Robust Identification of Uncertain Systems

The problem of identification of uncertain systems arises whenever we must choose a model from a model class that only approximately describes the underlying system. Identification schemes here need to overcome the distortions in the data due to undermodeling error in addition to random stochastic noise. We introduce the notion of robust consistency, which requires estimating the optimal approximation from corrupted data. We then provide fundamental limits for when consistency is achievable for a wide variety of contexts in both unstructured as well structured uncertain environments. Structured uncertainty pertains to problems where additional prior information such as frequency weighting or specific component-level uncertainties are known. We show the optimality of well-known instrument-variable techniques in several cases and characterize the instruments and inputs that lead to robust identification. We show that robust consistency is achievable if and only if there is an instrument that can uniformly de-correlate both the input and noise. The final part of the paper deals with the question of design of input sequences that satisfy these necessary and sufficient conditions. In this regard, we show that any sequence having a bounded strictly positive spectral density suffices. The rest of the paper then deals with the design of optimal deterministic sequences for robust identification. We develop fast-sine-sweeps in this context and by appealing to elementary continued fraction theory show that these have polynomially and uniformly decaying auto-correlations properties.