Regret-Optimal Defense Against Stealthy Adversaries: A System Level Approach

Abstract

Modern control designs in robotics, aerospace, and cyber-physical systems rely heavily on real-world data obtained through system outputs. However, these outputs can be compromised by system faults and malicious attacks, distorting critical system information needed for secure and reliable operation. In this paper, we introduce a novel regret-optimal control framework for designing controllers that make a linear system robust against stealthy attacks, including both sensor and actuator attacks. Specifically, we present (a) a convex optimization-based system metric to quantify the regret under the worst-case stealthy attack (the difference between actual performance and optimal performance with hindsight of the attack), which adapts and improves upon the $\mathcal{H}_2$ and $\mathcal{H}_{\infty}$ norms in the presence of stealthy adversaries, (b) an optimization problem for minimizing the regret of (a) in system-level parameterization, enabling localized and distributed implementation in large-scale systems, and (c) a rank-constrained optimization problem equivalent to the optimization of (b), which can be solved using convex rank minimization methods. We also present numerical simulations that demonstrate the effectiveness of our proposed framework.