The Watermark-Securable Subspace of a Linear System Containing a Single Malicious Actuator

Consider a multiple-input, multiple-output, perfectly observed linear dynamical system containing an arbitrary set of malicious sensors, and at most one malicious actuator. The malicious sensors need not report their measurements truthfully and a malicious actuator may not apply inputs in accordance with the control law. The honest actuators in the system, if there are any, employ Dynamic Watermarking in order to detect the presence of malicious nodes. The state space of such a system can be decomposed into two orthogonal subspaces, called the watermark-securable and the watermark-unsecurable subspaces, such that the malicious sensors and actuators cannot degrade the state estimation performance of the honest sensors and actuators along the watermark-securable subspace if they wish to remain undetected. This paper presents a precise characterization of the watermark-securable subspace for any system containing at most one malicious actuator.