A New Approach to Controlling Linear Dynamical Systems

2025-04-04

AI Summary

This paper studies online control of linear dynamical systems subject to adversarial disturbances and general convex cost functions. To address the limitations of existing approaches—namely, strong dependence on system stability margins and high computational complexity—we propose a novel convex relaxation framework based on spectral filters constructed from Hankel matrix eigenvectors. Our method improves the stability-margin dependence from polynomial to quasi-logarithmic, achieves the optimal $O(sqrt{T})$ regret bound, and reduces per-step runtime to near-linear complexity. Theoretically, our analysis is the first to guarantee performance for weakly stable (rather than strictly stable) systems and requires no prior knowledge of system parameters. Experiments demonstrate significant improvements in both robustness against adversarial perturbations and computational efficiency.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Multiagent Systems: Adversarial Agents
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We propose a new method for controlling linear dynamical systems under adversarial disturbances and cost functions. Our algorithm achieves a running time that scales polylogarithmically with the inverse of the stability margin, improving upon prior methods with polynomial dependence maintaining the same regret guarantees. The technique, which may be of independent interest, is based on a novel convex relaxation that approximates linear control policies using spectral filters constructed from the eigenvectors of a specific Hankel matrix.

Problem

• Control linear dynamical systems under adversarial disturbances
• Improve running time scaling with stability margin
• Use novel convex relaxation for linear control policies

Innovation

• Polylogarithmic running time with stability margin
• Novel convex relaxation for linear control
• Spectral filters from Hankel matrix eigenvectors