Notions of Centralized and Decentralized Opacity in Linear Systems

We formulate notions of opacity for cyberphysical systems modeled as discrete-time linear time-invariant systems. A set of secret states is k-ISO with respect to a set of nonsecret states if, starting from these sets at time 0, the outputs at time k are indistinguishable to an adversarial observer....

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Bibliographic Details
Published in:IEEE transactions on automatic control 2020-04, Vol.65 (4), p.1442-1455
Main Authors: Ramasubramanian, Bhaskar, Cleaveland, Rance, Marcus, Steven I.
Format: Article
Language:English
Subjects:
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Automata
Computers
Controllability
Cyber-physical systems
Discrete time systems
Discrete-event systems
Graph theory
Graphical representations
Linear systems
Nonlinear systems
nonsecret states
Observers
Opacity
output controllability
reachable sets
secret states
Time invariant systems
Trajectory
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Summary:We formulate notions of opacity for cyberphysical systems modeled as discrete-time linear time-invariant systems. A set of secret states is k-ISO with respect to a set of nonsecret states if, starting from these sets at time 0, the outputs at time k are indistinguishable to an adversarial observer. Necessary and sufficient conditions to ensure that a secret specification is k-ISO are established in terms of sets of reachable states. We also show how to adapt techniques for computing underapproximations and overapproximations of the set of reachable states of dynamical systems in order to soundly approximate k-ISO. Furthermore, we provide a condition for output controllability, if k-ISO holds, and show that the converse holds under an additional assumption. We extend the theory of opacity for single-adversary systems to the case of multiple adversaries and develop several notions of decentralized opacity. We study the following scenarios: first, the presence or lack of a centralized coordinator, and, second, the presence or absence of collusion among adversaries. In the case of colluding adversaries, we derive a condition for nonopacity that depends on the structure of the directed graph representing the communication between adversaries. Finally, we relax the condition that the outputs be indistinguishable and define a notion of \epsilon-opacity, and also provide an extension to the case of nonlinear systems.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2019.2920837