SIAM Journal on Scientific Computing, Volume 42, Issue 5, Page A3447-A3461, January 2020.
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.

中文翻译：

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数据驱动的McMillan度下界

SIAM科学计算杂志，第42卷，第5期，第A3447-A3461页，2020年1月。
在线性时不变系统的情况下，麦克米伦度规定了再现观察到的动力学的系统的最小可能尺寸。当这些观测结果采取冲激响应测量的形式时，系统在没有来自未知初始条件的输入的情况下演化而来，Ho和Kalman的结果表明麦克米伦度是根据这些测量建立的汉克矩阵的秩。不幸的是，在实验实践中使用该结果具有挑战性，因为测量总是受到噪声的污染，因此汉高矩阵几乎可以肯定是满级的。因此，从业人员通过手动设置一个阈值来估计此矩阵的秩（进而是McMillan度），该阈值将与无噪声汉克矩阵的非零奇异值相对应的大奇异值与与零摄动相对应的小奇异值分开无噪声汉克尔矩阵的奇异值。在这里，我们用韦尔定理指导的阈值代替此手动阈值。具体来说，假设测量结果受到加性高斯噪声的干扰，我们构造了一个概率上限，确定无噪声汉克矩阵的奇异值能被干扰多少；这提供了一个保守的阈值来估计排名，从而估计了麦克米兰学位。此结果来自具有正态分布项的随机Hankel矩阵的2范数上的新概率界。与现有的随机汉克尔矩阵结果不同，该边界没有未知常数，而且在经验观察到的边界的很小范围内。McMillan度上的此界限为更通用的模型订单选择技术（例如Akaike信息标准）提供了廉价的替代方案。