repetitive Control, Optimal Control, Polynomial Systems Theory. Abstract In this paper it is shown that discretisation of a class of well- known continuous-time repetitive control algorithms will de- stroy stability. In order to overcome this problem, a new op- timality based Repetitive Control algorithm is proposed for discrete-time systems. Under mild technical conditions on the plant the algorithm will result in asymptotic convergence for an arbitrary T -periodic reference signal and an arbitrary discrete- time linear time-invariant plant. Simulations highlight the dif- ferent theoretical findings in this paper. Copyright © 2003 IFAC 1 Introduction Many signals in engineering are periodic, or at least they can be accurately approximated by a periodic signal over a large time interval. This is true, for example, of most signals associated with engines, electrical motors and generators, converters, or machines performing a task over and over again. Hence it is an important control problem to try to track a periodic signal with the output of the plant or try to reject a periodic disturbance acting on a control system. In order to solve this problem, a relatively new research area called Repetitive Control has emerged in the control commu- nity. The idea is to use information from previous periods to modify the control signal so that the overall system would ’learn’ to track perfectly a given T -periodic reference signal. The first paper that uses this ideology seems to be (Inouye et al., 1981), where the authors use repetitive control to obtain a desired proton acceleration pattern in a proton synchotron mag- netic power supply. Since then repetitive control has found its way to several prac- tical applications, including robotics (Kaneko and Horowitz, 1997), motors (Kobayashi et al., 1999), rolling processes (Garimella and Srinivasan, 1996) and rotating mechanisms (Fung et al., 2000). However, most of the existing Repetitive Control algorithms are designed in continuous time, and they either don’t give perfect tracking or they require that original process is positive real. In order to overcome these limitations, in this paper a new optimality based Repetitive Control algo- rithm is introduced for linear time-invariant discrete-time sys- tems, which will result in perfect tracking under mild technical assumptions. 2 Problem definition and earlier work As a starting point in continuous-time Repetitive Control (RC) it is assumed that a mathematical model ˙ x(t)= Ax(t)+ Bu(t) y(t)= Cx(t)+ Du(t) (1) of the plant in question exists with x(0) = x 0 , t ∈ [0, ∞). Furthermore, A, B, C and D are finite-dimensional matrices of appropriate dimensions. From now on it is assumed that D =0, because in practise it very rare to find a system where the input function u(t) has an immediate effect on the output variable y(t). Furthermore, a reference signal r(t) is given, and it is known that r(t)= r(t + T ) for a given T (in other words the actual shape of r(t) is not necessarily known). The control design objective is to find a feedback controller that makes the system (1) to track the reference signal as accurately as pos- sible (i.e. lim t→∞ e(t)=0, e(t) := r(t) − y(t)), under the assumption that the reference signal r(t) is T -periodic. As was shown by (Francis and Wonhan, 1975), a necessary condition for asymptotic convergence is that a controller [Mu]t =[Ne](t) (2) where M and N are suitable operators, has to have an in- ternal model or the reference signal inside the operator M . Because r(t) is T -periodic, its internal model is 1 − σ T , where [σ T v](t)= v(t − T ) for v : R → R. Hence in (Yamamoto, 1993) it was suggested, that one possible (and obviously computationally simple) RC algorithm for the SISO case could be u(t)= u(t − T )+ e(t) (3) This algorithm has been analysed by several authors, see for example (Yamamoto, 1993), (Arimoto and Naniwa, 2000) and (Owens et al., 2001). It turns out that if the system (1) is pos- itive real (PR) then for e(·) ∈ L 2 [0, ∞) (this does not imply that lim t→∞ e(t) = 0)). The definition of a positive real sys- tem from (Anderson and Vongpanithred, 1973) is given in the following Definition 1 (A PR system - continuous-time case) Consider the transfer function matrix G(s) of the system (1) where G(s)= C(sI − A) −1 B + D. System (1) is positive real 1) Each element of the transfer function G(s) are analytic for Re[s] > 0