On estimation of the gain of a dynamical system

We analyze and compare methods to estimate the ℒ
₂
-gain (ℋ
_∞
-norm) of a stable linear dynamical system, that is the maximum of the absolute value of the corresponding frequency response. The standard approach is to estimate a parametric model of the system, which then is used for gain calculation. An asymptotic error variance expression for ℋ
_∞
-norm estimates based on finite impulse response (FIR) models is presented. We then study the problem of finding the minimum variance excitation signal that satisfies a given error variance bound for the FIR model gain estimate. It is shown that a sinusoidal signal with frequency equal to the peak frequency of the system is minimum variance optimal. The second approach to gain estimation is based on iterative experiments, and is inspired by numerical methods eigenvalue calculation. The asymptotic statistical properties of such gain estimation methods are compared to the FIR model approach. A transparent expression for the additional variance cost of using iterative experiments is derived. Finally, we present some ideas for input sequence estimation based on iterative experiments.