Discrete 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. In this paper, a 1D linear systems equivalent state space model description of the dynamics of so-called discrete non-unit memory linear repetitive processes is developed and used (with related approaches) to characterize key systems theoretic properties. \noindent {\bf Keywords:} 2D linear systems, repetitive dynamics, stability, 1D equivalent model, controllability, non-unit memory processes.

Discrete linear repetitive processes are a distinct class of two-dimensional linear systems with applications in areas ranging from long-wall coal cutting to iterative learning control schemes. In this paper, a one-dimensional linear systems equivalent state-space model description of the dynamics of discrete nonunit memory linear repetitive processes is developed and used (with related approaches) to characterize key systems theoretic properties.

Repetitive processes are a distinct class of two-dimensional linear systems of practical and theoretical interest. Most of the available control theory for them is for the case of linear dynamics and focuses on systems theoretic properties such as stability and controllability/observability. This paper uses an extension of standard, or one-dimensional, feedback control schemes to control a physically relevant subclass of these processes.

Repetitive processes are a distinct class of 2D systems of both practical and theoretical interest. Their essential characteristic is repeated sweeps, termed passes, through a set of dynamics defined over a finite duration with explicit interaction between the outputs, or pass profiles, produced as the process dynamics evolve. Experience has shown that these processes cannot be studied/controlled by direct application of existing theory. This fact, and the growing list of applications areas, has prompted an ongoing research programme into the development of a 'mature' systems theory for these processes for onward translation into reliable generally applicable controller design algorithms. This paper develops stability tests for a sub-class of so-called differential linear repetitive processes in the presence of a general set of initial conditions, where it is known that the structure of these conditions is critical to their stability properties.

This paper develops the basis of a so-called 2D Lyapunov equation based approach to the stability analysis of discrete linear repetitive processes. The key feature of this equation is that, in contrast to the so-called ID Lyapunov equation approach, it is defined in terms of matrices with constant entries and has a `similar' structure to the Lyapunov equation for standard, or ID, discrete linear systems. Here it is shown that the 2D Lyapunov equation gives, in general, a characterization of stability which is sufficient but not necessary. Despite this deficiency, it is also shown how the 2D Lyapunov equation can be used to characterize stability margins and robustness to uncertainty in the model description - very important topics for which few results are currently available. Some areas for short to medium term further research are also briefly noted.

The field of multidimensional systems theory suffers from the lack of a framework in which its many diverse strands could be unified. We propose the behavioural approach as such a framework. In the study of autoregressive systems, the use of behavioural tools is particularly useful since it allows the application of Oberst's duality theory, in which every AR nD behaviour is identified with a unique finitely generated module over the polynomial (Laurent polynomial) ring in n indeterminates. We have illustrated the efficacy of this approach by considering a fundamental algebraic concept, the annihilator of a module. We have shown that this concept relates to the autonomy of a behaviour, the idea of poles of an nD system, and notions of primeness and Bezout identities. We believe that the combination of the behavioural approach with algebraic techniques is suitable for the unification and solution of many problems in the study of multidimensional linear shift invariant systems