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$\begin{array}{ll}\mathcal{J}= & \max {u \in S(\Delta)} \int{t_{k}}^{t_{k+N}} l(\tilde{x}(t), u(t)) d t \ \text { s.t. } & \dot{\tilde{x}}(t)=f_{nn}(\tilde{x}(t), u(t)) \ & u(t) \in U, \quad \forall t \in\left[t_{k}, t_{k+N}\right) \ & \tilde{x}\left(t_{k}\right)=\bar{x}\left(t_{k}\right) \ & V(\tilde{x}(t)) \leq \rho_{e}, \quad \forall t \in\left[t_{k}, t_{k+N}\right) \ & \text { if } \bar{x}\left(t_{k}\right) \in \Omega_{\rho_{e}} \ & \dot{V}\left(\bar{x}\left(t_{k}\right), u\right) \leq \dot{V}\left(\bar{x}\left(t_{k}\right), \phi\left(\bar{x}\left(t_{k}\right)\right)\right. \ & \text { if } \bar{x}\left(t_{k}\right) \in \Omega_{\rho} \backslash \Omega_{\rho_{e}}\end{array}$
where $l(\tilde{x}(t), u(t)) = \int_{t_{k}}^{t_{k+N}}(x(t)^TQx(t) + u(t)^TRu(t)$), $Q$ and $R$ are hyperparameters which are required to tune. $f_{nn}$ is the predictive model of model predictive control (MPC), it is a black-box model and it is trained with large historical data.
No attack, model test
Experiments about the First Principle model
To test that the LMPC framework works in these experiments
System track using First Principle
System track using First Principle by using approximate gradient
Experiments about Data-driven model
System track using LSTM by using real sensor value from plant
System track using LSTM by using Digital Twin value to calculate control quantity
Attack starts from $i_0$, and stop after $L_a$ time points. It wishes to make the state far away from system stable point but keep the state in the bound and the stable range. $rho$ is the level set of the Lyapunov function $V(x)$ that characterizes the stability region of the closed-loop system.
Min-max Attacks, range:50-60, No resilient control
Min-max Attacks, range:50-60, Resilient control to maintain the control process
Geometric Attacks
$\bar{x}\left(t_{i}\right)=x\left(t_{i}\right)+\beta \times(1+\alpha)^{i-i_{0}}, \forall i \in\left[i_{0}, i_{0}+L_{a}\right]$
Attacks start from $i_0$, and stop after $L_a$ time points.
It wishes to deteriorate the closed-loop system stability
slowly at the beginning, then geometrically increase its impacts as time progresses, with its maximum damage achieved at the end of
the attack duration.
This attack is difficult to detect at first few steps.
Geometric Attacks, range:50-60, No resilient control
Geometric Attacks, range:50-60, Using resilient control,
No roll-back strategy
Geometric Attacks, range:50-60, Resilient control,
Using roll-back strategy
Surge Attacks
Surge attacks act similarly like min-max attacks initially to maximize the
disruptive impact for a short period of time, then they are reduced to
a lower value. In our case, the duration of the initial surge in terms of
sampling times is selected as $L_s [2,5]$ to differentiate itself from a
min-max attack. Moreover, the initial surge period $L_s$ is chosen to be
in this range such that the potential time delays on the detection
alarm will not be longer than $L_s$ where the impact is most severe, and
in turn, may cause a missed alarm from the detector. After the initial
surge, the reduced constant value—at which the attack stays, is
chosen considering the impact of the initial surge and the total duration
of the attack such that the cumulative error between state measurements
and their steady state values will not exceed the threshold
defined in some statistic-based detection methods (e.g., CUSUM).The
formulation of a surge attack is presented below:
$\bar{x}\left(t_{i}\right)=\min \left{\underset{x \in R^{n x}}{\operatorname{argmax}}\left{V\left(x\left(t_{i}\right)\right) \leq \rho\right} \underset{x \in R^{n^{x}}}{\operatorname{argmax}}\left{x\left(t_{i}\right) \in \chi\right}\right}$, if $i_{0} \leq i \leq i_{0}+L_{s}$$\bar{x}\left(t_{i}\right)=\underset{x \in R^{n x}}{\arg \max }\left{\left|x\left(t_{i}\right)\right|, 0 \leq i \leq i_{0}\right}$, if $i_{0}+L_{s}<i \leq i_{0}+L_{a}$
where $i_0$ is the start time of the attack, $L_s$ is the duration of the initial
surge, and $L_a$ is the total duration of the attack in terms of sampling
periods. After the initial surge, the attack is reduced to a lower constant
value, which is obtained by examining the secure state measurements
prior to the occurrence of the surge attack, and taking the
value that is furthest away from the origin.
Surge Attacks, range:30-60, No resilient control
Surge Attacks, range:30-60, Resilient control,
Using roll back strategy.
