Graph Topologies, Gap Metrics, and Robust Stability for Nonlinear Systems

Abstract

Abstract. Graph topologies for nonlinear operators which admit coprime factorisations are defined w.r.t. a gain function notion of stability in a general normed signal space setting. Several metrics are also defined and their relationship to the graph topologies are examined. In particular, relationships between nonlinear generalisations of the gap and graph metrics, Georgiou-type formulae and the graph topologies are established. Closed loop robustness results are given w.r.t. the graph topology, where the role of a coercivity condition on the nominal plant is emphasised.Key words. gap metric, graph metric, graph topology, robust stability, nonlinear systems AMS subject classifications. 93D09, 93D25, 93C101. Introduction. The theory of coprime factorisations of linear signal operators is well known to be a significant tool in the study of robustness of stability for linear feedback systems and has been extensively studied (see [5,16,20]). Perturbations to normalized co-prime factors form a good description of physically realistic deviations from nominal models, since they allow a unified treatment of both low and high frequency uncertainties [8]. In the linear theory, it is well known that the graph topology is the appropriate topological description for studying robustness of stability and that co-prime factor perturbations can be used to induce the graph topology. Furthermore, the graph topology is metrizable, and both the gap metric [3,21] and the graph metric [20] provide suitable metrizations, the former being more suitable for calculations by standard H ∞ optimizations, (although both metrics are topologically equivalent) [5,16,21]. There is thus a rich set of equivalences between the notions of co-prime factorisations, gap/graph metrics and topologies and their attendant robust stability theorems. Moreover, this framework is a cornerstone of modern robust linear control theory.Given the richness and importance of this framework in the linear setting, it is natural to seek extensions to the nonlinear case, and to alternative signal spaces. Indeed, by adopting a notion of stability corresponding to the existence of a linear gain, (typically either in an L 2 or L ∞ setting), a number of authors have previously considered a nonlinear theory of co-prime factorisation. Here we highlight three contributions of particular relevance to the context of this paper. In [18], Verma defined a notion of co-prime factorisation for nonlinear mappings and, presented a stability result for a nonlinear system. In [2], Anderson, James and Limebeer generalised the linear theory of normalized co-prime factor robustness optimisation to the case of affine input nonlinear systems and presented a optimal robustness margin. In [10], a new definition of "normalized" was introduced for left representation for the graph of a nonlinear system and different gap metrics were studied. Many further pointers to a growing literature on nonlinear co-prime factorisation can be found in the monograph [14] and the references therein.On the other ...

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