*2.2. FDIA*

The invaders attack the communication network, where invaders perfuse false data to messages or mensuration, which is shown in Figure 2. The FDI invader is able to perfuse the predetermined attack vector a by manipulating specific mensuration or messages. An invader is able to perfuse an attack vector to compromise the main mensuration which is shown in Equation (1); *e* represents the error vector; *z* defines the vector of measurements containing measurement readings from the sensor and ISO; *H* defines the Jacobian matrix with respect to *x*.

$$\mathbf{z}\_{\mathbf{a}} = \mathbf{z} + \mathbf{a} = H\mathbf{x} + \mathbf{a} + \mathbf{e} \tag{1}$$

where: a = (a1, a2, ..., am) <sup>T</sup> <sup>∩</sup> <sup>a</sup> <sup>=</sup> 0. According to [14], the attack vector <sup>a</sup> is able to be adjusted by Equation (2).

**Figure 2.** False data injection attack.

$$\mathbf{a} = \mathbf{H} \mathbf{c}\_{\prime} \tag{2}$$

where: c = (c1, c2, ..., cn) <sup>T</sup> <sup>∩</sup> <sup>c</sup> <sup>=</sup> 0 is an arbitrary vector. The estimated state vector *<sup>x</sup>*<sup>ˆ</sup> is altered in Equation (3) with the false data injection.

$$\pounds = \left(H^T \Lambda H\right)^{-1} H^T \Lambda z\_{\mathfrak{a}} = \mathfrak{X} + \mathfrak{c},\tag{3}$$

However, the residue *r* stays unaltered after attack and it still less than threshold τ;

$$\mathbf{r}\_{\mathbf{a}} = ||\mathbf{\hat{z}} - H\mathbf{\hat{x}}||\_2 \ = ||\mathbf{z} + \mathbf{a} - H(\mathbf{\hat{x}} + \mathbf{c})||\_2 \ = ||\mathbf{z} - H\mathbf{\hat{x}} + \mathbf{a} - H\mathbf{c}\rangle||\_2 \ = ||\mathbf{z} - H\mathbf{\hat{x}}||\_2 = r\_\star \tag{4}$$

So the FDI invaders are able to circumvent and be undiscoverable to the traditional dab data detection.
