Frequency Domain Gaussian Process Models for $H^\infty$ Uncertainties

TL;DR

This paper explores conditions under which complex-valued Gaussian processes in the frequency domain can serve as priors for robust control, linking Bayesian system identification with $H^$ uncertainties.

Contribution

It provides sufficient conditions for complex Gaussian processes to be $H^$ functions and offers an explicit covariance parameterization for stationary cases.

Findings

Identifies conditions for Gaussian processes to be $H^$ functions.

Provides an explicit covariance structure for stationary processes.

Bridges Bayesian system identification with robust control theory.

Abstract

Complex-valued Gaussian processes are used in Bayesian frequency-domain system identification as prior models for regression. If each realization of such a process were an $H_\infty$ function with probability one, then the same model could be used for probabilistic robust control, allowing for robustly safe learning. We investigate sufficient conditions for a general complex-domain Gaussian process to have this property. For the special case of processes whose Hermitian covariance is stationary, we provide an explicit parameterization of the covariance structure in terms of a summable sequence of nonnegative numbers.