This paper deals with the problem of H∞ fault detection for a class of linear discrete time-varying descriptor systems with missing measurements, and the missing measurements are described by a Bernoulli random binary switching sequence. We first translate the H∞ fault detection problem into an indefinite quadratic form problem. Then, a sufficient and necessary condition on the existence of the minimum is derived. Finally, an observer-based H∞ fault detection filter is obtained such that the minimum is positive and its parameter matrices are calculated recursively by solving a matrix differential equation. A numerical example is given to demonstrate the efficiency of the proposed method.
1. Introduction
During the last four decades, the fault diagnosis theory has received considerable attention, and many remarkable achievements have been obtained [1–6]. As mentioned in [2], it is well recognized that the model-based fault diagnosis techniques can be classified into three classical approaches: observer-based methods, parity space methods, and parameter identification based methods. Most of the achievements on fault diagnosis always assume that the observations contain the signal to be detected. However, in practice, the observation may contain the signal in a random manner. In this paper, the H∞ fault detection problem for a class of linear discrete time-varying descriptor systems with random missing measurements is investigated.
In practice, time variability is the inherent characteristic of most systems. Recently, research on fault diagnosis of linear systems with time-varying parameters has attracted more and more attention; see, for example, [7–22]. By dividing the filter gain matrix into two sections, an unknown input decoupling optimal filter for linear stochastic systems has been designed in [7], and its application on fault diagnosis has also been addressed. By employing the invariant subspace method and game theory, a game fault detection filter has been proposed in [8]. Based on parity space approach and stochastic signal processing methods, the fault detection and isolation have been studied for a class of linear discrete systems with stochastic inputs and deterministic disturbances and faults in [15, 21]. By applying adaptive observer method, a residual generator has been provided for linear discrete time-varying systems in [19, 22], and the residual adaptive threshold is derived by set-membership computations based on zonotopes. By assuming that the mean and variance of the fault and disturbances are known, the optimal fault detection filter for a class of stochastic systems has been developed in [10, 11, 13, 16]; however, in general, the prior information of the fault cannot be obtained. By solving a min-max problem with a generalized least-squares cost criterion, a generalized least-squares fault detection filter has been designed in [9, 12]. By using an adaptive observer method, a fault diagnosis technique for linear time-varying systems has been developed in [14, 17, 18]. However, to the best of authors’ knowledge, few reports on fault diagnosis problem of linear discrete time-varying descriptor systems have been published, which motivates the present study.
In classical fault diagnosis theories, all residual signals are obtained in the case that the measured output contains valid information. However, the data packet dropout is inevitable in navigation and guidance system, industrial control system, and network control system. Thus, in recent years, increasing attention has been paid to fault diagnosis problems for systems with missing measurements [23–25]. By employing the linear matrix inequality technique, both full-order and reduced-order fault detection filters have been considered for a class of linear discrete time-invariant systems with missing measurements and parameter uncertainty in [23]. In finite frequency domain, the fault detection problem has been concerned with systems with missing measurements in [24], and the fault detection scheme has been utilized to an aircraft model. In [25], the missing measurements are described by Markov random process, and the residual generator is presented as a discrete-time Markovian jump linear system. Note that the existing results mainly focus on the nondescriptor systems; there are few achievements on fault detection problem for descriptor systems with missing measurements. Therefore, in this paper, we aim to design the H∞ fault detection filter and residual evaluation scheme for a class of linear discrete time-varying descriptor systems with random missing measurements.
In this paper, based on the estimation method proposed in [26], a new fault detection filter design approach is developed for a class of linear discrete time-varying descriptor systems with random missing measurements. First, the H∞ fault detection problem is converted to the problem in which a certain indefinite quadratic form has a minimum and the fault detection filter parameter matrices are such that the minimum is positive. Then, by applying matrix analysis method, a necessary and sufficient condition for the indefinite quadratic form is analyzed. And by guaranteeing the positivity of the minimum, the parameter matrices of the fault detection filter are obtained. Moreover, the residual evaluation function and threshold are designed for the fault detection. Finally, a numerical example is provided to illustrate the performance of the H∞ fault detection filter and the residual evaluation scheme.
2. Problem Statement
Consider a discrete time-varying descriptor system described by the following model:(1)Mi+1xi+1=Aixi+Bfifi+wi,yi=λiCixi+Dfifi+vi,where x(i)∈Rn, w(i)∈Rn, y(i)∈Rm, v(i)∈Rm, and f(i)∈Rp are the state, external disturbance input, uncertain measurement output, measurement noise, and fault to be detected, respectively; f(i), w(i), and v(i) are bounded signals belonging to l2[0,N]; N is a positive integer; M(i), A(i), Bf(i), C(i), and Df(i) are known real-time-varying matrices with appropriate dimensions; and M(i) is a singular matrix with rank[M(i)]=n1, 0<n1<n; the random variable λ(i)∈R is a Bernoulli distributed white sequence taking the values of 0 and 1 with(2)Probλi=1=Eλ(i)=ρ,Probλi=0=1-Eλi=1-ρ,Eλ2i=ρ,E1-λi2=1-ρ,E1-λiλi=0,where E(·) denotes the mathematical expectation, Prob(·) denotes the probability distribution, and ρ∈R is a known positive scalar.
Hypothesis 1. The initial matrices of system (1) satisfy the condition that M(0)=A(-1)=Bf(-1)=I.
The H∞ fault detection problem under investigation in this paper can be stated as follows. Given a disturbance attenuation level γ>0, based on the measurement output sequence {y(i)}i=1k, find a residual signal r(k). If it exists, the following inequality is satisfied: (3)supx(0),d(i)i=0kE∑i=1kri-fi-1Tri-fi-1+∑i=0kdTidi-1kkkkkk×EM0x0-A-1x˘0TP0-1∑i=0kdTidikkkkkkikkkkkkkkkk×M0x0-A-1x˘0kkkkkkkikkkkkikki+∑i=0kdTidi-1<γ2,where(4)d(i)=fT(i)wT(i)vT(i)T,i=0,1,…,k-1,fT(i)0vT(i)T,i=k.P0 is a given positive definite matrix function which reflects the relative uncertainty of the initial state x(0) about the initial state estimate x˘(0). Without loss of generality, let x˘(0)=0.
Even if the measurement data are fully available, the valid information of fault f(i) is not contained in the measurement output {y(j)}j=0i when Df(i)=0. Thus, an H∞ one-step lag fault detection issue is defined as (3) to overcome this problem.
Define the following new variables: (5)x-i=xifixi-1fi-1,w-i=wifi+100.Thus, system (1) can be described as the following augmented model:(6)M-i+1x-i+1=A-ix-i+w-i,yi=λiC-ix-i+vi,where(7)M-i+1=Mi+10000I0000I0000I,A-i=AiBfi000000I0000I00,C-i=CiDfi00.
For the purpose of fault detection, the following observer-based fault detection filter is proposed as a residual generator:(8)x-^k+1=V1kx-^k+V2kyk,rk=Lkx-^k,x-^0=0,where L(i)=[000I]. Thus, the fault estimation problem can come down to the following: find the parameter matrices V1(k) and V2(k); if they exist, the performance index (3) is satisfied.
According to system (6), the performance index (3) is reexpressed as(9)supx0,w-i,vii=0kE∑i=1kri-Lix-iT+∑i=0kvTivi-1kkkkkkkkkkkkkkkk∑i=1kri-Lix-iT×ri-Lix-ikkkkkkkkiiik×EM-0x-0-A--1x-˘0TP-0-1∑i=0kvTiviikkkkkkkkkkk×M-0x-0-A--1x-˘0kkkkkkkkkkki+∑i=0k-1w-Tiw-ikkkkkkkkkkkkkkkkkkkkk+∑i=0kvTivi-1<γ2,where P-0=diag{P0,Ip,In,Ip} and x-˘(0)=[x˘T(0)000]T, and assume that x(-1)=0 and f(-1)=0.
Define(10)Jk=EM-0x-0-A--1x-˘0TP-0-1∑i=0k-1w-Tiw-i+∑i=0kvTivikkkkki×M-0x-0-A--1x-˘0kkkkkk+∑i=0k-1w-Tiw-i+∑i=0kvTivi-γ-2E∑i=1kri-Lix-iTri-Lix-i.Then, the H∞ fault detection problem is equivalent to the following:
Jk of (10) has minimum with respect to {x-(i)}i=0k and {w-(i)}i=0k;
{r(i)}i=0k can be chosen such that the value of Jk at its minimum is positive.
In the following, we will first discuss the mathematical expectation of Jk based on system (6). Then, the existence of the minimum Jk over {x-(i)}i=0k and {w-(i)}i=0k can be derived. Finally, a solution to parameter matrices V1(k) and V2(k) will be obtained such that Jk>0.
3. Main Results
From (6) and (10), we have (11)Jk=EM-0x-0-A--1x-˘0TP-0-1M-0x-0-A--1x-˘0Tkki×M-0x-0-A--1x-˘0+E∑i=0k-1M-i+1x-i+1-A-ix-iT∑i=0k-1M-i+1x-i+1-A-ix-iTkki×M-i+1x-i+1-A-ix-i+E∑i=0kyi-λiC-ix-iT∑i=0kyi-λiC-ix-iTkki×yi-λiC-ix-i-γ-2E∑i=1kri-Lix-iTri-Lix-i.Notice that(12)Ey(i)-λ(i)C-(i)x-(i)Ty(i)-λ(i)C-(i)x-(i)=E(i)C-T(i)C-(i)x-(i)yT(i)y(i)-λ(i)yT(i)C-(i)x-(i)-λ(i)x-T(i)kkkkkkj×C-T(i)y(i)+λ2(i)x-T(i)C-T(i)C-(i)x-(i)=yT(i)y(i)-ρyT(i)C-(i)x-(i)-ρx-T(i)C-T(i)y(i)+ρx-T(i)C-T(i)C-(i)x-(i)=yi-ρC-ix-iTyi-ρC-ix-i+ρ-ρ2x-T(i)C-T(i)C-(i)x-(i)=y(i)-C~(i)x-(i)Ty(i)-C~(i)x-(i)+Fix-iTFix-i,where(13)C~i=ρC-i,Fi=ρ-ρ2C-i.Then, we have (14)Jk=M-0x-0-A--1x-˘0TP-0-1×M-0x-0-A--1x-˘0+∑i=0k-1M-i+1x-i+1-A-ix-iTk×M-i+1x-i+1-A-ix-i+∑i=0kyi-C~ix-iTy(i)-C~(i)x-(i)+∑i=0kFix-iTF(i)x-(i)-γ-2∑i=1kri-Lix-iTri-Lix-i.Thus, Jk can be further expressed as(15)Jk=ΣkXk-YkTΠkΣkXk-Yk,where(16)Xk=x-kx-k-1x-k-2⋮x-2x-1x-0,Yk=y-ky-k-1y-k-2⋮y-2y-1y-0,Σk=ΩkΨk-1000000Ωk-1Ψk-2000000Ωk-2⋱000000⋱Ψ2000000Ω2Ψ1000000Ω1Ψ0000000Ω0,Πk=Φk0000Φk-10000⋱0000Φ0,Ωi=M-iFiC~iLi,Ω0=M-0F0C~0,Ψi-1=-A-i-1000,Φi=I0000I0000I0000-γ-2I,Φ0=P-0-1000I000I,y-i=00yiri,y-0=x-˘00y0,i=1,2,…,k.In virtue of the above variables, for all k>0, we have (17)Xk=x-(k)Xk-1,Yk=y-(k)Yk-1,Σk=Ω(k)α(k-1)0Σk-1,Πk=Φ(k)00Πk-1,αk-1=Ψk-10⋯0.
Lemma 1 (see [26]).
Consider matrices α, β, R, and x of appropriate dimensions, and R is symmetric. If and only if αTRα≥0 and Ker(αTRα)⊂Ker(Rα), for any β, we have (18)infxαx-βTRαx-β>-∞.If the minimum is attained, it is unique if and only if αTRα>0. Moreover, the optimal solution is derived by x^=(αTRα)-1αTRβ.
3.1. Existence Conditions of the Minimum
According to Lemma 1, Jk has the minimum if and only if ΣkTΠkΣk>0. When k=0, let(19)P0=Σ0TΠ0Σ0=M-0F0C~0TP-0-1000I000IM-0F0C~0=M-T0P-0-1M-0+C~T0C~0+FT0F0.Therefore, when k=0, J0 has the minimum if and only if P(0)>0.
Furthermore, for all k>0, we can obtain the following equation from (17):(20)ΣkTΠkΣk=ΩTkΦkΩkΩTkΦkαk-1αTk-1ΦkΩkΣk-1TΠk-1Σk-1+αTk-1Φkαk-1.
To ensure that (20) is positive definite, Σk-1TΠk-1Σk-1+αT(k-1)Φ(k)α(k-1) must be positive definite. Assume that Σk-1TΠk-1Σk-1>0 is satisfied, and note that(21)αT(k-1)Φ(k)α(k-1)=ΨT(k-1)0Φ(k)Ψ(k-1)0=-A-(k-1)000TI0000I0000I0000-γ-2I-A-(k-1)000000=A-Tk-1A-k-1000≥0.Therefore, Σk-1TΠk-1Σk-1+αT(k-1)Φ(k)α(k-1)>0. If and only if the Schur complement of Σk-1TΠk-1Σk-1+αT(k-1)Φ(k)α(k-1) in (20) is positive definite, we have ΣkTΠkΣk>0.
Define(22)Pk=ΩTkΦkΩk-ΩTkΦkαk-1×Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1×αTk-1ΦkΩk=ΩT(k)Φ(k)×I-αk-1αT(k-1)Φ(k)kkkko×Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1kkkko×αT(k-1)Φ(k)Ω(k)=ΩT(k)Φ(k)×I+αk-1Σk-1TΠk-1Σk-1-1αT(k-1)Φ(k)-1×Ωk.And notice that (23)Σk-1TΠk-1Σk-1=ΩTk-1Φk-1Ωk-1ΩTk-1Φk-1αk-2αTk-2Φk-1Ωk-1Σk-2TΠk-2Σk-2+αTk-2Φk-1αk-2=IΩTk-1Φk-1αk-2Σk-2TΠk-2Σk-2+αTk-2Φk-1αk-2-10I×Pk-100Σk-2TΠk-2Σk-2+αTk-2Φk-1αk-2×I0Σk-2TΠk-2Σk-2+αT(k-2)Φ(k-1)α(k-2)-1αT(k-2)Φ(k-1)Ω(k-1)I.Then, (24)αk-1Σk-1TΠk-1Σk-1-1αT(k-1)=Ψ(k-1)0×P(k-1)00Σk-2TΠk-2Σk-2+αT(k-2)Φ(k-1)α(k-2)-1×ΨT(k-1)0=Ψk-1P-1k-1ΨTk-1.Moreover, we obtain(25)Pk=ΩT(k)Φ(k)×I+αk-1Σk-1TΠk-1Σk-1-1αTk-1Φk-1×Ω(k)=ΩT(k)Φ(k)×I+Ψk-1P-1k-1ΨTk-1Φ(k)-1Ω(k)=M-(k)F(k)C~(k)L(k)TI0000I0000I0000-γ-2I×I+-A-(k-1)000P-1(k-1)-A-(k-1)000Tkkkkkkk×I0000I0000I0000-γ-2I-A-(k-1)000T-1M-(k)F(k)C~(k)L(k)=M-TkFTkC~Tk-γ-2LTk×I+A-k-1P-1k-1A-Tk-1-10000I0000I0000I×M-kFkC~kLk=M-T(k)I+A-k-1P-1k-1A-Tk-1-1M-(k)+C~TkC~k+FTkFk-γ-2LTkLk.Thus, it is readily known that Jk has a minimum if and only if P(k)>0.
In light of the above discussion, we have the following results.
Theorem 2.
Consider the linear discrete time-varying descriptor system (1), given a scalar γ>0; then Jk has a minimum over {x-(i)}i=0k and {w-(i)}i=0k if and only if P(k)>0(k=0,1,…,k), where(26)Pk=M-TkI+A-k-1P-1k-1A-Tk-1-1×M-(k)+C~T(k)C~(k)+FT(k)F(k)-γ-2LTkLk,P0=M-T0P-0-1M-0+C~T0C~0+FT0F0.
3.2. Design of the H∞ Fault Detection Filter
According to Lemma 1, it is known that if Jk has a minimum over {x-(i)}i=0k and {w-(i)}i=0k, the optimal solution is(27)X^k∣k=ΣkTΠkΣk-1ΣkTΠkYk.When k=0, we have(28)x-^0∣0=Σ0TΠ0Σ0-1Σ0TΠ0Y0=ΩT0Φ0Ω0-1ΩT0Φ0x-˘00y0=P-10M-0F0C~0TP-0-1000I000Ix-˘00y0=P-10M-T0P-0-1x-˘0+P-10C~T0y0.Furthermore, notice that(29)x-^k∣kX^k-1∣k=ΩT(k)Φ(k)Ω(k)ΩT(k)Φ(k)α(k-1)αT(k-1)Φ(k)Ω(k)Σk-1TΠk-1Σk-1+αT(k-1)Φ(k)α(k-1)-1×Ω(k)α(k-1)0Σk-1TΦ(k)00Πk-1y-(k)Yk-1=ΩT(k)Φ(k)Ω(k)ΩT(k)Φ(k)α(k-1)αT(k-1)Φ(k)Ω(k)Σk-1TΠk-1Σk-1+αT(k-1)Φ(k)α(k-1)-1×ΩTkΦky-kαTk-1Φky-k+Σk-1TΠk-1Yk-1=H11,kH12,kH21,kH22,k×ΩTkαTk-1Φky-k+0Σk-1TΠk-1Σk-1X^k-1k-1.Then, (30)x-^k∣k=H11,kΩTk+H12,kαTk-1Φ(k)y-(k)+H12,kΣk-1TΠk-1Σk-1X^k-1∣k-1,ΣkTΠkΣk-1=ΩT(k)Φ(k)Ω(k)ΩTkΦkαk-1αTk-1Φ(k)Ω(k)Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1=Φkαk-1-1IΩTkΦkαk-1Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-10Ikk×P(k)00Σk-1TΠk-1Σk-1+αTk-1Φkαk-1kk×I0Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1αTk-1Φ(k)Ω(k)I-1=I0-Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1αTk-1Φ(k)Ω(k)I×P-1(k)00Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1×I-ΩT(k)Φ(k)α(k-1)Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-10I=H11,kH12,kH21,kH22,k.Hence,(31)H11,k=P-1k,H12,k=-P-1kΩTkΦkαk-1×Σk-1TΠk-1Σk-1+αTk-1Φkαk-1-1=-P-1(k)ΩT(k)×Φ(k)-Φ(k)α(k-1)Σk-1TΠk-1Σk-1-1Σk-1TΠk-1Σk-1-1αT(k-1)-1kkkkk×αTk-1kkkkki×Φ-1k+αk-1Σk-1TΠk-1Σk-1-1kkkkkkkkk×Σk-1TΠk-1Σk-1-1αT(k-1)-1×α(k-1)Σk-1TΠk-1Σk-1-1=-P-1(k)ΩT(k)×Φ-1(k)+α(k-1)Σk-1TΠk-1Σk-1-1kkkkkkΣk-1TΠk-1Σk-1-1×αTk-1-1×α(k-1)Σk-1TΠk-1Σk-1-1=-P-1(k)ΩT(k)×Φ-1k+Ψk-1P-1k-1ΨTk-1-1×αk-1Σk-1TΠk-1Σk-1-1.From the above analysis, we obtain(32)H11,kΩT(k)Φ(k)+H12,kαTk-1Φ(k)=P-1(k)ΩT(k)Φ(k)-P-1kΩTk×Φ-1(k)+Ψk-1P-1k-1ΨTk-1-1×αk-1Σk-1TΠk-1Σk-1-1αTk-1Φk=P-1(k)ΩT(k)×ΦkΨk-1P-1k-1ΨTk-1Φkkkkkikk-Φ-1(k)+Ψk-1P-1k-1ΨTk-1-1kkkkkkk×Ψk-1P-1k-1ΨTk-1Φk=P-1kΩTk×Φ-1k+Ψk-1P-1k-1ΨTk-1-1.As such,(33)x-^(k∣k)=P-1(k)ΩT(k)×Φ-1(k)+Ψk-1P-1k-1ΨTk-1-1×y-(k)-Ψ(k-1)x-^(k-1∣k-1).Thus,(34)x-^k∣k=P-1kΩTk×Φ-1(k)+Ψk-1P-1k-1ΨTk-1-1×y-(k)-Ψk-1x-^k-1∣k-1=P-1(k)M-(k)F(k)C~(k)L(k)T×I0000I0000I0000-γ-2I-1+-A-(k-1)000-A-(k-1)000Tkkkkkkkk×P-1(k-1)-A-(k-1)000T-1×00ykrk--A-k-1000x-^k-1∣k-1=P-1(k)M-(k)F(k)C~(k)L(k)T×I+A-k-1P-1k-1A-Tk-1-10000I0000I0000-γ-2I×A-k-1x-^k-1∣k-10ykrk=P-1(k)M-T(k)×I+A-k-1P-1k-1A-Tk-1-1×A-(k-1)x-^k-1∣k-1+P-1(k)C~T(k)y(k)-γ-2P-1(k)LT(k)r(k).
Based on the above discussion, we present the main results of this paper.
Theorem 3.
Consider system (1), given a scalar γ>0; then the H∞ fault detection filter (8) that achieves (3) exists if, and only if, P(k)>0(k=0,1,…,k) and(35)V1k=P(k)+γ-2LT(k)L(k)-1M-T(k)×I+A-k-1P-1k-1A-Tk-1-1A-k-1V2k=Pk+γ-2LTkLk-1C~Tk,where P(k) is calculated by (26).
Proof.
Note that Jk has a minimum over {x-(i)}i=0k and {w-(i)}i=0k if and only if P(k)>0(k=0,1,…,k), and the minimum is (36)minxii=0kJk=M-0x-^0∣k-A--1x-˘0TP-0-1×M-(0)x-^0∣k-A-(-1)x-˘0+∑i=0k-1M-i+1x-^i+1∣k-A-ix-^i∣kT×M-(i+1)x-^i+1∣k-A-(i)x-^i∣k+∑i=0kyi-C~ix-^i∣kT×y(i)-C~(i)x-^i∣k+∑i=0kFix-^i∣kTFix-^i∣k-γ-2∑i=1kri-Lix-^i∣kT×ri-Lix-^i∣k.It is readily seen that a positive minimum of Jk is guaranteed by setting r(i)=L(i)x-^(i∣k), i=0,1,…,k. Furthermore, substituting r(k)=L(k)x-^(k∣k) into (34), we obtain(37)x-^k∣k=P(k)+γ-2LT(k)L(k)-1×M-T(k)I+A-k-1P-1k-1A-Tk-1-1kkkkkiI+A-k-1P-1k-1A-Tk-1-1×A-k-1x-^k-1∣k-1+C~T(k)y(k).In the light of the above equation, the parameter matrices of (8) can be given by (35). Hence, the theorem is proven.
From Theorem 3 in this section, the H∞ fault detection filter r(k) can be computed in the following steps.
Step 1.
Set γ>0, P0>0, x˘(0)=0, and k=0; calculate P-0 and x-˘(0) in (9).
Step 2.
Calculate P(0) and x-^(0∣0) using (26) and (28).
Step 3.
If P(0)>0, let r(0)=L(0)x-^(0∣0), and go to Step 4; otherwise, exit.
Step 4.
Let k=k+1; compute P(k) using (26).
Step 5.
If P(k)>0, compute V1(k), V2(k), and r(k) using (35) and (8), and go to Step 4; otherwise, exit.
Step 6.
Repeat Steps 4 to 5 till k=N.
Remark 4.
Note that the system augmentation has been applied to design the H∞ fault detection filter (8); it may lead to more expensive computational cost. Fortunately, an H∞ simultaneous state and unknown input estimator for descriptor system have been proposed in [27], and an H∞ fixed-lag smoother for missing measurements system has been given in [28], so it is possible, in the future, to design a new H∞ fault detection filter with lower computational cost by using the algorithm given in [27, 28].
4. Residual Evaluation
When the design of the H∞ fault detection filter has been completed, the next task is residual evaluation. First, the following residual evaluation function and the threshold are introduced to facilitate fault detection:(38)Jr,k=∑i=0krTiri,Jthk=supwi,vi∈l2,fi=0EJr,k.The following strategy is applied for fault detection:(39)Jr,k>Jthk⟹Thereisfault⟹alarmJr,k≤Jthk⟹Thereisnofault.If, for all 0≤i≤N, f(i)≡0, the system (1) can be redescribed as follows:(40)Mi+1xi+1=Aixi+wi,yi=λiCixi+vi.Employing a similar technical line of Section 3 in this paper, we can obtain the residual signal of system (40). For a given scalar γc>0, by employing performance index (3) and Theorem 3, we can judge whether the residual evaluation function with zero initial conditions achieves the following inequality: (41)Jr,k<γc2E∑i=0kwTiwi+∑i=0kvTivi.Note that w(i) and v(i) are bounded signals, so there exist m1(k) and m2(k) such that(42)∑i=0kwT(i)w(i)≤m1k,∑i=0kvTivi≤m2k.Suppose that the minimum γc achieving (41) is γcmin; then we have (43)Jr,k<γcmin2m1k+m2k.Thus, the residual threshold can be defined as(44)Jthk=γcmin2m1k+m2k.Finally, it can be judged based on the strategy given in (39).
5. A Numerical Example
Consider the discrete system (1) with the following parameters:(45)Mk=2.90002.5+0.3sink0000,Ak=2.1e-k001.6000-1.9,Bfk=3.22.74.5,Ck=2.601.8.Set γ=0.25 and P0=I. The unknown signals w(k), v(k), and f(k) are supposed to be(46)wk=0.3sink0.2cos3k0.4cos2k,f(k)=00≤k<40,1,40≤k≤60,vk=0.5sin4k,0,k>60.By applying Theorem 3, the H∞ fault detection filter is designed. Figure 1 shows the residual signal r(k) when ρ=1 and Df(k)=1.8, and Figure 2 shows the residual evaluation function J(r,k) and threshold Jth(k) when ρ=1 and Df(k)=1.8. Figure 3 shows the residual signal r(k) when ρ=1 and Df(k)=0, and Figure 4 shows the residual evaluation function J(r,k) and threshold Jth(k) when ρ=1 and Df(k)=0. It is shown that the tracking performance of H∞ fault detection filtering is good in the above two cases. Figure 5 shows the variation law of random parameter λ(k) when ρ=0.9. Figure 6 shows the residual signal r(k) when ρ=0.9 and Df(k)=1.8, and Figure 7 shows the residual evaluation function J(r,k) and threshold Jth(k) when ρ=0.9 and Df(k)=1.8. Figure 8 shows the residual signal r(k) when ρ=0.9 and Df(k)=0, and Figure 9 shows the residual evaluation function J(r,k) and threshold Jth(k) when ρ=0.9 and Df(k)=0. It is shown that the tracking performance of H∞ fault detection filtering is weakening when the system has missing measurements, but the threshold Jth(k) in Figures 7 and 9 has good performance.
Residual signal r(k) when ρ=1 and Df(k)=1.8.
Residual evaluation function J(r,k) and threshold Jth(k) when ρ=1 and Df(k)=1.8.
Residual signal r(k) when ρ=1 and Df(k)=0.
Residual evaluation function J(r,k) and threshold Jth(k) when ρ=1 and Df(k)=0.
Random parameters λ(k) when ρ=0.9.
Residual signal r(k) when ρ=0.9 and Df(k)=1.8.
Residual evaluation function J(r,k) and threshold Jth(k) when ρ=0.9 and Df(k)=1.8.
Residual signal r(k) when ρ=0.9 and Df(k)=0.
Residual evaluation function J(r,k) and threshold Jth(k) when ρ=0.9 and Df(k)=0.
6. Conclusions
In this paper, the Bernoulli random binary switching sequence has been applied to describe missing measurements, and it has been used as a basis on the study of the H∞ fault detection problem for linear discrete time-varying descriptor systems. The main contribution of this paper is to develop a novel approach on solving the H∞ fault detection problem for time-varying descriptor systems with missing measurements. An augmented system has been first obtained, and a relationship has been established between the H∞ fault detection problem and a certain indefinite quadratic form problem. By applying matrix analysis method, a necessary and sufficient existence condition and the explicit formula of the H∞ fault detection filter have been derived. The residual evaluation function and the threshold have been designed to facilitate fault detection. Finally, the proposed algorithm has been proved to be effective by a numerical example.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (nos. 61403061, 60774004, and 61273097), the Science and Technology Development Project of Dezhou, China (no. 2012B05), the Talent Introduction Project of Dezhou University, China (no. 311432), and the Solar Energy Special Project of Dezhou University, China (no. 14ZX03). The authors would like to thank the editor and reviewers for valuable comments and constructive suggestions aimed at improving the quality of this paper.
MahapatroA.KhilarP. M.Fault diagnosis in wireless sensor networks: a survey20131542000202610.1109/SURV.2013.030713.000622-s2.0-84888347505DingS. X.2008Berlin, GermanySpringerShenQ.JiangB.ShiP.Adaptive fault diagnosis for T-S fuzzy systems with sensor faults and system performance analysis20142222742852-s2.0-8489753844310.1109/TFUZZ.2013.2252355ZhangS.PattipatiK. R.HuZ.WenX.SankavaramC.Dynamic coupled fault diagnosis with propagation and observation delays20134361424143910.1109/TSMC.2013.22442092-s2.0-84898678662YaoL.QinJ.WangA.WangH.Fault diagnosis and fault-tolerant control for non-Gaussian non-linear stochastic systems using a rational square-root approximation model201371116124MR308819510.1049/iet-cta.2012.04662-s2.0-84877771823HoangN.-B.KangH.-J.A model-based fault diagnosis scheme for wheeled mobile robots201412363765110.1007/s12555-013-0012-12-s2.0-84900986524ChenJ.PattonR. J.Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances199614313136ChungW. H.SpeyerJ. L.A game-theoretic fault detection filter199843214316110.1109/9.661064MR16060012-s2.0-0031999814ChenR. H.SpeyerJ. L.Residual-sensitive fault detection filterProceedings of the 7th IEEE Mediterranean Conference on Control and Automation1999Haifa, Israel835851ChenR. H.SpeyerJ. L.Optimal stochastic fault detection filterProceedings of the American Control ConferenceJune 1999San Diego, Calif, USA91962-s2.0-0033283361ChenR. H.SpeyerJ. L.Optimal stochastic multiple-fault detection filterProceedings of the 38th Conference on Decision and Control1999Phoenix, Ariz, USA49654970ChenR. H.SpeyerJ. L.A generalized least-squares fault detection filter2000147747757ChenR. H.SpeyerJ. L.Robust multiple-fault detection filter200212867569610.1002/rnc.715MR19222852-s2.0-0037125250XuA. P.ZhangQ. H.Fault detection and isolation based on adaptive observers for linear time varying systemsProceedings of the 15th Triennial World Congress of the International Federation of Automatic Control2002Barcelona, SpainGustafssonF.Stochastic fault diagnosability in parity spacesProceedings of the 15th Triennial World Congress of the International Federation of Automatic Control2002Barcelona, SpainChenR. H.MingoriD. L.SpeyerJ. L.Optimal stochastic fault detection filter200339337739010.1016/S0005-1098(02)00245-5MR21377612-s2.0-0037361701XuA.ZhangQ.Residual generation for fault diagnosis in linear time-varying systems2004495767772MR20578132-s2.0-294259829010.1109/TAC.2004.825983ZhangQ. H.An adaptive observer for sensor fault estimation in linear time varying systemsProceedings of the 16th IFAC World Congress2005Prague, Czech RepublicCombastelC.ZhangQ.Robust fault diagnosis based on adaptive estimation and set-membership computationsProceedings of the 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS '06)September 2006Beijing, China127312782-s2.0-79961164248LiX.ZhouK.A time domain approach to robust fault detection of linear time-varying systemsProceedings of the 46th IEEE Conference on Decision and Control (CDC '07)December 2007New Orleans, La, USA1015102010.1109/CDC.2007.44342252-s2.0-62749120303GustafssonF.Statistical signal processing approaches to fault detection200731141542-s2.0-3424934171910.1016/j.arcontrol.2007.02.004CombastelC.ZhangQ. H.LalamiA.Fault diagnosis based on the enclosure of parameters estimated with an adaptive observerProceedings of the 17th World Congress the International Federation of Automatic Control2008Seoul, South Korea73147319GaoH.ChenT.WangL.Robust fault detection with missing measurements200881580481910.1080/00207170701684823MR2406887ZBL1152.933462-s2.0-41849117290LongY.YangG.-H.Fault detection in finite frequency domain for networked control systems with missing measurements20133509260526262-s2.0-8488420840810.1016/j.jfranklin.2013.01.015MR3146938ZhangP.DingS. X.FrankP. M.Fault detection of networked control systems with missing measurementsProceedings of the 5th Asian Control Conference2004New York, NY, USAIEEE Press12581263IshiharaJ. Y.TerraM. H.SilvaJ. P.H∞ estimation for rectangular discrete-time descriptor systemsProceedings of the American Control ConferenceJune 2006Minneapolis, Minn, USA565056542-s2.0-34047215708ZhaoH.ZhangC.XingG.H∞ filtering and smoothing for linear discrete time-varying descriptor systems with unknown inputs20126573474310.1049/iet-cta.2010.0565MR29529972-s2.0-84860523719ZhaoH.ZhangC.H∞ fixed-lag smoothing for linear discrete time-varying systems with uncertain observations2013224387397MR31276282-s2.0-8488475568110.1016/j.amc.2013.08.056