Resilient Tube-Based Model Predictive Control for Cyber-Physical Systems Under False Data Injection Attacks

Resilient Tube-Based Model Predictive Control for Cyber-Physical Systems Under False Data Injection AttacksNext, we introduce the following lemma:Lemma 1: If Assumption 3 holds, then the following recursive set dependency holds:$$X_N^/lambda /subseteq X_N^{/lambda-1} /subseteq /cdots /subseteq X_N^0 = X_N.$$Proof: Referring to the definition of $/mathcal{U}_N^/lambda$ in equation (/ref{可行控制集的定义}), we observe that the only difference between $/mathcal{U}_N^/lambda$ and $/mathcal{U}_N^{/lambda-1}$ is the $/lambda$th term. Note that if $u(/lambda) /in /mathcal{u}$, then it must also belong to $/mathcal{U}$. Hence, we have $/mathcal{U}_N^/lambda /subseteq /mathcal{U}_N^{/lambda-1}/subseteq /cdots /subseteq /mathcal{U}_N^0 = /mathcal{U}_N$. From equation (/ref{definition of X_N^LAM}), we can conclude that the recursive set dependency in equation (/ref{X_N recursive set dependency}) is valid.Hereafter, we investigate the input-to-state stability of the resilient MPC scheme under over-threshold FDI attacks. The main difference in the input-to-state stability analysis compared to the previous subsection is the contraction of the feasible region.Theorem 3: For a bounded uncertain cyber-physical system exposed to false-data-injection attacks modeled in equation (/ref{Mathematical model}), assuming all the aforementioned assumptions hold, the system is input-to-state stable under the resilient tube MPC scheme, and the region of attraction is $X_N^/lambda$.Proof: Firstly, we must ensure the feasibility of the resilient tube MPC scheme. As proven in Theorem 2, feasibility is achieved by letting the initial state $x_0$ lie in $X_N$ when the attack is within the threshold. Similarly, we can extend this conclusion to $x_0 /in X_N^/lambda$ when over-threshold attacks may occur.Based on the conclusions from the previous subsection, we can directly present the following useful inequality:$$/Upsilon^(x) - /Upsilon^(x^+) /geq /beta/|x_0^(x)/|^2 - /alpha/|/mathcal{A} /|.$$This inequality illustrates that despite the occurrence of over-threshold FDI attacks in the S-C channel, the optimal value function maintains its monotonic decreasing property, with a minimum decrement of $/beta/|x_0^(x)/|^2 - /alpha/|/mathcal{A} /|$.Next, we prove the exponential stability of $Z$ by contradiction. Assuming that, for an initial state lying in $X_N^/lambda$, it does not enter $/mathcal{X}_f /oplus Z$ in finite instants, we can find a $/bar{k} /in (0,/infty)$ such that $/Upsilon^(x_0) < /bar{k}(/beta/|x_0^(x)/|^2 - /alpha/|/mathcal{A}/|)$. Then, when $k>/bar{k}$, the optimal function decreases by more than $k(/beta/|x_0^*(x)/|^2 - /alpha/|/mathcal{A}/|)$ and becomes less than 0, contradicting its non-negativity. Therefore, the subsequent proof of the input-to-state stability of the set $Z$ follows a similar approach to that in the previous subsection.Algorithm 1 describes the structure of the proposed resilient tube MPC scheme.