Finite Time Identification in Unstable Linear Systems
Finite Time Identification in Unstable Linear Systems
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DOI：
10.1016/j.automatica.2018.07.008
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
Mohamad Kazem Shirani Faradonbeh;Ambuj Tewari;G. Michailidis
中科院分区：
文献类型：
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作者：
Mohamad Kazem Shirani Faradonbeh;Ambuj Tewari;G. Michailidis
稳定线性动力系统参数的识别是文献中一个经过充分研究的问题，无论是在低维还是高维设置中。然而，对于不稳定的情况几乎没有任何结果，特别是在有限的时间范围内。对于这种设置，动力学参数最小二乘估计的经典结果不适用，因此需要开发新的概念和技术方法来解决这个问题。不稳定的线性系统出现在控制理论、计量经济学和金融学的关键实际应用中。本研究为相当大一类重尾噪声分布的最小二乘估计的识别误差建立了有限时间界限，并建立了此类系统的转移矩阵。结果将估计所需的时间长度（样本）与问题维度的函数以及真实的底层转移矩阵和噪声分布的关键特征相关联。为了建立它们，利用随机矩阵和鞅差序列的适当浓度不等式。
Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensionalsettings. However, there are hardly any results for the unstable case, especially regardingfinite time bounds. For this setting, classical results on least-squares estimation of the dynamics parameters are not applicable and therefore new concepts and technical approaches need to be developed to address the issue. Unstable linear systems arise in key real applications in control theory, econometrics, and finance.This study establishes finite time bounds for the identification error of the least-squares estimates for a fairly large class of heavy-tailed noise distributions, and transition matrices of such systems. The results relate the time length (samples) required for estimation to a function of the problem dimension and key characteristics of the true underlying transition matrix and the noise distribution. To establish them, appropriate concentration inequalities for random matrices and for sequences of martingale differences are leveraged.