2D signal processing theory applied to learning control systems

A repetitive process is a 2D system characterised uniquely by a series of sweeps, termed passes, through a set of dynamics defined over a finite and fixed duration-termed the pass length and denoted by α. On each pass an output, termed the pass profile and denoted by Y_k(t),0⩽t⩽α, acts as a forcing function on, and hence contributes to, the new pass profile Y_k+1(t), 0⩽t⩽α, k⩽0. The unique control problem is that the output sequence {Y_k}_k⩾1, generated in response to an initial profile Y₀ and a sequence of current pass inputs/disturbances, can contain oscillations which increase in amplitude from pass to pass. Smyth (1992) describes in detail how this behaviour can arise in physically based examples, both in simulation and experiments on scaled models of industrial examples. The present authors describe the development of a rigorous stability theory, applicable to all linear dynamics constant pass length examples, which removes difficulties encountered in trying to apply directly standard linear theory and tests. The elements of this theory are summarised