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Stabilizing Linear Systems under Partial Observability: Sample Complexity and Fundamental Limits

Ziyi Zhang, Yorie Nakahira, Guannan Qu

TL;DR

The paper addresses stabilizing an unknown partially observable LTI system by isolating an unstable transfer component and learning it with a data-efficient lifted-Hankel SVD procedure. It combines a rank-$k$ Hankel factorization with robust $H_\infty$-based controller design, leveraging a small-gain condition on the stable subspace to guarantee full-system stability. The main result shows the data requirements scale with the number of unstable modes $k$ (as $O(k^2)$) and are independent of the full state dimension $n$, under standard controllability/observability and bound assumptions, with a steady treatment of non-diagonalizable unstable blocks. Simulations validate the approach, showing significant improvements in data efficiency for high-dimensional systems with many stable modes.

Abstract

We study the problem of stabilizing an unknown partially observable linear time-invariant (LTI) system. For fully observable systems, leveraging an unstable/stable subspace decomposition approach, state-of-art sample complexity is independent from system dimension $n$ and only scales with respect to the dimension of the unstable subspace. However, it remains open whether such sample complexity can be achieved for partially observable systems because such systems do not admit a uniquely identifiable unstable subspace. In this paper, we propose LTS-P, a novel technique that leverages compressed singular value decomposition (SVD) on the ''lifted'' Hankel matrix to estimate the unstable subsystem up to an unknown transformation. Then, we design a stabilizing controller that integrates a robust stabilizing controller for the unstable mode and a small-gain-type assumption on the stable subspace. We show that LTS-P stabilizes unknown partially observable LTI systems with state-of-the-art sample complexity that is dimension-free and only scales with the number of unstable modes, which significantly reduces data requirements for high-dimensional systems with many stable modes.

Stabilizing Linear Systems under Partial Observability: Sample Complexity and Fundamental Limits

TL;DR

The paper addresses stabilizing an unknown partially observable LTI system by isolating an unstable transfer component and learning it with a data-efficient lifted-Hankel SVD procedure. It combines a rank- Hankel factorization with robust -based controller design, leveraging a small-gain condition on the stable subspace to guarantee full-system stability. The main result shows the data requirements scale with the number of unstable modes (as ) and are independent of the full state dimension , under standard controllability/observability and bound assumptions, with a steady treatment of non-diagonalizable unstable blocks. Simulations validate the approach, showing significant improvements in data efficiency for high-dimensional systems with many stable modes.

Abstract

We study the problem of stabilizing an unknown partially observable linear time-invariant (LTI) system. For fully observable systems, leveraging an unstable/stable subspace decomposition approach, state-of-art sample complexity is independent from system dimension and only scales with respect to the dimension of the unstable subspace. However, it remains open whether such sample complexity can be achieved for partially observable systems because such systems do not admit a uniquely identifiable unstable subspace. In this paper, we propose LTS-P, a novel technique that leverages compressed singular value decomposition (SVD) on the ''lifted'' Hankel matrix to estimate the unstable subsystem up to an unknown transformation. Then, we design a stabilizing controller that integrates a robust stabilizing controller for the unstable mode and a small-gain-type assumption on the stable subspace. We show that LTS-P stabilizes unknown partially observable LTI systems with state-of-the-art sample complexity that is dimension-free and only scales with the number of unstable modes, which significantly reduces data requirements for high-dimensional systems with many stable modes.

Paper Structure

This paper contains 17 sections, 19 theorems, 161 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Background on $H_\infty$ control
  4. Algorithm Idea
  5. Decomposition of Dynamics
  6. Approximate low-rank factorization of the lifted Hankel matrix
  7. Robust stabilizing controller design
  8. Algorithm
  9. Main Results
  10. Proof Outline
  11. Simulations
  12. Proof of Step 1: Bounding Hankel Matrix Error
  13. Proof of Step 2: Bounding Dynamics Error
  14. Proof of Step 3
  15. System identification when $D \neq 0$
  16. ...and 2 more sections

Key Result

Lemma 3.1

Given $\gamma>0$, the closed loop transfer function defined in eq:full_closed_form is internally stable for any $\Vert \Delta(z)\Vert_{H_{\infty}} \leq \gamma$ if and only if $\Vert K(z) F_K(z)\Vert_{H_{\infty}}<\frac{1}{\gamma}.$

Figures (2)

  • Figure 1: The above shows the length of rollouts needed to identify and stabilize an unstable system with the unstable dimension $k=5$. The solid line shows the average length of rollouts the learner takes to stabilize the system. The shaded area shows the standard variation of the length of rollouts. The proposed method requires the shortest rollouts and has the smallest standard variation.
  • Figure 2: The above shows the number of rollouts needed to identify and stabilize an unstable system with the unstable dimension $k=5$. The solid line shows the average number of rollouts the learner takes to stabilize the system. The shaded area shows the standard variation of the number of rollouts. The proposed method requires the least number rollouts and has small standard variation.

Theorems & Definitions (34)

  • Definition 2.1: $\mathcal{H}_{\infty}$-space
  • Lemma 3.1: Theorem 8.1 of robust
  • Theorem 5.3
  • Lemma 6.1
  • Lemma 6.2
  • Lemma 6.3
  • Lemma 6.4
  • Proposition 6.5
  • Lemma A.1
  • proof
  • ...and 24 more