Nonparametric Finite Time LTI System Identification
Abstract
We address the problem of learning the parameters of a stable linear time invariant (LTI) system or linear dynamical system (LDS) with unknown latent space dimension, or order, from a single time–series of noisy input-output data. We focus on learning the best lower order approximation allowed by finite data. Motivated by subspace algorithms in systems theory, where the doubly infinite system Hankel matrix captures both order and good lower order approximations, we construct a Hankel-like matrix from noisy finite data using ordinary least squares. This circumvents the non-convexities that arise in system identification, and allows accurate estimation of the underlying LTI system. Our results rely on careful analysis of self-normalized martingale difference terms that helps bound identification error up to logarithmic factors of the lower bound. We provide a data-dependent scheme for order selection and find an accurate realization of system parameters, corresponding to that order, by an approach that is closely related to the Ho-Kalman subspace algorithm. We demonstrate that the proposed model order selection procedure is not overly conservative, i.e., for the given data length it
中文速览
从单段带噪声的时间序列中识别线性时不变（LTI）系统参数，同时又不知道系统的真实阶次，这是控制理论和时间序列分析中长期悬而未决的难题。本文的核心思路是：用普通最小二乘法（OLS）从有限数据中构造一个类Hankel矩阵来近似无限系统Hankel矩阵，再借助Ho-Kalman子空间算法提取系统参数，从而绕开了传统方法中阶次未知时带来的非凸优化困难。通过对自归一化鞅差分项（self-normalized martingale difference）的精细分析，作者给出了识别误差的有限时间上界，精确到对数因子与信息论下界吻合，并设计了一套数据驱动的阶次选择方案，能在给定数据量下自适应地找到最优低阶近似。实验和理论双双表明，所选阶次不会过于保守——数据不足以支撑更高阶估计时，算法会如实反映这一局限而非盲目过拟合。这项工作首次将单段有限噪声数据下的LTI系统辨识与自适应阶次选择统一在一个有严格统计保证的框架里，对控制、机器人和经济学中的实际建模需求具有重要意义。
原文 arXiv:1902.01848;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1902.01848v6