*3.1. Stability Analysis*

In this subsection, the following improvements will be made in the stability analysis process to reduce the system conservativeness. First, a new Lyapunov–Krasovskii function with fourfold integration is constructed; second, Wirtinger's inequality is applied to process the integral term, which is in the time derivative of the Lyapunov–Krasovskii function; third, a relaxation matrix is introduced to deal with the premise variable mismatch problem.

**Theorem 1.** *For given scalars* 0 < *ε* < 1*,* 0 < *τm* ≤ *τM, γ* > 0*, and the membership functions satisfying w*\$*j* − *ψj<sup>m</sup>*\$*j* ≥ 0(0 < *ψj* ≤ <sup>1</sup>)*, if IT2 FRS (15) is asymptotically stable, and achieving the expected H*∞ *performance level γ, then there exists parameter matrix P* > 0*, Qi* (*i* = 1, <sup>2</sup>)*, Si* > 0 (*i* = 1, <sup>2</sup>)*, Ri* > 0 (*i* = 1, 2, <sup>3</sup>)*, Ti* > 0 (*i* = 1, <sup>2</sup>)*,* Λ*i* > 0 (*i* = 1, <sup>2</sup>)*, A*ˆ*j, B*ˆ*j, C*ˆ *j, D*ˆ *j and Wi* > 0*,* (*i* = 1, 2, ... ,*<sup>r</sup>*)*, meanwhile, the following inequalities exist in the appropriate dimensions:*

$$
\Delta\_{ij} - \mathcal{W}\_i < 0 \tag{20}
$$

⎥

⎥

⎥

*,*

⎥

$$
\psi\_i \Xi\_{ii} - \psi\_i \mathcal{W}\_i + \mathcal{W}\_i \preccurlyeq 0 \tag{21}
$$

$$
\Psi\_{\dot{j}} \Xi\_{\dot{i}j} + \Psi\_{\dot{i}} \Xi\_{\dot{j}i} - \Psi\_{\dot{i}} \mathcal{W}\_{\dot{j}} - \Psi\_{\dot{j}} \mathcal{W}\_{\dot{i}} + \mathcal{W}\_{\dot{i}} + \mathcal{W}\_{\dot{j}} < 0, \ \dot{i} < \dot{j} \tag{22}
$$

*for* <sup>Ξ</sup>*ij* = Ξ11*ij* Ξ12*ij* ∗ Ξ22*ij , in which* Ξ11*ij* = <sup>Φ</sup>11*ij* <sup>Φ</sup>12*ij* ∗ <sup>Φ</sup>22*ij , where* <sup>Φ</sup>11*ij* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ Φ11 *<sup>H</sup>TR*1 0 0 *<sup>H</sup>TR*3 ∗ −2*R*<sup>1</sup> *R*1 0 0 ∗ ∗ Φ33 *R*2 0 ∗ ∗ ∗−2*R*2 0 ∗ ∗ ∗ ∗−2*R*3 ⎤ ⎦ <sup>Φ</sup>12*ij* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 *PBij* − *PBeij* − *PBωij* 00 0 0 00 0 0 *R*2 00 0 *R*3 00 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , <sup>Φ</sup>22*ij* = ⎡ ⎢ ⎢ ⎣ Φ66 0 00 ∗ *εC<sup>T</sup> i* Λ2*Ci* 0 Φ79 ∗ ∗− Λ1 0 ∗∗ ∗ Φ99 ⎤ ⎥ ⎥ ⎦*, P Aij* + *ATijP* + *<sup>H</sup><sup>T</sup>*(*Q*1 + *Q*2)*<sup>H</sup>* − *<sup>H</sup><sup>T</sup>*(*<sup>R</sup>*1 + *<sup>R</sup>*3)*H,* Φ33 = − *Q*1 − *R*1 − *R*2*,* Φ66 = − *Q*2 − *R*3*,* =*εC<sup>T</sup> i* Λ2 0 0 *Di ,* Δ*τ* = *τM* − *<sup>τ</sup>m,*Φ<sup>99</sup> = − *γ*2 *I* + *ε* 0 0 *Di T* Λ2 0 0 *Di ,* =⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ √*τm* 2 *S<sup>T</sup>* 1 *ϕ*1 √Δ*τ* 2 *S<sup>T</sup>* 2 *ϕ*1 *<sup>τ</sup>mR<sup>T</sup>* 1 *ϕ*1 Δ*τR<sup>T</sup>* 2 *ϕ*1 *τM R<sup>T</sup>* 3 *ϕ*1 *τ*2 √*m* 6*T<sup>T</sup>* 1 *ϕ*1 Δ*τ*<sup>2</sup> √6 *T<sup>T</sup>* 2 *ϕ*1 *Cij* 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 √*τm* 2 *S<sup>T</sup>* 1 *ϕ*2 √Δ*τ* 2 *S<sup>T</sup>* 2 *ϕ*2 *<sup>τ</sup>mR<sup>T</sup>* 1 *ϕ*2 Δ*τR<sup>T</sup>* 2 *ϕ*2 *τM R<sup>T</sup>* 3 *ϕ*2 *τ*2 √*m* 6*T<sup>T</sup>* 1 *ϕ*2 Δ*τ*<sup>2</sup> √6 *T<sup>T</sup>* 2 *ϕ*2 *Dij* − √*τm* 2 *S<sup>T</sup>* 1 *ϕ*3 − √Δ*τ* 2 *S<sup>T</sup>* 2 *ϕ*3 <sup>−</sup>*τmR<sup>T</sup>* 1 *ϕ*3 − Δ*τR<sup>T</sup>* 2 *ϕ*3 −*τM R<sup>T</sup>* 3 *ϕ*3 − *τ*2 √*m* 6*T<sup>T</sup>* 1 *ϕ*3 −Δ*τ*<sup>2</sup> √6 *T<sup>T</sup>* 2 *ϕ*3 − *Deij* √*τm* 2 *S<sup>T</sup>* 1 *ϕ*4 √Δ*τ* 2 *S<sup>T</sup>* 2 *ϕ*4 *<sup>τ</sup>mR<sup>T</sup>* 1 *ϕ*4 Δ*τR<sup>T</sup>* 2 *ϕ*4 *τM R<sup>T</sup>* 3 *ϕ*4 *τ*2 √*m* 6*T<sup>T</sup>* 1 *ϕ*4 Δ*τ*<sup>2</sup> √6 *T<sup>T</sup>* 2 *ϕ*4 *<sup>D</sup>ωij* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ =*diag*" −*S*<sup>1</sup> −*S*<sup>2</sup> − *R*1 − *R*2 − *R*3 − *T*1 − *T*2 −*I* # *, H Aij, ϕ*2= *HBij, ϕ*3= *HBeij, ϕ*4= *HBωij*.

.

**Proof.** For the FRS (15), construct the following Lyapunov–Krasovskii function:

$$V(t) = V\_1(t) + V\_2(t) + V\_3(t) + V\_4(t) + V\_5(t)\tag{23}$$

*<sup>V</sup>*1(*t*) = *ξ<sup>T</sup>*(*t*)*Pξ*(*t*), *<sup>V</sup>*2(*t*) = 2 *tt*−*τm ξ<sup>T</sup>*(*s*)*HTQ*1*Hξ*(*s*)*ds* + 2 *tt*−*τ<sup>M</sup> ξ<sup>T</sup>*(*s*)*HTQ*2*Hξ*(*s*)*ds*, *<sup>V</sup>*3(*t*) = *<sup>τ</sup>m*2 *tt*−*τm* 2 *ts* .*<sup>ξ</sup><sup>T</sup>*(*v*)*HTR*1*<sup>H</sup>* .*ξ*(*v*)*dvds* + (*<sup>τ</sup>M* − *<sup>τ</sup>m*)2 *t*−*τm t*−*τ<sup>M</sup>* 2 *ts* .*<sup>ξ</sup><sup>T</sup>*(*v*)*HTR*2*<sup>H</sup>* .*ξ*(*v*)*dvds* <sup>+</sup>*τM*2 *tt*−*τ<sup>M</sup>* 2 *ts* .*<sup>ξ</sup><sup>T</sup>*(*v*)*HTR*3*<sup>H</sup>* .*ξ*(*v*)*dvds* , *<sup>V</sup>*4(*t*) = 2 0<sup>−</sup>*τm* 2 0*θ* 2 *tt*+*λ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTS*1*<sup>H</sup>* .*ξ*(*s*)*dsdλd<sup>θ</sup>* + 2 <sup>−</sup>*τm* −*τM* 2 0*θ* 2 *tt*+*λ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTS*2*<sup>H</sup>* .*ξ*(*s*)*dsdλdθ*, *<sup>V</sup>*5(*t*) = *<sup>τ</sup>m*2 0<sup>−</sup>*τm* 2 0*θ* 2 0*λ* 2 *tt*+*k* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTT*1*<sup>H</sup>* .*ξ*(*s*)*dsdkdλd<sup>θ</sup>* +(*<sup>τ</sup>M* − *<sup>τ</sup>m*)2 <sup>−</sup>*τm* −*τM* 2 0*θ* 2 0*λ* 2 *tt*+*k* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTT*2*<sup>H</sup>* .*ξ*(*s*)*dsdkdλd<sup>θ</sup>* . and *P* = *P<sup>T</sup>* > 0, *Qi* > 0, *Si* > 0, *Ti* > 0, *i* = 1, 2, *Rj* > 0, *j* = 1, 2, 3. Along the trajectory of the FRS (15), the time derivative of *V*(*t*) is:

$$V(t) = V\_1(t) + V\_2(t) + V\_3(t) + V\_4(t) + V\_5(t)\tag{24}$$

where

.

where

*<sup>V</sup>*1(*t*) = 2*ξ<sup>T</sup>*(*t*)*P ξ*(*t*), . *<sup>V</sup>*2(*t*) = *ξ<sup>T</sup>*(*t*)*H<sup>T</sup>*(*Q*1 + *Q*2)*Hξ*(*t*) − *ξ<sup>T</sup>*(*t* − *<sup>τ</sup>m*)*HTQ*1*Hξ*(*<sup>t</sup>* − *<sup>τ</sup>m*) − *ξ<sup>T</sup>*(*t* − *<sup>τ</sup>M*)*HTQ*2*Hξ*(*<sup>t</sup>* − *<sup>τ</sup>M*), . *<sup>V</sup>*3(*t*) = . *ξ T* (*t*)*H<sup>T</sup>τ*2*mR*<sup>1</sup> + (*<sup>τ</sup>M* − *<sup>τ</sup>m*)<sup>2</sup>*R*<sup>2</sup> + *<sup>τ</sup>*2*MR*3*<sup>H</sup>* .*ξ*(*t*) − *<sup>τ</sup>m*2 *tt*−*τm* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTR*1*<sup>H</sup>* .*ξ*(*s*)*ds* −(*<sup>τ</sup><sup>M</sup>* − *<sup>τ</sup>m*)2 *t*−*τm t*−*τ<sup>M</sup>* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTR*2*<sup>H</sup>* .*ξ*(*s*)*ds* − *<sup>τ</sup>M*2 *tt*−*τ<sup>M</sup>* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTR*3*<sup>H</sup>* .*ξ*(*s*)*ds* , . *<sup>V</sup>*4(*t*) = *τ*2*m*2 .*<sup>ξ</sup><sup>T</sup>*(*t*)*HTS*1*<sup>H</sup>* .*ξ*(*t*) + (*<sup>τ</sup>M*−*τm*)<sup>2</sup> 2 .*<sup>ξ</sup><sup>T</sup>*(*t*)*HTS*2*<sup>H</sup>* .*ξ*(*t*) − 2 0<sup>−</sup>*τm* 2 *tt*+*θ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTS*1*<sup>H</sup>* .*ξ*(*s*)*dsd<sup>θ</sup>* −2 <sup>−</sup>*τm* −*τM* 2 *tt*+*θ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTS*2*<sup>H</sup>* .*ξ*(*s*)*dsd<sup>θ</sup>* , . *<sup>V</sup>*5(*t*) = *τ*4*m*6 .*<sup>ξ</sup><sup>T</sup>*(*t*)*HTT*1*<sup>H</sup>* .*ξ*(*t*) + (*<sup>τ</sup>M*−*τm*)<sup>4</sup> 6 .*<sup>ξ</sup><sup>T</sup>*(*t*)*HTT*2*<sup>H</sup>* .*ξ*(*t*) <sup>−</sup>*τm*2 0<sup>−</sup>*τm* 2 0*θ* 2 *tt*+*λ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTT*1*<sup>H</sup>* .*ξ*(*s*)*dsdλd<sup>θ</sup>* − (*<sup>τ</sup>M* − *<sup>τ</sup>m*)2 <sup>−</sup>*τm* −*τM* 2 0*θ* 2 *tt*+*λ* .*<sup>ξ</sup><sup>T</sup>*(*s*)*HTT*2*<sup>H</sup>* .*ξ*(*s*)*dsdλd<sup>θ</sup>* . The integral term in . *<sup>V</sup>*3(*t*), which we treat by applying Lemma 3, yields

$$-\tau\_m \int\_{t-\tau\_m}^t \dot{\xi}^T(s) H^T R\_1 H \dot{\xi}(s) ds \le \begin{bmatrix} H\_\theta^x(t) \\ H\_\theta^x(t-\tau\_1(t)) \\ H\_\theta^x(t-\tau\_m) \end{bmatrix}^T \begin{bmatrix} -R\_1 & R\_1 & 0 \\ \* & -2R\_1 & R\_1 \\ \* & \* & -R\_1 \end{bmatrix} \begin{bmatrix} H\_\theta^x(t) \\ H\_\theta^x(t-\tau\_1(t)) \\ H\_\theta^x(t-\tau\_m) \end{bmatrix} \tag{25}$$

$$- \left( \boldsymbol{\pi}\_{\mathsf{M}} - \boldsymbol{\pi}\_{\mathsf{m}} \right) \int\_{t-\mathsf{T}\_{\mathsf{M}}}^{t-\mathsf{T}\_{\mathsf{M}}} \dot{\boldsymbol{\xi}}^{T} (\mathsf{s}) H^{T} \mathsf{R}\_{2} H \dot{\boldsymbol{\xi}}(\mathsf{s}) ds \leq \begin{bmatrix} H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{m}}) \\ H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{s}}(t)) \\ H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{M}}) \end{bmatrix}^{T} \begin{bmatrix} -\mathsf{R}\_{2} & \mathsf{R}\_{2} & \mathbf{0} \\ \ast & -2\mathsf{R}\_{2} & \mathsf{R}\_{2} \\ \ast & \ast & -\mathsf{R}\_{2} \end{bmatrix} \begin{bmatrix} H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{m}}) \\ H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{s}}(t)) \\ H \boldsymbol{\xi}(t-\mathsf{T}\_{\mathsf{M}}) \end{bmatrix} \tag{26}$$

$$-\tau\_{\rm M} \int\_{t-\tau\_{\rm M}}^{t} \dot{\boldsymbol{\xi}}^{T}(\mathbf{s}) H^{T} \mathbf{R} \mathbf{J} \dot{\mathbf{J}}(\mathbf{s}) d\mathbf{s} \leq \begin{bmatrix} H^{x}\_{\rm S}(t) \\ H^{x}\_{\rm S}(t-\tau\_{\rm M}(t)) \\ H^{y}\_{\rm S}(t-\tau\_{\rm M}) \end{bmatrix}^{T} \begin{bmatrix} -R\_{3} & R\_{3} & 0 \\ \* & -2R\_{3} & R\_{3} \\ \* & \* & -R\_{3} \end{bmatrix} \begin{bmatrix} H^{y}\_{\rm S}(t) \\ H^{y}\_{\rm S}(t-\tau\_{\rm M}(t)) \\ H^{y}\_{\rm S}(t-\tau\_{\rm M}) \end{bmatrix} \tag{27}$$

Furthermore, in a bid to obtain stability conditions with low conservativeness, the following slack matrix is introduced:

$$\sum\_{i=1}^{r} \sum\_{j=1}^{r} \tilde{m}\_i (\tilde{m}\_j - \tilde{w}\_j) W\_i = 0,\\ W\_i = W\_i^T, (i = 1, 2, \dots, r) \tag{28}$$

*r*

*r*

From (23) to (28), we can obtain

$$\begin{aligned} &\sum\_{i=1}^{r}\sum\_{j=1}^{r}\tilde{m}\_{i}\tilde{w}\_{j}\Sigma\_{ij} \\ &=\sum\_{i=1}^{r}\sum\_{j=1}^{r}\tilde{m}\_{i}(\tilde{m}\_{j}-\tilde{w}\_{j}+\Psi\_{j}\tilde{m}\_{j}-\Psi\_{j}\tilde{m}\_{j})\mathcal{W}\_{i}+\sum\_{i=1}^{r}\sum\_{j=1}^{s}\tilde{m}\_{i}\tilde{w}\_{j}\Sigma\_{ij} \\ &=\sum\_{i=1}^{r}\tilde{m}\_{i}^{2}(\Phi\_{i}\Sigma\_{ii}-\Psi\_{i}\mathcal{W}\_{i}+\mathcal{W}\_{i}) \\ &+\sum\_{i=1}^{r-1}\sum\_{j=i+1}^{r}\tilde{m}\_{i}\tilde{m}\_{j}(\Phi\_{j}\Sigma\_{ij}-\Psi\_{j}\mathcal{W}\_{i}+\mathcal{W}\_{i}+\Psi\_{i}\Xi\_{ji}-\Psi\_{i}\mathcal{W}\_{j}+\mathcal{W}\_{j}) +\sum\_{i=1}^{r}\sum\_{j=1}^{r}\tilde{m}\_{i}(\tilde{w}\_{j}-\Psi\_{j}\tilde{m}\_{j})(\Xi\_{ij}-\mathcal{W}\_{i}) \end{aligned} \tag{29}$$

under *w*\$*j* − *ψj<sup>m</sup>*\$*j* ≥ 0 for all *j*. Combined with the event-triggering mechanism (5), we can derive

$$\dot{V}(t) + r\_\varepsilon^T(t)r\_\varepsilon(t) - \gamma^2 \tilde{\omega}^T(t)\tilde{\omega}(t) \le \sum\_{i=1}^r \sum\_{j=1}^r \tilde{m}\_i \tilde{w}\_j \tilde{\omega}^T(t) \Xi\_{ij} \tilde{\varsigma}(t) \tag{30}$$

where

$$\begin{aligned} \zeta^T(t) &= \begin{bmatrix} \eta\_1(t) & \eta\_2(t) \end{bmatrix}, \zeta\_1(t) = \begin{bmatrix} \ \xi^T(t) & \xi^T(t-\tau\_1(t)) & \xi^T(t-\tau\_{\text{m}})H^T & \xi^T(t-\tau\_2(t)) \end{bmatrix}, \\\ \zeta\_2(t) &= \begin{bmatrix} \ \xi^T(t-\tau\_3(t)) & \xi^T(t-\tau\_{\text{M}})H^T & \xi^T(t-\tau(t))H^T & \epsilon\_k^T(t) & \overline{\omega}^T(t) \end{bmatrix}. \\\ \text{By using Schur complement, } \Xi\_{ij} &\le 0 \text{, hence, we have} \end{aligned}$$

$$
\dot{V}(t) + r\_\varepsilon^T(t)r\_\varepsilon(t) - \gamma^2 \tilde{\omega}^T(t)\tilde{\omega}(t) \le 0 \tag{31}
$$

Integrating from *0* to ∞ simultaneously on the left and right sides of (30), we can obtain: 

$$\int\_0^\infty r\_\varepsilon^T(t) r\_\varepsilon(t) dt < \gamma^2 \int\_0^\infty \widetilde{\omega}^T(t) \widetilde{\omega}(t) dt\tag{32}$$

Equation (32) representative *re*(*t*)2 < *<sup>γ</sup>ω*\$(*t*)2 holds for any nonzero *ω*\$(*t*) ∈ *<sup>L</sup>*2[0, <sup>∞</sup>). Thus, the FRS (15) is under the restriction of Theorem 1 is asymptotically stable and satisfies the given *H*∞ performance index *γ*. -

**Remark 4.** *The Lyapunov–Krasovskii function (23) constructed contains multiple integrals, such as triple, quadruple integrals. The more system and time delay information are considered, and the amplification of the integral term processing is avoided effectively. Convergence of global asymptotic stability is guaranteed. Moreover, more recently, the introduction of the relaxation matrix (28) makes the obtained stability criterion with less conservative.*

### *3.2. Fault Diagnosis Filter Design*

In this section, solving the parameters of the FD filter is transformed into the problem of matrix convex optimization, which can be solved by MATLAB. Using the matrix transformation and deformation, the proposed filter design method is implemented.

**Theorem 2.** *For given scalars* 0 < *ε* < 1*,* 0 < *τm* ≤ *τM, γ* > 0*, and the membership functions satisfying w*\$*j* − *ψj<sup>m</sup>*\$*j* ≥ 0*,* (0 < *ψj* ≤ <sup>1</sup>)*, if the IT2 FRS (15) is asymptotically stable and meets the expected H*∞ *performance level γ, then there exists parameter matrix P* > 0*, Qi* > 0 (*i* = 1, <sup>2</sup>)*, Si* > 0 (*i* = 1, <sup>2</sup>)*, Ri* > 0 (*i* = 1, 2, <sup>3</sup>)*, Ti* > 0 (*i* = 1, <sup>2</sup>)*,* Λ*i* > 0 (*i* = 1, <sup>2</sup>)*, A* \$ *j, B* \$ *j, C* \$ *j, D* \$ *j and W* \$ *T i*= *W* \$ *i have suitable dimensions satisfying the following inequality:*

$$
\vec{\Xi}\_{ij} - \vec{\mathcal{W}}\_i < 0 \tag{33}
$$

$$
\Psi\_i \check{\Xi}\_{i\bar{i}} - \Psi\_{\bar{i}} \check{\Psi}\_{\bar{i}} + \check{\mathcal{W}}\_{\bar{i}} < 0 \tag{34}
$$

$$
\psi\_j \check{\Xi}\_{ij} + \psi\_i \check{\Xi}\_{ji} - \psi\_i \check{\mathsf{W}}\_j - \psi\_j \check{\mathsf{W}}\_i + \check{\mathsf{W}}\_i + \check{\mathsf{W}}\_j < 0, \ i < j \tag{35}
$$

*for* Ξ \$ *ij* = Ξ\$11*ij* Ξ\$12*ij* ∗ Ξ\$22*ij , in which* Ξ \$11 *ij* = Φ\$11*ij* Φ\$12*ij* ∗ Φ\$22*ij , where* Φ\$11*ij* = ⎡ ⎢⎢⎢⎢⎢⎣ Φ \$ 11 Φ \$ 12 *R*1 0 0 *R*3 ∗ Φ \$ 22 000 0 ∗ ∗−2*R*1 *R*1 0 0 ∗∗ ∗ Φ \$ 44 *R*2 0 ∗ ∗ ∗ ∗−2*R*2 0 ∗ ∗ ∗ ∗ ∗−2*R*3 ⎤ ⎥⎥⎥⎥⎥⎦*,* Φ\$12 *ij* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ 0 *B* \$ *jCi* −*B* \$ *j P*1*Bi <sup>P</sup>*1*Bf i B* \$ *jDi* 0 *B* \$ *jCi* −*B* \$ *j YBi YBf i B* \$ *jDi* 00 0 0 0 0 00 0 0 0 0 *R*2 000 0 0 *R*3 000 0 0 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦, Φ\$22*ij* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ Φ \$ 77 00 0 0 0 ∗ Φ \$ 88 00 0 Φ \$ 812 ∗ ∗−Λ1 000 ∗ ∗ ∗−*γ*<sup>2</sup> *I* 0 0 ∗ ∗ ∗ ∗−*γ*<sup>2</sup> *I* 0 ∗∗ ∗ ∗ ∗ Φ \$ 1212 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦*,* Φ \$ 11 = *P*1*Ai* + *P*1Δ*A* + *ATi P*1 + *ATi* Δ*P* + *Q*1 + *Q*2 − *R*1 − *R*3*,* <sup>Φ</sup>\$12 = *ATi Y* + *A*\$*j* + Δ*ATY,* Φ \$ 22 = *A* \$ *j* + *A* \$*T j ,* Φ \$ 44 = −*Q*<sup>1</sup> − *R*1 − *R*2*,* Φ \$ 77 = −*Q*<sup>2</sup> − *R*3*,* Φ \$ 88 = *ε<sup>C</sup>Ti* Λ2*Ci,* Φ \$ 812 = *ε<sup>C</sup>Ti* Λ2*Di,* <sup>Φ</sup>\$1212 = −*γ*<sup>2</sup> *I* + *ε<sup>D</sup>Ti* Λ2*Di*. Ξ12*ij* = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ √ *τm* 2*S*1*Ai* <sup>√</sup><sup>Δ</sup>*τ*2*S*2*Ai <sup>τ</sup>mR*1*Ai* Δ*τR*2*Ai <sup>τ</sup>MR*3*Ai τ*2 <sup>√</sup>*<sup>m</sup>*6*T*1*Ai* Δ*τ*<sup>2</sup> √6 *T*2*Ai* 0 0000 00 0 *C* \$ *j* 0 . . . 0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0...0 ⎫⎪⎬⎪⎭5 0000 00 0 *D* \$ *jCi* 0000 00 0 −*D* \$ *j* √ *τm* 2*S*1*Bi* <sup>√</sup><sup>Δ</sup>*τ*2*S*2*Bi <sup>τ</sup>mR*1*Bi* Δ*τR*2*Bi <sup>τ</sup>MR*3*Bi τ*2 <sup>√</sup>*<sup>m</sup>*6*T*1*Bi* Δ*τ*<sup>2</sup> √6 *T*2*Bi* 0 0000 00 0 −*I* 0000 00 0 *D* \$ *jDi* ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, Ξ \$22 *ij* = *diag*" −*S*<sup>1</sup> −*S*<sup>2</sup> −*R*<sup>1</sup> −*R*<sup>2</sup> −*R*<sup>3</sup> −*T*<sup>1</sup> −*T*<sup>2</sup> −*I* #.

Based on the above condition for the establishment of linear matrix inequality, the filter parameter matrix is obtained as follows

$$
\vec{A}\_{\dot{j}} = \Upsilon^{-1} \vec{A}\_{\dot{j}\prime} \,\, \mathcal{B}\_{\dot{j}} = \Upsilon^{-1} \vec{B}\_{\dot{j}\prime} \,\, \mathcal{C}\_{\dot{j}} = \vec{\mathcal{C}}\_{\dot{j}\prime} \,\, \mathcal{D}\_{\dot{j}} = \vec{\mathcal{D}}\_{\dot{j}}.\tag{36}
$$

**Proof.** On the basis of Theorem 1, we set *P* = -*P*1 *P*2 ∗ *P*3 , *J*1 = *diag*1*<sup>I</sup>*, *<sup>P</sup>*2*P*−<sup>1</sup> 3 3, *J*2 = *diag*{*J*1, *I* ... *I* } 18 }.

Then, we have to multiply the left and right sides of Equations (20)–(22) by *J*2 and *JT*2 . It yields that

$$
\tilde{\Xi}\_{ij} - \mathcal{W}\_i + \Sigma\_1^T \overline{\Delta}\_f \Sigma\_2 + \Sigma\_2^T \overline{\Delta}\_f \Sigma\_1 < 0 \tag{37}
$$

The application of Lemma 2 achieves the conversion of (37) to (38).

$$
\varepsilon\_{\overline{\omega}j}^{\overline{\omega}} - \mathcal{W}\_i + \varepsilon\_1^{-1} \Sigma\_1^T \delta^2 \Sigma\_1 + \varepsilon\_1 \Sigma\_2^T \Sigma\_2 < 0 \tag{38}
$$

To facilitate the simplification and operation of the matrix, the following expression is made: \$

$$
\widetilde{\mathcal{W}}\_i = f\_2 \mathcal{W}\_i f\_2^T,\ \ Y = P\_2 P\_3^{-1} P\_2^T \ \sigma
$$

$$
\widetilde{A}\_{\dot{\jmath}} = P\_2 \widehat{A}\_{\dot{\jmath}} P\_3^{-1} P\_2^T, \\
\widetilde{B}\_{\dot{\jmath}} = P\_2 \widehat{B}\_{\dot{\jmath}}, \\
\widetilde{C}\_{\dot{\jmath}} = \widehat{C}\_{\dot{\jmath}} P\_3^{-1} P\_2^T, \\
\widetilde{D}\_{\dot{\jmath}} = \widehat{D}\_{\dot{\jmath}} P\_3^{-1} P\_2^T.
$$

Bringing them into Equations (20)–(22), we can obtain Equations (33)–(35).

By using Schur Complement Lemma, the matrix *P* is equivalent to *P*1 − *<sup>P</sup>*2*P*−<sup>1</sup> 3 *PT*2 = *P*1 − *Y* > 0. Furthermore, equivalently under transformation *PT*2 *<sup>P</sup>*3*xf*(*t*), the parameters of the fault detection filter can be yielded as follows:

$$\begin{aligned} \mathcal{A}\_{\dot{j}} &= P\_2^{-T} P\_3 (P\_2^{-1} \tilde{A}\_{\dot{j}} P\_2^{-T} P\_3) P\_3^{-1} P\_2^T = Y^{-1} \tilde{A}\_{\dot{j}}, \mathcal{B}\_{\dot{j}} = P\_2^{-T} P\_3 (P\_2^{-1} \tilde{B}\_{\dot{j}}) = Y^{-1} \tilde{B}\_{\dot{j}}, \\ \mathcal{C}\_{\dot{j}} &= (\tilde{\mathcal{C}}\_{\dot{j}} P\_2^{-T} P\_3) P\_3^{-1} P\_2^T = \tilde{\mathcal{C}}\_{\dot{j}}, \mathcal{D}\_{\dot{j}} = (\tilde{\mathcal{D}}\_{\dot{j}} P\_2^{-T} P\_3) P\_3^{-1} P\_2^T = \tilde{\mathcal{D}}\_{\dot{j}}. \end{aligned}$$

According to Theorem 2, we determine the FD filter parameters by solving the convex optimization problems:

min *γ* subject to the inequalities (33)–(35). The proof is completed. -
