On the Watermark-Securable Subspace of a Linear Stochastic System

We consider a multiple-input, multiple-output perfectly observed linear dynamical system with an arbitrary set of malicious sensors and actuators. The honest actuators watermark their control inputs and perform certain tests to detect the presence of malicious sensors. 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 no matter what attack strategy the malicious sensors and actuators employ, they 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 characterizes a certain subset of the watermark-securable subspace.