Online Optimization of LTI Systems Under Persistent Attacks: Stability, Tracking, and Robustness

Abstract

We study the stability properties of the interconnection of an LTI dynamical plant and a feedback controller that generates control signals that are compromised by a malicious attacker. We consider two classes of controllers: a static output-feedback controller, and a dynamical gradient-flow controller that seeks to steer the output of the plant towards the solution of a convex optimization problem. We analyze the stability of the closed-loop system under a class of switching attacks that persistently modify the control inputs generated by the controllers. The stability analysis leverages the framework of hybrid dynamical systems, Lyapunov-based arguments for switching systems with unstable modes, and singular perturbation theory. Our results reveal that under a suitable time-scale separation, the stability of the interconnected system can be preserved when the attack occurs with "sufficiently low frequency" in any bounded time interval. We present simulation results in a power-grid example that corroborate the technical findings