Limit sets and switching strategies in parameter-optimal iterative learning control

Abstract

This paper characterizes the existence and form of the possible limit error signals in typical parameter-optimal Iterative Learning Control. The set of limit errors has attracting and repelling components and the behaviour of the algorithm in the vicinity of these sets can be associated with the undesirable properties of apparent (but in fact temporary) convergence or permanent slow convergence properties in practice. The avoidance of these behaviours in practice is investigated using novel switching strategies. Deterministic strategies are analysed to prove the feasibility of the concept by proving that each of a number of such strategies is guaranteed to produce global convergence of errors to zero independent of the details of plant dynamics. For practical applications a random switching strategy is proposed to replace these approaches and shown, by example, to produce substantial potential improvements when compared with the non-switching case. The work described in this paper is covered by pending patent applications in the UK and elsewhere.

Iterative Learning Control — Monotonicity and Optimization

“…In summary, a zero tracking error in the limit is guaranteed when G e + G T e is a positive-definite matrix, but in other cases the limiting tracking error can be (and can be seen to be) a non-zero vector. A more detailed convergence analysis of this algorithm can be found in (Owens and Feng, 2003) and, more recently, as part of the publication (Owens and Daley, 2008).…”

“…The recent publication (Owens and Daley, 2008) provided considerable insight into the issue of convergence to nonzero errors in parameter optimal ILC. The essential structure of the results obtained can be summarised (where M = G for consistency with the notation in the reference) as a decomposition of the limit set into two subsets, one S − ∞ , which attracts iterative trajectories, and one S + ∞ , which repels such trajectories.…”

“…In contrast, if the algorithm leads to an error time series on some iteration that is close to the repelling component, the algorithm will appear to be converging slowly but, in reality, it is simply "pausing" before it is repelled and further monotonic convergenceclearly non-desirable in practice as slow convergence is a waste of resource whether that is time or experimental cost. Examples can be found in the reference (Owens and Daley, 2008) which indicate that even simple non-positive plants can exhibit such behaviour.…”

“…A conceptual solution to the problem is presented in (Owens and Daley, 2008) using switching. The essence of the idea is that, using the ILC law…”

“…The paper (Owens and Daley, 2008) provides a complete description of such deterministic switching algorithms and illustrates the wide range of successful switching strategies that can be considered. It also presents a statistical analysis of the algorithms that suggest that random switching is beneficial although the nonlinear nature of the problem makes exact theoretical analysis difficult.…”