On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory

Yang Hu, Adam Wierman, Guannan Qu

Keywords: linear time-invariant systems, sample complexity, stability, learning-based control

TL;DR: We propose a new spectral-decomposition-based algorithm that stabilizes an LTI system with $k$ unstable eigenvalues on a single trajectory, using only $O(k \log n)$ samples, and thus incurring a sub-exponential state norm.

Abstract: Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (LTI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in $n$, the state dimension, which incurs a state norm that blows up exponentially in $n$. We propose a novel algorithm based on spectral decomposition that only needs to learn ``a small part'' of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our algorithm stabilizes an LTI system on a single trajectory with $O(k \log n)$ samples, where $k$ is the instability index of the system. This represents the first sub-linear sample complexity result for the stabilization of LTI systems under the regime when $k = o(n)$.