Strong practical stability and stabilization of differential linear repetitive processes

Differential linear repetitive processes evolve over a subset of the upper-right quadrant of the 2D plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t ∈ [ 0 , α ] where α is termed the pass length or duration. On each pass an output, termed the pass profile, is produced, which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. The result can be oscillations in the pass-to-pass direction that cannot be controlled by direct application of standard, or 1D linear systems theory. The existing stability theory demands a bounded-input bounded-output property uniformly, which in the case of the along-the-pass dynamics means for t ∈ [ 0 , ∞ ] and for ( k , t ) ∈ [ 0 , ∞ ] × [ 0 , ∞ ] ⊃ [ 0 , ∞ ] × [ 0 , α ] where the integer k ≥ 0 denotes the pass number or index. The pass length is always finite, however, and hence this stability theory could well be too strong in many cases and, in particular, impose very strong conditions in terms of control law design. This paper develops an alternative in such cases by relaxing the requirement for the bounded-input bounded-output property to hold when k → ∞ and t → ∞ simultaneously, provides an explanation of the implications of this in the frequency domain, and then develops control law design algorithms.