1D models for class of 2D linear systems

The basic unique control problem for repetitive processes arises from the explicit interaction between successive pass profiles. In particular, the output sequence of pass profiles can contain oscillations that increase in amplitude in the pass to pass direction. Such behaviour is easily generated in simulation studies and in experiments on scaled models of industrial examples such as long-wall coal cutting. In long-wall coal cutting this problem appears as severe undulations in the newly cut floor profile which means that cutting operations (i.e. productive work) must be suspended to enable their manual removal. This problem is one of the key factors behind the stop/start cutting pattern of a typical working cycle in a coal mine. In general, this problem cannot be removed by standard, i.e. 1D, control action. The basic reason for this is that such an approach essentially ignores their inherent 2D systems structure. Motivated by this key fact, Rogers and Owens (1992) have developed a stability theory for repetitive processes with linear dynamics and a constant pass length. This theory is based on an abstract model in a Banach space setting which includes all such processes as special cases. The results of applying this theory to a range of special cases are also known including discrete linear repetitive processes which are the subject of this paper