*2.2. Event-Triggered FD Filter*

Next, an event-triggering mechanism is introduced within the system, which is between the considered system and FD Filter as shown in Figure 1.

**Figure 1.** Framework of IT2 T-S NNSs with event-triggered scheme.

The current sampled signal must reach the trigger threshold of the event monitoring terminal before it can be transmitted to the next node. Similar to [34], we can define the event-triggering mechanism as:

$$
\varepsilon\_k^T(t) \Lambda v\_k(t) \supset \varepsilon y^T(i\_k h) \Lambda y(i\_k h) \tag{5}
$$

where *ε* ∈ [0, <sup>1</sup>), *ek*(*t*) is the threshold error, which is the key factor that determines whether the event trigger mechanism occurs, and is obtained by subtracting current sampled data *y*(*tkh*) from the latest transmitted data *y*(*ikh*). Λ denotes the positive triggering parameters.

ZOH provides information about the last transmitted data continuously, the input signal received by the filter can be described as

$$\overline{y}(t\_k h) = y(t\_k h), \ t \in [t\_k h + \tau\_{t\_k}, t\_{k+1} h + \tau\_{t\_{k+1}}) \tag{6}$$

The system can be transformed into a new time lag system, which can be directly analyzed with time lag system theory. Without loss of generality, the holding region of ZOH is expressed as:

$$
\Omega = \left[ t\_k h + \pi\_{l\_k} t\_{k+1} h + \pi\_{l\_{k+1}} \right) = \overset{\text{m}}{\underset{\bullet}{\text{d}}} \Omega\_l \tag{7}
$$

$$\begin{cases} \Omega\_0 = \left[ t\_k h + \pi\_{t\_k}, t\_k h + h + \overline{\pi} \right] \\ \Omega\_i = \left[ t\_k h + ih + \overline{\pi}, t\_k h + (i+1)h + \overline{\pi} \right), \ i = 1, 2, \dots, m-1 \\ \Omega\_{\text{ill}} = \left[ t\_k h + mh + \overline{\pi}, t\_{k+1} h + \pi\_{t\_{k+1}} \right) \end{cases} \tag{8}$$

Define *τ*(*t*) = *t* − *ikh*, where *ikh* = *tkh* + *lh*, *l* = 0, 1, . . . , *m*, and then we can obtain:

$$0 < \tau\_m \le \tau(t) \le h + \overline{\tau} = \tau\_M \tag{9}$$

Based on the above, *y*(*t*) can be rewritten as:

$$\overline{y}(t\_k h) = \left[ y(t - \tau(t)) - \varepsilon\_k(t) \right] \tag{10}$$

**Remark 1.** *The introduction of the event triggering mechanism (5) reduces redundant transmission data and saves network resources.*

Summarizing the previous discussion, the IT2 fuzzy FD filter is modeled by IT2 T–S fuzzy rules:

*Filter Rule j*: IF *ϕ*1(*x*(*t*)) is *O* \$ *j*1, *ϕ*2(*x*(*t*)) is *O* \$ *j*2, . . . . . . , and *<sup>ϕ</sup>q*(*x*(*t*)) is *O* \$ *jq*, THEN

$$\begin{cases}
\dot{\mathbf{x}}\_F(t) = \boldsymbol{\mathring{A}}\_j \mathbf{x}\_F(t) + \boldsymbol{\mathring{B}}\_j \overline{\mathbf{y}}(t) \\
 r\_F(t) = \boldsymbol{\mathring{C}}\_j \mathbf{x}\_F(t) + \boldsymbol{\mathring{D}}\_j \overline{\mathbf{y}}(t)
\end{cases} \tag{11}$$

in which, *A* ˆ *j*, *B* ˆ *j*, *C* ˆ *j*, and *D* ˆ *j* are FD filter gain matrices. *xF*(*t*) ∈ *Rnx* , *y*(*t*) ∈ *Rny* , and *rF*(*t*) ∈ *Rnr* represent the state vector, the output, and residual output vector of the eventtriggered FD filter. The fuzzy set is *<sup>O</sup>*\$*jβ*, *j* = 1, 2, ... ,*s*, *β* = 1, 2, ... , *q*, *q* is the number of fuzzy sets. *ϕ*(*x*(*t*)) = *<sup>ϕ</sup>*1(*x*(*t*)), *ϕ*2(*x*(*t*)),..., *<sup>ϕ</sup>q*(*x*(*t*))*<sup>T</sup>* are the premise variables. The firing strength of *jth* rule is expressed by interval sets:

$$K\_j(\mathbf{x}(t)) = [\underline{\mathbf{x}}\_j(\mathbf{x}(t)), \overline{\mathbf{x}}\_j(\mathbf{x}(t))] \tag{12}$$

with *<sup>κ</sup>j*(*x*(*t*)) = *q* Π *<sup>β</sup>*=<sup>1</sup>*μO*\$*j<sup>β</sup>* (*ϕβ*(*x*(*t*))) ≥ 0, *<sup>κ</sup>j*(*x*(*t*)) = *q* Π *<sup>β</sup>*=<sup>1</sup>*μO*\$*j<sup>β</sup>* (*ϕβ*(*x*(*t*))) ≥ 0, *<sup>μ</sup>N*\$*jλ* (*ϕλ*(*x*(*t*))) ≥ *<sup>μ</sup>N*\$*jλ* (*ϕλ*(*x*(*t*))) ≥ 0, *<sup>κ</sup>j*(*x*(*t*)) ≥ *<sup>κ</sup>j*(*x*(*t*)) ≥ 0, *<sup>κ</sup>j*(*x*(*t*)) and *<sup>κ</sup>j*(*x*(*t*)) represent, the bounds of membership, where *<sup>μ</sup>N*\$*jλ* (*ϕλ*(*x*(*t*))) and *<sup>μ</sup>N*\$*jλ* (*ϕλ*(*x*(*t*))) represent the bounds of the membership function, respectively. The event-triggered FD filter is designed as:

$$\begin{cases} \dot{\mathbf{x}}\_F(t) = \sum\_{j=1}^r \tilde{\phi}\_j(\mathbf{x}(t)) [\hat{A}\_j \mathbf{x}\_F(t) + \hat{B}\_j \overline{\mathbf{y}}(t)] \\\ r\_F(t) = \sum\_{j=1}^r \tilde{\phi}\_j(\mathbf{x}(t)) [\hat{\mathbb{C}}\_j \mathbf{x}\_F(t) + \hat{\mathcal{D}}\_j \overline{\mathbf{y}}(t)] \end{cases} \tag{13}$$

where *φ* \$ *j*(*x*(*t*)) = *<sup>φ</sup>j*(*x*(*t*))*<sup>κ</sup>j*(*x*(*t*)) + *φj*(*x*(*t*))*<sup>κ</sup>j*(*x*(*t*)) ≥ 0, *r* ∑ *j*=1 *φ* \$ *j*(*x*(*t*)) = 1, while *φj*(*x*(*t*)) ≥ 0 and *<sup>φ</sup>j*(*x*(*t*)) ≥ 0 are nonlinear functions used to represent the uncertainty of theFDfilter,satisfying

$$
\underline{\Phi}\_j(\mathbf{x}(t)) + \overline{\Phi}\_j(\mathbf{x}(t)) = 1 \tag{14}
$$

For the convenience of the following writing, using *ρ*\$*i*, *φ* \$ *j* instead of *ρ*\$*i*(*x*(*t*)), *φ* \$ *<sup>j</sup>*(*x*(*t*)).

**Remark 2.** *The FD filter (13) proposed has two advantages. Firstly, the model has higher accuracy by using IT2 T-S fuzzy theory to describe uncertainty effectively. Secondly, the FD filter is more general as the object's affiliation function and the fuzzy rules are not shared with the FD filter.*

### *2.3. Fault Residual System (FRS)*

In this section, the fault residual system is developed based on models (3) and (13). The fault diagnosis problem is simplified to the problem of asymptotic tracking of residuals and faults. Combination the ETS (5), and defining with *ξ*(*t*) = *x<sup>T</sup>*(*t*) *xTF* (*t*) *T*,

*ω*(*t*) = *ω<sup>T</sup>*(*t*) *f <sup>T</sup>*(*t*) *ω<sup>T</sup>*(*<sup>t</sup>* − *d*(*t*)) *T* , *re*(*t*) = *rF*(*t*) − *f*(*t*), the FRS can be represented as: ⎧ ⎪⎪⎨ ⎪⎪⎩ . *ξ*(*t*) = *r* ∑ *i*=1 *r* ∑ *j*=1 *<sup>ρ</sup>*\$*iφ*\$*j Aijξ*(*t*) + *BijHξ*(*t* − *τ*(*t*)) + *<sup>B</sup>ωij<sup>ω</sup>*(*t*) − *Beijek*(*t*) *re*(*t*) = *r* ∑ *i*=1 *r* ∑ *j*=1 *<sup>ρ</sup>*\$*iφ*\$*j Cijξ*(*t*) + *DijHξ*(*t* − *τ*(*t*)) + *<sup>D</sup>ωij<sup>ω</sup>*(*t*) − *Deijek*(*t*) (15) *Aij* = - *Ai* 0 0 *A*ˆ*j* , *Bij* = - 0 *<sup>B</sup>*<sup>ˆ</sup>*jCi* , *<sup>B</sup>ωij* = - *Bi Bf i* 0 0 0 *<sup>B</sup>*<sup>ˆ</sup>*jDi* , *Beij* = - 0 *B*ˆ*j* ,*Cij* = 0 *C*ˆ *j* , *Dij* = - *D*ˆ *jCi* 0 , *<sup>D</sup>ωij* = 0 −*I D*ˆ *jDi* , *Deij* = *D*ˆ *j*, *H* = *I* 0 .

> The target of this section is to design the FD filter (13) and triggering mechanism (5) such that the FRS (15) satisfies asymptotically stable with the *H*∞ performance indicators. In the meantime, the following conditions are satisfied:

