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

Modern control designs in robotics, aerospace, and cyber-physical systems heavily depend on real-world data obtained through the system outputs. In the face of system faults and malicious attacks, however, these outputs can be compromised to misrepresent some portion of the system information that critically affects their secure and trustworthy operation. In this paper, we introduce a novel regret-optimal control framework for designing controllers that render a linear system robust against stealthy attacks, including sensor and actuator attacks, as well as external disturbances. In particular, we establish (a) a convex optimization-based system metric to quantify the regret with the worst-case stealthy attack (the true performance minus the optimal performance in hindsight with the knowledge of the stealthy attack), which improves and adaptively interpolates $\mathcal{H}2$ and $\mathcal{H}{\infty}$ norms in the presence of stealthy adversaries, (b) an optimization problem for minimizing the regret of 1 expressed in the system level parameterization, which is useful for its localized and distributed implementation in large-scale systems, and (c) a rank-constrained optimization problem (i.e., optimization with a convex objective subject to convex constraints and rank constraints) equivalent to the optimization problem of (b). Finally, we conduct a numerical simulation which showcases the effectiveness of our approach.