Function theory related to H∞ control

Abstract

We define T(E), a subset of C3, related to the structured singular value μ of 2×2 matrices. μ is used to analyse performance and robustness of linear feedback systems in control engineering. We find a characterisation for the elements of T(E) and establish a necessary and sufficient condition for the existence of an analytic function from the unit disc into T(E) satisfying an arbitrary finite number of interpolation conditions. We prove a Schwarz Lemma for T(E) when one of the points in T(E) is (0, 0, 0), then we show that in this case, the Carath´eodory and Kobayashi distances between the two points in T(E) coincide. We also give a characterisation of the interior, the topological boundary and the distinguished boundary of T(E), then we define T(E)-inner functions and show that if there exists an analytic function from the unit disc into T(E) that satisfies the interpolating conditions, then there is a rational T(E)-inner function that interpolates.We define T(E), a subset of C3, related to the structured singular value μ of 2×2 matrices. μ is used to analyse performance and robustness of linear feedback systems in control engineering. We find a characterisation for the elements of T(E) and establish a necessary and sufficient condition for the existence of an analytic function from the unit disc into 􀀀E satisfying an arbitrary finite number of interpolation conditions. We prove a Schwarz Lemma for T(E) when one of the points in 􀀀E is (0, 0, 0), then we show that in this case, the Carath´eodory and Kobayashi distances between the two points in T(E) coincide. We also give a characterisation of the interior, the topological boundary and the distinguished boundary of T(E), then we define T(E)-inner functions and show that if there exists an analytic function from the unit disc into T(E) that satisfies the interpolating conditions, then there is a rational T(E)-inner function that interpolates.