Opacity and its trade-offs with security in linear dynamical systems

includes bibliographical references and index

MTech (Res); 2023; Electrical Communication Engineering

Opacity is a notion of privacy that is well-studied in computer science and discrete-event
systems. In our work, we extend the opacity notion to linear dynamical systems. Opacity
describes an eavesdropper’s inability to estimate a system’s “secret” states by observing
the system’s outputs. We consider four opacity classes - initial-state, current-state, Kstep, and infinite-step opacity - and show that they are fundamentally connected with
two subspaces of the linear system - the weakly unobservable subspace and the weakly
unconstructible subspace. Further, we establish that a trade-off exists between opacity
and security in the system. We show this in two ways – (i) we prove that an opaque
system always permits undetectable attacks, (ii) we show that expanding the set of
opaque states in the system always expands the set of undetectable attacks. We also
propose optimization algorithms to minimally perturb a non-opaque system to make it
opaque. We demonstrate our results on a smart grid system. Our work is the first to
study opacity in such generality for linear dynamical systems, and provides necessary
mathematical foundation for system designers to develop and build opaque systems,
while ensuring adequate security