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Initial-Value Privacy of Linear Dynamical Systems

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Document pages: 29 pages

Abstract: This paper studies initial-value privacy problems of linear dynamicalsystems. We consider a standard linear time-invariant system with randomprocess and measurement noises. For such a system, eavesdroppers having accessto system output trajectories may infer the system initial states, leading toinitial-value privacy risks. When a finite number of output trajectories areeavesdropped, we consider a requirement that any guess about the initial valuescan be plausibly denied. When an infinite number of output trajectories areeavesdropped, we consider a requirement that the initial values should not beuniquely recoverable. In view of these two privacy requirements, we definedifferential initial-value privacy and intrinsic initial-value privacy,respectively, for the system as metrics of privacy risks. First of all, weprove that the intrinsic initial-value privacy is equivalent tounobservability, while the differential initial-value privacy can be achievedfor a privacy budget depending on an extended observability matrix of thesystem and the covariance of the noises. Next, the inherent network nature ofthe considered linear system is explored, where each individual statecorresponds to a node and the state and output matrices induce interaction andsensing graphs, leading to a network system. Under this network systemperspective, we allow the initial states at some nodes to be public, andinvestigate the resulting intrinsic initial-value privacy of each individualnode. We establish necessary and sufficient conditions for such individual nodeinitial-value privacy, and also prove that the intrinsic initial-value privacyof individual nodes is generically determined by the network structure. Theseresults may be extended to linear systems with time-varying dynamics under thesame analysis framework.

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