**4. Simulation**

In this section, we provide several examples to illustrate the usefulness of the designed IT2 fuzzy FD approach and to compare it with the existing results in [44,45] to show the advantages of our method.

Two rules have been considered in the following IT2 fuzzy system (system parameters are borrowed from [46])

$$\begin{cases} \dot{\mathbf{x}}(t) = \sum\_{i=1}^{2} \widetilde{\rho}\_{i}(\mathbf{x}(t)) [A\_{i}\mathbf{x}(t) + B\_{i}\omega(t) + B\_{f}\mathbf{i}(t)] \\ \quad y(t) = \sum\_{i=1}^{2} \widetilde{\rho}\_{i}(\mathbf{x}(t)) [\mathbf{C}\_{i}\mathbf{x}(t) + D\_{i}\omega(t)] \end{cases} \tag{39}$$

with *A*1 = - −1 0.2 −0.9 0.15 , *A*2 = - −0.4 0.2 −0.8 −1.10 , *B*1 = - 0.1 0.2 , *B*1 = - 0.4 0.9 , *Bf* 1 = - −0.1 0.01 , *B*1 = - −0.1 0.01 , *C*1 = 0.1 0.1 , *C*2 = 0.1 0.2 , *D*1 = *D*2 = 0.01. The membership functions of the plant and fault detection filter are depicted in Table 2. The nonlinear functions are chosen as, i.e., *<sup>ρ</sup>i*(*<sup>x</sup>*1(*t*) = sin(*x*21(*t*)), *<sup>ρ</sup>i*(*<sup>x</sup>*1(*t*) = 1 − sin(*x*21(*t*)), *i* = 1, 2, and *<sup>φ</sup>j*(*x*(*t*)) = *φj*(*x*(*t*)) = 0.5 for *j* = 1, 2.

**Table 2.** Membership functions for plant and filter.


In order to derive the gain matrices of the FD filter in (7), we assume the parameter sets (*<sup>τ</sup>m*, *τM*,*ε*, *l*, -2) = (0.01, 0.1, 0.5, 0.7, 0.5). Then by solving the conditions in Theorem 2, we can obtain

$$
\hat{A}\_1 = \begin{bmatrix} -1.6738 & 0.1545 \\ -0.5992 & -0.3587 \end{bmatrix}, \hat{A}\_2 = \begin{bmatrix} -0.6885 & -0.1969 \\ 0.8140 & -2.3963 \end{bmatrix},
$$

$$
\mathcal{B}\_1 = \begin{bmatrix} -2.8318 \times 10^{-12} \\ 9.3801 \times 10^{-13} \end{bmatrix}, \mathcal{B}\_2 = \begin{bmatrix} -1.7555 \times 10^{-12} \\ -9.6037 \times 10^{-13} \end{bmatrix},
$$

$$
\begin{aligned}
\mathring{C}\_1 &= \begin{bmatrix} 0.1087 & -0.0306 \end{bmatrix}, \mathring{C}\_2 = \begin{bmatrix} 0.0980 & -0.0180 \end{bmatrix}, \\
\mathring{D}\_1 &= 1.2609 \times 10^{-12}, \mathring{D}\_2 = 1.6357 \times 10^{-12}, \Lambda = 5.3637 \times 10^{-12}.
\end{aligned}
$$

Besides, the *H*∞ performance is calculated as *γ* = 2.4227. According to the FD mechanism, we set the fault signal as

$$f(t) = \begin{cases} \ 2,20 < t < 30\\ \ 0,0others \end{cases} \tag{40}$$

and the external disturbance *ω*(*t*) is stochastic noise that belongs to standard normal distribution. Let the initial states be *x*0 = *x*ˆ0 = 0 0 *T*. Then, we can derive Figures 2–4. Specifically, Figure 2 depicts the actual transmission instants and intervals under the event-triggered scheme. In the simulation time (50 s) and sampling period (0.1 s), only 20.0% of sampled data are transmitted over the wireless network. Clearly, it saves many communication resources. Figures 3 and 4, respectively, show the trajectories of the error *re*(*t*) without/with fault.

**Figure 2.** Transmission instants and intervals.

**Figure 3.** The trajectories of *re*(*t*) without fault.

**Figure 4.** The trajectories of *re*(*t*) with fault.

Moreover, the threshold *Jth* can be calculated without fault, i.e., *Jth* = 4.0711 × 10−13. Then, it is not hard to obtain that *J*(*t*) = 12 24.9 0 *rT F* (*s*)*rF*(*s*)*ds* 31 2 = 4.0826 × 10−<sup>13</sup> > *Jth*. This means that the fault can be detected after 4.9 s. Further, Figure 5 illustrates the fault detection results demonstrating that the proposed FD approach is effective.

**Figure 5.** The trajectories of evaluation function with/without fault.

Following the above steps, considering the different types of faults, we performed three sets of simulations. Then, we produced Table 3 and derived Figures 6 and 7.


**Table 3.** Verification for different types of faults.

**Figure 6.** Transmission instants and intervals for experiment (**<sup>a</sup>**–**<sup>c</sup>**).

**Figure 7.** The trajectories of evaluation function with/without fault for experiment (**<sup>a</sup>**–**<sup>c</sup>**).

Experiment a uses the same system parameters and fault types as those the in the literature [44]. During the simulation time (100 s) with the sampling period (0.1 s), the cycle triggering time is 1000, and the events triggering time is 239. Simultaneously, the results show that the proposed method obtains a faster detection time. In experiment b, the step signal is used to represent the sudden fault. The final time is 10 s, and the sampling period is 0.1 s. With the same experimental conditions, the proposed method has fewer triggers and a faster detection speed. It can be seen that the structure of the event triggering mechanism we used is simpler. More recently, in order to discuss the effectiveness of the method for time-varying faults. Experiment c was performed by considering an inverted pendulum on a cart. It readjusts that the experimental time is 30 s and sampling period is 0.01 s, and only 26.3% of sampled data is transmitted over the wireless network. In Figure 7, one can see that the fault can be detected after 0.6 s.
