On the Limits of Robust Control Under Adversarial Disturbances

TL;DR

This paper investigates the fundamental limits of robust control for constrained linear systems under bounded adversarial disturbances, providing explicit conditions where robust control becomes impossible.

Contribution

It introduces novel algebraic conditions linking disturbance size, system spectral properties, and constraint geometry to the infeasibility of robust control.

Findings

Derived closed-form expressions for control infeasibility conditions.

Identified key spectral and geometric factors affecting robustness.

Provided practical criteria for actuator and sensor design considerations.

Abstract

This paper addresses a fundamental and important question in control: under what conditions does there fail to exist a robust control policy that keeps the state of a constrained linear system within a target set, despite bounded disturbances? This question has practical implications for actuator and sensor specification, feasibility analysis for reference tracking, and the design of adversarial attacks in cyber-physical systems. While prior research has predominantly focused on using optimization to compute control-invariant sets to ensure feasible operation, our work complements these approaches by characterizing explicit sufficient conditions under which robust control is fundamentally infeasible. Specifically, we derive novel closed-form, algebraic expressions that relate the size of a disturbance set -- modelled as a scaled version of a basic shape -- to the system's spectral properties and the geometry of the constraint sets.