# Fault Detection for a Class of Uncertain Linear Systems

Authors: Qingyu Su; Jian Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Design problem of the fault detection filter for a class of uncertain feedback systems is discussed in this paper. The system under consideration is the model with the nonstandard parameter uncertain regions. The fault detection filter design problem is reduced to an optimized filter design problem, which maximizes the sensitivity of the fault in the fault cases and meets the disturbance attenuation performance in both fault-free cases and fault cases, simultaneously. A numerical example is used to demonstrate the effectiveness of the proposed design method.


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## Body

## 1. Introduction

In recent years, fault detection technique is given more and more attentions for the higher demands of safety and reliability of the control systems. A considerable sum of results on fault detection, both in the light of the data-driven and the model-based detection techniques, have been achieved (see [1–3]). Among the model-based techniques, a number of results which depend on the linear matrix inequality (LMI) techniques have appeared to deal with the fault detection problem. References [4–6] investigate the estimation problem of the fault by designing a filter. Recently, a so-called H∞/H- design approach for the fault detection problem has received considerable attentions. Further, the fault detection problem associated with some certain H∞/H- performance indices is discussed in [7–10]. The fault models considered in the above papers are classical and can describe a large class of the fault types. However, it should be pointed out that the actuator stuck fault also needs to be investigated when the amplitude of the faults are not larger enough to be detected.At the same time, most of the models are obtained by the identification experiments with some limited input/output data. Identification experiment in the prediction error (PE) framework delivers an uncertain region which contains the true system at some (user-chosen) probability level [11]. The uncertain region can result in an identified parametric ellipsoid set which is regarded as a nonstandard parameter uncertain structure in the related classical papers. The ellipsoidal uncertain structure is studied by some results; see [12–15] which reduce the uncertain region by employing the robust tools. Compared with the stability and the stabilization problems for this particular uncertainty, there are no results that to pay attention to deal with the fault detection problem with the models which have the ellipsoidal uncertainty. Moreover, since the existing approaches about fault detection (FD) are not appropriate for FD problem for this kind of uncertainty, the new FD technique should solve this case.Motivated by the above reasons, the optimization design problem of the FD filter for uncertain linear feedback system is studied. The design objective is to maximize the sensitivity of the fault in the fault cases, which subjects to the disturbances attenuation performance in both fault-free cases and the fault cases. In practical, the frequency of the disturbance signals and the fault signals are usually in bound. Thus, a notion of the finite-frequency performance indices is introduced to describe the performance indices; LMIs-based sufficient conditions are provided using the finite-frequency approach proposed in [16, 17]. The optimal solution of gain matrices of the FD filter can be obtained.The main contribution of this paper that the direct design approach of the FD filter proposed to the linear model with elliptical uncertain structure. Unlike other fault detection approaches for the uncertain model, in which the FD filter is achieved not only depending on the known lower and known upper bounds of the uncertainty but also extracting convex polygons of the uncertain region, a general parametrization for the set of multipliers is introduced to decrease the conservatism. To the best of our knowledge, such a framework for the fault detection with the elliptical parameter uncertainty structure has not been reported in the literature.The paper is organized as follows. Section2 introduces the problem under consideration and presents the design objectives. Section 3 illustrates a FD filter design approach in detail. The algorithm is given in Section 4. An illustrative example is given in Section 5 to demonstrate the proposed method. Conclusions to this work are given in the last section.Notation. For a matrix A, AT denotes its transpose. For a symmetric matrix, A>0 (A≥0) and A<0 (A≤0) denote positive-definiteness (positive semidefinite matrix) and negative-definiteness (negative semi-definite matrix), respectively. The Hermitian part of a square matrix M is denoted by He(M):=M+MT. The symbol * within a matrix represents the symmetric entries. σmin(G) denotes minimum singular values of the transfer matrix G. The symbol ⊗ denotes the Kronecker product, and 0 denotes zero matrix with appropriate dimension. Both I and ℐ denote identity matrix with appropriate dimension. ℐ0 and ℐ1 denote the matrices [10;00] and [1;0] of appropriate dimensions, respectively.


## 2. Problem Formulation

### 2.1. System Model

Consider the following uncertain linear systems:(1)x˙(t)=Ax(t)+B1d(t)+B2f(t)+L1w(t),y(t)=Cx(t)+D1d(t)+D2f(t)+L2w(t),w(t)=Δz(t),z(t)=Hx(t)Δ∈Δ,
where x(t) is the state y(t) is the output, u(t) is the input, and d(t) is an energy-bounded disturbance; r0 is the reference input, x(t) is the state; A, B1, Bu, C, D, L1, L2, and H are known constant matrices of appropriate dimensions. Consider that Δ∈Δ, where Δ is defined as
(2)Δ=diag(θ,…,θ︸n0),θ=[θ1T,…,θnθT]T∈Rnθ,
where n0 is the number of θ in Δ, then the dimension of the Δ is nθn0×n0, and Δ is the parameter set of Δ. The uncertain parameter θ with a certain use-chosen probability (𝒳) is at the uncertain set 𝒰. The uncertain set 𝒰 is an ellipsoid in the parameter space:
(3)𝒰={θ∣θTℛθ<𝒳},
where ℛ describes the form of the ellipsoid and 𝒳 is a constant computed by the desired probability in model identification.Remark 1.
The parameter uncertainty region in (1) is ellipsoidal. This kind of particular uncertainty is derived from classical prediction error identification, and it is a typical parameter uncertainty of the model which is identified. Although the parameter uncertainty can turn into the polytopic uncertainty by extracting convex polygons of the ellipsoidal region, this transformation method enlarges the the uncertain region and increases the conservatism. In this paper, the ellipsoidal uncertainty is disposed directly.To detect the actuator stuck faults, the FD filter is designed such that the residualr(t) can be obtained as
(4)x˙f(t)=Afxf(t)+Bfy(t),r(t)=Cfxf(t)+Dfy(t),
where xf is the state of the FD filter. Consider that Af∈Rnf×nf, Bf∈Rnf×nm, Cf∈Rnf, Df∈Rm.By combining (1) and (4), we have the following augmented systems:
(5)ξ(t)=A-ξ(t)+B-dd(t)+L-1w(k)+B-ff(t),r(t)=C-ξ(t)+D-dd(t)+D-ff(t)+L-2w(t),w(t)=Δz(t)z(t)=H-ξ(t)Δ∈Δ,
where ξ(t)=[x(t)T,xf(t)T]T,
(6)[A-B-dB-fL-1C-D-dD-fL-2]=[A0B1B2L1BfCAfBfD1BfD2BfL2DfCCfDfD1DfD2DfL2]H-=[H0].


### 2.2. Problem Formulation

The design problem of the FD filter to be addressed in this paper can be expressed as follows.The design objective: consider a class of uncertain linear systems (1); the FD filter (4) is designed such that the augmented model (5) is stable; the disturbance effects on the residual are minimized in both the fault-free case and the fault cases, while the fault effects on the residual are maximized in the fault cases. To detect actuator faults, our design objective of the FD filter can now be formulated as the following optimization problem:
(7)maxβ(8)s.t.δmax(Grd(jω))<γ,γ>0,∀|ω|≤ω-1,Δ∈Δ(9)δmin(Grf(jω))>ββ>0,∀|ω|≤ω-2,Δ∈Δ,
where ω-1 is the upper bounds on the frequency for the disturbance and ω-2 is the upper bounds on the frequency for the fault. Grd is the transfer function from the disturbance d to the residual signal r with fault-free case and fault cases. Grf is the transfer function from the fault f to the residual signal r with fault cases.Remark 2.
Condition (8) describes the disturbance attenuation condition in both the fault-free case and fault cases. Condition (9) is formulated as the sensitiveness performance in the fault cases.Before ending this section, the following lemmas and proposition shall be recalled to prove our main results for the FD filter design.Lemma 3 (see [18]).
Given a symmetric matrixΘ∈Rn×n and two matrices ℳ and ℋ of column dimension n, there exists matrix ℱ that satisfies Θ+ℳTℱℋ+ℋTℱTℳ<0 if and only if the following two conditions hold:
(10)𝒩ℳTΘ𝒩ℳ<0𝒩ℋΘ𝒩ℋT<0,
where 𝒩ℳ and 𝒩ℋ denote arbitrarily bases of null space of ℳ and ℋ, respectively.Lemma 4 (see [19]).
Letξ∈Rn, 𝒩∈Rn×n, and ℳ∈Rn×m such that rank (𝒩)<n. The following statements are equivalent:(1)
ξTℳξ<0, for ∀ξ≠0, subject to 𝒩ξ=0;(2)
𝒩⊥Tℳ𝒩⊥<0, where 𝒩⊥ is the kernel of 𝒩;(3)
ℳ-ε𝒩T𝒩<0, for some scalar ε∈R;(4)
ℳ+XT𝒩+𝒩TX<0, for some matrix X∈Rn×m.Proposition 5 (see [11]).
Consider the uncertaintyΔ with θ∈𝒰 defined in (3), Γ∈Γ is defined as
(11)Γ:={Γ=[ΔI]∣Δ∈Δ}.
Restrict the parametrization of matrices Λ defined as
(12)Λ:={Λ∣Λ=[Λ11Λ12Λ12TΛ22]}
with Λ22 as a positive complex Hermitian matrix of dimension n×n. Consider that Λ11=-(Λ22⊗(R/𝒳)+Λ1)∈Rnθn×nθn and
(13)Λ12=(0p12⋯p1n-p120⋯p2n⋮⋱⋱⋮-p1n^⋯⋯0)∈Rnθn×n,
where ℛ describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (Λa,pij) can take any values provided. Consider that Λa∈Rnθn×nθn has the following structure that:
(14)Λ1∈Λ1:{[0k12⋯k1n-k120⋯⋮⋮⋱⋱k(n-1)n-k1n⋯-k(n-1n)0]∣kkkkkkkkkkij=-kijT∈Rnθ×nθ[0k12⋯k1n-k120⋯⋮⋮⋱⋱k(n-1)n-k1n⋯-k(n-1n)0]}
and pij∈Rn, i=1,…,n, j=1,…,n. For Λ∈Λ, ΓTΛΓ>0, ∀Γ∈Γ.


## 2.1. System Model

Consider the following uncertain linear systems:(1)x˙(t)=Ax(t)+B1d(t)+B2f(t)+L1w(t),y(t)=Cx(t)+D1d(t)+D2f(t)+L2w(t),w(t)=Δz(t),z(t)=Hx(t)Δ∈Δ,
where x(t) is the state y(t) is the output, u(t) is the input, and d(t) is an energy-bounded disturbance; r0 is the reference input, x(t) is the state; A, B1, Bu, C, D, L1, L2, and H are known constant matrices of appropriate dimensions. Consider that Δ∈Δ, where Δ is defined as
(2)Δ=diag(θ,…,θ︸n0),θ=[θ1T,…,θnθT]T∈Rnθ,
where n0 is the number of θ in Δ, then the dimension of the Δ is nθn0×n0, and Δ is the parameter set of Δ. The uncertain parameter θ with a certain use-chosen probability (𝒳) is at the uncertain set 𝒰. The uncertain set 𝒰 is an ellipsoid in the parameter space:
(3)𝒰={θ∣θTℛθ<𝒳},
where ℛ describes the form of the ellipsoid and 𝒳 is a constant computed by the desired probability in model identification.Remark 1.
The parameter uncertainty region in (1) is ellipsoidal. This kind of particular uncertainty is derived from classical prediction error identification, and it is a typical parameter uncertainty of the model which is identified. Although the parameter uncertainty can turn into the polytopic uncertainty by extracting convex polygons of the ellipsoidal region, this transformation method enlarges the the uncertain region and increases the conservatism. In this paper, the ellipsoidal uncertainty is disposed directly.To detect the actuator stuck faults, the FD filter is designed such that the residualr(t) can be obtained as
(4)x˙f(t)=Afxf(t)+Bfy(t),r(t)=Cfxf(t)+Dfy(t),
where xf is the state of the FD filter. Consider that Af∈Rnf×nf, Bf∈Rnf×nm, Cf∈Rnf, Df∈Rm.By combining (1) and (4), we have the following augmented systems:
(5)ξ(t)=A-ξ(t)+B-dd(t)+L-1w(k)+B-ff(t),r(t)=C-ξ(t)+D-dd(t)+D-ff(t)+L-2w(t),w(t)=Δz(t)z(t)=H-ξ(t)Δ∈Δ,
where ξ(t)=[x(t)T,xf(t)T]T,
(6)[A-B-dB-fL-1C-D-dD-fL-2]=[A0B1B2L1BfCAfBfD1BfD2BfL2DfCCfDfD1DfD2DfL2]H-=[H0].


## 2.2. Problem Formulation

The design problem of the FD filter to be addressed in this paper can be expressed as follows.The design objective: consider a class of uncertain linear systems (1); the FD filter (4) is designed such that the augmented model (5) is stable; the disturbance effects on the residual are minimized in both the fault-free case and the fault cases, while the fault effects on the residual are maximized in the fault cases. To detect actuator faults, our design objective of the FD filter can now be formulated as the following optimization problem:
(7)maxβ(8)s.t.δmax(Grd(jω))<γ,γ>0,∀|ω|≤ω-1,Δ∈Δ(9)δmin(Grf(jω))>ββ>0,∀|ω|≤ω-2,Δ∈Δ,
where ω-1 is the upper bounds on the frequency for the disturbance and ω-2 is the upper bounds on the frequency for the fault. Grd is the transfer function from the disturbance d to the residual signal r with fault-free case and fault cases. Grf is the transfer function from the fault f to the residual signal r with fault cases.Remark 2.
Condition (8) describes the disturbance attenuation condition in both the fault-free case and fault cases. Condition (9) is formulated as the sensitiveness performance in the fault cases.Before ending this section, the following lemmas and proposition shall be recalled to prove our main results for the FD filter design.Lemma 3 (see [18]).
Given a symmetric matrixΘ∈Rn×n and two matrices ℳ and ℋ of column dimension n, there exists matrix ℱ that satisfies Θ+ℳTℱℋ+ℋTℱTℳ<0 if and only if the following two conditions hold:
(10)𝒩ℳTΘ𝒩ℳ<0𝒩ℋΘ𝒩ℋT<0,
where 𝒩ℳ and 𝒩ℋ denote arbitrarily bases of null space of ℳ and ℋ, respectively.Lemma 4 (see [19]).
Letξ∈Rn, 𝒩∈Rn×n, and ℳ∈Rn×m such that rank (𝒩)<n. The following statements are equivalent:(1)
ξTℳξ<0, for ∀ξ≠0, subject to 𝒩ξ=0;(2)
𝒩⊥Tℳ𝒩⊥<0, where 𝒩⊥ is the kernel of 𝒩;(3)
ℳ-ε𝒩T𝒩<0, for some scalar ε∈R;(4)
ℳ+XT𝒩+𝒩TX<0, for some matrix X∈Rn×m.Proposition 5 (see [11]).
Consider the uncertaintyΔ with θ∈𝒰 defined in (3), Γ∈Γ is defined as
(11)Γ:={Γ=[ΔI]∣Δ∈Δ}.
Restrict the parametrization of matrices Λ defined as
(12)Λ:={Λ∣Λ=[Λ11Λ12Λ12TΛ22]}
with Λ22 as a positive complex Hermitian matrix of dimension n×n. Consider that Λ11=-(Λ22⊗(R/𝒳)+Λ1)∈Rnθn×nθn and
(13)Λ12=(0p12⋯p1n-p120⋯p2n⋮⋱⋱⋮-p1n^⋯⋯0)∈Rnθn×n,
where ℛ describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (Λa,pij) can take any values provided. Consider that Λa∈Rnθn×nθn has the following structure that:
(14)Λ1∈Λ1:{[0k12⋯k1n-k120⋯⋮⋮⋱⋱k(n-1)n-k1n⋯-k(n-1n)0]∣kkkkkkkkkkij=-kijT∈Rnθ×nθ[0k12⋯k1n-k120⋯⋮⋮⋱⋱k(n-1)n-k1n⋯-k(n-1n)0]}
and pij∈Rn, i=1,…,n, j=1,…,n. For Λ∈Λ, ΓTΛΓ>0, ∀Γ∈Γ.


## 3. Fault Detection Design

In this section, Lemma6 and Proposition 7 are first given, and inequality conditions for performance indices (8) and (9) are also given. Based on these conditions, an algorithm based on LMIs is presented.Lemma 6 (see [16]).
Consider the following uncertain system:(15)x˙(t)=(A+B1Δ(I-D11Δ)-1C1)x(t)+B2u(t),y(t)=(C2+D21Δ(I-D11Δ)-1C1)x(t)+D22u(t),
with Δ∈Δ. Let a real symmetric matrix Π and a positive scalar ϖ be given. The following robust finite-frequency condition
(16)[G1(jω)TI]Π[G1(jω)TI]T<0|ω|≤ϖ,Δ∈Δ
holds if there exist real symmetric matrices 𝒫, 𝒬, Φ∈Φ, and Λ∈Λ such that
(17)[𝒜ℬℐ0]T[-𝒬𝒫𝒫ϖ2𝒬][𝒜ℬℐ0]<[𝒞T𝒟T]Φ[𝒞𝒟],(18)𝒬≥𝒞1TΛ𝒞1,
where
(19)Φ:={[Ψ00Π]:[Γ00Γ]TΨ[Γ00Γ]≤0,∀Γ∈Γ},[𝒜ℬ𝒞1𝒟1𝒞2𝒟2]:=[AB1B20000I0I00C1D1100C2D21D22000I0][𝒞𝒟]:=[𝒞11𝒟11𝒞2𝒟2]=[𝒞10𝒞1𝒜𝒞1ℬ𝒞2𝒟2].Proposition 7.
Consider the uncertaintyΔ with θ∈𝒰 defined in (3) and Γ∈Γ defined in (11). Restrict the parametrization of matrices Ψ defined as
(20)Ψ=[Ψ11Ψ12Ψ12TΨ22]=[Ψ111Ψ112Ψ121Ψ122Ψ112TΨ113Ψ122TΨ123Ψ121TΨ122Ψ221Ψ222Ψ122TΨ123TΨ222TΨ223],
where [Ψ113Ψ123*Ψ223] are positive complex Hermitian matrices of dimension 2n×2n; Ψ111, Ψ121, and Ψ221 of the dimension nθn×nθn have the structure with -(Ψ113⊗(R/𝒳)+Ψ1), -(Ψ123⊗(R/𝒳)+Ψ2), and -(Ψ223⊗(R/𝒳)+Ψ3), respectively. Moreover(21)Ψp2=(0pp2⋯ppn-pp20⋯ppn⋮⋱⋱⋮-ppn^⋯⋯0)∈Rnθn×n,p=11,12,22,
where ℛ describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (Ψ1, Ψ2, Ψ3, p1ij, p2ij,  p3ij, i=1,…,n, j=1,…,n) can take any values provided. For t=1,2,3, i=1,…,n, and j=1,…,n, Ψt∈Rnθn×nθn has the following structure:
(22)Ψt∈Ψt:{[0kt12⋯kt1n-kt120⋯⋮⋮⋱⋱kt(n-1)n-kt1n⋯-kt(n-1)n0]∣kkkkkkkkkkktij=-ktijT∈Rnθ×nθ[0kt12⋯kt1n-kt120⋯⋮⋮⋱⋱kt(n-1)n-kt1n⋯-kt(n-1)n0]},
and ptij∈Rn. Then, for Ψ∈Ψ, it has
(23)[Γ00Γ]TΨ[Γ00Γ]>0,∀Γ∈Γ,
where Γ is defined in (11).Proof.
Let allΨp2,Ψt,p∈{11,12,22}, t∈{1,2,3} have the structure in the statement of the proposition:
(24)Ψp2TΔ+ΔTΨp2=0,ΔTΨtΔ=0.
Therefore, for every Ψ∈Ψ, we can have(25)[Γ00Γ]TΨ[Γ00Γ]=[Δ00IΔ00I]TΨ[Δ00IΔ00I]=[ΔTΨ111ΔT+Ψ112TΔ+ΔTΨ112+Ψ113ΔTΨ121ΔT+Ψ122TΔ+ΔTΨ122+Ψ123*ΔTΨ221ΔT+Ψ222TΔ+ΔTΨ222+Ψ223]=[(1-θTR𝒳θ)Ψ111(1-θTR𝒳θ)Ψ121*(1-θTR𝒳θ)Ψ221].When θ∈𝒰 and [Ψ113Ψ123*Ψ223]>0, we can have
(26)[Γ00Γ]TΨ[Γ00Γ]>0,∀Γ∈Γ.


### 3.1. The Disturbance Attenuation Condition

Theorem 8.
Consider the system in (5) with Δ∈Δ. A real symmetric matrix Πa=[100-γ2I] is given. Condition
(27)δmax(Grd(jω))<γ,|ω|<ω-2,Δ∈Δ
holds if there exist matrix variables A^, B^, C^, D^, M, R, Ya, Ea, La, Na, Wa, Sa,
(28)Pa=[Pa11Pa12Pa13Pa21Pa22Pa23Pa31Pa32Pa33],Qa=[Qa11Qa12Qa13Qa21Qa22Qa23Qa31Qa32Qa33],Ψa=[Ψa111Ψa112Ψa121Ψa122Ψa112TΨa113Ψa122TΨa123Ψa121TΨa122Ψa221Ψa222Ψa122TΨa123TΨa222TΨa223],Λa=[Λa11Λa12Λa12TΛa22]
of appropriate dimensions satisfying Qa>0, Ψa∈Ψ,  and Λa∈Λ such that(29)[-Qa11-Qa12-Qa13Pa11-YaTPa12+EaTPa13-LaTa17a280*-Qa22-Qa23Pa21-MTPa22-MTPa23a27a280**-Qa33Qa31-WaTPa32-SaTPa33-NaTa37a380***a44a45a46a47a48(D^C)T****a55a56a57a58C^T*****a66a67a68(D^L2)T******a77a78(D^D1)T*******a880********I-He(R)]<0(30)[Qa11-HTΛa22HQa12Qa13-HTΛa12T*Qa22Qa23**Qa33-Λa11]>0,where
(31)a17=-YaTv1,a27=-MTv1,a37=-WaTv1,a18=-YaTv2,a28=-MTv2,a38=-WaTv2,a44=ω-12Qa11+HTΨa113H+ATHTΨa223HA+He(YaA+B^C+HTΨa123HA),a45=ω-12Qa12+A^-ATEaT+CTB^T,a46=ω-12Qa13+HTΨa112T+ATHTΨa122T+HTΨa123HL1+ATHTΨa223HL1+YaL1+B^L2+ATLaT,a55=ω-12Qa22+He(A^)a56=ω-21Qa23-EaL1+B^L2a66=ω-12Qa33+Ψa111+L1THTΨa223HL1+He(Ψa122HL1+LaL1),a47=HTΨa123HB1+ATHTΨa223HB1+YaB1+B^D1+(ATYaT+CTB^T)v1,a48=HTΨa122T+ATHTΨa222T+Wa+(ATYaT+CTB^T)v2,a57=-EaB1+B^D1+A^Tv1,a58=Sa+A^Tv2,a67=Ψa122HB1+L1THTΨa223HB1+LaB1+(L1TYT+L2TB^T)v1,a68=Ψa121+L1THTΨa222T+N+(L1TYaT+L2TB^T)v2,a77=-γ2I+B1THTΨa223HB1+He((B1TYaT+D1TB^T)v1),a78=B1THTΨa222T+v1TWa+(B1TYaT+D1TB^T)v2,a88=Ψa221+He(v2TWa),Λ:={Λa:satisfy(12)inProposition5},Ψ:={Ψa:satisfy(20)inProposition7},Proof.
SubstitutingΠa into (17), Grd(jω)TGrd(jω)<γ2 which is equivalent to (27). By applying Lemma 6, (27) holds if
(32)[𝒜ℬaℐ0]T[-𝒬a𝒫a𝒫aω-2𝒬a][𝒜ℬaℐ0]<[𝒞T𝒟T][Ψa00Πa][𝒞𝒟](33)𝒬a>𝒞1TΛa𝒞1,
where
(34)[𝒜ℬa𝒞1𝒟1𝒞2𝒟2]:=[A-L-1B-d0000I0I00H-000C-L-2D-d000I0],[𝒞𝒟]:=[𝒞11𝒟11𝒞2𝒟2]=[𝒞10𝒞1𝒜𝒞1ℬa𝒞2𝒟2].
Let(35)Υa=[-𝒬a𝒫a0𝒫aω-2𝒬a+𝒞11TΨa𝒞11+𝒞2TΠa𝒞2𝒞11TΨa𝒟11+𝒞2TΠa𝒟20𝒟11TΨa𝒞11+𝒟2TΠa𝒞2𝒟11TΨa𝒟11+𝒟2TΠa𝒟2]
(32) is equivalent to
(36)[𝒜ℬaI00I]TΥa[𝒜ℬaI00I]<0.
And on the other hand,
(37)[I00]Υa[I00]T=-𝒬a<0.
By combining (36) with (37), and applying Lemma 3, (32) holds if and only if
(38)Υa+[-I𝒜TℬaT]𝒳a[0I000I]+[0I000I]T𝒳aT[-I𝒜TℬaT]T<0,
where 𝒳a introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳a is defined as 𝒳a=[𝒳a1𝒳a1V1] and partition 𝒳a1 into [xa11-xa12xa13x21x220xa31xa32xa33] where x21 and x22 are nonsingular matrices variable. V1 is a given matrix, defined as V1=[v1T00v2T00]T.
LetJ=diag{I,x22-1x21,I}, and define the linearizing change of the control variables as follows:(39)Qa=JT𝒬aJ,Xa1=[YaT-EaTLaTMTMT0WaTSaTNaT]=[x11-x12x22-1x21x13x21Tx22-Tx21x21Tx22-Tx210x31x32x22-1x21x33],Pa=JT𝒫aJ,[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I]
By Pre- and postmultiplying (38) by diag{JT,JT,I} and diag{J,J,I} and choos Ψa satisfying Proposition 5, (38) is equivalent to(40)
[
-
Q
a
P
a
-
X
a
1
-
X
a
1
V
1
*
ω
-
2
Q
a
+
𝒞
11
T
Ψ
a
𝒞
11
+
𝒞
2
T
Π
a
𝒞
2
+
He
(
𝒜
T
X
a
1
)
𝒞
11
T
Ψ
a
𝒟
11
+
𝒞
2
T
Π
a
𝒟
2
+
𝒜
T
X
a
1
V
1
+
X
a
1
T
ℬ
a
*
*
𝒟
11
T
Ψ
a
𝒟
11
+
𝒟
2
T
Π
a
𝒟
2
+
He
(
ℬ
a
T
X
a
1
V
1
)
]
<
0
,where𝒜, ℬa, 𝒞11, 𝒞2 are the matrices which use the new defined variables A^f, B^f,  C^f in the matrices 𝒜, ℬa, 𝒞11, 𝒞2.By simple calculation, we can obtain that(41)[𝒞2TΠa𝒞2𝒞2TΠa𝒟2*𝒟2TΠa𝒟2]=[0000000000-γ2I00000]+[ℭTL-2TD-2T0][ℭL-2D-20]=[000-γ2ℐ0]+[ℭ1T𝔇2T][ℭ1𝔇2],
where ℭ=[DfCC^f], ℭ1=[ℭL-2], 𝔇2=[D-20]. By applying the Schur complement formula, the following can be obtained:(42)[-QaPa-Xa1-Xa1V10*ω-2Qa+𝒞11TΨa𝒞11+He(𝒜TXa1)𝒞11TΨa𝒟11+𝒜TXa1V1+Xa1Tℬaℭ1T**𝒟11TΨa𝒟11+He(ℬaTXa1V1)-γ2ℐ0𝔇2T***-I]<0.Then, pre-and postmultiplydiag{I,I,I,RT} to (42). Due to -RTR≤I-He(R) and defineing A^=MA^f, B^=MB^, C^=RTC^f, D^=RTDf, (29) hold. In addition, and pre- and postmultiplying (33) by J1T and J1, respectively, (33) becomes (30).Hence, if conditions (29) and (30) holds, the augmented uncertain system (5) is stable and guarantees the H∞ performance (27), which completes the proof.


### 3.2. The Fault Sensitiveness Condition for Faulty Case

Theorem 9.
Consider the system in (5) with Δ∈Δ. A real symmetric matrix Πb=[-100β2I]. The following condition
(43)δmin(Grf(jω))>β,|ω|<ω-2,Δ∈Δ
holds if there exist matrix variables A^, B^, C^, D^, M, R, Yb, Eb, Lb, Nb, Wb, Sb,
(44)Pb=[Pb11Pb12Pb13Pb21Pb22Pb23Pb31Pb32Pb33],Qb=[Qb11Qb12Qb13Qb21Qb22Qb23Qb31Qb32Qb33]Ψb=[Ψb111Ψb112Ψb121Ψb122Ψb112TΨb113Ψb122TΨb123Ψb121TΨb122Ψb221Ψb222Ψb122TΨb123TΨb222TΨb223],Λb=[Λb11Λb12Λb12TΛb22]
of appropriate dimensions satisfying Qb>0, Ψb∈Ψ, and  Λb∈Λ such that(45)[-Qb11-Qb12-Qb13Pb11-YbTPb12-EbTPb13-LbT-YbTv3-YbTv40*-Qb22-Qb23Pb21-MTPb22-MTPb23-MTv3-MTv40**-Qb33Pb31-WbTPb32-SbTPb33-NbT-WbTv3-WbTv40***c44b45b46b47b48-ℐ1RT****b55B56b57b58-ℐ1RT*****b66b67b69-ℐ1RT******b77b780*******b880******00-I]<0,(46)[Qb11-HTΛb22HQb12Qb13-HTΛb12T*Qb22Qb23**Qb33-Λb11]>0,where
(47)b44=ω-22Qb11+HTΨ113H+ATHTΨb223HA+He(YbA+B^C+ℐ1D^C+HTΨb123HA),b45=ω-22Qb12+A^+ATEbT+CTB^T+CTB^T+(ℐ1D^C)T+ℐ1C^,b46=ω-22Qb13+HTΨb112T+ATHTΨb122T+HTΨb123HL1+ATHTΨb223HL1+B^L2+ATLbT+ℐ1D^L2+CTD^Tℐ1T,b55=ω-22Qb22+He(A^+ℐ1C^),b56=ω-22Qb23+EL1+B^L2+ℐ1D^L2+C^Tℐ1T,b66=ω-22Qb33+Ψb111+L1THTΨb223HL1+He(Ψb122HL1+LbL1+ℐ1D^L2),b47=HTΨb123HB2+ATHTΨb223HB2+YbB2+B^D2+(YbA+B^C)Tv3+ℐ1D^D2,b48=HTΨb122T+ATHTΨb222T+Wb+(YbA+B^C)Tv4,b57=EbB2+B^D2+A^Tv3+ℐ1D^D2,b58=Sb+A^Tv4,b67=Ψb122HB2+L1THTΨb223HB2+LbB2+(YL1+B^L2)Tv3+ℐ1D^D2,b68=Ψb121+L1THTΨb222T+Nb+(YbL1+B^L2)Tv4,b77=β2I+B2THTΨa223HB2+He((YbB2+B^D2)Tv3),b78=B2THTΨb222T+v3TWb+(YbB2+B^D2)Tv4,b88=Ψb221+He(v4TWb),Λ:={Λb:satisfy(12)inProposition5},Ψ:={Ψb:satisfy(20)inProposition7},Proof.
SubstitutingΠb into (17), Grr0i(jω)TGrr0i(jω)<β2I which is equivalent to (43). Applying Lemma 6, (43) holds if
(48)[𝒜ℬbℐ0]T[-𝒬b𝒫b𝒫bω-2𝒬b][𝒜ℬbℐ0]<[𝒞T𝒟T][Ψb00Πb][𝒞𝒟](49)𝒬b>𝒞1TΛb𝒞1
hold.
Where(50)[𝒜ℬb𝒞1𝒟1𝒞2𝒟2]:=[A-L-1B-f0000I0I00H-000C-L-2D-f000I0][𝒞𝒟]:=[𝒞11𝒟11𝒞2𝒟2]=[𝒞10𝒞1𝒜𝒞1ℬb𝒞2𝒟2].
Let(51)Υb=[-𝒬b𝒫b0𝒫bω-2𝒬b+𝒞11TΨb𝒞11+𝒞2TΠb𝒞2𝒞11TΨb𝒟11+𝒞2TΠb𝒟20𝒟11TΨb𝒞11+𝒟2TΠb𝒞2𝒟11TΨb𝒟11+𝒟2TΠb𝒟2]
(48) is equivalent to
(52)[𝒜ℬbI00I]TΥb[𝒜ℬbI00I]<0,
and on the other hand,
(53)[I00]Υb[I00]T=-𝒬b<0.
Combining (52) with (53) and applying Lemma 3, (48) holds if and only if
(54)Υb+[-I𝒜TℬbT]𝒳b[0I000I]+[0I000I]T𝒳bT[-I𝒜TℬbT]T<0,
where 𝒳b introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳b is defined as 𝒳b=[𝒳b1𝒳b1V2], where 𝒳b1=[xb11-xb12xb13x21x220xb31xb32xb33], where x21 and x22 are the same as in Theorem 8. V2 is an given matrix, defined as V2=[v3T00v4T00]T. Pre- and postmultiply (54) by diag{JT,JT,I} and diag{J,J,I}, where J is defined as Theorem 8 and choose Ψb satisfying Proposition 5, (54) is equivalent to(55)[-QbPb-Xb1-Xb1V2*ω-2Qb-𝒞b2T𝒞b2+𝒞11TΨb𝒞11+He(Xb1T𝒜-b)-𝒞b2T𝒟b2+𝒞11TΨb𝒟11+Xb1Tℬ-b+𝒜-bXb1V2**-𝒟b2T𝒟b2+𝒟11TΨb𝒟11+β2ℐ0+He(ℬ-bTXb1V2)]<0,where
(56)[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I],𝒳b1=JTX1J=[YbTEbTLbTMTMT0WbTSbTNbT],𝒜-b=J-1𝒜J=[A0L1B^fCA^fB^fL2000],ℬ-b=J-1ℬb=[B20B^fD200I],𝒞b2=[DfCC^fDfL2],𝒟b2=[DfD20],
Condition (55) is nonlinear owing to the product terms -Cb2TCb2. To solve this problem, condition (55) is equivalent to
(57)Ξb2TΘΞb2<0,
where(58)Ξb2=[I000I000I0𝒞b2𝒟b2],Θ=[-QbPb-X1-X1V20*𝒞11TΨb𝒞11+He(X1T𝒜-bi)X1Tℬ-b+𝒜-biX1V2+𝒞11TΨb𝒟110**𝒟11TΨb𝒟11+β2ℐ0+He(ℬ-bTXb1V2)0000-I].
DefiningG=[0𝒞b2𝒟b2-I] satisfies GΞb2=0. Using Lemma 4, and given a matrix T, (57) is equivalent to
(59)Θ+TG+(TG)T<0,
where one notes T as
(60)T=[0ℛ00]T,ℛ=[Rℐ1TRℐ1TRℐ1T].
Defining A^=MA^f, B^=MB^, C^=RTC^f, and D^=RTDf and after some matrix manipulation, (59) becomes (45). By pre- and postmultiplying (49) by JT and J, respectively, (49) becomes (46).Remark 10.
Theorem8 considers the attenuation performance for the disturbance d(t), and the conservatism comes from the special structure of 𝒳a1 and the same one in 𝒳a. Theorem 9 considers the sensitiveness performance for the fault f(t), and the conservatism comes from the special structure of 𝒳b1 and the same one in 𝒳b. Then, the conservatism for Theorem 9 comes as the same as Theorem 8. However, in order to obtain the better detection performance, the sensitiveness performance is considered over the conservatism.


### 3.3. Stability Condition for the Filter

Due to the fault tolerant controller, the closed-loop system is stable both in the fault-free cases and in the fault cases. Then, the condition that the augmented systems (1) is stable can translate into the condition that the FD filter is stable.Lemma 11 (see [20]).
By considering the FD filter (4), the FD filter is stable, if there exist matrix variables M, A^ and Ps>0 satisfying
(61)[-(M+MT)A^+PsM*-Ps0**-Ps]<0.


### 3.4. Algorithm

In the precious sections, Theorems8 and 9 and Lemma 11 have formulated the inequality conditions for the performance indexes (8) and (9) and the stable condition, respectively. Summarily, we have the following theorem.Theorem 12.
By considering the uncertain system model (1), there exists a FD filter (4) such that the augmented system model (5) is stable and satisfies the performance indices (8) and (9) if inequality conditions (29), (30), (45), (46), and (61) hold.Proof.
Combining Theorems8–9 and Lemma 11, it is obviously that the theorem holds.Remark 13.
In Theorems8–9, if the matrices vi, i∈{1,…,4} are given, all conditions (29), (30), (45), and (46) are the LMI conditions. Specially, if the disturbance and the fault are a scalar, respectively, vi is vector, and it can be easily chosen, which has been further confirmed in the simulink. In addition, Propositions 5 and 7 are used to directly deal with the ellipsoidal uncertainty; the variable structure is complex. However, all the variables in Propositions 5 and 7 is composed of linear matrices, and can be used for LMI conditions.The gain matricesAf, Bf, Cf, and Df defining the FD filter can be derived by the means of a standard procedure. Denote A^, B^, C^, D^, and M is the optimal solution of (7) with condition (29), (30), (45), (46), and (61):(1)
compute thex21, x22 by solving the following factorization problem M=x21Tx22-1x21;(2)
computeA^f, B^f by A^, B^ and M. Compute C^f, Df by C^, D^ and R;(3)
finally, obtain the gain matrices of the FD filter(62)[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I].


## 3.1. The Disturbance Attenuation Condition

Theorem 8.
Consider the system in (5) with Δ∈Δ. A real symmetric matrix Πa=[100-γ2I] is given. Condition
(27)δmax(Grd(jω))<γ,|ω|<ω-2,Δ∈Δ
holds if there exist matrix variables A^, B^, C^, D^, M, R, Ya, Ea, La, Na, Wa, Sa,
(28)Pa=[Pa11Pa12Pa13Pa21Pa22Pa23Pa31Pa32Pa33],Qa=[Qa11Qa12Qa13Qa21Qa22Qa23Qa31Qa32Qa33],Ψa=[Ψa111Ψa112Ψa121Ψa122Ψa112TΨa113Ψa122TΨa123Ψa121TΨa122Ψa221Ψa222Ψa122TΨa123TΨa222TΨa223],Λa=[Λa11Λa12Λa12TΛa22]
of appropriate dimensions satisfying Qa>0, Ψa∈Ψ,  and Λa∈Λ such that(29)[-Qa11-Qa12-Qa13Pa11-YaTPa12+EaTPa13-LaTa17a280*-Qa22-Qa23Pa21-MTPa22-MTPa23a27a280**-Qa33Qa31-WaTPa32-SaTPa33-NaTa37a380***a44a45a46a47a48(D^C)T****a55a56a57a58C^T*****a66a67a68(D^L2)T******a77a78(D^D1)T*******a880********I-He(R)]<0(30)[Qa11-HTΛa22HQa12Qa13-HTΛa12T*Qa22Qa23**Qa33-Λa11]>0,where
(31)a17=-YaTv1,a27=-MTv1,a37=-WaTv1,a18=-YaTv2,a28=-MTv2,a38=-WaTv2,a44=ω-12Qa11+HTΨa113H+ATHTΨa223HA+He(YaA+B^C+HTΨa123HA),a45=ω-12Qa12+A^-ATEaT+CTB^T,a46=ω-12Qa13+HTΨa112T+ATHTΨa122T+HTΨa123HL1+ATHTΨa223HL1+YaL1+B^L2+ATLaT,a55=ω-12Qa22+He(A^)a56=ω-21Qa23-EaL1+B^L2a66=ω-12Qa33+Ψa111+L1THTΨa223HL1+He(Ψa122HL1+LaL1),a47=HTΨa123HB1+ATHTΨa223HB1+YaB1+B^D1+(ATYaT+CTB^T)v1,a48=HTΨa122T+ATHTΨa222T+Wa+(ATYaT+CTB^T)v2,a57=-EaB1+B^D1+A^Tv1,a58=Sa+A^Tv2,a67=Ψa122HB1+L1THTΨa223HB1+LaB1+(L1TYT+L2TB^T)v1,a68=Ψa121+L1THTΨa222T+N+(L1TYaT+L2TB^T)v2,a77=-γ2I+B1THTΨa223HB1+He((B1TYaT+D1TB^T)v1),a78=B1THTΨa222T+v1TWa+(B1TYaT+D1TB^T)v2,a88=Ψa221+He(v2TWa),Λ:={Λa:satisfy(12)inProposition5},Ψ:={Ψa:satisfy(20)inProposition7},Proof.
SubstitutingΠa into (17), Grd(jω)TGrd(jω)<γ2 which is equivalent to (27). By applying Lemma 6, (27) holds if
(32)[𝒜ℬaℐ0]T[-𝒬a𝒫a𝒫aω-2𝒬a][𝒜ℬaℐ0]<[𝒞T𝒟T][Ψa00Πa][𝒞𝒟](33)𝒬a>𝒞1TΛa𝒞1,
where
(34)[𝒜ℬa𝒞1𝒟1𝒞2𝒟2]:=[A-L-1B-d0000I0I00H-000C-L-2D-d000I0],[𝒞𝒟]:=[𝒞11𝒟11𝒞2𝒟2]=[𝒞10𝒞1𝒜𝒞1ℬa𝒞2𝒟2].
Let(35)Υa=[-𝒬a𝒫a0𝒫aω-2𝒬a+𝒞11TΨa𝒞11+𝒞2TΠa𝒞2𝒞11TΨa𝒟11+𝒞2TΠa𝒟20𝒟11TΨa𝒞11+𝒟2TΠa𝒞2𝒟11TΨa𝒟11+𝒟2TΠa𝒟2]
(32) is equivalent to
(36)[𝒜ℬaI00I]TΥa[𝒜ℬaI00I]<0.
And on the other hand,
(37)[I00]Υa[I00]T=-𝒬a<0.
By combining (36) with (37), and applying Lemma 3, (32) holds if and only if
(38)Υa+[-I𝒜TℬaT]𝒳a[0I000I]+[0I000I]T𝒳aT[-I𝒜TℬaT]T<0,
where 𝒳a introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳a is defined as 𝒳a=[𝒳a1𝒳a1V1] and partition 𝒳a1 into [xa11-xa12xa13x21x220xa31xa32xa33] where x21 and x22 are nonsingular matrices variable. V1 is a given matrix, defined as V1=[v1T00v2T00]T.
LetJ=diag{I,x22-1x21,I}, and define the linearizing change of the control variables as follows:(39)Qa=JT𝒬aJ,Xa1=[YaT-EaTLaTMTMT0WaTSaTNaT]=[x11-x12x22-1x21x13x21Tx22-Tx21x21Tx22-Tx210x31x32x22-1x21x33],Pa=JT𝒫aJ,[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I]
By Pre- and postmultiplying (38) by diag{JT,JT,I} and diag{J,J,I} and choos Ψa satisfying Proposition 5, (38) is equivalent to(40)
[
-
Q
a
P
a
-
X
a
1
-
X
a
1
V
1
*
ω
-
2
Q
a
+
𝒞
11
T
Ψ
a
𝒞
11
+
𝒞
2
T
Π
a
𝒞
2
+
He
(
𝒜
T
X
a
1
)
𝒞
11
T
Ψ
a
𝒟
11
+
𝒞
2
T
Π
a
𝒟
2
+
𝒜
T
X
a
1
V
1
+
X
a
1
T
ℬ
a
*
*
𝒟
11
T
Ψ
a
𝒟
11
+
𝒟
2
T
Π
a
𝒟
2
+
He
(
ℬ
a
T
X
a
1
V
1
)
]
<
0
,where𝒜, ℬa, 𝒞11, 𝒞2 are the matrices which use the new defined variables A^f, B^f,  C^f in the matrices 𝒜, ℬa, 𝒞11, 𝒞2.By simple calculation, we can obtain that(41)[𝒞2TΠa𝒞2𝒞2TΠa𝒟2*𝒟2TΠa𝒟2]=[0000000000-γ2I00000]+[ℭTL-2TD-2T0][ℭL-2D-20]=[000-γ2ℐ0]+[ℭ1T𝔇2T][ℭ1𝔇2],
where ℭ=[DfCC^f], ℭ1=[ℭL-2], 𝔇2=[D-20]. By applying the Schur complement formula, the following can be obtained:(42)[-QaPa-Xa1-Xa1V10*ω-2Qa+𝒞11TΨa𝒞11+He(𝒜TXa1)𝒞11TΨa𝒟11+𝒜TXa1V1+Xa1Tℬaℭ1T**𝒟11TΨa𝒟11+He(ℬaTXa1V1)-γ2ℐ0𝔇2T***-I]<0.Then, pre-and postmultiplydiag{I,I,I,RT} to (42). Due to -RTR≤I-He(R) and defineing A^=MA^f, B^=MB^, C^=RTC^f, D^=RTDf, (29) hold. In addition, and pre- and postmultiplying (33) by J1T and J1, respectively, (33) becomes (30).Hence, if conditions (29) and (30) holds, the augmented uncertain system (5) is stable and guarantees the H∞ performance (27), which completes the proof.


## 3.2. The Fault Sensitiveness Condition for Faulty Case

Theorem 9.
Consider the system in (5) with Δ∈Δ. A real symmetric matrix Πb=[-100β2I]. The following condition
(43)δmin(Grf(jω))>β,|ω|<ω-2,Δ∈Δ
holds if there exist matrix variables A^, B^, C^, D^, M, R, Yb, Eb, Lb, Nb, Wb, Sb,
(44)Pb=[Pb11Pb12Pb13Pb21Pb22Pb23Pb31Pb32Pb33],Qb=[Qb11Qb12Qb13Qb21Qb22Qb23Qb31Qb32Qb33]Ψb=[Ψb111Ψb112Ψb121Ψb122Ψb112TΨb113Ψb122TΨb123Ψb121TΨb122Ψb221Ψb222Ψb122TΨb123TΨb222TΨb223],Λb=[Λb11Λb12Λb12TΛb22]
of appropriate dimensions satisfying Qb>0, Ψb∈Ψ, and  Λb∈Λ such that(45)[-Qb11-Qb12-Qb13Pb11-YbTPb12-EbTPb13-LbT-YbTv3-YbTv40*-Qb22-Qb23Pb21-MTPb22-MTPb23-MTv3-MTv40**-Qb33Pb31-WbTPb32-SbTPb33-NbT-WbTv3-WbTv40***c44b45b46b47b48-ℐ1RT****b55B56b57b58-ℐ1RT*****b66b67b69-ℐ1RT******b77b780*******b880******00-I]<0,(46)[Qb11-HTΛb22HQb12Qb13-HTΛb12T*Qb22Qb23**Qb33-Λb11]>0,where
(47)b44=ω-22Qb11+HTΨ113H+ATHTΨb223HA+He(YbA+B^C+ℐ1D^C+HTΨb123HA),b45=ω-22Qb12+A^+ATEbT+CTB^T+CTB^T+(ℐ1D^C)T+ℐ1C^,b46=ω-22Qb13+HTΨb112T+ATHTΨb122T+HTΨb123HL1+ATHTΨb223HL1+B^L2+ATLbT+ℐ1D^L2+CTD^Tℐ1T,b55=ω-22Qb22+He(A^+ℐ1C^),b56=ω-22Qb23+EL1+B^L2+ℐ1D^L2+C^Tℐ1T,b66=ω-22Qb33+Ψb111+L1THTΨb223HL1+He(Ψb122HL1+LbL1+ℐ1D^L2),b47=HTΨb123HB2+ATHTΨb223HB2+YbB2+B^D2+(YbA+B^C)Tv3+ℐ1D^D2,b48=HTΨb122T+ATHTΨb222T+Wb+(YbA+B^C)Tv4,b57=EbB2+B^D2+A^Tv3+ℐ1D^D2,b58=Sb+A^Tv4,b67=Ψb122HB2+L1THTΨb223HB2+LbB2+(YL1+B^L2)Tv3+ℐ1D^D2,b68=Ψb121+L1THTΨb222T+Nb+(YbL1+B^L2)Tv4,b77=β2I+B2THTΨa223HB2+He((YbB2+B^D2)Tv3),b78=B2THTΨb222T+v3TWb+(YbB2+B^D2)Tv4,b88=Ψb221+He(v4TWb),Λ:={Λb:satisfy(12)inProposition5},Ψ:={Ψb:satisfy(20)inProposition7},Proof.
SubstitutingΠb into (17), Grr0i(jω)TGrr0i(jω)<β2I which is equivalent to (43). Applying Lemma 6, (43) holds if
(48)[𝒜ℬbℐ0]T[-𝒬b𝒫b𝒫bω-2𝒬b][𝒜ℬbℐ0]<[𝒞T𝒟T][Ψb00Πb][𝒞𝒟](49)𝒬b>𝒞1TΛb𝒞1
hold.
Where(50)[𝒜ℬb𝒞1𝒟1𝒞2𝒟2]:=[A-L-1B-f0000I0I00H-000C-L-2D-f000I0][𝒞𝒟]:=[𝒞11𝒟11𝒞2𝒟2]=[𝒞10𝒞1𝒜𝒞1ℬb𝒞2𝒟2].
Let(51)Υb=[-𝒬b𝒫b0𝒫bω-2𝒬b+𝒞11TΨb𝒞11+𝒞2TΠb𝒞2𝒞11TΨb𝒟11+𝒞2TΠb𝒟20𝒟11TΨb𝒞11+𝒟2TΠb𝒞2𝒟11TΨb𝒟11+𝒟2TΠb𝒟2]
(48) is equivalent to
(52)[𝒜ℬbI00I]TΥb[𝒜ℬbI00I]<0,
and on the other hand,
(53)[I00]Υb[I00]T=-𝒬b<0.
Combining (52) with (53) and applying Lemma 3, (48) holds if and only if
(54)Υb+[-I𝒜TℬbT]𝒳b[0I000I]+[0I000I]T𝒳bT[-I𝒜TℬbT]T<0,
where 𝒳b introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳b is defined as 𝒳b=[𝒳b1𝒳b1V2], where 𝒳b1=[xb11-xb12xb13x21x220xb31xb32xb33], where x21 and x22 are the same as in Theorem 8. V2 is an given matrix, defined as V2=[v3T00v4T00]T. Pre- and postmultiply (54) by diag{JT,JT,I} and diag{J,J,I}, where J is defined as Theorem 8 and choose Ψb satisfying Proposition 5, (54) is equivalent to(55)[-QbPb-Xb1-Xb1V2*ω-2Qb-𝒞b2T𝒞b2+𝒞11TΨb𝒞11+He(Xb1T𝒜-b)-𝒞b2T𝒟b2+𝒞11TΨb𝒟11+Xb1Tℬ-b+𝒜-bXb1V2**-𝒟b2T𝒟b2+𝒟11TΨb𝒟11+β2ℐ0+He(ℬ-bTXb1V2)]<0,where
(56)[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I],𝒳b1=JTX1J=[YbTEbTLbTMTMT0WbTSbTNbT],𝒜-b=J-1𝒜J=[A0L1B^fCA^fB^fL2000],ℬ-b=J-1ℬb=[B20B^fD200I],𝒞b2=[DfCC^fDfL2],𝒟b2=[DfD20],
Condition (55) is nonlinear owing to the product terms -Cb2TCb2. To solve this problem, condition (55) is equivalent to
(57)Ξb2TΘΞb2<0,
where(58)Ξb2=[I000I000I0𝒞b2𝒟b2],Θ=[-QbPb-X1-X1V20*𝒞11TΨb𝒞11+He(X1T𝒜-bi)X1Tℬ-b+𝒜-biX1V2+𝒞11TΨb𝒟110**𝒟11TΨb𝒟11+β2ℐ0+He(ℬ-bTXb1V2)0000-I].
DefiningG=[0𝒞b2𝒟b2-I] satisfies GΞb2=0. Using Lemma 4, and given a matrix T, (57) is equivalent to
(59)Θ+TG+(TG)T<0,
where one notes T as
(60)T=[0ℛ00]T,ℛ=[Rℐ1TRℐ1TRℐ1T].
Defining A^=MA^f, B^=MB^, C^=RTC^f, and D^=RTDf and after some matrix manipulation, (59) becomes (45). By pre- and postmultiplying (49) by JT and J, respectively, (49) becomes (46).Remark 10.
Theorem8 considers the attenuation performance for the disturbance d(t), and the conservatism comes from the special structure of 𝒳a1 and the same one in 𝒳a. Theorem 9 considers the sensitiveness performance for the fault f(t), and the conservatism comes from the special structure of 𝒳b1 and the same one in 𝒳b. Then, the conservatism for Theorem 9 comes as the same as Theorem 8. However, in order to obtain the better detection performance, the sensitiveness performance is considered over the conservatism.


## 3.3. Stability Condition for the Filter

Due to the fault tolerant controller, the closed-loop system is stable both in the fault-free cases and in the fault cases. Then, the condition that the augmented systems (1) is stable can translate into the condition that the FD filter is stable.Lemma 11 (see [20]).
By considering the FD filter (4), the FD filter is stable, if there exist matrix variables M, A^ and Ps>0 satisfying
(61)[-(M+MT)A^+PsM*-Ps0**-Ps]<0.


## 3.4. Algorithm

In the precious sections, Theorems8 and 9 and Lemma 11 have formulated the inequality conditions for the performance indexes (8) and (9) and the stable condition, respectively. Summarily, we have the following theorem.Theorem 12.
By considering the uncertain system model (1), there exists a FD filter (4) such that the augmented system model (5) is stable and satisfies the performance indices (8) and (9) if inequality conditions (29), (30), (45), (46), and (61) hold.Proof.
Combining Theorems8–9 and Lemma 11, it is obviously that the theorem holds.Remark 13.
In Theorems8–9, if the matrices vi, i∈{1,…,4} are given, all conditions (29), (30), (45), and (46) are the LMI conditions. Specially, if the disturbance and the fault are a scalar, respectively, vi is vector, and it can be easily chosen, which has been further confirmed in the simulink. In addition, Propositions 5 and 7 are used to directly deal with the ellipsoidal uncertainty; the variable structure is complex. However, all the variables in Propositions 5 and 7 is composed of linear matrices, and can be used for LMI conditions.The gain matricesAf, Bf, Cf, and Df defining the FD filter can be derived by the means of a standard procedure. Denote A^, B^, C^, D^, and M is the optimal solution of (7) with condition (29), (30), (45), (46), and (61):(1)
compute thex21, x22 by solving the following factorization problem M=x21Tx22-1x21;(2)
computeA^f, B^f by A^, B^ and M. Compute C^f, Df by C^, D^ and R;(3)
finally, obtain the gain matrices of the FD filter(62)[A^fB^fC^f0]=[x21Tx2200I][AfBfCf0][x22Tx2100I].


## 4. Thresholds Computation

After the parameter matrices of the FD filterAf, Bf, Cf, and Df are designed, similar to that proposed in [21], the residual evaluation function Jr(t) can be selected as
(63)Jr(t)=1t∫0trT(τ)r(τ)dτ.Under fault-free condition, the residualr(t) becomes
(64)r(t)=Grd(jω)d(t),
and namely, the Parseval Theorem (see [22]), one has that
(65)Jr(t)≤∥Grd0∥∞∥d∥rms≤∥Grd0∥∞d¯,
where d- is the upper bound on the disturbance. It is assumed that d- is the known. As a consequence, the following threshold results:
(66)Jth=∥Grd0∥∞d¯.Based on this, the occurrence of faults can be detected by the following logic rule.(67)∥Jr∥≤Jth,noalarm,∥Jr∥>Jth,withalarm.


## 5. Example

In this section, an example is given to illustrate the effectiveness of our design method. Consider the uncertain linear system by(68)A=[0.51-3-3],B1=[0.4-0.5],B2=[0.5-1],L1=[0.200.100000.2],C=[11],D1=0.5,D2=0.5,L2=[0000],H=[1001],Δ=[θ00θ],θ=[θ1θ2],
and the uncertain parameters [θ1Tθ2T]∈𝒰 satisfy {θ∣θTℛθ<1}, as shown in Figure 1, where ℛ=[1.250.250.251.25]. It is worth pointing out that if the method in this paper is adopted, the uncertain region is only in the ellipsoid (solid lines). Otherwise, the ellipsoid has to translate into the convex bounded polyhedral domain (dashed line) for the existing method to deal with the uncertainty, which enlarges the uncertain region and results in increasing the conservatism.Figure 1
Ellipsoidal uncertainty.Given the matrices(69)v1=[0-0.2],v2=[00000000],v3=[-0.5-0.5],v4=[1.51.51.5-1.511-1-1],the disturbance isd(t)=0.8sin(0.1t)exp(-0.05t). Then, the upper bounds on the frequency for the disturbance and the reference input can choose ω-1=0.5 and ω-2=0, respectively. Given γ=1, for the practical needs, we obtain the sensitivity performance β=0.4234 by solving the optimization problem (16) with the conditions in Theorem 12 and the FD filter with gain matrices as
(70)Af=[-3.53040.0850-6.0843-0.2812],Bf=[0.06962.5117],Cf=[-0.0539-0.2526],Df=-1.3339.
Choose θ1=0.2 and θ1=-0.6. Considering the case that the fault occurs between 70sec and 150sec, the residual signal and the residual evaluation function Jr(t) and the threshold Jth(t) are shown in Figure 2. Consider that Jr(t)>Jth(t) at 92.5sec, which means that the fault alarms when the outage fault occurs. Hence, the fault can be detected.The residual signal and the residual evaluation functionJr.
(a)
(b)


## 6. Conclusions

In this paper, the design problem of the FD filter for uncertain linear systems has been addressed. A method for the design problem is given in terms of the linear matrix inequality (LMI) optimization techniques. The new proposed method has two main advantages. The first one is to resolve the problem of designing FD filters subjected to the model with ellipsoidal parametric uncertainty region, while this problem cannot be directly resolved using the existing method. The second one is that the finite-frequency performances of the fault and the disturbance have been detected.



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*Source: 856914-2013-11-28.xml*