A Data-Driven McMillan Degree Lower Bound

In the context of linear time-invariant systems, the McMillan degree prescribes the smallest possible dimension of a system that reproduces the observed dynamics. When these observations take the form of impulse response measurements where the system evolves without input from an unknown initial condition, a result of Ho and Kalman reveals the McMillan degree as the rank of a Hankel matrix built from these measurements. Unfortunately, using this result in experimental practice is challenging as measurements are invariably contaminated by noise and hence the Hankel matrix will almost surely be full rank. Hence practitioners estimate the rank of this matrix---and thus the McMillan degree---by manually setting a threshold separating large singular values corresponding to the nonzero singular values of the noise-free Hankel matrix and small singular values corresponding to perturbation of zero singular values of the noise-free Hankel matrix. Here we replace this manual threshold with a threshold guided by Weyl's theorem. Specifically, assuming measurements are perturbed by additive Gaussian noise we construct a probabilistic upper bound on how much the singular values of the noise-free Hankel matrix can be perturbed; this provides a conservative threshold for estimating the rank and hence the McMillan degree. This result follows from a new probabilistic bound on the 2-norm of a random Hankel matrix with normally distributed entries. Unlike existing results for random Hankel matrices, this bound features no unknown constants and, moreover, is within a small factor of the empirically observed bound. This bound on the McMillan degree provides an inexpensive alternative to more general model order selection techniques such as the Akaike information criteria.