From: On the use of watermark-based schemes to detect cyber-physical attacks
A | : | State matrix. |
B | : | Input matrix. |
C | : | Output matrix. |
W(z) | : | Process noise. |
Q | : | Process noise variance. |
V(z) | : | Output noise. |
R | : | Output noise variance. |
X or X(z) | : | Vector of state variables. |
U or U(z) | : | Control input vector. |
Y or Y(z) | : | Vector of the sensors measurements. |
U ∗ or U ∗(z) | : | Optimal control input vector. |
Δ U or Δ U(z) | : | Watermark. |
Δ U(z)(i) | : | Multi-watermark. |
Y ΔU(z) | : | Output due to the watermark. |
Y ′ or Y ′(z) | : | Measurements injected by the adversary. |
U ′ or U ′(z) | : | Control inputs injected by the adversary. |
\(\hat {X}\) or \(\hat {X}(z)\) | : | Vector of estimated state variables. |
\(\hat {X}^{(-)}\) or \(\hat {X}^{(-)}(z)\) | : | Vector of estimated state variables before applying the rectification. |
\(\hat {X}^{(+)}\) or \(\hat {X}^{(+)}(z)\) | : | Vector of estimated state variables after applying the rectification. |
K f | : | Kalman gain. |
P (−) | : | A priori error covariance. |
P (+) | : | A posteriori error covariance. |
L | : | Feedback grain. |
S | : | Riccati equation solution. |
J | : | Quadratic cost. |
Δ J s | : | Increment of quadratic cost due to the single-watermark. |
Δ J m | : | Increment of quadratic cost due to the multi-watermark. |
E[Δ u] | : | Offset of Δ u. |
Var[Δ u] | : | Variance of Δ u. |
\(\mathcal {W}\) | : | LMS weight matrix. |
DR | : | Detection ratio. |
g t | : | Alarm signal. |
\(\hat {T}\) | : | Samples eavesdropped by the adversary. |
\(\mathcal {P}\) | : | Co-variance of the i.i.d. Gaussian signal. |
r(z) | : | Residue. |
γ | : | Detection threshold. |
Γ and Ω | : | Ponderation matrices. |
(n 0... n m ) | : | Weight of the polinomial N(z). |
(d 0... d n ) | : | Weight of the polinomial D(z). |
FN | : | False negatives. |
FP | : | False positives. |
AD | : | Samples detected. |
SA | : | Samples under attack. |