The robust stability of iterative learning control

Abstract

This thesis examines the notion of the long term robust stability of iterative learning control (ILC) systems engaged in trajectory tracking, using a robust stability theorem based on a biased version of the nonlinear gap metric. This is achieved through two main results:

The first concerns the establishment of a nonlinear robust stability theorem, where signals are measured relative to a given trajectory. Although primarily motivated by ILC, the theorem provided is applicable to a wider range of problems. This is due to its development being made independently of any particular signal space, provided the space is furnished with a definition of causality. The theorem's formulation therefore permits its implementation on single- or multi-dimensional problems in a variety of different settings. Necessitated by an ILC constraint concerning reference signals, the trajectory that stability is measured relative to must often lie outside the signal space that is chosen. The robust stability theorem is therefore devised to address signals that lie in extended signal spaces throughout. Additionally, the theorem is applicable to nonlinear systems, and it is shown that the biased gap measure collapses to the standard nonlinear gap measure when the bias is set to zero. It is also shown to collapse to the classical linear gap when restricting the analysis to certain linear systems.

The second result applies the robust stability theorem to ILC using a 2D signal space. Initially the subject of ILC is reviewed and some of the problems associated with controllers are described; in particular the issues of long-term stability and the criteria for convergence. ILC algorithms expressed in the 'lifted system' or 'supervector' formulation are discussed and then analysed using the biased robust stability theorem. Results are presented regarding the use of filtering in ILC algorithms to aid robustness, and also the robustness of inverse model-based techniques. The robust stability tool applied to ILC in this thesis is not restricted in its analysis to algorithms which are 'causal' (in an ILC sense); and is based on a general unstructured uncertainty model in contrast to the existing literature, whereby uncertainties are typically constrained to additive, multiplicative or parametric models.

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