Minimax Lower Bounds for H_∞-Norm Estimation

The problem of estimating the H_∞-norm of an LTI system from noisy input/output measurements has attracted recent attention as an alternative to parameter identification for bounding unmodeled dynamics in robust control. In this paper, we study lower bounds for H_∞-norm estimation under a query model where at each iteration the algorithm chooses a bounded input signal and receives the response of the chosen signal corrupted by white noise. We prove that when the underlying system is an FIR filter, H_∞-norm estimation is no more efficient than model identification for passive sampling. For active sampling, we show that norm estimation is at most a factor of logr more sample efficient than model identification, where r is the length of the filter. We complement our theoretical results with experiments which demonstrate that a simple non-adaptive estimator of the norm is competitive with state-of-the-art adaptive norm estimation algorithms.