Opacity Versus Security in Linear Dynamical Systems
Varkey M. John, Vaibhav Katewa · IEEE Transactions on Automatic Control · 2024
Opacity is a notion of privacy that is well studied in computer science and discrete-event systems. It describes an eavesdropper's inability to infer a system's “secret” states by observing the system's outputs. In this article, we consider opacity in linear dynamical systems and study four opacity classes—initial-state, current-state,$K$-step and infinite-step opacity, and show that they are fundamentally connected to two subspaces of the linear system—the weakly unobservable subspace and the weakly unconstructible subspace. With these subspaces, we derive conditions for the opacity of secret states under constrained and unconstrained state and input sets. Further, we establish that a tradeoff exists between opacity and security in the system. We show this in two ways: 1) We prove that an opaque system always permits undetectable attacks. 2) We show that expanding the set of opaque states in the system expands the set of undetectable attacks. Our work provides the necessary mathematical foundation for system designers to build opaque systems, while ensuring adequate security.