Approximability Models and Optimal System Identification

This article considers the problem of optimally recovering stable linear time-invariant systems observed via linear measurements made on their transfer functions. A common modeling assumption is replaced here by the related assumption that the transfer functions belong to a model set described by approximation capabilities. Capitalizing on recent optimal-recovery results relative to such approximability models, we construct some optimal algorithms and characterize the optimal performance for the identification and evaluation of transfer functions in the framework of the Hardy Hilbert space and of the disc algebra. In particular, we determine explicitly the optimal recovery performance for frequency measurements taken at equispaced points on an inner circle or on the torus.