[Paper Review] Finite Time Identification in Unstable Linear Systems

Mohamad Kazem Shirani Faradonbeh, Ambuj Tewari+1 more|arXiv (Cornell University)|Oct 5, 2017

Control Systems and Identification17 references4 citations

TL;DR

This paper establishes finite-time identification bounds for least-squares estimation of the transition matrix in unstable linear dynamical systems, even under heavy-tailed noise. By leveraging concentration inequalities for random matrices and anti-concentration properties of martingale differences, it derives non-asymptotic error bounds that scale with system dimension, eigenvalue characteristics of the transition matrix, and noise tail behavior.

ABSTRACT

Identification of the parameters of stable linear dynamical systems is a well-studied problem in the literature, both in the low and high-dimensional settings. However, there are hardly any results for the unstable case, especially regarding finite 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.

Motivation & Objective

• Address the lack of finite-time identification results for unstable linear dynamical systems, where classical least-squares methods fail due to state explosion.
• Provide non-asymptotic error bounds for least-squares estimation of the transition matrix in unstable systems under general heavy-tailed noise distributions.
• Characterize the sample complexity required for accurate identification in terms of system dimension, spectral properties of the transition matrix, and noise tail behavior.
• Extend theoretical tools from random matrix theory and martingale difference sequences to unstable dynamics, overcoming exponential growth in state norms.

Proposed method

• Decompose the state vector dynamics into stable and unstable subspaces using spectral decomposition of the transition matrix $A_0$.
• Apply concentration inequalities for random matrices to control the behavior of the Gram matrix of the state trajectory in the unstable subspace.
• Use anti-concentration properties of martingale difference sequences to bound the maximum singular value of the noise-weighted state matrix.
• Introduce a regularization-like structure via the inverse square root of the Gram matrix $\Sigma_n^{-1/2}$ to stabilize estimation in high-dimensional, unstable settings.
• Derive high-probability bounds on the $\ell_2$ estimation error of the least-squares estimator by decomposing the error into components across stable and unstable subspaces.
• Utilize spectral norms and eigenvalue decay/growth rates (e.g., $|\lambda_{\min}(A_2)|^{-n/3}$) to quantify the influence of unstable dynamics on estimation error.

Experimental results

Research questions

• RQ1 What is the finite-time error bound for least-squares estimation of the transition matrix in unstable linear systems with heavy-tailed noise?
• RQ2 How does the required sample size scale with the system dimension and the spectral properties of the transition matrix?
• RQ3 Can concentration inequalities for random matrices and martingale differences be adapted to handle unstable dynamics with exponentially growing state norms?
• RQ4 What role do the eigenvalues of the transition matrix and the tail behavior of the noise distribution play in determining the identification time horizon?
• RQ5 How does the estimation error depend on the structure of the unstable subspace, particularly in terms of eigenvalue magnitudes and multiplicities?

Key findings

• The paper establishes a high-probability finite-time bound on the $\ell_2$ estimation error of the least-squares estimator for unstable linear systems, valid under general heavy-tailed noise.
• The required sample size $n$ scales with the system dimension $p$, the spectral radius of the unstable part of the transition matrix, and the tail thickness of the noise distribution.
• The error bound decays as $O(n^{-1/2})$ under favorable spectral conditions, but the rate is modulated by the inverse of the smallest eigenvalue of the unstable component.
• For the unstable subspace, the estimation error is controlled via a term scaling as $n^{\mu(A_2) - 1/2} |\lambda_{\min}(A_2)|^{-2n/3}$, reflecting exponential growth in the state norm.
• The analysis shows that accurate identification is still possible in finite time even when the system is unstable, provided the noise is sub-Gaussian or heavy-tailed with finite moments.
• The derived bounds are non-asymptotic and explicitly quantify the trade-off between system instability, noise characteristics, and sample complexity.