Input-to-State Stability of Cyber-Physical Systems Under Threshold-Bounded FDI Attacks

/subsection{Input-to-State Stability Analysis for FDIs under Threshold}In this subsection, we analyze the input-to-state stability (ISS) of the set $Z$ in cyber-physical systems under the condition that False Data Injection (FDI) attacks are restricted within a predefined threshold.Theorem 2: For a bounded uncertain cyber-physical system subjected to FDI attacks within a defined threshold, the set $Z$ maintains robust input-to-state stability, with the feasible domain $X_N$ serving as the attractive domain.Proof: To validate Theorem 2, we introduce assumptions concerning the stage cost and terminal cost of the value function //cite{Sunqi_phd}, a common practice in Model Predictive Control (MPC) studies for guaranteeing input-to-state stability and robustness.Assumption 1:/begin{equation}/label{Assumption for ISpS}/mathcal{V}f(f(x,/kappa^(x)+/mathcal{a},w)) - /mathcal{V}_f(x) /leq -/mathcal{l}(x,/kappa^(x))+ /alpha/|/mathcal{A}/|/end{equation}In this equation, $/| /mathcal{A}/| /triangleq /text{sup}{/mathcal{a} /in /mathcal{A}}/|/mathcal{a}/|$ represents the supremum of the within-threshold FDI attacks, effectively signifying the threshold of FDI attacks discussed in this subsection.**Remark:**In this constrained attack scenario, $/|/mathcal{A}/| = A_{th}$. This assumption stems from the decrement property of the terminal cost. When the FDI attack is absent ($/|/mathcal{A}/|=0$), it aligns with the conventional axiom in MPC literature, which asserts that the terminal cost function strictly decreases by the amount of the single-step stage cost. However, under an attack, this property may be weakened by a factor related to the supremum of $/mathcal{a}$. Therefore, to maintain the assumption's validity, the attack's energy level (represented by $/mathcal{a}$) needs to be limited, aligning with the scenario examined in this subsection.| Modification | Rationale ||---|---|| Changed 'Base on this scenario' to 'Based on this scenario' | Corrected grammatical error. || Changed 'false-data-injection attack within threshold' to 'FDI attacks within a defined threshold' | Improved clarity and consistency by using the abbreviation FDI and specifying a 'defined threshold.' || Changed '$Z$ remains robustly input-to-state stable' to 'the set $Z$ maintains robust input-to-state stability' | Enhanced sentence flow and clarity. || Changed 'attractive domain is the feasible domain $X_N$' to 'feasible domain $X_N$ serving as the attractive domain' | Improved sentence structure and readability. || Changed 'To make the theorem 2 valid' to 'To validate Theorem 2' | More concise and aligned with academic style. || Changed 'we need to make some assumptions on the stage cost and terminal cost of the value function' to 'we introduce assumptions concerning the stage cost and terminal cost of the value function' | More formal and academic writing style. || Changed 'This is common sense in most MPC studies' to 'This is a common practice in most Model Predictive Control (MPC) studies' | Provides more context and avoids using informal language like 'common sense.' || Changed 'In other words, $/| /mathcal{A}/|$ can represent the threshold of FDI attacks discussed in this subsection.' to 'In this equation, $/| /mathcal{A}/| //triangleq //text{sup}_{//mathcal{a} //in //mathcal{A}}//|//mathcal{a}//|$ represents the supremum of the within-threshold FDI attacks, effectively signifying the threshold of FDI attacks discussed in this subsection.' | Provides a more precise and formal definition of $//| //mathcal{A}//|$. || Reworded the Remark section for better clarity and flow. | The original wording was convoluted and difficult to follow. The rewritten version is more concise and easier to understand. || Changed all double quotes to single quotes throughout the text. | Following the user's instruction. |