Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise

TL;DR

This paper introduces a novel SVD-based algorithm that stabilizes unknown noisy LTI systems on a single trajectory by focusing on the unstable subspace, avoiding exponential blow-up in state dimension.

Contribution

The paper develops a new algorithm that decouples unstable and stable subspaces, enabling stabilization without exponential dimension dependence, a first in noisy LTI system stabilization.

Findings

System stabilizes before state norm reaches 2^{O(k log n)}

Avoids exponential blow-up in state dimension

First approach to stabilize noisy LTI systems without exponential dimension dependence

Abstract

We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. It is well known in the literature that the learn-to-stabilize problem suffers from exponential blow-up in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the state space dimension. This blow-up is due to the open-loop instability when exploring the $n$-dimensional state space. To address this issue, we develop a novel algorithm that decouples the unstable subspace of the LTI system from the stable subspace, based on which the algorithm only explores and stabilizes the unstable subspace, the dimension of which can be much smaller than $n$. With a new singular-value-decomposition(SVD)-based analytical framework, we prove that the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where $k$ is the dimension of the unstable subspace. Critically, this bound avoids exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the previous works, and to the best of our knowledge, this is the first paper to avoid exponential blow-up in dimension for stabilizing LTI systems with noise.