*2.4. FD Mechanism*

Define the following FD mechanism.

$$\begin{aligned} J(t) &= \left\{ \int\_0^t r\_F^T(s) r\_F(s) ds \right\}^{\frac{1}{2}} \\ J\_{th} &= \sup\_{w \in L\_2, f = 0} \left\{ \int\_0^{T\_d} r\_F^T(s) r\_F(s) ds \right\}^{\frac{1}{2}} \end{aligned} \tag{16}$$

where *J*(*t*) is the residual evaluation function, and *Jth* is the threshold, *Td* represents the limited length of evaluation time. The fault detection mechanism is as follows:

$$\begin{cases} \quad J(t) > J\_{th} \Rightarrow \text{with } faults \Rightarrow alarm \\\quad J(t) \le J\_{th} \Rightarrow no \; faults. \end{cases} \tag{17}$$

**Lemma 1.** *(Schur complement)* [41] *For the given matrix S* = - *S*11 *S*12 *S*21 *S*22 < 0*, where SRr*∗*r thethreesetsconditionsandholdand*


**Lemma 2.** *[42] For real matrices Z*, *X*,*Y with appropriate dimensions, in which the is symmetric, then*

$$Z + XK(t)Y + Y^T K(t)X^T < 0\tag{18}$$

*for all K<sup>T</sup>*(*t*)*K*(*t*) ≤ *I, there exists ε* > 0*, such that:*

$$Z + \varepsilon XX^T + \varepsilon^{-1}Y^TY < 0$$

**Lemma 3.** *[43] Given a symmetric and positive matrix R, inequality (18) holds:* \$

$$-\int\_{t-\overline{\pi}}^{t} \dot{\theta}^{T}(s) \tilde{\mathcal{W}} \dot{\theta}(s) ds \leq \frac{1}{\overline{\pi}} \begin{bmatrix} \theta(t) \\ \theta(t-\tau(t)) \\ \theta(t-\overline{\pi}) \end{bmatrix}^{T} \begin{bmatrix} -\check{\mathsf{R}} & \check{\mathsf{R}} & 0 \\ \* & -2\check{\mathsf{R}} & \check{\mathsf{R}} \\ \* & \* & -\check{\mathsf{R}} \end{bmatrix} \begin{bmatrix} \theta(t) \\ \theta(t-\tau(t)) \\ \theta(t-\overline{\pi}) \end{bmatrix} \tag{19}$$

Φ11

Φ79

Ξ12*ij*

> Ξ22*ij*

*ϕ*1 =

=

**Remark 3.** *It is worth noting that the fault residual system is built via IT2 T-S fuzzy model, considering the event-triggered communication mechanisms, disturbances and network time delays. In the existing work, there is less research on the IT2 T-S fuzzy network control system FD filtering with event triggering, which is one of the innovative points in this section.*
