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Abstract-Discrete linear repetitive processes are a distinct class of two-dimensional (2-D) linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The feature which makes them distinct from other classes of 2-D linear systems is that information propagation in one of the two distinct directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper, an LMI approach is used to produce highly significant new results on the stability analysis of these processes and the design of control schemes for them. These results are, in the main, for processes with singular dynamics and for those with so-called dynamic boundary conditions. Unlike other classes of 2-D linear systems, these feedback control laws have a firm physical basis, and the LMI setting is also shown to provide a (potentially) very powerful setting in which to characterize the robustness properties of these processes.

Abstract-This paper considers differential linear repetitive processes which are a distinct class of two-dimensional continuous-discrete linear systems of both physical and systems theoretic interest. The substantial new results are on the application of linear-matrix-inequality-based tools to stability analysis and controller design for these processes, where the class of control laws used has a well defined physical basis. It is also shown that these tools extend naturally to cases when there is uncertainty in the state-space model of the underlying dynamics.Index Terms-Controller design, linear matrix inequality (LMI) design, repetitive dynamics, uncertainty. I. INTRODUCTIONThe essential unique characteristic of a repetitive, or multipass, process is a series of sweeps-termed passes-through a set of dynamics defined over a fixed finite duration known as the pass length. On each pass, an output-termed the pass profile-is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This, in turn, leads to the unique control problem for these processes in that the output sequence of pass Manuscript received December 8, 2002; revised February 21, 2003. Please note that the review process for this paper was handled exclusively by the TCAS-I editorial office and that the paper has been published in TCAS-II for logistical reasons. This paper was recommended by Associate Editor C. Xiao.K. Galkowski and W. Paszke are with the Institute of Control and Computation Engineering, University of Zielona Gora, Zielona Gora, Poland.E. Rogers is with the Department of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K.S. Xu and J. profiles generated can contain oscillations that increase in amplitude in the pass-to-pass direction.To introduce a formal definition, let < +1 denote the pass length (assumed constant). Then in a repetitive process the pass profile y k (t), 0 t , generated on pass k acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile y k+1 (t), 0 t , k 0.Physical examples of repetitive processes include long-wall coal cutting and metal rolling operations; see, for example, [1]. Also, in recent years, applications have arisen where adopting a repetitive process setting for analysis has distinct advantages over alternatives. Examples of these so-called algorithmic applications of repetitive processes include classes of iterative learning control (ILC) schemes [5]and iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle [6]. In the case of ILC for the linear dynamics case, the stability theory for differential and discrete linear repetitive processes is the essential basis for a rigorous stability/convergence analysis of such schemes.Attempts to control these processes using standard, termed one-dimensional (1-D) here, systems theory/algorithms fail (except in a few very restrictive special cases) precisely because such an approach ignores t...

Multidimensional control systems have been the subject of much productive research over more than three decades. In contrast to standard control systems, there has been much less reported on applications where the multidimensional setting is the only possible setting for design or produces implementations that perform to at least the same level. This paper addresses the latter area where case studies focusing on control law design and evaluation, including experimental results in one case, are reported. These demonstrate that movement towards the actual deployment of multidimensional control systems is increasing.

SUMMARYDiscrete linear repetitive processes are a distinct class of 2D linear systems with applications in areas ranging from long-wall coal cutting through to iterative learning control schemes. The main feature which makes them distinct from other classes of 2D linear systems is that information propagation in one of the two independent directions only occurs over a finite duration. This, in turn, means that a distinct systems theory must be developed for them. In this paper, the major new development is that an LMI based reformulation of the stability conditions can used to enable the design of a family of control laws which have a well defined physical basis. It is also noted that this setting can be used to investigate robustness aspects.