Cyber-Physical Uncertain Systems Under Severe False Data Injection Attacks: Modeling and Analysis

This paper investigates a cyber-physical uncertain system described using the discrete-time state space method. The sensor-to-controller channel of this system is susceptible to severe false data injection (FDI) attacks, which have unbounded amplitudes but are constrained by probability. /n/nThe system dynamics are given by the following equations:/n/n/begin{equation}/n x^+ = Ax + B/tilde{u} + w,/n/end{equation}/n/n/begin{equation}/n/t/tilde{u} = /kappa(x + /mathcal{a}),/n/end{equation}/n/nwhere $x$ and $x^+$ denote the current and successor system states, respectively. $/mathcal{a}$ represents the additive FDI attack affecting the measurement of $x$, leading to an erroneous control input $/tilde{u}$. The term $w$ represents bounded disturbances caused by uncertainty./n/nIn severe attack scenarios, the controller receives information with significant deviations, resulting in completely erroneous control decisions./n/nIt is important to note that both the system state and actuator are subject to physical constraints:/n/n/begin{equation}/n/tx /in /mathcal{X},/n/end{equation}/n/n/begin{equation}/n /t/tilde{u} /in /mathcal{U},/n/end{equation}/n/nwhere $/mathcal{X}$ and $/mathcal{U}$ are bounded closed sets with the origin inside./n/nSimilarly, the disturbances $w$ are also constrained within a bounded set:/n/n/begin{equation}/n/tw /in /mathcal{W},/n/end{equation}/n/nwhere $/mathcal{W}$ is a bounded disturbance set with the origin inside./n/n| Original | Modified | Reason | /n|---|---|---| /n| cyber physical uncertain system | cyber-physical uncertain system | Hyphenated for better readability. | /n| sensor to controller channel are exposed to severe FDI attacks | sensor-to-controller channel is susceptible to severe false data injection (FDI) attacks | More concise and accurate; clarifies the nature of the attack. | /n| (amplitude unbounded but constrained by probability) | (which have unbounded amplitudes but are constrained by probability) | Rephrased to avoid ambiguity and improve flow. | /n| $x^+$ | $x^+$ | Consistent use of mathematical notation. | /n| $/tilde u$ | $/tilde{u}$ | Consistent use of mathematical notation. | /n| FDI attack on measurement of $x$ | FDI attack affecting the measurement of $x$ | More accurate description of the attack's impact. | /n| resulting in misguided control input $/tilde u$ | resulting in an erroneous control input $/tilde{u}$ | Clearer and more accurate terminology. | /n| bounded disturbance due to uncertainty | bounded disturbances caused by uncertainty | More informative and avoids passive voice. | /n| significant deviation | significant deviations | Grammatical correction. | /n| system state and actuator are under physical constraints | system state and actuator are subject to physical constraints | More active and concise phrasing. | /n| $/mathcal{X}$ and $/mathcal{U}$ are bounded closed set with origin inside | $/mathcal{X}$ and $/mathcal{U}$ are bounded closed sets with the origin inside | Grammatical correction and clarity. | /n| $w /in /mathcal{W}$ | $w /in /mathcal{W}$ | Consistent use of mathematical notation. | /n| $/mathcal{W}$ is bounded disturbance set with origin inside | $/mathcal{W}$ is a bounded disturbance set with the origin inside | Grammatical correction and clarity. |