Many physical systems, data analysis procedures, computational algorithms and (more recently) learning algorithms have a natural (and underexploited) two-dimensional (2D) structure due to the presence of more than one spatial variable, the combined effect of space and time or the combined effect of a spatial/time variable and an integer index representing iteration, pass or trial number. Physical examples of such systems include bench mining systems, metal rolling, automatic ploughing aids and vehicle convoy coordination on motorways whilst algorithmic examples include image processing, discrete models of spatial behaviour, point mapping algorithms and recursive learning schemes as illustrated by trajectory learning in iterative learning control or recursive/feedback neural networks.
Repetitive, or multipass processes are uniquely characterised by a series of sweeps, termed passes, through a set of dynamics defined over a finite and fixed 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 new pass profile. Industrial example include long-wall coal cutting and metal rolling operations. Recently, a new applications area for the theory of repetitive processes has emerged from the fact that they form a natural basis for the analysis of iterative learning control schemes. The basic feature of such a scheme is that the controller learns the input necessary to generate a desired output by repeated trials, where the control signal for the current trial is constructed as a function of the previous outputs and (possibly) inputs. A typical industrial example is a robotic manipulator which repetitively performs the same task, such as following a geometric trajectory.
Repetitive processes have clear well defined structural links with 2D linear systems represented by, for example, the well known Roesser model. At a basic level, these links arise from the fact that two 'co-ordinates' are required to specify a repetitive process variable, i.e. pass number k and position t along a pass of the fixed and finite length.
The aim of the lecture would be to sketch existing theory and applications for nD systems as well as repetitive processes and to highlight the current open problems.
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