## 3 Zero-Dynamics Attack on Wind Turbine
The attack signal \(a_{k}\) of ZDA is generated from a dynamic system given by  
\[a_{k}=-\frac{1}{g_{\text{d}}}\psi_{\text{d}}^{\top}z_{k},\qquad z_{k+1}=S_{ \text{d}}z_{k}, \tag{11}\]  
where \(z_{k}\in\mathbb{R}^{2}\) is the state of attack generator. It is noted that the attack is constructed using the system parameters such as \(S_{\text{d}}\), \(\psi_{\text{d}}\), and \(g_{\text{d}}\). Now, we investigate the behavior of the closed-loop system under ZDA. Firstly, the controller \(C_{\text{Pl}}(z)\) represented in the state space  
\[ \begin{split} c_{k+1}&=c_{k}+e_{k},\ e_{k}=-\hat{y }_{k}\\ \hat{u}_{k}&=k_{1}c_{k}+k_{P}e_{k},\end{split} \]  
has been designed so that the closed-loop system under no ZDA is stable, as demonstrated in Section 2.3, namely, the matrix \(\hat{A}_{\text{CL}}\) shown below is Schur. \[\hat{A}_{\text{CL}}= \begin{bmatrix}S_{\text{d}}&P_{\text{d}}&0\\ \psi_{\text{d}}^{\top}&\phi_{\text{d}}-g_{\text{d}}k_{P}&g_{\text{d}}k_{I}\\ 0&-1&1\end{bmatrix} . \tag{12}\]  
When the attack \(a_{k}\) generated by (11) is injected into system (10) (it is equivalent to system (9)), we can derive  
Figure 4: Wind turbine control system under the zero–dynamics attack at the actuator side. ZOH—zero-order hold; SS—simple sampler; PI—Proportional–Integral. from which one has  
\[ \begin{bmatrix}\eta_{k+1}-z_{k+1}\\ \xi_{k+1}\\ c_{k+1}\end{bmatrix} =\hat{A}_{\text{CL}} \begin{bmatrix}\eta_{k}-z_{k}\\ \xi_{k}\\ c_{k}\end{bmatrix} .\]  
Since \(\hat{A}_{\text{CL}}\) given in (12) is Schur, there exist \(\kappa>0\) and \(\lambda\) with \(|\lambda|<1\) such that  
\[\left\| \begin{bmatrix}\eta_{k}-z_{k}\\ \xi_{k}\\ c_{k}\end{bmatrix} \right\|\leq\kappa\lambda^{k}\left\| \begin{bmatrix}\eta_{0}- z_{0}\\ \xi_{0}\\ c_{0}\end{bmatrix} \right\|. \tag{13}\]  
This relation implies that under a ZDA, the internal state \(\eta_{k}\) converges to the state of ZDA, while other states \(\xi_{k}\) and \(c_{k}\) converge to zero. It is remarkable that the relation (13) holds regardless of the stability of the zero dynamics. Hence, if \(S_{d}\) is unstable, then the internal state \(\eta_{k}\) diverges whenever \(z_{0}\) depends on an unstable eigenvector of \(S_{d}\), while this behavior cannot be observed by monitoring the signal \((\xi_{k},c_{k})\). Unfortunately, the wind turbine system (8) has an unstable zero when the sample time \(T_{s}\) belongs to some region. Figure 4(a) shows the locus of the zeros with respect to the sampling time from \(T_{s}=0.001\) s to \(T_{s}=0.1\) s. When \(T_{s}=0.001\) s, two zeros are located near 1, as depicted by the crosses. When \(T_{s}\) increases and then becomes 0.08 s, \(z_{2}\) is located outside the unit circle. The blue and red circles on the real axis in Figure 4(a) indicate the location of two zeros when \(T_{s}=0.1\) s. As shown in Figure 4(b), if \(T_{s}\in[0.08,0.13]\) s, at least one zero is located outside the unit circle. Suppose that \(T_{s}\) belongs to the region where the wind turbine has an unstable zero. When the attack (11) is injected, the rotor angular velocity \(\omega_{r}\) and the (spring) torsional torque \(Q_{s}\) will diverge while the generator angular velocity \(\omega_{g}\) and the controller state \(c_{k}\) converge to zero. This can be interpreted so that the attack intentionally moves two components (\(\omega_{r}\) and \(Q_{s}\)) of the operating point but leaves \(\omega_{g}\) at the normal operating point and deceives the controller as if all components remain unchanged. Under the situation described in Figure 4, the effect of ZDA on the wind turbine system is presented. Since the system is of nonminimum phase when \(T_{s}=0.1\) s, the attack signal diverges for an appropriately chosen initial condition as shown in Figure 6, and it is expected that the internal states also diverge. Suppose that the hacker injected \(a_{k}\) through the communication network at \(t=0\).