Table 1 Notations used in this paper

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.