Robust Asymptotic Stability Analysis of Uncertain Systems under FDI Attacks

Since our proposed scheme is a generalization of robust methods in FDI attack scenarios, it is necessary to review some relevant results from previous research [1] [2].Theorem 1: For bounded uncertain systems, the set Z exhibits robust asymptotic stability, with the feasible domain X_N serving as the attractive region.Proof: We will utilize the preliminary results of stability analysis outlined below./begin{equation}/label{pre1} /Upsilon^(x) /geq /sigma_1||x_0^(x)||^2, /quad /forall x /in X_N./end{equation}Remark: This inequality implies that the optimal value function Υ(x) has a lower bound associated with the quadratic form of the current optimal initial state ||x₀(x)||², as long as the state lies within the feasible region X_N./begin{equation}/label{pre2} /begin{aligned} &/Upsilon^(x^+)-/Upsilon^(x) /leq -/sigma_1||x_0^(x)||^2, // &/forall x /in X_N, /forall x^+ /in (Ax + B/kappa^(x)) /oplus W . /end{aligned}/end{equation}Remark: As the system state progresses, the optimal value function strictly decreases, with a magnitude associated with ||x₀(x)||². This inequality holds when the system state lies within the feasible region X_N and the successor state is determined by the optimal control rule, despite the presence of disturbances./begin{equation}/label{pre3} /Upsilon^(x) /leq /sigma_2 ||x_0^(x)||^2, /quad /forall x /in X_f /oplus Z./end{equation}Remark: This inequality indicates that for system states within the extended terminal region, the optimal value function Υ(x) has an upper bound related to ||x₀(x)||².Henceforth, we can establish the exponential convergence of the optimal value function Υ(x), as well as the absolute value of the optimal initial state ||x₀(x)||.Firstly, it is worth noting that the comparison constant σ₂ is greater than σ₁, enabling us to define the base of exponential convergence as ϱ ≜ 1−σ₁ / σ₂ ∈ (0,1).Secondly, we denote the solution of the iterative equation of the system x⁺= Ax+Bκ(x)+w as the system trajectory x(i) for all i ∈ 𝒩. Since our aim is to prove the robust asymptotic stability of the set Z, which represents the augmented origin of the attacked system, we are solely concerned with system trajectories where the initial state lies in the terminal region, namely X_f ⊕ Z. Therefore, we define the state set as follows:/begin{equation}/label{small region} /Omega_a /triangleq x | /{ /Upsilon^(x) /leq a, /quad /forall a > 0 /}./end{equation}Based on the definition of Z, it follows that when a=0, the set Ω_a= Ω₀=Z. By gradually increasing a, the set Ω_a expands accordingly. We can always find an a such that Ω_a ⊂ Z ⊕ X_f, where inequality (3) holds. It is evident that within this region, x ∈ X_N always holds true because X_f ⊕ Z ⊂ X_N, satisfying the conditions of (1) and (2).References[1] Mayne, D. Q. (2005). Model predictive control: Recent developments and future promise. Automatica, 41(2), 297-308.[2] Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789-814.