**2. Problem Formulation**

*2.1. IT2 T-S Nonlinear Networked Systems*

An NNSs is modeled by IT2 T–S fuzzy rules by using state-space representation, its parameter uncertainty and external perturbations are described.

*Plant rule i*: IF *<sup>ι</sup>*1(*x*(*t*)) is *<sup>G</sup>*\$*i*1, *<sup>ι</sup>*2(*x*(*t*)) is *<sup>G</sup>*\$*i*2, . . . . . . , and *<sup>ι</sup>p*(*x*(*t*)) is *<sup>G</sup>*\$*ip*, THEN

$$\begin{cases}
\dot{\mathbf{x}}(t) = A\_i \mathbf{x}(t) + B\_i \omega(t) + B\_{fi} f(t) \\
y(t) = C\_i \mathbf{x}(t) + D\_i \omega(t)
\end{cases} \tag{1}$$

In the IT2 T-S NNSs, *Ai*, *Bi*, *Bf i*, *Ci*, and *Di* are system matrices. Separately, *x*(*t*) ∈ *Rnx* , *y*(*t*) ∈ *Rny* , *f*(*t*) ∈ *Rnf* represents the state vector, measured output, and the fault signal waiting to be detected, in particular, *ω*(*t*) ∈ *Rnω* is the external disturbance which belongs to *<sup>L</sup>*2[0, <sup>∞</sup>). Define *ι*(*x*(*t*)) = *<sup>ι</sup>*1(*x*(*t*)), *<sup>ι</sup>*2(*x*(*t*)),..., *<sup>ι</sup>p*(*x*(*t*))*<sup>T</sup>* stands for premise variable, the number of fuzzy sets is *p*, the IT2 fuzzy set is described as *<sup>G</sup>*\$*i<sup>α</sup>*, where *i* = 1, 2, ... ,*r*, and *α* = 1, 2, . . . , *p*, the firing strength of *ith* rule is defined as follows [39]:

$$\mathcal{W}\_i(\mathbf{x}(t)) = [\underline{\underline{\phi}}\_i(\mathbf{x}(t)), \bar{\underline{\phi}}\_i(\mathbf{x}(t))] \tag{2}$$

where *i*(*x*(*t*)) = *p* Π *<sup>α</sup>*=1 *<sup>μ</sup>G*\$*iα* (*ια*(*x*(*t*))) ≥ 0, \_ *i*(*x*(*t*)) = *p* Π *<sup>α</sup>*=1 \_ *μG*\$*iα* (*ια*(*x*(*t*))) ≥ 0, \_ *μG*\$*iα* (*ια*(*x*(*t*))) ≥ *<sup>μ</sup>G*\$*iα* (*ια*(*x*(*t*))) ≥ 0, \_ *i*(*x*(*t*)) ≥ *i*(*x*(*t*)) ≥ 0. We can ge<sup>t</sup> the IT2 fuzzy model after weighting, as follows: 

$$\begin{cases}
\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \tilde{\rho}\_{i}(\mathbf{x}(t)) [A\_{i}\mathbf{x}(t) + B\_{i}\omega(t) + B\_{fi}f(t)] \\
\quad y(t) = \sum\_{i=1}^{r} \tilde{\rho}\_{i}(\mathbf{x}(t)) [\mathbb{C}\_{i}\mathbf{x}(t) + D\_{i}\omega(t)]
\end{cases} \tag{3}$$

where *ρ*\$*i*(*x*(*t*)) = *<sup>ρ</sup>i*(*x*(*t*)*i*(*x*(*t*)) + *<sup>ρ</sup>i*(*x*(*t*))*i*(*x*(*t*)) ≥ 0, meanwhile *r* ∑ *i*=1 *ρ* \$*i*(*x*(*t*)) = 1, *<sup>ρ</sup>i*(*x*(*t*)) and *<sup>ρ</sup>i*(*x*(*t*)) are greater than zero, which represent the weighting functions and satisfying:

$$
\underline{\rho}\_i(\mathbf{x}(t)) + \overline{\rho}\_i(\mathbf{x}(t)) = 1 \tag{4}
$$

Obviously, in the process of NNSs modeling, we define a fuzzy set for the membership function to describe its uncertainty, which provides a basis for the subsequent design of a low conservation fault diagnosis filter.
