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

Ziyi Zhang, yorie nakahira, Guannan Qu

Keywords: Linear dynamical system, Learning-based control, Learn-to-stabilize, Sample complexity, Unknown system

TL;DR: We design a novel SVD-based algorithm to stabilize an unknown noisy LTI system with stability guarantee and avoid exponential blow-up in state norm.

Abstract: We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. The state-of-the-art guarantees that the system is stabilized before the system state reaches $2^{O(k \log n)}$ in $L^2$-norm, where $n$ is the state dimension, and $k$ is the dimension of the unstable subspace. However, this bound only holds in *noiseless* LTI systems that have a control input dimension at least as large as the dimension of unstable subspace, making it impractical in many real-life scenarios. In noisy systems, unknown noise is not only amplified by unstable system modes but also imposes significant difficulty in estimating the system dynamics or bounding the estimation errors. Furthermore, the aforementioned complexity is only achievable when the system has a number of control inputs that are at least as many as the dimension of the unstable subspace. To address these issues, we develop a novel algorithm with a singular-value-decomposition(SVD)-based analytical framework and show that the system is stabilized with the same complexity guarantee with the state-of-the-art in a noisy environment. With the SVD-based framework, we can bound the error of system identification with Davis-Kahan Theorem and design a controller that does not require the invertibility of the control matrix, making it possible to apply this algorithm in under-actuated settings. To the best of our knowledge, this paper is the first to achieve learning-to-stabilize unknown LTI system without exponential blow-up in noisy and under-actuated systems. We further demonstrate the advantage of the proposed algorithm in under-actuated settings.

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Submission Number: 262