## 4 Computer Simulation Examples
(**a**) Comparison of output y(t). (**b**) Disturbance. Figure 3: Simulation results for Theorem 1. (**a**) Comparison of randomly occurring uncertainties \(\bar{x}\). (**b**) Controller (2) compared with actuator attack and [19]. Figure 2: State trajectories of AUV for Theorem 1. **Example 2**. : _We will adopt the continuous-time system as the one considered in [20], and the system matrices \(A,B,C,A_{r}\), and \(C_{r}\) are defined in Example 1, and uncertain measurements \(M_{1}=M_{2}= \begin{bmatrix}0&0\\ 0&0\\ 0.00865&0.00692\end{bmatrix} \), \(N_{1}=N_{2}= \begin{bmatrix}1&0&0.4\\ 0&1&0\end{bmatrix} \), \(\Delta(t)= \begin{bmatrix}sin(2\pi T_{s})&0\\ 0&sin(2\pi T_{s})\end{bmatrix} \). The initial states are \(x(0)=[-1\) m \(0\) m \(-2^{\circ}]^{\mathsf{T}}\), and time-varying delay is defined as \(\tau(t)=0.98+0.95 sin(t)\)._  
\begin{table} \begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Settling Time(s)**} \\ \hline
**Method** & **Surge** & **Sway** & **Yaw** & **Figure** \\ \hline Theorem 1 & 14.23 & 9.84 & 19.68 & Figure 2 \\ [PERSON]’s Method & 20 & 19 & 22 & Figure 8 in [19] \\ \hline \hline \end{tabular}   
In Table 3, the lower bound, upper bound, and controller settling time are compared with reference [19]. \begin{table} \begin{tabular}{c c c c} \hline \hline
**Method** & **Lower Bound** & **Upper Bound** & \multicolumn{2}{c}{**Controller**} \\ \hline Theorem 1 (Figure 4b) & 0.01 & 1.96 & 8.2 \\ Controller (43) in [19] & 0.01 & 1.8 & 14.97 \\ Controller (60) in [19] & 0.01 & 1.8 & 14.97 \\ \hline \hline \end{tabular}   
* (Auxiliary state dynamics system I) For the simulation of Theorem 3, the AUV model in [19] uses the reference model matrices \(A_{r}\), \(C_{r}\), as defined in Table 1, and we use \(G=[0.8002\ \ 0.1480;\ \ 0.0100\ \ 0.4000;\ \ 0.4053\ -2.0530]\). The initial state of the reference model is \(x_{r}(0)=[0.5\) m; \(0\) m], which represents the surge and sway initial positions. Moreover, Theorem 3 obtained the control gain matrix \(K=[53.9297-64.1506-10.4171]\) by using the same parameters as Theorem 1. The surge and sway positions, as well as the yaw angle of the auxiliary system state trajectories, are shown in Figure 5.  
Figure 5: Simulation results for Theorem 3. The first three cases present the simulation results for Theorem 2, and the last case presents the simulation results for Theorem 4. 1. (AUV system II with time-varying delays) For the simulation of Theorem 2, we used the following parameter values: time \(T_{s}=0.01s\), \(H_{\infty}\) performance level minimum \(\gamma=0.005\), lower bound \(\tau_{m}=0.01\), upper bound \(\tau_{M}=2.1\), \(\delta=\tau_{M}-\tau_{m}\), the probability of an random event occurs \(\bar{\alpha}=0.7\), the possibility of an actuator attack \(\bar{\beta}=0.9\), the possibility of a sensor attack \(\bar{\gamma}=0.1\), the probability of an actuator attack \(\chi_{a}=0.2\), the probability of a sensor attack \(\bar{\chi}_{s}=0.2\), \(\mu=0.1\), \(\rho=2.8\), and \(\kappa=0.1\). We obtained the controller gain matrix \(K=[6.6535\,-6.1005\,-2.1372]\) for the AUV system (6). Thus, the state trajectories simulation result is presented in Figure 6.
2. (Actuator attacks and controller) When actuator attacks occur with \(\bar{\chi}_{a}=0.1,0.2,\)\(0.4,0.5,0.7,0.8\) in system (6), the simulation results of these different probabilities of actuator attacks are shown in Figure 7a. The performance of the controller \(u(t)\), compared with that of the actuator attack, is presented in Figure 7b. 3. (Sensor attacks and measured output) The simulation comparison of the measured output \(y(t)\) with a sensor attack is shown in Figure 8a. Additionally, the time-varying delay \(\tau(t)\) graph is presented in Figure 8b. 4. (Auxiliary state dynamics system II) Theorem 4 for the AUV auxiliary system yields the controller gain matrix \(K=[5.0621\,-5.2353\,-1.7397]\), using the same parameter values as in Theorem 2.