Computation of the Trajectories Generated by 2D Discrete Model Approximations of the Dynamics of Differential Linear Repetitive Processes

Abstract

Differential linear repetitive processes are a class of 2D linear systems which can be used, for example, to model industrial processes such as long-wall coal cutting. Also they can be used to study the properties of classes of iterative learning control schemes and the convergence properties of iterative algorithms for solving nonlinear dynamic optimal control problems based on the maximum principle. The key unique feature of interest in this paper is the fact that information propagation in one of the two separate directions evolves continuously over a fixed finite interval and in the other it is, in effect, discrete. This paper describes the development of discrete approximations for these processes, resulting in 2D linear systems state space models of the well known Fornasini Marchesini form on which to base further analysis. In this context, the remainder of this paper develops formulas for computing the trajectories generated by these 2D representations which, by analogy with the standard (1D) case, can be expected to play a key role in characterising basic systems theoretic properties such as controllability and observability. Some on-going work and areas for further development in these and related areas will also be briefly discussed.