Covertly Controlling a Linear System

TL;DR

This paper investigates the limits of covert control of linear systems, showing that unstable systems cannot be covertly stabilized, but stable ones can be covertly manipulated, revealing a three-way trade-off among information, control, and covertness.

Contribution

It formally defines the covert control problem for linear systems and characterizes the fundamental limits and trade-offs involved, especially for AR(1) systems.

Findings

Unstable systems cannot be covertly stabilized.

Stable systems can be covertly controlled, including parameter changes and resets.

A three-fold trade-off exists among information, control performance, and covertness.

Abstract

Consider the problem of covertly controlling a linear system. In this problem, Alice desires to control (stabilize or change the behavior of) a linear system, while keeping an observer, Willie, unable to decide if the system is indeed being controlled or not. We formally define the problem, under a model where Willie can only observe the system's output. Focusing on AR(1) systems, we show that when Willie observes the system's output through a clean channel, an inherently unstable linear system can not be covertly stabilized. However, an inherently stable linear system can be covertly controlled, in the sense of covertly changing its parameter or resetting its memory. Moreover, we give positive and negative results for two important controllers: a minimal-information controller, where Alice is allowed to use only $1$ bit per sample, and a maximal-information controller, where Alice is allowed to view the real-valued output. Unlike covert communication, where the trade-off is between rate and covertness, the results reveal an interesting \emph{three--fold} trade--off in covert control: the amount of information used by the controller, control performance and covertness.