Linear Repetitive Processes
Repetitive processes have been defined as those involving the processing of a material or workpiece by a sequence of passes of the processing tool. Industrial examples include metal rolling, long wall coal cutting and bench mining operations, and algorithmic examples such as classes of iterative learning control schemes.
On each sweep an output is produced called the pass profile, which then contributes to the dynamics of the next sweep. It is this interaction between passes which leads to the unique control problem associated with these processes in that oscillations can occur in the sequence of output pass profiles which increase in amplitude from pass to pass.
In the situation where it is the previous pass profile only which contributes (explicitly) to the current pass, the process is called unit memory, whereas if the previous M pass profiles contribute to the current one we have a non-unit memory process, with M as the memory length (for example in the bench mining operation M = 10).
A linear repetitive process is inherently 2D in nature since two pieces of information are required to specify each point - the time or the distance along the sweep, and the sweep number.
Rogers and Owens have developed a rigorous stbility theory for the class of linear repetitive processes with constant pass length. This theory is based on an abstract model in a Banach space setting, and includes all such processes as special cases. Two such special cases are the classes of differential and discrete linear repetitive processes.
The stability theory consists of two separate concepts - asymptotic stability and the stronger condition of stability along the pass.