# H−/L∞ Fault Detection Observer Design for Uncertain Lipschitz Nonlinear Systems

Authors: Yufeng Qin; Xianjun Shi; Yufeng Long; Jiapeng Lv
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper deals with the robust fault detection problem in the finite frequency domain for a class of Lipschitz nonlinear systems with uncertain parameters. The design conditions ofH−/L∞ fault detection observers are proposed. These conditions make the error estimation system stable, and the residual has L∞ disturbance attenuation performance for external disturbances and uncertain parameters. In addition, considering that fault occurs in the finite frequency domain, the residual also has H− fault sensitivity performance. To ease of solution, the observer design conditions are further transformed into linear matrix inequalities. On this basis, a dynamic threshold design method based on L∞ analysis is presented. Finally, simulation results demonstrate the superiority of the proposed method.


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## Body

## 1. Introduction

The structure and function of the control system are becoming more complicated, which requires the system has better reliability and testability. Therefore, fault diagnosis techniques are also constantly developed [1–3]. Fault detection based on state observer is one of the most widely used fault detection methods. Generating residuals by comparing the state of the observer and the system and determining whether a fault occurs according to residual exceeds the threshold or not.Since it is impossible to obtain a complete and accurate mathematical model, therefore, the residual must be robust to external disturbance and uncertain parameters. Patton and Chen proposed a disturbance decoupling method based on characteristic structure configuration [4]. Dong et al. investigated the robust fault detection filter design problem for a class of discrete-time conic-type nonlinear Markov jump systems with jump fault signals [5]. Yin et al. investigated the problem of robust fault diagnosis of stochastic discrete-event systems against model uncertainty [6]. However, our fundamental purpose is to detect the fault accurately and timely, so a balance needs to be struck between robustness and fault sensitivity. Hou and Patton used H− index to measure the sensitivity of residual to fault and proposed the concept of H−/H∞ observer [7]. Estrada and Wei designed H−/H∞ fault detection observer for uncertain linear systems, respectively [8, 9]. This method has been applied to uncertain nonlinear systems [10–13].All the above methods design the observer in full frequency domain. In fact, faults usually occur in finite frequency domain. For example, slow-varying fault usually occurs in low-frequency bandwidth, and the frequency of abrupt fault is zero. In addition, designing an observer in finite frequency domain can reduce the design conservatism [14]. Iwasaki and Hara proposed the generalized Kalman-Yakubovich-Popov lemma and solved the H− performance analysis problem in finite frequency domain [15]. Subsequently, many works have been done on the design of H−/H∞ fault detection observer for linear systems in finite frequency domain [16–19]. Zhai et al. proposed a robust H−/H∞ fault detection method for spatial interconnected systems (SISs) with polyhedral uncertainties in the finite frequency domain [20]. Wang considers parametric uncertainties and applies the method to uncertain linear systems [21].However, as a gain measure of energy-to-energy, theH∞ norm has some defects when applied to residual evaluation. The reason is that the disturbance signal is unknown, and it is difficult to calculate the energy of the disturbance signal in the whole time domain. In addition, the periodic signal and the step signal are energy unbounded. While L∞ norm describes the peak value of the signal, it is easy to obtain and more suitable for residual calculation. Some achievements have been made on the H−/L∞ fault detection observer for linear systems [22–25]. In practical applications, most systems are nonlinear, and the above methods are not applicable. For a class of Lipschitz nonlinear systems, Zhou proposed an H−/L∞ observer design method, but this method does not consider the uncertainty of model parameters [26]. Uncertain parameters and the nonlinear term bring great challenges to the index analysis of H− and L∞ in finite frequency domain. Therefore, this paper proposes an H−/L∞ fault detection observer design method in finite frequency domain for a class of Lipschitz nonlinear systems with uncertain parameters.


### 1.1. Notation

This paper uses following mathematical notations:ℝn×m denotes n×m dimensional real matrices. 0 and I denote the zero matrix and identity matrix with appropriate dimensions, respectively. For a matrix A, AT and A∗ denote the transpose and the complex conjugate transpose of A. HeA=A+A∗ denotes the sum of matrix A and its complex conjugate transpose. For a matrix P, P>0 or P<0 denote P is a positive definite matrix or a negative definite matrix. λminP and λmaxP denote the minimum and maximum eigenvalue of P. For a signal xt∈ℝn, the L2 norm is defined as xt=xtTxt, the L∞ norm is defined as xt∞=supt≥0xt, and ℱxt denote the Fourier transform of xt.


## 1.1. Notation

This paper uses following mathematical notations:ℝn×m denotes n×m dimensional real matrices. 0 and I denote the zero matrix and identity matrix with appropriate dimensions, respectively. For a matrix A, AT and A∗ denote the transpose and the complex conjugate transpose of A. HeA=A+A∗ denotes the sum of matrix A and its complex conjugate transpose. For a matrix P, P>0 or P<0 denote P is a positive definite matrix or a negative definite matrix. λminP and λmaxP denote the minimum and maximum eigenvalue of P. For a signal xt∈ℝn, the L2 norm is defined as xt=xtTxt, the L∞ norm is defined as xt∞=supt≥0xt, and ℱxt denote the Fourier transform of xt.


## 2. Problem Description

Considering the following system(1)x˙t=A+ΔAtxt+But+φxt+D1dt+Fft,yt=Cxt+D2dt,where xt∈ℝn, yt∈ℝm, ut∈ℝr denote the state, output, and input vectors, respectively. dt∈ℝp denote the external disturbance and measurement noise. ft∈ℝq denotes the fault. A, B, C, D1, D2, F are known matrices with appropriate dimensions, respectively. ΔAt denotes uncertain parameters. φxt is the nonlinear term and satisfies the Lipschitz condition(2)φx1t−φx2t≤γx1t−x2t,where γ is a known scalar representing the Lipschitz constant.Assumption 1.
The time-varying matrixΔAtsatisfiesΔAt=M∑tN, whereM,Nare known.∑tis unknown and satisfies∑tT∑t≤I.Assumption 2.
dtsatisfiesdt∞≤d¯, whered¯is a known scalar.
Without causing confusion,xt is denoted as x for simplicity, and the other variables are denoted in the same way. Constructing the following fault detection observer as(3)x^˙=Ax^+Bu+φx^+Ly−Cx^,r=y−Cx^,where x^ is the state vector of the observer, r is the residual vector, and L is a matrix that needs to be designed. The state estimate error is denoted as(4)e=x−x^.
So we can get(5)e˙=A−LCe+ΔAx+Δφ+D1−LD2d+Ff,r=Ce+D2d.where Δφ=φx−φx^. The finite frequency domain can be expressed by [25](6)Ω≔ω∈ℝ|τω−ω1ω−ω2≤0,where ω is the frequency of fault signal, ω1 and ω2 are known scalars. When τ=1, −ω1≤ω2≤ωl, we can get ω≤ωl, it means that the fault occurs in the low-frequency bandwidth, where ωl denotes the boundary of low-frequency bandwidth. When τ=1, ω1<ω2, we can get ω1≤ω≤ω2, and it means that the fault occurs in the middle frequency bandwidth. When τ=−1, −ω1=ω2=ωh, we can get ω≥ωh, and it means that the fault occurs in the high-frequency bandwidth, where ωh denotes the boundary of high-frequency bandwidth. The observer design conditions are proposed as follows.(i)
The state estimation error system (5) is stable.(ii)
Under zero initial condition, state estimation error system (5) has L∞ performance μ(7)r2≤μ2d¯2+χe0,t+μ1ξ,(i)
whereμ=μ1+μ2, μ1 and μ2 are scalars need to be designed. χe0,t is a time-varying function related to the initial estimate error value e0, when t⟶∞, χe0,t⟶0. ξ satisfies(8)ξ=α∫0te−αt−τNx^dτ,whereα>0 is a known scalar.(i)
When a fault occurs in the finite frequency domain (6), the state estimate error e satisfies(9)∫0∞τω1e+je˙ω2e+je˙∗dt≤0.
And the state estimation error system (5) has H− performance β(10)∫0∞rTrdt≥β2∫0∞fTfdt.


## 3.H−/L∞ Fault Detection Observer Design

### 3.1. Observer Design Method

The state estimation error system (5) can be rewritten as(11)e˙=Al+ΔAe+M∑Nx^+Δφ+D1−LD2d+Ff,r=Ce+D2d,where Al=A−LC. When only considering the influence of external disturbance and initial conditions on the residual, (11) can be represented as(12)e˙d=Al+ΔAed+M∑Nx^+Δφ+D1−LD2d,rd=Ced+D2d.When only considering the influence of fault on the residual, (11) can be represented as(13)e˙f=Al+ΔAef+Δφ+Ff,rf=Cef.


### 3.2. Stability andL∞ Disturbance Attenuation Performance

Theorem 1.
Given scalarsγ>0, α>0, μ1>0, the state estimation error system (5) satisfies (a) and (b) if there exist scalars δ1>0, μ2>0, and matrices P1=P1T>0∈ℝn×n, L∈ℝn×m such that(14)Ξ11P1MP1MP1D1−LD2P1⋆−I000⋆⋆−αI00⋆⋆⋆−αI0⋆⋆⋆⋆−δ1I<0,(15)CTC−μ1P1CTD2⋆−μ2I+D2TD2<0,where Ξ11=HeP1Al+NTN+δ1γ2I+αP1.Proof.
We take the following Lyapunov function(16)Vd=edTP1ed.
Then, we can obtain(17)V˙d=e˙dTP1ed+edTP1e˙d.
According to (12), V˙d is obtained as(18)V˙d=edTHeP1Aled+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx=edTHeP1Aled+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx^+2edTP1M∑Ned.Lemma 1.
Suppose matricesV, S have appropriate dimensions, if matrix Σ satisfies ΣTΣ≤I, then the following inequality can be established for any scalar ϵ>0 and vector x,y∈ℝn [27].(19)2xTV∑Sy≤1ϵxTVVTx+ϵyTSTSy.
According to (19), set ϵ=1, and we can get(20)2edTP1M∑Ned≤edTP1MP1MTed+edTNTNed.
The following inequality can be derived as(21)V˙d≤edTHeP1Al+NTN+P1MP1MTed+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx^.
IfV˙d satisfies(22)V˙d≤−αVd+αdTd+αx¯Tx¯.where x¯=ΣNx^. We derive the following inequality if we integrate (22)(23)Vd≤Vd0e−αt+α∫0te−αt−τdTddτ+α∫0te−αt−τx¯Tx¯dτ.
According to assumption2, (23) is equivalent to the following formula:(24)Vd≤Vd0e−αt+d¯2∫0tαe−αt−τdτ+α∫0te−αt−τx¯Tx¯dτ.
According to assumption1, we can obtain the following formula:(25)x¯Tx¯=∑Nx^T∑Nx^=x^TNTΣT∑Nx^≤Nx^TNx^.
Then,(26)Vd≤Vd0e−αt+1−e−αtd¯2+α∫0te−αt−τNx^TNx^dτ≤Vd0e−αt+d¯2+α∫0te−αt−τNx^2dτ.
Therefore, whend=0, x^=0, Vd satisfies Vd≤Vd0e−αt. We obtain(27)λminP1ed2≤edTP1ed≤edT0P1ed0e−αt≤λmaxP1ed02e−αt.
It follows thated2≤λmaxP1/λminP1ed02e−αt. When t⟶∞, so that ed⟶0, namely the state estimation error system (5) satisfies (a). According to Lipschitz condition, δ1ΔφTΔφ≤δ1γ2edTed. If the following inequality is satisfied,(28)V˙d+αVd−αdTd−αx¯Tx¯+δ1γ2edTed−δ1ΔφTΔφ≤0.
Then, the inequality (22) can be derived. Substituting (16) and (21) into (28), it follows that(29)edx¯dΔφTΓ11P1MP1D1−LD2dP1⋆−αI00⋆⋆−αI0⋆⋆⋆−δ1Iedx¯dΔφ<0.where Γ11=HeP1Al+NTN+P1MP1MT+αP1+δ1γ2I. According to Schur’s complement lemma, (29) is equivalent to (14). Besides, the following inequality can be obtained from (15)(30)eddTCTC−μ1P1CTD2⋆−μ2I+D2TD2edd<0.
(30) is equivalent to the following formula:(31)rdTrd≤μ1Vd+μ2dTd.
Combining (26) with (31), the following formula can be obtained:(32)rd2≤μ1Vd0e−αt+μ1d¯2+μ1ξ+μ2dTd≤μ2d¯2+χed0,t+μ1ξ,where χed0,t=μ1edT0P1ed0e−αt, namely the state estimation error system (5) satisfies (b).


### 3.3.H− Fault Sensitive Performance

Theorem 2.
Given a scalarγ>0, the state estimation error system (5) satisfies (c) if there exist scalars δ2>0, β>0, and matrices P2=P2T>0∈ℝn×n, L∈ℝn×m, Q=QT∈ℝn×n satisfies τQ>0 such that(33)Θ11Θ12Θ13★Θ22Θ23★★Θ33<0,whereΘ11=HeP2Al+ΔA−ω1ω2Q−CTC−Al+ΔATQAl+ΔA+jω0Al+ΔATQ−jω0QAl+ΔA+δ2γ2I,Θ12=P2F−Al+ΔATQF−jω0QF,Θ13=P2−Al+ΔATQ−jω0Q,Θ22=β2I−FTQF,Θ23=−FTQ,Θ33=−Q−δ2I,ω0=ω1+ω2/2.Proof.
We take the following Lyapunov function(34)Vf=efTP2ef.
Then, we derive the following formula:(35)V˙f=efTHeP2Al+ΔAef+2efTP2Ff+2efTP2Δφ.
Considering the fault occurs in finite frequency domain, and theH− performance index is expressed as(36)J_=∫0∞V˙f+β2fTf−rfTrf−trHeω1ef+je˙fω2ef+je˙f∗Q2dt.
Under zero initial condition, ifJ−≤0, it follows that(37)∫0∞β2fTf−rfTrf−trHeω1ef+je˙fω2ef+je˙f∗Q2dt≤0.
DefineΦ=∫0∞ω1ef+je˙fω2ef+je˙f∗dt. According to Parseval’s theorem [28],(38)Φ=12Π∫−∞+∞Fω1ef+je˙fFω2ef+je˙f∗dω=12Π∫−∞+∞ω1−ωω2−ωFefFefTdω.
It can be seen thatτΦ≤0. If Q satisfies τQ>0, then trΦQ≤0. As a result, we obtain(39)trHeω1ef+je˙fω2ef+je˙f∗Q2=trΦQ2+trΦ∗Q2≤0.
Combining (37) and (39), it follows that ∫0∞β2fTf−rfTrfdt≤0, namely the estimation error system (5) satisfies (c). According to Lipschitz condition, δ2ΔφTΔφ≤δ2γ2efTef. Therefore, the following inequality is a sufficient condition of J−≤0:(40)V˙f+β2fTf−rfTrf+δ2γ2efTef−δ2ΔφTΔφ−trHeω1ef+je˙fω2ef+je˙f∗Q/2≤0.
Furthermore,(41)trHeω1ef+je˙fω2ef+je˙f∗Q2=trω1ω2efefTQ+e˙fe˙fTQ+jω0e˙fefTQ−jω0efe˙fTQ=ω1ω2efTQef+e˙fTQe˙f+jω0efTQe˙f−jω0e˙fTQef.
Substitute (13) into (41), the following formula can be derived as(42)trHeω1ef+je˙fω2ef+je˙f∗Q2=ω1ω2efTQef+efTAl+ΔATQAl+ΔAef+efTAl+ΔATQFf+efTAl+ΔATQΔφ+fTFTQAl+ΔAef+fTFTQFf+fTFTQΔφ+ΔφTQAl+ΔAef+ΔφTQFf+ΔφTQΔφ+jω0efTQAl+ΔAef+jω0efTQFf+jω0efTQΔφ−jω0efTAl+ΔATQef−jω0fTFTQef−jω0ΔφTQef.
Combining (35), (40) and (42), it follows that(43)effΔφTΘ11Θ12Θ13⋆Θ22Θ23⋆⋆Θ33effΔφ<0.
And (43) is equivalent to (33).
As can be seen that the inequalities given in Theorem1 and Theorem 2 are nonlinear, which are not conducive to the solution, therefore, the following theorem is proposed to convert (14) and (33) into linear matrix inequalities.Theorem 3.
Given scalarsγ>0, α>0, μ1>0, α1>0, α2>0, matrix K∈ℝn×q, the observer (3) is an H−/L∞ fault detection observer in finite frequency domain for system (1) if there exist scalars δ1>0, δ2>0, μ2>0, β>0 and matrices P1=P1T>0∈ℝn×n, P2=P2T>0∈ℝn×n, G∈ℝn×n, H∈ℝn×m, L=G−1H∈ℝn×m, Q=QT∈ℝn×n satisfies τQ>0 such that(44)minμ2−β,(45)Π11GMGMGD1−HD2GΠ16⋆−I000α1MTGT⋆⋆−αI00α1MTGT⋆⋆⋆−αI0α1GD1−HD2T⋆⋆⋆⋆−δ1Iα1GT⋆⋆⋆⋆⋆−α1G−α1GT<0,(46)CTC−μ1P1CTD2⋆−μ2I+D2TD2<0,(47)Λ11Λ12Λ13α2Gα2GM⋆β2I+HeKTGF−KTG+FTGTKTGKTGM⋆⋆−Q−G−GTGGM⋆⋆⋆−δ2I0⋆⋆⋆⋆−I<0,whereΠ11=HeGA−HC+NTN+δ1γ2I+αP1,Π16=−G+P1+α1GA−HCT,Λ11=Heα2GA−HC−ω1ω2Q−CTC+NTN+δ2γ2I,Λ12=α2GF+GA−HCTK,Λ13=−α2G+GA−HCT+P2−jω0Q.Proof.
(14) can be expressed as(48)A+ℰN+NTET<0.where A=A110000⋆−I000⋆⋆−α1I00⋆⋆⋆−α1I0⋆⋆⋆⋆−δ1I, N=A−LCMMD1−LD2I, E=P10000, A11=αP1+δ1γ2I+NTN. Furthermore,(49)A+ℰN+NTET=INTA+PN+NTST−S+E+NTGT⋆−G−GTIN.
Then, (48) is equivalent to the following inequality:(50)A+PN+NTST−S+E+NTGT⋆−G−GT<0.
We chooseS=GT0000T, G=α1G, then (50) is equivalent to (45). According to Schur’s complement lemma, (47) can be rewritten as(51)Λ11−NTNΛ12Λ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I+NT000N000+α2GMKTGMGM0α2GMKTGMGM0T<0.
According to Lemma1,(52)Λ11−NTNΛ12Λ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I+NT000ΣTα2GMKTGMGM0T+α2GMKTGMGM0∑NT000T<0.
We obtain(53)Ψ11Ψ12Ψ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I<0,where Ψ11=Heα2GAl+ΔA−ω1ω2Q−CTC+δ2γ2I, Ψ12=α2GF+Al+ΔATGTK, and Ψ13=−α2G+Al+ΔATGT+P2−jω0Q. If the left and right sides of Ψ11Ψ12Ψ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I are multiplied by I0Al+ΔAT00IFT000II and its transpose matrix, respectively, then we can obtain (33). □


### 3.4. Dynamic Threshold Design Method

It is necessary to design an appropriate residual evaluation function and the detection threshold. In this paper,L2 norm of the residual r is selected as the residual evaluation function(54)J=r.The following inequality holds when no fault occurs:(55)χe0,t=μ1e0TP1e0e−αt≤μ1λmaxP1e02e−αt.According to (32), it follows that(56)r2≤μd¯2+μ1ξ+μ1λmaxP1e02e−αt.Therefore, we select the following function as the dynamic detection threshold(57)Jthf=0=μd¯2+μ1ξ+μ1λmaxP1e02e−αt.J>Jth means that a fault occurs, and J≤Jth means that no fault occurs.


## 3.1. Observer Design Method

The state estimation error system (5) can be rewritten as(11)e˙=Al+ΔAe+M∑Nx^+Δφ+D1−LD2d+Ff,r=Ce+D2d,where Al=A−LC. When only considering the influence of external disturbance and initial conditions on the residual, (11) can be represented as(12)e˙d=Al+ΔAed+M∑Nx^+Δφ+D1−LD2d,rd=Ced+D2d.When only considering the influence of fault on the residual, (11) can be represented as(13)e˙f=Al+ΔAef+Δφ+Ff,rf=Cef.


## 3.2. Stability andL∞ Disturbance Attenuation Performance

Theorem 1.
Given scalarsγ>0, α>0, μ1>0, the state estimation error system (5) satisfies (a) and (b) if there exist scalars δ1>0, μ2>0, and matrices P1=P1T>0∈ℝn×n, L∈ℝn×m such that(14)Ξ11P1MP1MP1D1−LD2P1⋆−I000⋆⋆−αI00⋆⋆⋆−αI0⋆⋆⋆⋆−δ1I<0,(15)CTC−μ1P1CTD2⋆−μ2I+D2TD2<0,where Ξ11=HeP1Al+NTN+δ1γ2I+αP1.Proof.
We take the following Lyapunov function(16)Vd=edTP1ed.
Then, we can obtain(17)V˙d=e˙dTP1ed+edTP1e˙d.
According to (12), V˙d is obtained as(18)V˙d=edTHeP1Aled+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx=edTHeP1Aled+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx^+2edTP1M∑Ned.Lemma 1.
Suppose matricesV, S have appropriate dimensions, if matrix Σ satisfies ΣTΣ≤I, then the following inequality can be established for any scalar ϵ>0 and vector x,y∈ℝn [27].(19)2xTV∑Sy≤1ϵxTVVTx+ϵyTSTSy.
According to (19), set ϵ=1, and we can get(20)2edTP1M∑Ned≤edTP1MP1MTed+edTNTNed.
The following inequality can be derived as(21)V˙d≤edTHeP1Al+NTN+P1MP1MTed+2edTP1Δφ+2edTP1D1−LD2d+2edTP1M∑Nx^.
IfV˙d satisfies(22)V˙d≤−αVd+αdTd+αx¯Tx¯.where x¯=ΣNx^. We derive the following inequality if we integrate (22)(23)Vd≤Vd0e−αt+α∫0te−αt−τdTddτ+α∫0te−αt−τx¯Tx¯dτ.
According to assumption2, (23) is equivalent to the following formula:(24)Vd≤Vd0e−αt+d¯2∫0tαe−αt−τdτ+α∫0te−αt−τx¯Tx¯dτ.
According to assumption1, we can obtain the following formula:(25)x¯Tx¯=∑Nx^T∑Nx^=x^TNTΣT∑Nx^≤Nx^TNx^.
Then,(26)Vd≤Vd0e−αt+1−e−αtd¯2+α∫0te−αt−τNx^TNx^dτ≤Vd0e−αt+d¯2+α∫0te−αt−τNx^2dτ.
Therefore, whend=0, x^=0, Vd satisfies Vd≤Vd0e−αt. We obtain(27)λminP1ed2≤edTP1ed≤edT0P1ed0e−αt≤λmaxP1ed02e−αt.
It follows thated2≤λmaxP1/λminP1ed02e−αt. When t⟶∞, so that ed⟶0, namely the state estimation error system (5) satisfies (a). According to Lipschitz condition, δ1ΔφTΔφ≤δ1γ2edTed. If the following inequality is satisfied,(28)V˙d+αVd−αdTd−αx¯Tx¯+δ1γ2edTed−δ1ΔφTΔφ≤0.
Then, the inequality (22) can be derived. Substituting (16) and (21) into (28), it follows that(29)edx¯dΔφTΓ11P1MP1D1−LD2dP1⋆−αI00⋆⋆−αI0⋆⋆⋆−δ1Iedx¯dΔφ<0.where Γ11=HeP1Al+NTN+P1MP1MT+αP1+δ1γ2I. According to Schur’s complement lemma, (29) is equivalent to (14). Besides, the following inequality can be obtained from (15)(30)eddTCTC−μ1P1CTD2⋆−μ2I+D2TD2edd<0.
(30) is equivalent to the following formula:(31)rdTrd≤μ1Vd+μ2dTd.
Combining (26) with (31), the following formula can be obtained:(32)rd2≤μ1Vd0e−αt+μ1d¯2+μ1ξ+μ2dTd≤μ2d¯2+χed0,t+μ1ξ,where χed0,t=μ1edT0P1ed0e−αt, namely the state estimation error system (5) satisfies (b).


## 3.3.H− Fault Sensitive Performance

Theorem 2.
Given a scalarγ>0, the state estimation error system (5) satisfies (c) if there exist scalars δ2>0, β>0, and matrices P2=P2T>0∈ℝn×n, L∈ℝn×m, Q=QT∈ℝn×n satisfies τQ>0 such that(33)Θ11Θ12Θ13★Θ22Θ23★★Θ33<0,whereΘ11=HeP2Al+ΔA−ω1ω2Q−CTC−Al+ΔATQAl+ΔA+jω0Al+ΔATQ−jω0QAl+ΔA+δ2γ2I,Θ12=P2F−Al+ΔATQF−jω0QF,Θ13=P2−Al+ΔATQ−jω0Q,Θ22=β2I−FTQF,Θ23=−FTQ,Θ33=−Q−δ2I,ω0=ω1+ω2/2.Proof.
We take the following Lyapunov function(34)Vf=efTP2ef.
Then, we derive the following formula:(35)V˙f=efTHeP2Al+ΔAef+2efTP2Ff+2efTP2Δφ.
Considering the fault occurs in finite frequency domain, and theH− performance index is expressed as(36)J_=∫0∞V˙f+β2fTf−rfTrf−trHeω1ef+je˙fω2ef+je˙f∗Q2dt.
Under zero initial condition, ifJ−≤0, it follows that(37)∫0∞β2fTf−rfTrf−trHeω1ef+je˙fω2ef+je˙f∗Q2dt≤0.
DefineΦ=∫0∞ω1ef+je˙fω2ef+je˙f∗dt. According to Parseval’s theorem [28],(38)Φ=12Π∫−∞+∞Fω1ef+je˙fFω2ef+je˙f∗dω=12Π∫−∞+∞ω1−ωω2−ωFefFefTdω.
It can be seen thatτΦ≤0. If Q satisfies τQ>0, then trΦQ≤0. As a result, we obtain(39)trHeω1ef+je˙fω2ef+je˙f∗Q2=trΦQ2+trΦ∗Q2≤0.
Combining (37) and (39), it follows that ∫0∞β2fTf−rfTrfdt≤0, namely the estimation error system (5) satisfies (c). According to Lipschitz condition, δ2ΔφTΔφ≤δ2γ2efTef. Therefore, the following inequality is a sufficient condition of J−≤0:(40)V˙f+β2fTf−rfTrf+δ2γ2efTef−δ2ΔφTΔφ−trHeω1ef+je˙fω2ef+je˙f∗Q/2≤0.
Furthermore,(41)trHeω1ef+je˙fω2ef+je˙f∗Q2=trω1ω2efefTQ+e˙fe˙fTQ+jω0e˙fefTQ−jω0efe˙fTQ=ω1ω2efTQef+e˙fTQe˙f+jω0efTQe˙f−jω0e˙fTQef.
Substitute (13) into (41), the following formula can be derived as(42)trHeω1ef+je˙fω2ef+je˙f∗Q2=ω1ω2efTQef+efTAl+ΔATQAl+ΔAef+efTAl+ΔATQFf+efTAl+ΔATQΔφ+fTFTQAl+ΔAef+fTFTQFf+fTFTQΔφ+ΔφTQAl+ΔAef+ΔφTQFf+ΔφTQΔφ+jω0efTQAl+ΔAef+jω0efTQFf+jω0efTQΔφ−jω0efTAl+ΔATQef−jω0fTFTQef−jω0ΔφTQef.
Combining (35), (40) and (42), it follows that(43)effΔφTΘ11Θ12Θ13⋆Θ22Θ23⋆⋆Θ33effΔφ<0.
And (43) is equivalent to (33).
As can be seen that the inequalities given in Theorem1 and Theorem 2 are nonlinear, which are not conducive to the solution, therefore, the following theorem is proposed to convert (14) and (33) into linear matrix inequalities.Theorem 3.
Given scalarsγ>0, α>0, μ1>0, α1>0, α2>0, matrix K∈ℝn×q, the observer (3) is an H−/L∞ fault detection observer in finite frequency domain for system (1) if there exist scalars δ1>0, δ2>0, μ2>0, β>0 and matrices P1=P1T>0∈ℝn×n, P2=P2T>0∈ℝn×n, G∈ℝn×n, H∈ℝn×m, L=G−1H∈ℝn×m, Q=QT∈ℝn×n satisfies τQ>0 such that(44)minμ2−β,(45)Π11GMGMGD1−HD2GΠ16⋆−I000α1MTGT⋆⋆−αI00α1MTGT⋆⋆⋆−αI0α1GD1−HD2T⋆⋆⋆⋆−δ1Iα1GT⋆⋆⋆⋆⋆−α1G−α1GT<0,(46)CTC−μ1P1CTD2⋆−μ2I+D2TD2<0,(47)Λ11Λ12Λ13α2Gα2GM⋆β2I+HeKTGF−KTG+FTGTKTGKTGM⋆⋆−Q−G−GTGGM⋆⋆⋆−δ2I0⋆⋆⋆⋆−I<0,whereΠ11=HeGA−HC+NTN+δ1γ2I+αP1,Π16=−G+P1+α1GA−HCT,Λ11=Heα2GA−HC−ω1ω2Q−CTC+NTN+δ2γ2I,Λ12=α2GF+GA−HCTK,Λ13=−α2G+GA−HCT+P2−jω0Q.Proof.
(14) can be expressed as(48)A+ℰN+NTET<0.where A=A110000⋆−I000⋆⋆−α1I00⋆⋆⋆−α1I0⋆⋆⋆⋆−δ1I, N=A−LCMMD1−LD2I, E=P10000, A11=αP1+δ1γ2I+NTN. Furthermore,(49)A+ℰN+NTET=INTA+PN+NTST−S+E+NTGT⋆−G−GTIN.
Then, (48) is equivalent to the following inequality:(50)A+PN+NTST−S+E+NTGT⋆−G−GT<0.
We chooseS=GT0000T, G=α1G, then (50) is equivalent to (45). According to Schur’s complement lemma, (47) can be rewritten as(51)Λ11−NTNΛ12Λ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I+NT000N000+α2GMKTGMGM0α2GMKTGMGM0T<0.
According to Lemma1,(52)Λ11−NTNΛ12Λ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I+NT000ΣTα2GMKTGMGM0T+α2GMKTGMGM0∑NT000T<0.
We obtain(53)Ψ11Ψ12Ψ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I<0,where Ψ11=Heα2GAl+ΔA−ω1ω2Q−CTC+δ2γ2I, Ψ12=α2GF+Al+ΔATGTK, and Ψ13=−α2G+Al+ΔATGT+P2−jω0Q. If the left and right sides of Ψ11Ψ12Ψ13α2G⋆Λ22Λ23KTG⋆⋆Λ33G⋆⋆⋆−δ2I are multiplied by I0Al+ΔAT00IFT000II and its transpose matrix, respectively, then we can obtain (33). □


## 3.4. Dynamic Threshold Design Method

It is necessary to design an appropriate residual evaluation function and the detection threshold. In this paper,L2 norm of the residual r is selected as the residual evaluation function(54)J=r.The following inequality holds when no fault occurs:(55)χe0,t=μ1e0TP1e0e−αt≤μ1λmaxP1e02e−αt.According to (32), it follows that(56)r2≤μd¯2+μ1ξ+μ1λmaxP1e02e−αt.Therefore, we select the following function as the dynamic detection threshold(57)Jthf=0=μd¯2+μ1ξ+μ1λmaxP1e02e−αt.J>Jth means that a fault occurs, and J≤Jth means that no fault occurs.


## 4. Simulation

In this section, we use the state-space model of a single-link robot to demonstrate the effectiveness of the proposed method. The state-space model [14] is given as(58)A=0100−48.612.548.60000101.950−1.950,B=021.600,C=100001000010,F=021.600,φx=000−0.333sinx3,D1=00000100000000001000,D2=000.1000000.1000000.1.Supposing there exists uncertain parametersM=00.800.4T, N=0.250.30.250, Σ is a time-varying random number between −1,1, and d is a random vector with values between −0.1,0.1. It is considered that the fault occurs in low-frequency bandwidth ω≤0.1. We choose scalars γ=0.333, α=1, μ1=0.8, α1=0.1, α2=1, d¯=0.1, matrix K=−5F. Supposing the initial value of the system state is x=0.10−0.10T, the initial value of observer state is set as x^=0000T. By solving Theorem 3, the L∞ and H− performance indexes are calculated as μ2=0.5249, μ=μ1+μ2=1.151, β=3.3322, and the gain matrix of the H−/L∞ fault detection observer is obtained as(59)L=23.60630.84250.1837−48.78762.345751.41120.11030.505939.10941.96430.328125.9156.Considering the system occurs abrupt faultf1 and slow-varying fault f2, respectively. (60)f1=00≤t<100.110≤t<70070≤t<80,f2=00≤t<100.1+0.05sin0.1t−1010≤t<70070≤t<80.The simulation results are shown in Figure1 and Figure 2. JH−/L∞ is the residual evaluation result generated by H−/L∞ observer, and JthH−/L∞ is the dynamic threshold. According to the simulation results, we can obtain that JH−/L∞ remains near 0 when no fault occurs, indicating that JH−/L∞ is robust to external disturbance and uncertain parameters. When faults occur at 10 s, JH−/L∞ quickly exceeds the detection threshold, and when faults disappeared at 70 s, JH−/L∞ also quickly fell below the detection threshold, indicating that the proposed H−/L∞ observer has good sensitivity to fault. The simulation proves that the proposed method can be used for fault detection of nonlinear systems with uncertain parameters. To illustrate the superiority of the proposed method, we have made the following comparisons: (a) comparison of the proposed method with L∞ observer in finite frequency domain. Supposing the system occurs abrupt fault and slow-varying fault f1 and f2, respectively. The L∞ performance index of L∞ fault detection observer is calculated as μ2=0.0215, μ=μ1+μ2=0.9064, and the gain matrix is obtained as(61)L=1.45750.25761.0510−3.980946.678824.20071.336910.534314.26410.40445.18808.6268.Figure 1
Fault detection result by the proposed method forf1.Figure 2
Fault detection result by the proposed method forf2.The comparison results are shown in Figure3 and Figure 4.Figure 3
Comparison result of the proposed method withL∞ observer in finite frequency domain for f1.Figure 4
Comparison result of the proposed method withL∞ observer in finite frequency domain for f2.JL∞ is the residual evaluation result generated by L∞ observer, and JthL∞ is the dynamic threshold for observer. As can be seen from Figures 3 and 4 that the L∞ observer cannot detect f1 and f2 although JL∞ has changed when the fault occurs, it is not sensitive enough to fault. The reason is that the robustness and fault sensitivity are contradictory. The L∞ disturbance attenuation performance index of L∞ fault detection observer is better than the observer we proposed. However, it also inhibits the fault sensitivity. In contrast, we design the H− index to improve the sensitivity of residual to fault, which illustrates the advantages of the proposed method. (b) Comparison of the proposed method with H−/L∞ observer in full frequency domain. Supposing the system occurs abrupt fault and slow-varying fault f1 and f2, respectively. The L∞ and H− performance indexes of H−/L∞ fault detection observer in full frequency domain are calculated as μ2=0.0155, μ=μ1+μ2=0.9031, β=1.4142, and the gain matrix is obtained as(62)L=6.56490.94880.1753−53.734012.382154.39090.41090.189030.87592.09010.070812.1824.The comparison results are shown in Figure5 and 6.Figure 5
Comparison result of the proposed method withH−/L∞ observer in full frequency domain for f1.Figure 6
Comparison result of the proposed method withH−/L∞ observer in full frequency domain for f2.As can be seen from Figures5 and 6 that the H−/L∞ observer in full frequency domain cannot detect fault f1. For the fault f2, it can only be detected within 10s−40s, indicating that the H−/L∞ observer in full frequency domain is not sensitive enough to the fault. The reason is that it is conservative to design an fault detection observer in full frequency domain without considering the frequency of fault. If the fault amplitude is low, the observer may not be able to detect the fault adequately. In this paper, the frequency domain of fault is considered in the observer design process, so that the observer design limitation is reduced and improved the sensitivity of the residual to fault. (c) Comparison of the proposed method with the method proposed in [14]. Considering the system occurs the following abrupt fault f1 and slow-varying fault f2, respectively. (63)f1=00≤t<100.110≤t<70070≤t<80,f2=00≤t<100.15+0.05sin0.1t−1010≤t<70070≤t<80.Supposingd is a random vector with values between −0.5,0.5. The comparison results are shown in Figure 7 and Figure 8.Figure 7
Comparison result of the proposed method with the method proposed in [14] for f1.Figure 8
Comparison result of the proposed method with the method proposed in [14] for f2.As can be seen from Figures7 and 8, although both the proposed method and the method proposed in [14] can detect the occurrence of faults, the fault detection effect of the proposed method is better: the residual evaluation result obtained by the proposed method completely exceeds the threshold, while the residual evaluation result obtained by the method of [14] does not fully exceed the threshold. The reason is that [14] does not consider the model parameters uncertainty, so it needs to take the model parameters uncertainty as additional noise, which increases the average energy of the noise signal over a time interval. And the fault detection threshold is designed based on the maximum value of the average energy of the noise signal. Therefore, it needs to raise the fault detection threshold. When the noise intensity increases, the fault may not be adequately detected by residual evaluation results. The method proposed in the paper considers the influence of model parameters uncertainty in the design process of the observer and dynamic threshold and can fully detect the occurrence of faults, which shows the superiority of the proposed method.


## 5. Conclusion

In this paper, we derive and analyze a robust fault detection scheme for uncertain Lipschitz nonlinear systems in finite frequency domain. A design method ofH−/L∞ fault detection observer is proposed, and the superiority of the proposed method is verified by a numerical example. However, only the matrix A has time-varying uncertain parameters, and the other matrices are not considered in the paper. The problem of robust fault detection in finite frequency domain for nonlinear systems with more uncertain parameters is the focus of future work.



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*Source: 6596746-2022-04-28.xml*