Adaptive control and robustness in the gap metric

We consider the construction of adaptive controllers for minimum phase linear systems which achieve non-zero robustness margins in the sense of the gap metric. The gap perturbation margin may be more constrained for larger disturbances and for larger parametric uncertainties. Working in an L2 setting, and within the framework of the nonlinear gap metric, universal adaptive controllers are first given achieving stabilisation for first order nominal plants, and the results are then generalised to relative degree one nominal plants. Necessary asymptotic properties of the robustness margins are derived for the class of controllers considered. Extensions to the Linfty setting are also developed where two alternative designs are given. A notion of a semi universal control design is introduced, which is the property that a bound on performance exists which is independent of the a-priori known uncertainty level, and a characterisation is given for when semi-universal designs outperform the class of memoryless controllers and the class of LTI controllers. Robust semi-universal adaptive control designs are given for nominal plants under the classical assumptions of adaptive control in both the L2 and Linfty settings. The results are applied throughout to explicit classes of unmodelled dynamics including the Rohrs example