Fault Detection for a Class of Uncertain Linear Systems

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.

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. Section 2 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, 𝐴^𝑇 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+𝑀^𝑇. The symbol ∗ within a matrix represents the symmetric entries. 𝜎_min(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: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵₁𝑑(𝑡)+𝐵₂𝑓(𝑡)+𝐿₁𝑤(𝑡),𝑦(𝑡)=𝐶𝑥(𝑡)+𝐷₁𝑑(𝑡)+𝐷₂𝑓(𝑡)+𝐿₂𝑤(𝑡),𝑤(𝑡)(𝑡),𝑧(𝑡)(𝑡)=Δ𝑧=𝐻𝑥Δ∈Δ, (1) 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, 𝐵_𝑢, 𝐶, 𝐷, 𝐿₁, 𝐿₂, and 𝐻 are known constant matrices of appropriate dimensions. Consider that Δ∈Δ, where Δ is defined as Δ=diag𝜃,…,𝜃_𝑛0,𝜃=𝜃^𝑇₁,…,𝜃^𝑇_𝑛𝜃^𝑇∈𝑅^𝑛𝜃, (2) where 𝑛₀ is the number of 𝜃 in Δ, then the dimension of the Δ is 𝑛_𝜃𝑛₀×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: 𝒰=𝜃∣𝜃^𝑇ℛ𝜃<𝒳, (3) 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 residual 𝑟(𝑡) can be obtained as 𝑥̇_𝑓(𝑡)=𝐴_𝑓𝑥_𝑓(𝑡)+𝐵_𝑓𝑦(𝑡),𝑟(𝑡)=𝐶_𝑓𝑥_𝑓(𝑡)+𝐷_𝑓𝑦(𝑡), (4) where 𝑥_𝑓 is the state of the FD filter. Consider that 𝐴_𝑓∈𝑅^{𝑛𝑓×𝑛𝑓}, 𝐵_𝑓∈𝑅^{𝑛𝑓×𝑛𝑚}, 𝐶_𝑓∈𝑅^𝑛𝑓, 𝐷_𝑓∈𝑅^𝑚.

By combining (1) and (4), we have the following augmented systems: 𝜉(𝑡)=𝐴𝜉(𝑡)+𝐵_𝑑𝑑(𝑡)+𝐿₁𝑤(𝑘)+𝐵_𝑓𝑓(𝑡),𝑟(𝑡)=𝐶𝜉(𝑡)+𝐷_𝑑𝑑(𝑡)+𝐷_𝑓𝑓(𝑡)+𝐿₂𝑤(𝑡),𝑤(𝑡)(𝑡)𝑧(𝑡)==Δ𝑧𝐻𝜉(𝑡)Δ∈Δ, (5) where 𝜉(𝑡)=[𝑥(𝑡)^𝑇,𝑥_𝑓(𝑡)^𝑇]^𝑇, 𝐴𝐵_𝑑𝐵_𝑓𝐿₁𝐶𝐷_𝑑𝐷_𝑓𝐿₂=𝐴0𝐵₁𝐵₂𝐿₁𝐵_𝑓𝐶𝐴_𝑓𝐵_𝑓𝐷₁𝐵_𝑓𝐷₂𝐵_𝑓𝐿₂𝐷_𝑓𝐶𝐶_𝑓𝐷_𝑓𝐷₁𝐷_𝑓𝐷₂𝐷_𝑓𝐿₂𝐻=𝐻0. (6)

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: max𝛽s.t.𝛿_max𝐺_𝑟𝑑𝑗𝜔<𝛾,𝛾>0,∀|𝜔|≤𝜔₁𝛿,Δ∈Δ_min𝐺_𝑟𝑓𝑗𝜔>𝛽𝛽>0,∀|𝜔|≤𝜔₂,Δ∈Δ, (7)(8)(9) where 𝜔₁ is the upper bounds on the frequency for the disturbance and 𝜔₂ is the upper bounds on the frequency for the fault. 𝐺_𝑟𝑑 is the transfer function from the disturbance 𝑑 to the residual signal 𝑟 with fault-free case and fault cases. 𝐺_𝑟𝑓 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 matrix Θ∈𝑅^𝑛×𝑛 and two matrices ℳ and ℋ of column dimension 𝑛, there exists matrix ℱ that satisfies Θ+ℳ^𝑇ℱℋ+ℋ^𝑇ℱ^𝑇ℳ<0 if and only if the following two conditions hold: 𝒩^𝑇_ℳΘ𝒩_ℳ<0𝒩_ℋΘ𝒩^𝑇_ℋ<0, (10) where 𝒩_ℳ and 𝒩_ℋ denote arbitrarily bases of null space of ℳ and ℋ, respectively.

Lemma 4 (see [19]). Let 𝜉∈𝑅^𝑛, 𝒩∈𝑅^𝑛×𝑛, and ℳ∈𝑅^𝑛×𝑚 such that rank (𝒩)<𝑛. The following statements are equivalent:(1)𝜉^𝑇ℳ𝜉<0, for ∀𝜉≠0, subject to 𝒩𝜉=0;(2)𝒩^⟂𝑇ℳ𝒩^⟂<0, where 𝒩^⟂ is the kernel of 𝒩;(3)ℳ−𝜀𝒩^𝑇𝒩<0, for some scalar 𝜀∈𝑅;(4)ℳ+𝑋^𝑇𝒩+𝒩^𝑇𝑋<0, for some matrix 𝑋∈𝑅^𝑛×𝑚.

Proposition 5 (see [11]). Consider the uncertainty Δ with 𝜃∈𝒰 defined in (3), Γ∈Γ is defined as Δ𝐼Γ∶=Γ=∣Δ∈Δ. (11) Restrict the parametrization of matrices Λ defined as ΛΛ∶=Λ∣Λ=¹¹Λ₁₂Λ^𝑇₁₂Λ₂₂ (12) with Λ₂₂ as a positive complex Hermitian matrix of dimension 𝑛×𝑛. Consider that Λ₁₁=−(Λ₂₂⊗(𝑅/𝒳)+Λ₁)∈𝑅^{𝑛𝜃𝑛×𝑛𝜃𝑛} and Λ₁₂=0𝑝₁₂⋯𝑝_1𝑛−𝑝₁₂0⋯𝑝_2𝑛⋮⋱⋱⋮−𝑝₁𝑛⋯⋯0∈𝑅^{𝑛𝜃𝑛×𝑛}, (13) where ℛ describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (Λ_𝑎,𝑝_𝑖𝑗) can take any values provided. Consider that Λ_𝑎∈𝑅^{𝑛𝜃𝑛×𝑛𝜃𝑛} has the following structure that: Λ₁∈Λ_𝟏∶0𝑘₁₂⋯𝑘_1𝑛−𝑘₁₂0⋯⋮⋮⋱⋱𝑘_{(𝑛−1)𝑛}−𝑘_1𝑛⋯−𝑘_{(𝑛−1𝑛)}0∣𝑘_𝑖𝑗=−𝑘^𝑇_𝑖𝑗∈𝑅^{𝑛𝜃×𝑛𝜃} (14) and 𝑝_𝑖𝑗∈𝑅^𝑛, 𝑖=1,…,𝑛, 𝑗=1,…,𝑛. For Λ∈Λ, Γ^𝑇ΛΓ>0, ∀Γ∈Γ.

3. Fault Detection Design

In this section, Lemma 6 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: 𝑥̇(𝑡)=𝐴+𝐵₁Δ𝐼−𝐷₁₁Δ⁻¹𝐶₁𝑥(𝑡)+𝐵₂𝑢(𝑡),𝑦(𝑡)=𝐶₂+𝐷₂₁Δ𝐼−𝐷₁₁Δ⁻¹𝐶₁𝑥(𝑡)+𝐷₂₂𝑢(𝑡), (15) with Δ∈Δ. Let a real symmetric matrix Π and a positive scalar 𝜛 be given. The following robust finite-frequency condition 𝐺₁𝑗𝜔^𝑇𝐼Π𝐺₁𝑗𝜔^𝑇𝐼^𝑇<0|𝜔|≤𝜛,Δ∈Δ (16) holds if there exist real symmetric matrices 𝒫, 𝒬, Φ∈Φ, and Λ∈Λ such that 𝒜ℬℐ0^𝑇−𝒬𝒫𝒫𝜛²𝒬<𝒜ℬ𝒞ℐ0𝒟^𝑇𝑇Φ𝒞𝒟,𝒬≥𝒞^𝑇₁Λ𝒞₁, (17)(18) where Φ∶=∶Ψ00ΠΓ00Γ^𝑇Ψ,𝒞Γ00Γ≤0,∀Γ∈Γ𝒜ℬ₁𝒟₁𝒞₂𝒟₂∶=𝐴𝐵₁𝐵₂0000𝐼0𝐼𝐶00₁𝐷₁₁00𝐶₂𝐷₂₁𝐷₂₂000𝒞𝐼0𝒞𝒟∶=¹¹𝒟₁₁𝒞₂𝒟₂=𝒞₁0𝒞₁𝒜𝒞₁ℬ𝒞₂𝒟₂. (19)

Proposition 7. Consider the uncertainty Δ with 𝜃∈𝒰 defined in (3) and Γ∈Γ defined in (11). Restrict the parametrization of matrices Ψ defined as ΨΨ=¹¹Ψ₁₂Ψ^𝑇₁₂Ψ₂₂=Ψ₁₁₁Ψ₁₁₂Ψ₁₂₁Ψ₁₂₂Ψ^𝑇₁₁₂Ψ₁₁₃Ψ^𝑇₁₂₂Ψ₁₂₃Ψ^𝑇₁₂₁Ψ₁₂₂Ψ₂₂₁Ψ₂₂₂Ψ^𝑇₁₂₂Ψ^𝑇₁₂₃Ψ^𝑇₂₂₂Ψ₂₂₃, (20) where Ψ₁₁₃Ψ₁₂₃∗Ψ₂₂₃ are positive complex Hermitian matrices of dimension 2𝑛×2𝑛; Ψ₁₁₁, Ψ₁₂₁, and Ψ₂₂₁ of the dimension 𝑛_𝜃𝑛×𝑛_𝜃𝑛 have the structure with −(Ψ₁₁₃⊗(𝑅/𝒳)+Ψ₁), −(Ψ₁₂₃⊗(𝑅/𝒳)+Ψ₂), and −(Ψ₂₂₃⊗(𝑅/𝒳)+Ψ₃), respectively. MoreoverΨ_𝑝2=0𝑝_𝑝2⋯𝑝_𝑝𝑛−𝑝_𝑝20⋯𝑝_𝑝𝑛⋮⋱⋱⋮−𝑝_𝑝𝑛⋯⋯0∈𝑅^{𝑛𝜃𝑛×𝑛},𝑝=11,12,22, (21) where ℛ describes the form of the ellipsoid for uncertainty in (3), and the elements of this parametrization (Ψ₁, Ψ₂, Ψ₃, 𝑝_1𝑖𝑗, 𝑝_2𝑖𝑗,  𝑝_3𝑖𝑗, 𝑖=1,…,𝑛, 𝑗=1,…,𝑛) can take any values provided. For 𝑡=1,2,3, 𝑖=1,…,𝑛, and 𝑗=1,…,𝑛, Ψ_𝑡∈𝑅^{𝑛𝜃𝑛×𝑛𝜃𝑛} has the following structure: Ψ_𝑡∈Ψ_𝐭∶0𝑘_𝑡12⋯𝑘_𝑡1𝑛−𝑘_𝑡120⋯⋮⋮⋱⋱𝑘_{𝑡(𝑛−1)𝑛}−𝑘_𝑡1𝑛⋯−𝑘_{𝑡(𝑛−1)𝑛}0∣𝑘_𝑡𝑖𝑗=−𝑘^𝑇_𝑡𝑖𝑗∈𝑅^{𝑛𝜃×𝑛𝜃}, (22) and 𝑝_𝑡𝑖𝑗∈𝑅^𝑛. Then, for Ψ∈Ψ, it has Γ00Γ^𝑇ΨΓ00Γ>0,∀Γ∈Γ, (23) where Γ is defined in (11).

Proof. Let all Ψ_𝑝2,Ψ_𝑡,𝑝∈{11,12,22}, 𝑡∈{1,2,3} have the structure in the statement of the proposition: Ψ^𝑇_𝑝2Δ+Δ^𝑇Ψ_𝑝2=0,Δ^𝑇Ψ_𝑡Δ=0. (24) Therefore, for every Ψ∈Ψ, we can haveΓ00Γ^𝑇Ψ=Γ00ΓΔ00𝐼Δ00𝐼^𝑇Ψ=ΔΔ00𝐼Δ00𝐼_𝑇Ψ₁₁₁Δ^𝑇+Ψ^𝑇₁₁₂Δ+Δ^𝑇Ψ₁₁₂+Ψ₁₁₃Δ^𝑇Ψ₁₂₁Δ^𝑇+Ψ^𝑇₁₂₂Δ+Δ^𝑇Ψ₁₂₂+Ψ₁₂₃∗Δ^𝑇Ψ₂₂₁Δ^𝑇+Ψ^𝑇₂₂₂Δ+Δ^𝑇Ψ₂₂₂+Ψ₂₂₃=1−𝜃^𝑇𝑅𝒳𝜃Ψ₁₁₁1−𝜃^𝑇𝑅𝒳𝜃Ψ₁₂₁∗1−𝜃^𝑇𝑅𝒳𝜃Ψ₂₂₁. (25)When 𝜃∈𝒰 and ^Ψ113Ψ123_∗Ψ223>0, we can have Γ00Γ^𝑇ΨΓ00Γ>0,∀Γ∈Γ. (26)

3.1. The Disturbance Attenuation Condition

Theorem 8. Consider the system in (5) with Δ∈Δ. A real symmetric matrix Π_𝑎=¹⁰_0−𝛾²_𝐼 is given. Condition 𝛿_max𝐺_𝑟𝑑𝑗𝜔<𝛾,|𝜔|<𝜔₂,Δ∈Δ (27) holds if there exist matrix variables 𝐴, 𝐵, 𝐶, 𝐷, 𝑀, 𝑅, 𝑌_𝑎, 𝐸_𝑎, 𝐿_𝑎, 𝑁_𝑎, 𝑊_𝑎, 𝑆_𝑎, 𝑃_𝑎=𝑃_𝑎11𝑃_𝑎12𝑃_𝑎13𝑃_𝑎21𝑃_𝑎22𝑃_𝑎23𝑃_𝑎31𝑃_𝑎32𝑃_𝑎33,𝑄_𝑎=𝑄_𝑎11𝑄_𝑎12𝑄_𝑎13𝑄_𝑎21𝑄_𝑎22𝑄_𝑎23𝑄_𝑎31𝑄_𝑎32𝑄_𝑎33,Ψ_𝑎=Ψ_𝑎111Ψ_𝑎112Ψ_𝑎121Ψ_𝑎122Ψ^𝑇_𝑎112Ψ_𝑎113Ψ_𝑎122^𝑇Ψ_𝑎123Ψ^𝑇_𝑎121Ψ_𝑎122Ψ_𝑎221Ψ_𝑎222Ψ^𝑇_𝑎122Ψ^𝑇_𝑎123Ψ^𝑇_𝑎222Ψ_𝑎223,Λ_𝑎=Λ_𝑎11Λ_𝑎12Λ^𝑇_𝑎12Λ_𝑎22 (28) of appropriate dimensions satisfying 𝑄_𝑎>0, Ψ_𝑎∈Ψ,  and Λ_𝑎∈Λ such that−𝑄_𝑎11−𝑄_𝑎12−𝑄_𝑎13𝑃_𝑎11−𝑌^𝑇_𝑎𝑃_𝑎12+𝐸^𝑇_𝑎𝑃_𝑎13−𝐿^𝑇_𝑎𝑎₁₇𝑎₂₈0∗−𝑄_𝑎22−𝑄_𝑎23𝑃_𝑎21−𝑀^𝑇𝑃_𝑎22−𝑀^𝑇𝑃_𝑎23𝑎₂₇𝑎₂₈0∗∗−𝑄_𝑎33𝑄_𝑎31−𝑊^𝑇_𝑎𝑃_𝑎32−𝑆^𝑇_𝑎𝑃_𝑎33−𝑁^𝑇_𝑎𝑎₃₇𝑎₃₈0∗∗∗𝑎₄₄𝑎₄₅𝑎₄₆𝑎₄₇𝑎₄₈𝐷𝐶^𝑇∗∗∗∗𝑎₅₅𝑎₅₆𝑎₅₇𝑎₅₈𝐶^𝑇∗∗∗∗∗𝑎₆₆𝑎₆₇𝑎₆₈𝐷𝐿₂^𝑇∗∗∗∗∗∗𝑎₇₇𝑎₇₈𝐷𝐷₁^𝑇∗∗∗∗∗∗∗𝑎₈₈0(𝑅)𝑄∗∗∗∗∗∗∗∗𝐼−He<0_𝑎11−𝐻^𝑇Λ_𝑎22𝐻𝑄_𝑎12𝑄_𝑎13−𝐻^𝑇Λ^𝑇_𝑎12∗𝑄_𝑎22𝑄_𝑎23∗∗𝑄_𝑎33−Λ_𝑎11>0, (29)(30)where 𝑎₁₇=−𝑌^𝑇_𝑎𝑣₁,𝑎₂₇=−𝑀^𝑇𝑣₁,𝑎₃₇=−𝑊^𝑇_𝑎𝑣₁,𝑎₁₈=−𝑌^𝑇_𝑎𝑣₂,𝑎₂₈=−𝑀^𝑇𝑣₂,𝑎₃₈=−𝑊^𝑇_𝑎𝑣₂,𝑎₄₄=𝜔²₁𝑄_𝑎11+𝐻^𝑇Ψ_𝑎113𝐻+𝐴^𝑇𝐻^𝑇Ψ_𝑎223𝑌𝐻𝐴+He_𝑎𝐴+𝐵𝐶+𝐻^𝑇Ψ_𝑎123,𝑎𝐻𝐴₄₅=𝜔²₁𝑄_𝑎12+𝐴−𝐴^𝑇𝐸^𝑇_𝑎+𝐶^𝑇𝐵^𝑇,𝑎₄₆=𝜔²₁𝑄_𝑎13+𝐻^𝑇Ψ^𝑇_𝑎112+𝐴^𝑇𝐻^𝑇Ψ^𝑇_𝑎122+𝐻^𝑇Ψ_𝑎123𝐻𝐿₁+𝐴^𝑇𝐻^𝑇Ψ_𝑎223𝐻𝐿₁+𝑌_𝑎𝐿₁+𝐵𝐿₂+𝐴^𝑇𝐿^𝑇_𝑎,𝑎₅₅=𝜔²₁𝑄_𝑎22+He𝐴𝑎₅₆=𝜔²₁𝑄_𝑎23−𝐸_𝑎𝐿₁+𝐵𝐿₂𝑎₆₆=𝜔²₁𝑄_𝑎33+Ψ_𝑎111+𝐿^𝑇₁𝐻^𝑇Ψ_𝑎223𝐻𝐿₁+HeΨ_𝑎122𝐻𝐿₁+𝐿_𝑎𝐿₁,𝑎₄₇=𝐻^𝑇Ψ_𝑎123𝐻𝐵₁+𝐴^𝑇𝐻^𝑇Ψ_𝑎223𝐻𝐵₁+𝑌_𝑎𝐵₁+𝐵𝐷₁+𝐴^𝑇𝑌^𝑇_𝑎+𝐶^𝑇𝐵^𝑇𝑣₁,𝑎₄₈=𝐻^𝑇Ψ^𝑇_𝑎122+𝐴^𝑇𝐻^𝑇Ψ^𝑇_𝑎222+𝑊_𝑎+𝐴^𝑇𝑌^𝑇_𝑎+𝐶^𝑇𝐵^𝑇𝑣₂,𝑎₅₇=−𝐸_𝑎𝐵₁+𝐵𝐷₁+𝐴^𝑇𝑣₁,𝑎₅₈=𝑆_𝑎+𝐴^𝑇𝑣₂,𝑎₆₇=Ψ_𝑎122𝐻𝐵₁+𝐿^𝑇₁𝐻^𝑇Ψ_𝑎223𝐻𝐵₁+𝐿_𝑎𝐵₁+𝐿^𝑇₁𝑌^𝑇+𝐿^𝑇₂𝐵^𝑇𝑣₁,𝑎₆₈=Ψ_𝑎121+𝐿^𝑇₁𝐻^𝑇Ψ^𝑇_𝑎222+𝑁+𝐿^𝑇₁𝑌^𝑇_𝑎+𝐿^𝑇₂𝐵^𝑇𝑣₂,𝑎₇₇=−𝛾²𝐼+𝐵^𝑇₁𝐻^𝑇Ψ_𝑎223𝐻𝐵₁+He𝐵^𝑇₁𝑌^𝑇_𝑎+𝐷^𝑇₁𝐵^𝑇𝑣₁,𝑎₇₈=𝐵^𝑇₁𝐻^𝑇Ψ^𝑇_𝑎222+𝑣^𝑇₁𝑊_𝑎+𝐵^𝑇₁𝑌^𝑇_𝑎+𝐷^𝑇₁𝐵^𝑇𝑣₂,𝑎₈₈=Ψ_𝑎221+He𝑣^𝑇₂𝑊_𝑎,Λ∶=Λ_𝑎(),Ψ∶𝑠𝑎𝑡𝑖𝑠𝑓𝑦12𝑖𝑛𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛5Ψ∶=_𝑎∶𝑠𝑎𝑡𝑖𝑠𝑓𝑦(20)𝑖𝑛𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛7, (31)

Proof. Substituting Π_𝑎 into (17), 𝐺_𝑟𝑑(𝑗𝜔)^𝑇𝐺_𝑟𝑑(𝑗𝜔)<𝛾² which is equivalent to (27). By applying Lemma 6, (27) holds if 𝒜ℬ_𝑎ℐ0^𝑇−𝒬_𝑎𝒫_𝑎𝒫_𝑎𝜔²𝒬_𝑎𝒜ℬ_𝑎<𝒞ℐ0_𝑇𝒟^𝑇Ψ_𝑎00Π_𝑎𝒬𝒞𝒟_𝑎>𝒞^𝑇₁Λ_𝑎𝒞₁, (32)(33) where 𝒜ℬ_𝑎𝒞₁𝒟₁𝒞₂𝒟₂∶=𝐴𝐿₁𝐵_𝑑0000𝐼0𝐼00𝐻000𝐶𝐿₂𝐷_𝑑000,𝒞𝐼0𝒞𝒟∶=¹¹𝒟₁₁𝒞₂𝒟₂=𝒞₁0𝒞₁𝒜𝒞₁ℬ_𝑎𝒞₂𝒟₂. (34)
Let Υ_𝑎=^{−𝒬𝑎𝒫𝑎0𝒫}_𝑎^𝜔₂^𝒬_𝑎^+𝒞𝑇₁₁^Ψ_𝑎^𝒞₁₁^+𝒞𝑇₂^Π_𝑎^𝒞₂^𝒞𝑇₁₁^Ψ_𝑎^𝒟₁₁^+𝒞𝑇₂^Π_𝑎^𝒟_{20𝒟𝑇11Ψ𝑎𝒞11+𝒟𝑇2Π𝑎𝒞2𝒟𝑇11Ψ𝑎𝒟11+𝒟𝑇2Π𝑎𝒟2} (35) (32) is equivalent to 𝒜ℬ_𝑎𝐼00𝐼^𝑇Υ_𝑎𝒜ℬ_𝑎𝐼00𝐼<0. (36) And on the other hand, 𝐼00Υ_𝑎𝐼00^𝑇=−𝒬_𝑎<0. (37)
By combining (36) with (37), and applying Lemma 3, (32) holds if and only if Υ_𝑎+𝒜−𝐼_𝑇ℬ^𝑇_𝑎𝒳_𝑎+0𝐼000𝐼0𝐼000𝐼^𝑇𝒳^𝑇_𝑎𝒜−𝐼_𝑇ℬ^𝑇_𝑎^𝑇<0, (38) where 𝒳_𝑎 introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳_𝑎 is defined as 𝒳_𝑎=𝒳_𝑎1𝒳_𝑎1𝑉₁ and partition 𝒳_𝑎1 into 𝑥_𝑎11−𝑥_𝑎12𝑥_𝑎13𝑥₂₁𝑥₂₂0𝑥_𝑎31𝑥_𝑎32𝑥_𝑎33 where 𝑥₂₁ and 𝑥₂₂ are nonsingular matrices variable. 𝑉₁ is a given matrix, defined as 𝑉₁=^𝑣𝑇1_𝑣⁰⁰_𝑇200^𝑇.
Let 𝐽=diag{𝐼,𝑥⁻¹₂₂𝑥₂₁,𝐼}, and define the linearizing change of the control variables as follows:𝑄_𝑎=𝐽^𝑇𝒬_𝑎𝑋𝐽,_𝑎1=𝑌^𝑇_𝑎−𝐸^𝑇_𝑎𝐿^𝑇_𝑎𝑀^𝑇𝑀^𝑇0𝑊^𝑇_𝑎𝑆^𝑇_𝑎𝑁^𝑇_𝑎=𝑥₁₁−𝑥₁₂𝑥⁻¹₂₂𝑥₂₁𝑥₁₃𝑥^𝑇₂₁𝑥^−𝑇₂₂𝑥₂₁𝑥^𝑇₂₁𝑥^−𝑇₂₂𝑥₂₁0𝑥₃₁𝑥₃₂𝑥⁻¹₂₂𝑥₂₁𝑥₃₃,𝑃_𝑎=𝐽^𝑇𝒫_𝑎𝐴𝐽,_𝑓𝐵_𝑓𝐶_𝑓0=𝑥^𝑇₂₁𝑥₂₂0𝐴0𝐼^𝑓𝐵_𝑓𝐶_𝑓0𝑥^𝑇₂₂𝑥₂₁00𝐼 (39)
By Pre- and postmultiplying (38) by diag{𝐽^𝑇,𝐽^𝑇,𝐼} and diag{𝐽,𝐽,𝐼} and choos Ψ_𝑎 satisfying Proposition 5, (38) is equivalent to

^{−𝑄𝑎𝑃𝑎−𝑋𝑎1−𝑋𝑎1𝑉1}_{∗𝜔2𝑄𝑎+𝒞}^𝑇_{11Ψ𝑎𝒞11+𝒞}^𝑇_{2Π𝑎𝒞2+He𝒜}^𝑇_{𝑋𝑎1𝒞}^𝑇_{11Ψ𝑎𝒟11+𝒞}^𝑇_{2Π𝑎𝒟2+𝒜}^𝑇_{𝑋𝑎1𝑉1+𝑋}^𝑇_{𝑎1ℬ𝑎∗∗𝒟𝑇11Ψ𝑎𝒟11+𝒟𝑇2Π𝑎𝒟2+Heℬ𝑇𝑎𝑋𝑎1𝑉1}<0, (40)

where 𝒜, ℬ_𝑎, 𝒞₁₁, 𝒞₂ are the matrices which use the new defined variables 𝐴_𝑓, 𝐵_𝑓,  𝐶_𝑓 in the matrices 𝒜, ℬ_𝑎, 𝒞₁₁, 𝒞₂.

By simple calculation, we can obtain that 𝒞^𝑇₂Π_𝑎𝒞₂𝒞^𝑇₂Π_𝑎𝒟₂∗𝒟^𝑇₂Π_𝑎𝒟₂=0000000000−𝛾²𝐼000+ℭ00_𝑇𝐿^𝑇₂𝐷^𝑇₂0ℭ𝐿₂𝐷₂0=000−𝛾²ℐ₀+ℭ^𝑇₁𝔇^𝑇₂ℭ₁𝔇₂, (41) where ℭ=𝐷_𝑓𝐶𝐶_𝑓, ℭ₁=ℭ𝐿₂, 𝔇₂=𝐷₂0. By applying the Schur complement formula, the following can be obtained:−𝑄_𝑎𝑃_𝑎−𝑋_𝑎1−𝑋_𝑎1𝑉₁0∗𝜔₂𝑄_𝑎+𝒞^𝑇₁₁Ψ_𝑎𝒞₁₁+He𝒜^𝑇𝑋_𝑎1𝒞^𝑇₁₁Ψ_𝑎𝒟₁₁+𝒜^𝑇𝑋_𝑎1𝑉₁+𝑋^𝑇_𝑎1ℬ_𝑎ℭ^𝑇₁∗∗𝒟^𝑇₁₁Ψ_𝑎𝒟₁₁+Heℬ^𝑇_𝑎𝑋_𝑎1𝑉₁−𝛾²ℐ₀𝔇^𝑇₂∗∗∗−𝐼<0. (42)

Then, pre-and postmultiply diag{𝐼,𝐼,𝐼,𝑅^𝑇} to (42). Due to −𝑅^𝑇𝑅≤𝐼−He(𝑅) and defineing 𝐴=𝑀𝐴_𝑓, 𝐵=𝑀𝐵, 𝐶=𝑅^𝑇𝐶_𝑓, 𝐷=𝑅^𝑇𝐷_𝑓, (29) hold. In addition, and pre- and postmultiplying (33) by 𝐽^𝑇₁ 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 Π_𝑏=⁻¹⁰_0𝛽²_𝐼. The following condition 𝛿_min𝐺_𝑟𝑓𝑗𝜔>𝛽,|𝜔|<𝜔₂,Δ∈Δ (43) holds if there exist matrix variables 𝐴, 𝐵, 𝐶, 𝐷, 𝑀, 𝑅, 𝑌_𝑏, 𝐸_𝑏, 𝐿_𝑏, 𝑁_𝑏, 𝑊_𝑏, 𝑆_𝑏, 𝑃_𝑏=𝑃_𝑏11𝑃_𝑏12𝑃_𝑏13𝑃_𝑏21𝑃_𝑏22𝑃_𝑏23𝑃_𝑏31𝑃_𝑏32𝑃_𝑏33,𝑄_𝑏=𝑄_𝑏11𝑄_𝑏12𝑄_𝑏13𝑄_𝑏21𝑄_𝑏22𝑄_𝑏23𝑄_𝑏31𝑄_𝑏32𝑄_𝑏33Ψ_𝑏=Ψ_𝑏111Ψ_𝑏112Ψ_𝑏121Ψ_𝑏122Ψ^𝑇_𝑏112Ψ_𝑏113Ψ_𝑏122^𝑇Ψ_𝑏123Ψ^𝑇_𝑏121Ψ_𝑏122Ψ_𝑏221Ψ_𝑏222Ψ^𝑇_𝑏122Ψ^𝑇_𝑏123Ψ^𝑇_𝑏222Ψ_𝑏223,Λ_𝑏=Λ_𝑏11Λ_𝑏12Λ^𝑇_𝑏12Λ_𝑏22 (44) of appropriate dimensions satisfying 𝑄_𝑏>0, Ψ_𝑏∈Ψ, and  Λ_𝑏∈Λ such that−𝑄_𝑏11−𝑄_𝑏12−𝑄_𝑏13𝑃_𝑏11−𝑌^𝑇_𝑏𝑃_𝑏12−𝐸^𝑇_𝑏𝑃_𝑏13−𝐿^𝑇_𝑏−𝑌^𝑇_𝑏𝑣₃−𝑌^𝑇_𝑏𝑣₄0∗−𝑄_𝑏22−𝑄_𝑏23𝑃_𝑏21−𝑀^𝑇𝑃_𝑏22−𝑀^𝑇𝑃_𝑏23−𝑀^𝑇𝑣₃−𝑀^𝑇𝑣₄0∗∗−𝑄_𝑏33𝑃_𝑏31−𝑊^𝑇_𝑏𝑃_𝑏32−𝑆^𝑇_𝑏𝑃_𝑏33−𝑁^𝑇_𝑏−𝑊^𝑇_𝑏𝑣₃−𝑊^𝑇_𝑏𝑣₄0∗∗∗𝑐₄₄𝑏₄₅𝑏₄₆𝑏₄₇𝑏₄₈−ℐ₁𝑅^𝑇∗∗∗∗𝑏₅₅𝐵₅₆𝑏₅₇𝑏₅₈−ℐ₁𝑅^𝑇∗∗∗∗∗𝑏₆₆𝑏₆₇𝑏₆₉−ℐ₁𝑅^𝑇∗∗∗∗∗∗𝑏₇₇𝑏₇₈0∗∗∗∗∗∗∗𝑏₈₈0𝑄∗∗∗∗∗∗00−𝐼<0,_𝑏11−𝐻^𝑇Λ_𝑏22𝐻𝑄_𝑏12𝑄_𝑏13−𝐻^𝑇Λ^𝑇_𝑏12∗𝑄_𝑏22𝑄_𝑏23∗∗𝑄_𝑏33−Λ_𝑏11>0, (45)(46)where 𝑏₄₄=𝜔²₂𝑄_𝑏11+𝐻^𝑇Ψ₁₁₃𝐻+𝐴^𝑇𝐻^𝑇Ψ_𝑏223𝑌𝐻𝐴+He_𝑏𝐴+𝐵𝐶+ℐ₁𝐷𝐶+𝐻^𝑇Ψ_𝑏123,𝑏𝐻𝐴₄₅=𝜔²₂𝑄_𝑏12+𝐴+𝐴^𝑇𝐸^𝑇_𝑏+𝐶^𝑇𝐵^𝑇+𝐶^𝑇𝐵^𝑇+ℐ₁𝐷𝐶^𝑇+ℐ₁𝑏𝐶,₄₆=𝜔²₂𝑄_𝑏13+𝐻^𝑇Ψ^𝑇_𝑏112+𝐴^𝑇𝐻^𝑇Ψ^𝑇_𝑏122+𝐻^𝑇Ψ_𝑏123𝐻𝐿₁+𝐴^𝑇𝐻^𝑇Ψ_𝑏223𝐻𝐿₁+𝐵𝐿₂+𝐴^𝑇𝐿^𝑇_𝑏+ℐ₁𝐷𝐿₂+𝐶^𝑇𝐷^𝑇ℐ^𝑇₁,𝑏₅₅=𝜔²₂𝑄_𝑏22+He𝐴+ℐ₁𝐶,𝑏₅₆=𝜔²₂𝑄_𝑏23+𝐸𝐿₁+𝐵𝐿₂+ℐ₁𝐷𝐿₂+𝐶^𝑇ℐ^𝑇₁,𝑏₆₆=𝜔²₂𝑄_𝑏33+Ψ_𝑏111+𝐿^𝑇₁𝐻^𝑇Ψ_𝑏223𝐻𝐿₁+HeΨ_𝑏122𝐻𝐿₁+𝐿_𝑏𝐿₁+ℐ₁𝐷𝐿₂,𝑏₄₇=𝐻^𝑇Ψ_𝑏123𝐻𝐵₂+𝐴^𝑇𝐻^𝑇Ψ_𝑏223𝐻𝐵₂+𝑌_𝑏𝐵₂+𝐵𝐷₂+𝑌_𝑏𝐴+𝐵𝐶^𝑇𝑣₃+ℐ₁𝐷𝐷₂,𝑏₄₈=𝐻^𝑇Ψ^𝑇_𝑏122+𝐴^𝑇𝐻^𝑇Ψ^𝑇_𝑏222+𝑊_𝑏+𝑌_𝑏𝐴+𝐵𝐶^𝑇𝑣₄,𝑏₅₇=𝐸_𝑏𝐵₂+𝐵𝐷₂+𝐴^𝑇𝑣₃+ℐ₁𝐷𝐷₂,𝑏₅₈=𝑆_𝑏+𝐴^𝑇𝑣₄,𝑏₆₇=Ψ_𝑏122𝐻𝐵₂+𝐿^𝑇₁𝐻^𝑇Ψ_𝑏223𝐻𝐵₂+𝐿_𝑏𝐵₂+𝑌𝐿₁+𝐵𝐿₂^𝑇𝑣₃+ℐ₁𝐷𝐷₂,𝑏₆₈=Ψ_𝑏121+𝐿^𝑇₁𝐻^𝑇Ψ^𝑇_𝑏222+𝑁_𝑏+𝑌_𝑏𝐿₁+𝐵𝐿₂^𝑇𝑣₄,𝑏₇₇=𝛽²𝐼+𝐵^𝑇₂𝐻^𝑇Ψ_𝑎223𝐻𝐵₂+He𝑌_𝑏𝐵₂+𝐵𝐷₂^𝑇𝑣₃,𝑏₇₈=𝐵^𝑇₂𝐻^𝑇Ψ^𝑇_𝑏222+𝑣^𝑇₃𝑊_𝑏+𝑌_𝑏𝐵₂+𝐵𝐷₂^𝑇𝑣₄,𝑏₈₈=Ψ_𝑏221+He𝑣^𝑇₄𝑊_𝑏,Λ∶=Λ_𝑏(),Ψ∶𝑠𝑎𝑡𝑖𝑠𝑓𝑦12𝑖𝑛𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛5Ψ∶=_𝑏∶𝑠𝑎𝑡𝑖𝑠𝑓𝑦(20)𝑖𝑛𝑃𝑟𝑜𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛7, (47)

Proof. Substituting Π_𝑏 into (17), 𝐺_{𝑟𝑟0𝑖}(𝑗𝜔)^𝑇𝐺_{𝑟𝑟0𝑖}(𝑗𝜔)<𝛽²𝐼 which is equivalent to (43). Applying Lemma 6, (43) holds if 𝒜ℬ_𝑏ℐ0^𝑇−𝒬_𝑏𝒫_𝑏𝒫_𝑏𝜔²𝒬_𝑏𝒜ℬ_𝑏<𝒞ℐ0_𝑇𝒟^𝑇Ψ_𝑏00Π_𝑏𝒬𝒞𝒟_𝑏>𝒞^𝑇₁Λ_𝑏𝒞₁ (48)(49) hold.
Where 𝒜ℬ_𝑏𝒞₁𝒟₁𝒞₂𝒟₂∶=𝐴𝐿₁𝐵_𝑓0000𝐼0𝐼00𝐻000𝐶𝐿₂𝐷_𝑓000𝒞𝐼0𝒞𝒟∶=¹¹𝒟₁₁𝒞₂𝒟₂=𝒞₁0𝒞₁𝒜𝒞₁ℬ_𝑏𝒞₂𝒟₂. (50)
Let Υ_𝑏=^{−𝒬𝑏𝒫𝑏0𝒫}_𝑏^𝜔₂^𝒬_𝑏^+𝒞𝑇₁₁^Ψ_𝑏^𝒞₁₁^+𝒞𝑇₂^Π_𝑏^𝒞₂^𝒞𝑇₁₁^Ψ_𝑏^𝒟₁₁^+𝒞𝑇₂^Π_𝑏^𝒟_{20𝒟𝑇11Ψ𝑏𝒞11+𝒟𝑇2Π𝑏𝒞2𝒟𝑇11Ψ𝑏𝒟11+𝒟𝑇2Π𝑏𝒟2} (51) (48) is equivalent to 𝒜ℬ_𝑏𝐼00𝐼^𝑇Υ_𝑏𝒜ℬ_𝑏𝐼00𝐼<0, (52) and on the other hand, 𝐼00Υ_𝑏𝐼00^𝑇=−𝒬_𝑏<0. (53)
Combining (52) with (53) and applying Lemma 3, (48) holds if and only if Υ_𝑏+𝒜−𝐼_𝑇ℬ^𝑇_𝑏𝒳_𝑏+0𝐼000𝐼0𝐼000𝐼^𝑇𝒳^𝑇_𝑏𝒜−𝐼_𝑇ℬ^𝑇_𝑏^𝑇<0, (54) where 𝒳_𝑏 introduced by Lemma 3 is the variable matrix of appropriate dimensions. Here, 𝒳_𝑏 is defined as 𝒳_𝑏=𝒳_𝑏1𝒳_𝑏1𝑉₂, where 𝒳_𝑏1=^{𝑥𝑏11−𝑥𝑏12𝑥𝑏13𝑥}₂₁^𝑥₂₂⁰_{𝑥𝑏31𝑥𝑏32𝑥𝑏33}, where 𝑥₂₁ and 𝑥₂₂ are the same as in Theorem 8. 𝑉₂ is an given matrix, defined as 𝑉₂=^𝑣𝑇3_𝑣⁰⁰_𝑇400^𝑇. Pre- and postmultiply (54) by diag{𝐽^𝑇,𝐽^𝑇,𝐼} and diag{𝐽,𝐽,𝐼}, where 𝐽 is defined as Theorem 8 and choose Ψ_𝑏 satisfying Proposition 5, (54) is equivalent to^{−𝑄𝑏𝑃𝑏−𝑋𝑏1−𝑋𝑏1𝑉2}_{∗𝜔2𝑄𝑏−𝒞}^𝑇_{𝑏2𝒞𝑏2+𝒞}^𝑇_{11Ψ𝑏𝒞11+He𝑋}^𝑇_{𝑏1𝒜𝑏−𝒞}^𝑇_{𝑏2𝒟𝑏2+𝒞}^𝑇_{11Ψ𝑏𝒟11+𝑋}^𝑇_{𝑏1ℬ𝑏+𝒜𝑏𝑋𝑏1𝑉2∗∗−𝒟𝑇𝑏2𝒟𝑏2+𝒟𝑇11Ψ𝑏𝒟11+𝛽2ℐ0+Heℬ𝑇𝑏𝑋𝑏1𝑉2}<0, (55)where 𝐴_𝑓𝐵_𝑓𝐶_𝑓0=𝑥^𝑇₂₁𝑥₂₂0𝐴0𝐼^𝑓𝐵_𝑓𝐶_𝑓0𝑥^𝑇₂₂𝑥₂₁0,𝒳0𝐼_𝑏1=𝐽^𝑇𝑋₁𝑌𝐽=^𝑇𝑏𝐸^𝑇_𝑏𝐿^𝑇_𝑏𝑀^𝑇𝑀^𝑇0𝑊^𝑇_𝑏𝑆^𝑇_𝑏𝑁^𝑇_𝑏,𝒜_𝑏=𝐽⁻¹𝒜𝐽=𝐴0𝐿₁𝐵_𝑓𝐶𝐴_𝑓𝐵_𝑓𝐿₂,000ℬ_𝑏=𝐽⁻¹ℬ_𝑏=𝐵₂0𝐵_𝑓𝐷₂0,𝒞0𝐼_𝑏2=𝐷_𝑓𝐶𝐶_𝑓𝐷_𝑓𝐿₂,𝒟_𝑏2=𝐷_𝑓𝐷₂0, (56)
Condition (55) is nonlinear owing to the product terms −𝐶^𝑇_𝑏2𝐶_𝑏2. To solve this problem, condition (55) is equivalent to Ξ^𝑇_𝑏2ΘΞ_𝑏2<0, (57) whereΞ_𝑏2=𝐼000𝐼000𝐼0𝒞_𝑏2𝒟_𝑏2,Θ=−𝑄_𝑏𝑃_𝑏−𝑋₁−𝑋₁𝑉₂0∗𝒞^𝑇₁₁Ψ_𝑏𝒞₁₁+He𝑋^𝑇₁𝒜_𝑏𝑖𝑋^𝑇₁ℬ_𝑏+𝒜_𝑏𝑖𝑋₁𝑉₂+𝒞^𝑇₁₁Ψ_𝑏𝒟₁₁0∗∗𝒟^𝑇₁₁Ψ_𝑏𝒟₁₁+𝛽²ℐ₀+Heℬ^𝑇_𝑏𝑋_𝑏1𝑉₂0.000−𝐼 (58)
Defining 𝐺=0𝒞_𝑏2𝒟_𝑏2−𝐼 satisfies 𝐺Ξ_𝑏2=0. Using Lemma 4, and given a matrix 𝑇, (57) is equivalent to Θ+𝑇𝐺+(𝑇𝐺)^𝑇<0, (59) where one notes 𝑇 as 𝑇=0ℛ00^𝑇,ℛ=𝑅ℐ^𝑇₁𝑅ℐ^𝑇₁𝑅ℐ^𝑇₁. (60) Defining 𝐴=𝑀𝐴_𝑓, 𝐵=𝑀𝐵, 𝐶=𝑅^𝑇𝐶_𝑓, and 𝐷=𝑅^𝑇𝐷_𝑓 and after some matrix manipulation, (59) becomes (45). By pre- and postmultiplying (49) by 𝐽^𝑇 and 𝐽, respectively, (49) becomes (46).

Remark 10. Theorem 8 considers the attenuation performance for the disturbance 𝑑(𝑡), and the conservatism comes from the special structure of 𝒳_𝑎1 and the same one in 𝒳_𝑎. Theorem 9 considers the sensitiveness performance for the fault 𝑓(𝑡), and the conservatism comes from the special structure of 𝒳_𝑏1 and the same one in 𝒳_𝑏. 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 𝑃_𝑠>0 satisfying −𝑀+𝑀^𝑇𝐴+𝑃_𝑠𝑀∗−𝑃_𝑠0∗∗−𝑃_𝑠<0. (61)

3.4. Algorithm

In the precious sections, Theorems 8 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 Theorems 8–9 and Lemma 11, it is obviously that the theorem holds.

Remark 13. In Theorems 8–9, if the matrices 𝑣_𝑖, 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, 𝑣_𝑖 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 matrices 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, and 𝐷_𝑓 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 the 𝑥₂₁, 𝑥₂₂ by solving the following factorization problem 𝑀=𝑥^𝑇₂₁𝑥⁻¹₂₂𝑥₂₁;(2)compute 𝐴_𝑓, 𝐵_𝑓 by 𝐴, 𝐵 and 𝑀. Compute 𝐶_𝑓, 𝐷_𝑓 by 𝐶, 𝐷 and 𝑅; (3)finally, obtain the gain matrices of the FD filter 𝐴_𝑓𝐵_𝑓𝐶_𝑓0=𝑥^𝑇₂₁𝑥₂₂0𝐴0𝐼^𝑓𝐵_𝑓𝐶_𝑓0𝑥^𝑇₂₂𝑥₂₁00𝐼. (62)

4. Thresholds Computation

After the parameter matrices of the FD filter 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, and 𝐷_𝑓 are designed, similar to that proposed in [21], the residual evaluation function 𝐽_𝑟(𝑡) can be selected as 𝐽_𝑟(𝑡)=1𝑡^𝑡₀𝑟^𝑇(𝜏)𝑟(𝜏)𝑑𝜏. (63)

Under fault-free condition, the residual 𝑟(𝑡) becomes 𝑟(𝑡)=𝐺_𝑟𝑑𝑗𝜔𝑑(𝑡), (64) and namely, the Parseval Theorem (see [22]), one has that 𝐽_𝑟(𝑡)≤𝐺_𝑟𝑑0∞‖𝑑‖_rms≤𝐺_𝑟𝑑0∞𝑑, (65) where 𝑑 is the upper bound on the disturbance. It is assumed that 𝑑 is the known. As a consequence, the following threshold results: 𝐽_th=𝐺_𝑟𝑑0∞𝑑. (66)

Based on this, the occurrence of faults can be detected by the following logic rule. 𝐽_𝑟≤𝐽_th𝐽,noalarm,_𝑟>𝐽_th,withalarm. (67)

5. Example

In this section, an example is given to illustrate the effectiveness of our design method. Consider the uncertain linear system by 𝐴=0.51−3−3,𝐵₁=0.4−0.5,𝐵₂=,𝐿0.5−1₁=,𝐷0.200.100000.2,𝐶=11₁=0.5,𝐷₂𝐿=0.5,₂=,𝜃0000,𝐻=1001Δ=𝜃00𝜃,𝜃=¹𝜃₂, (68) and the uncertain parameters ^𝜃𝑇1_𝜃𝑇2∈𝒰 satisfy {𝜃∣𝜃^𝑇ℛ𝜃<1}, as shown in Figure 1, where ℛ=^1.250.25_0.251.25. 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 𝑣₁=0−0.2,𝑣₂=,𝑣00000000₃=−0.5−0.5,𝑣₄=,1.51.51.5−1.511−1−1 (69)

the disturbance is 𝑑(𝑡)=0.8sin(0.1𝑡)exp(−0.05𝑡). Then, the upper bounds on the frequency for the disturbance and the reference input can choose 𝜔₁=0.5 and 𝜔₂=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 𝐴_𝑓=−3.53040.0850−6.0843−0.2812,𝐵_𝑓=,𝐶0.06962.5117_𝑓=−0.0539−0.2526,𝐷_𝑓=−1.3339. (70) 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 𝐽_𝑟(𝑡) and the threshold 𝐽_th(𝑡) are shown in Figure 2. Consider that 𝐽_𝑟(𝑡)>𝐽_th(𝑡) at 92.5sec, which means that the fault alarms when the outage fault occurs. Hence, the fault can be detected.

(a)

(b)

Figure 2 

The residual signal and the residual evaluation function 𝐽𝑟.

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.