Analyzing System Stability under Consecutive Over-Threshold FDI Attacks

In the previous subsection, we established the stability of the system when it experiences FDI attacks below the threshold. In this case, we can identify a feasible region $X_N$ for the initial states, ensuring that there exists a control sequence 'u' that satisfies the control input constraint. To distinguish it from the newly proposed feasible region in this section, we refer to this region as $X_N^0$. For all states $x$ in $X_N^0$, any sequence 'u' can form an admissible control input set $//mathcal{U}_N^0$. To clarify:/n/n/[X_N^0 //triangleq /{x| U_N^0 //neq //emptyset /},/]/n/[//mathcal{U}_N^0 = /{ 'u' | u(i) //in //mathcal{U},//quad x^(i,'u') //in //mathcal{X} ,//quad x^(N,'u') //in //mathcal{X}f /},/]/n/nwhere $i //in //mathcal{N}{[0,N-1]}$./n/nIt is important to note that $X_N^0$ represents the feasible region when all FDI attacks are under the preset threshold and is not applicable in cases where random over-threshold attacks occur. Hence, we introduce a new region denoted as $X_N^//lambda$ to represent the admissible initial state set in the presence of at most $//lambda$ consecutive over-threshold FDI attacks./n/nFrom the proposed algorithm, we can identify a control sequence 'u' = { u(1), u(2),//cdots u(//lambda),u^(//lambda+1),//cdots u^(N) }. This sequence contains the first $//lambda$ feasible control inputs solved at historical instants and $N-//lambda$ optimal control inputs solved at the present. Both the feasible and optimal control inputs satisfy the constraints./n/nBased on this control sequence, we define a new control input set as follows:/n/n/[/n//begin{aligned}/n//mathcal{U}_N^//lambda = /{ 'u' | &u(i) //in //mathcal{u},//forall i //in [1,//lambda] ///n&u(i) //in //mathcal{U},//forall i //in [//lambda+1,N] ///n&x^(i,'u') //in //mathcal{X},//forall i //in [0,N-1]///n&x^(N,'u') //in //mathcal{X}_f /},/n//end{aligned}/n/]/n/nwhere $//mathcal{u}$ represents the control input sequence stored in the control buffer. This means that even in the worst case scenario of consecutive $//lambda$ over-threshold attacks, we can still find a feasible control input sequence to guide the system's state from the initial region to the terminal set./n/nThe corresponding feasible state set is defined as:/n/n/[X_N^//lambda //triangleq /{x| //mathcal{U}_N^//lambda //neq //emptyset/}./]/n/nAssumption 3:/nConsidering the FDI attack scenario modeled in equation (//ref{Mathematical model}), the initial feasible region $X_N^//lambda$ is not empty for each calculated $//lambda$./n/nRemark:/nThis assumption ensures that the control problem is feasible even in the worst-case scenario. The reason why this situation is considered the worst-case is that the maximum number of consecutive occurrences ($//lambda$) happens at the beginning of the horizon $N$, when the system is farthest from the equilibrium steady state.