On the Operational Significance of the Securable Subspace for Partially Observed Linear Stochastic Systems

We address the problem of security for general partially observed linear stochastic systems, where some of the sensors and actuators may be malicious. We consider multiple-input-multiple-output linear stochastic systems that are under attack, where an arbitrary subset of its sensors and actuators are “malicious.” A malicious sensor need not report its measurements truthfully, and a malicious actuator need not apply inputs in accordance with the prescribed control policy. For any such system, we show that there exists a decomposition of the state space into two orthogonal subspaces, called the securable and the unsecurable subspaces, and design a test that can be used by the honest sensors and actuators, such that if the malicious activity is to remain undetected by this test, then the covariance of the projection of the state estimation error of the honest nodes on the securable subspace remains at its designed value regardless of what attack strategy the malicious sensors and actuators choose to employ. This test therefore guarantees that the malicious nodes can degrade the state estimation performance only along the unsecurable subspace of the linear dynamical system.